Solution Manual For Mechanics Of Materials, 7th Edition

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Solutions Manualto accompanyMechanics of MaterialsSeventh EditionFerdinand P. BeerLate of Lehigh UniversityE. Russell Johnston, Jr.Late of University of ConnecticutJohn T. DeWolfUniversity of ConnecticutDavid F. MazurekUnited States Coast Guard AcademyPrepared byAmy Mazurek

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vDESCRIPTION OF THE MATERIAL CONTAINED INMECHANICS OF MATERIALS,Seventh EditionChapter 1Introduction–Concept of StressThe main purpose of this chapter is to introduce the concept of stress. After a short review ofStatics in Section 1.1 emphasizing the use of free-body diagrams, Sections 1.2 through 1.2discussnormal stressesunder an axial loading,shearing stresses—with applications to pins andbolts in single and double shear—andbearing stresses. This section also introduces the studentto the concepts of analysis and design. Section 1.2A emphasizes the fact that stresses areinherently statically indeterminate and that, at this point, normal stresses under an axial loadingcan only be assumed to be uniformly distributed. Moreover, such an assumption requires that theaxial loading be centric.Section 1.2D is devoted to the application of these concepts to the analysis of asimple structure.Section 1.2E describes how students should approach the solution of a problem in mechanics ofmaterials using the SMART methodology: Strategy, Modeling, Analysis and Reflect & Think.Section 1.2E also discusses thenumerical accuracyto be expected in such a solution. Problemsincluded in the first lesson also serve as a review of the methods of analysis of trusses, frames,and mechanisms learned in statics.Section 1.3 discusses the determination of normal and shearing stresseson oblique planesunderan axial loading, while Section 1.4 introduces the components of stress under general loadingconditions. This section emphasizes the fact that the components of the shearing stresses exertedon perpendicular planes, such asτxyandτyx, must be equal. It also introduces the students to theconcept of transformation of stress. However, the study of the computational techniquesassociated with the transformation of stress at a point is delayed until Chapter 7, after studentshave discovered for themselves the need for such techniques.Section 1.5 is devoted to design considerations. It introduces the concepts ofultimate load,ultimate stress, andfactor of safety. It also discusses the reasons for the use of factors of safety inengineering practice. The section ends with an optional presentation of an alternative method ofdesign,Load and Resistance Factor Design.Chapter 2Stress and Strain–Axial LoadingThis chapter is devoted to the analysis and design of members under a centric axial loading.Section 2.1A introduces the concept ofnormal strain, while Section 2.1B describes the generalproperties of the stress-strain diagrams of ductile and brittle materials and defines theyieldstrength, ultimate strength,andbreaking strengthof a material. Section 21C, which is optional,definestrue stressandtrue strain. Section 2.1D introducesHooke’s law, themodulus ofelasticity,and theproportional limitof a material. It defines asisotropicthose materials whosemechanical properties are independent of the direction considered and asanisotropicthose

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viwhose mechanical properties depend upon that direction. Among the latter arefiber-reinforcedcomposite materials,that are described in this section.Section 2.1E discusses the elastic and the plastic behavior of a material and defines itselasticlimit, while Section 2.1F is devoted to fatigue and the behavior of materials under repeatedloadings. The first lesson of Chapter 2 ends with Section 2.1G, which shows how Hooke’s lawcan be used to determine the deformation of a rod of uniform or variable cross section under oneor several loads, and introduces the concept ofrelative displacement. Section 2.2 discussesstatically indeterminate problemsinvolving members under an axial load. As indicated in thepreface of the text and in the introduction to this manual, the authors believe it is important tointroduce the students at an early stage to the concept of statical indeterminacy and to show themhow the analysis of deformations can be used in the solution of problems that cannot be solvedby the methods of statics alone. It will also help them realize that stresses, being staticallyindeterminate, can be computed only by considering the corresponding distribution of strains.Section 2.3 discusses thethermal expansionof rods and shows how to determine stresses instatically indeterminate members subjected to temperature changes.Section 2.4 introduces the concept oflateral strainfor an isotropic material and definesPoisson’s ratio. Section 2.5 discusses themultiaxial loadingof a structural element and derivesthe generalized Hooke’s law for such a loading. Since this derivation is based on the applicationof theprinciple of superposition, this principle is also introduced in Section 2.5, and theconditions under which it can be used are clearly stated. Section 2.6 is optional. It discusses thechange in volume of a material under a multiaxial loading and defines thedilatationand thebulkmodulusormodulus of compressionof a given material.Section 2.7 introduces the concept ofshearing strain. It should be noted that the authors definethe shearing strain as the change in the angle formed by the faces of the element of materialconsidered, and not as the angle through which one of these faces rotates. Hooke’s law forshearing stress and strain and themodulus of rigidityare also introduced in this section, as well asthegeneralized Hooke’s lawfor a homogeneous, isotropic material under the most general stressconditions. Section 2.8 points out that strains, just as stresses, depend upon the orientation of theplanes considered. It also establishes the fact that the constantsE,v, andGare not independentfrom each other and derives Eq. (2.35), that expresses the relation among these three constants.Section2.9,whichisoptional,extendsthestress-strainrelationshipstofiber-reinforcedcomposite materials.The relations obtained are expressed by Eqs. (2.37) and (2.39) and involvethree different values of the modulus of elasticity and six different values of Poisson’s ratio.Section 2.10 discusses thedistribution of the normal stressesunder a centric axial loading andshows 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 ofstresses can be assumed uniform. This result verifies Saint-Venant’s principle. Section 2.11discussesstress concentrationsnear circular holes and fillets in flat bars under axial loading.Section 2.12 is devoted to theplastic deformationof members under centric axial loads andintroduces the concept of anelastoplastic material. As stated in the preface of the text, theauthors believe that students should be exposed to the concept of plastic deformation in the first

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viicourse in mechanics of materials, if only to let them realize the limitations of the assumption of alinear stress-strain relation in engineering applications. By introducing this concept early in thecourse in connection with axial loading, rather than later with torsion or bending, one makes iteasier for the students to understand and accept it. For the same reason,residual stressesarediscussed in Section 2.13 in connection with axial loading. However, since some instructors maynot want to include the concept of residual stresses in an elementary course, this section isoptional and can be omitted without any prejudice to the understanding of the rest of the text.Chapter 3TorsionThe Introduction introduces this type of loading, while Section 3.1 establishes the relation thatmust be satisfied, on the basis of statics, by the shearing stresses in a given section of a shaftsubjected to a torque. This condition, however, does not suffice to determine the stresses, andone must analyze the deformations that occur in the shaft. This is done in Section 3.1A, where itis proved that the distribution of shearing strains in a circular shaft is linear. It should be notedthat the discussion presented in See. 3.1B is based solely on the assumption of rigid end plates,rather than on arbitrary and gratuitous assumptions regarding the deformations of a shaft. Theresults obtained in this and the following sections clearly depend upon the validity of thisassumption, but can be extended to other loading conditions through the application of Saint-Venant’s principle.Section 3.1C is devoted to the analysis of the shearing stresses in the elastic range and presentsthe derivation of theelastic torsion formulasfor circular shafts. The section ends with remarkson the transformation of stresses in torsion and the comparison between the failures of ductileand brittle materials in torsion.The formula for theangle of twistof a shaft in the elastic range is derived in Section 3.2. Thissection also contains various applications involving the twisting of single and gear-connectedshafts. Section 3.3 deals with the solution of problems involvingstatically indeterminate shafts.Section 3.4 is devoted to thedesign of transmission shaftsand begins with the determination ofthe torque required to transmit a given power at a given speed, both in SI and U.S. customaryunits. Note that the effect of bending on the design of transmission shafts will be discussed inSection 8.2, which is optional. Section 3.5 discussesstress concentrationsat fillets in circularshafts.Sections 3.6 through 3.7 deal with theplastic deformationsandresidual stressesin circularshafts and are optional. Since a similar presentation of the plastic deformations and residualstresses of members in pure bending is given in Chapter 4, the instructor may decide to includeonly one of these presentations in the course. Section 3.6 describes the general method for thedetermination of the torque corresponding to a given maximum shearing stress in a shaft made ofa material with anonlinear stress-strain diagram, while Sections 3.7 and 3.8 deal, respectively,with the deformations and the residual stresses in shafts made of anelastoplastic material.Sections 3.9 and 3.10 are also optional. They are devoted, respectively, to the torsion of solidmembers and thin-walled hollow shafts ofnoncircular section.

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viiiChapter 4Pure BendingThe Introduction defines this type of loading and shows how the results obtained in the followingsections can be applied to the analysis of other types of loadings as well, namely, eccentric axialloadings and transverse loadings. Sections 4.1 and 4.1A establish the relation that must besatisfied, on the basis of statics, by thenormal stressesin a given section of a member subjectedto pure bending. This condition, however, does not suffice to determine the stresses, and onemust analyze thedeformationsthat occur in the member. This is done in Section 4.1B, where it isproved 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 ofthe member, except that the couples should be applied in such a way that the ends of the memberremain plane. Whether this can actually be accomplished is discussed at the end of Section 4.3.Section 4.2 is devoted to the analysis of the normal stresses in the elastic range and presents thederivation of theelastic flexure formulas. It also defines the elastic section modulus and endswith the derivation of the formula for thecurvatureof an elastic beam. Section 4.3 discusses theanticlastic curvatureof members in pure bending and also states the loading conditions requiredfor the ends of the member to remain plane.Section 4.4 discusses the determination of stresses in members made of several materials anddefines thetransformed sectionof such members. It also shows how the transformed section canbe used to determine the radius of curvature of the member. The section ends with a discussionof the stresses inreinforced-concrete beams. Section 4.5 deals with the stress concentrations atfillets and grooves in flat bars under pure bending.Section 4.6 is optional. This section discusses the plastic deformations and residual stresses inmembers subjected to pure bending in much the same way that these were discussed in Sections 3.6through 3.8 in the case of members in torsion. Section 4.6 describes the general method for thedetermination of the bending moment corresponding to a given maximum normal stress in amember possessing two planes of symmetry and made of a material with a nonlinear stress-straindiagram. Section 4.6A deals with members made of anelastoplastic materialand derivesformulas relating the thickness of the elastic core and the radius of curvature with the appliedbending moment in the case of members with a rectangular cross section. It also defines theshape factorand theplastic section modulusof members with a nonrectangular section. Section4.6B deals with the determination of the plastic moment of members made of an elastoplasticmaterial and possessing a single plane of symmetry, while Section 4.6C discussesresidualstresses.Section 4.7 shows how the stresses due to atwo-dimensional eccentric axial loadingcan beobtained by replacing the given eccentric load by a centric load and a couple, and superposingthe corresponding stresses. Attention is called to the fact that the neutral axisdoes not passthrough the centroid of the section.Section 4.8 deals with theunsymmetric bendingof elastic members. It is first shown that theneutral axis of a cross section will coincide with the axis of the bending couple if, and only if, the

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ixaxis of the couple is directed along one of the principal centroidal axes of the cross section. It isthen shown that stresses due to unsymmetric bending can always be determined by resolving thegiven bending couple into two component couples directed along the principal axes of thesection and superposing the corresponding stresses. Sample Problem 4.10 has been included inthis section to provide an opportunity for students to use material on Mohr’s Circle to determinestresses in non-symmetrical sections. The material is based on what students will have learned ina study of Statics where the transformation of moments of inertia is covered for problems whereit is necessary to look at the rotation of coordinate axes. Problems 4.141 through 4.143 are alsoasterisked and are best solved with the use of Mohr’s Circle.This method of analysis is extended in Section 4.9 to the determination of the stresses due to aneccentric axial loading in three-dimensional space. The eccentric load is replaced by an equivalentsystem consisting of a centric load and two bending couples, and the corresponding stresses aresuperposed.Section 4.10 is optional; it deals with the bending ofcurved members.Chapter 5Analysis and Design of Beamsfor BendingIn the Introduction beams are defined as slender prismatic members subjected to transverse loadsand are classified according to the way in which they are supported. It is shown that the internalforces in any given cross section are equivalent to a shear forceVand a bending coupleM.Thebending coupleMcreatesnormal stressesin the section, while the shear forceVcreatesshearingstresses.The former is determined in this chapter, using the flexure formula (5.1), while the latterwill be discussed in Chapter 6.Since the dominant criterion in the design of beams for strength is usually the bending stresses inthe beam, the determination of the maximum value of the bending moment in the beam is themost important factor to be considered. To facilitate the determination of the bending moment inany given section of the beam, the concept ofshear and bending-moment diagramswill beintroduced in Section 5.1, using free-body diagrams of various portions of the beam.An alternative method for the determination of shear and bending-moment diagrams, based onrelations among load, shear, and bending moment, is presented in Section 5.2. To maintain theinterest of the students, most of the problems to be assigned are focused on the engineeringapplications of these methods and call for the determination, not only of the shear and bendingmoment, but also of thenormal stresses in the beam.Section 5.3 is devoted to thedesign of prismatic beamsbased on the allowable normal stress forthe material used. Sample Problems and problems to be assigned include wooden beams ofrectangular cross section, as well as rolled-steel W and S beams. An optional paragraph on page 372describes the application ofLoad and Resistance Factor Designto beams under transverse loading.Section 5.4 introduces the concept ofsingularity functionsand shows how these functions canprovide an alternative and effective method for the determination of the shear and bending

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xmoment at any point of a beam under the most general loading condition. While this section isoptional, it should be included in the lesson schedule if singularity functions are to be used laterfor the determination of the slope and deflection of a beam (Section 9.3). It is pointed out onpage 387 that singularity functions are particularly well suited to theuse of computers, andseveral optional problems requiring the use of a computer (Probs. 5.118 through 5.125) havebeen included in this assignment.Section 5.5, which is optional, is devoted tononprismatic beams, such as forged or cast beamsdesigned to be of constant strength, and rolled-steel beams reinforced with cover plates.Chapter 6Shearing Stresses in Beams and Thin-Walled MembersIt is shown in the Introduction that a transverse load createsshearing stressesas well as normalstresses in a beam. Considering first the horizontal face of a beam element, it is shown inSections 61A that the horizontal shear per unit lengthq, orshear flow, is equal toVQ/I.Thisresult is applied in Concept Application 6.1 to the determination of the shear force in the nailsconnecting three planks forming a wooden beam, as well as in Probs. 6.1 through 6.4. Probs. 6.5through 6.8 apply the same concepts to steel beams made of sections bolted togetherIn Section 6.1B theaverage shearing stressτaveexerted on the horizontal face of the beamelement isobtained by dividing the shear flowqby the widthtof the beam:aveVQIt(6.6)Note that since the shearing stressesτxyandτyxexerted at a given point are equal, the expressionobtained also represents the average shearing stress exerted at a given height on a vertical sectionof the beam. This formula is used to determine shearing stresses in a beam made of glued planksin Sample Prob. 6.1 and to design a timber beam in Sample Prob. 6.2. Problems 6.9 through 6.12and 6.21 and 6.22 call for the determination of shearing stresses in various types of beams.Section 6.1C explores shearing stresses in common beam types In Concept Applications 6.2 and6.3 the designs obtained on the basis of normal stresses, respectively, for a timber beam inSample Prob. 5.7 and for a rolled-steel beam in Sample Prob. 5.8 are checked and found to beacceptable from the point of view of shearing stresses.Section 6.2 is optional and discusses the distribution of stresses in a narrow rectangular beam.In Section 6.3 the expressionq=VQ/Iobtained on Section 6.1A for the shear flow on thehorizontal face of a beam element is shown to remain valid for the curved surface of a beamelement of arbitrary shape. It is then applied in Concept Application 6.4 and in Probs. 6.29through 6.33 to the determination of the shearing forces and shearing stresses in nailed and gluedvertical surfaces.

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xiSection 6.4 deals with the determination of shearing stresses inthin-walled membersand showsthat Eq. (6.6) can be applied to the determination of the average shearing stress in a section ofarbitrary orientation.Section 6.5, which is optional, describes the formation ofplastic zonesin beams subjected totransverse loads.Section 6.6, which is also optional, deals with theunsymmetric loadingof thin-walled members,the determination of theshear center, and the computation of the shearing stresses caused by ashearing force exerted at the shear center.Chapter 7Transformations of Stress and StrainAfter a short introduction, formulas for thetransformation of plane stressunder a rotation ofaxes are derived in Section 7.1A, while the principal planes of stress, principal stresses, andmaximum shearing stress are determined in Section 7.1B.Section 7.2 is devoted to the use ofMohr’s circle. It should be noted that the convention used inthe text provides for a rotation on Mohr’s circle in thesame senseas the corresponding rotationof the element; in other words, this convention is the same as that used in statics for thetransformation of moments and products of inertia. Attention is called to the statement at thebottom of page 493 of the text and the accompanying Fig. 7.15.Section 7.3 discusses the general (three-dimensional) state of stress and establishes the fact thatthree principal axes of stress and three principal stresses exist. Section 7.4 shows how threedifferent Mohr’s circles can be used to represent the transformations of stress associated withrotations of the element about the principal axes. The results obtained are used to show that in astate of plane stress, the maximum shearing stress does not necessarily occur in the plane ofstress.Section 7.5 is optional. Section 7.5A presents the two criteria most commonly used to predictwhether a ductile material will yieldunder a given state of plane stress, while Section 7.5Bdiscusses the two criteria used to predict thefracture of brittle materials.Section 7.6 deals with stresses inthin-walled pressure vessels; it is limited to the analysis ofcylindrical and spherical pressure vessels.The second part of the chapter (Sections 7.7 through 7.9) deals withtransformations of strainand is optional. Section 7.7A presents the derivation of the formulas for the transformation of strainunder a rotation of axes. It should be noted that this derivation is based on the consideration of anoblique triangle (Fig. 7.51) and the use of the law of cosines, and that the determination of theshearing strain is facilitated by the use of Eq. (7.43), which relates it to the normal strain alongthe coordinate axes and their bisector.

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xiiSection 7.7B introducesMohr’scircle forplane strain, and Section 7.8 discusses the three-dimensional analysis of strain and its application to the determination of the maximum shearingstrain in states of plane strain and of plane stress. Section 7.9 deals with the use of strain rosettesfor the determination of states of plane strain.Chapter 8Principal Stresses under a Given LoadingThis chapter is devoted to the determination of theprincipal stressesandmaximum shearingstressin beams, transmission shafts subjected to transverse loads as well as to torques, andbodies of arbitrary shape under combined loadings.In the Introduction it is shown that, while only normal stresses occur on a square element withhorizontal and vertical faces located at the surface of a beam,shearing stresses will occur if theelement is rotated through 45o(Fig. 8.1). The reverse situation is observed for an element withhorizontal and vertical faces subjected only to shearing stresses (Fig. 8.2). The analysis of beams,therefore, should include the determination of the principal stresses and maximum shearingstress at various points.This is done in Section 8.1for cantilever beams of various rectangular sections subjected to asingle concentrated load at their free end. It is found that the principal stressσmaxdoes not exceedthe maximum normal stressσmdetermined by the method of Chapter 5 except very close to theload. While this result holds for most beams of nonrectangular section, it may not be valid forrolled-steel W and S beams, and the analysis and design of such beams should include thedetermination of the principal stressσmaxat the junction of the web with the flanges of the beam.(See Sample Probs. 8.1 and 8.2, and Probs. 8.1 through 8.14).Section 8.2 is devoted to the analysis and design oftransmission shaftsusing gears or sprocketwheels to transmit power to and from the shaft. These shafts are subjected to transverse loads aswell as to torques. The design of such shafts is the subject of Sample Prob. 8.3 and Probs. 8.15through 8.30.The determination of the stresses at a given pointKof a body due to acombined loadingis thesubject of Section 8.3. First, the loading is reduced to an equivalent system of forces and couplesin a section of the body containingK. Next, the normal and shearing stresses are determined atK.Finally, using one of the methods of transformation of stresses presented in Chapter 7, theprincipal planes, principal stresses, and maximum shearing stress may be determined atK. Thisprocedure is illustrated in Concept Application 8.1 and Sample Probs. 8.4 and 8.5.

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xiiiChapter 9Deflection of BeamsThe relation derived in Chapter 4 between the curvature of a beam and the bending moment isrecalled in Section 9.1 and used to predict the variation of the curvature along the beam. InSection 9.1A, the equation of theelastic curvefor a beam is obtained through two successiveintegrations, after the bending moment has been expressed as a function of the coordinatex.Concept Applications 9.1 and 9.2 show how the boundary conditions can be used to determinethe two constants of integration in the cases of a cantilever beam and of a simply supportedbeam. Concept Application 9.3 indicates how to proceed when the bending moment must berepresented by two different functions ofx.Section 9.1B is optional; it shows in the case of a beam supporting a distributed load, how theequation of the elastic curve can be obtained directly from the function representing the loaddistribution through the use of four successive integrations.Section 9.2 is devoted to the analysis ofstatically indeterminate beamsand to the determinationof the reactions at their supports. It is suggested that a minimum of two lessons be spent onSections 9.1 through 9.2 if neither the use of singularity functions (Section 9.3) nor themoment-area method (Sections 9.5 through 9.6) are to be covered in the course.Section 9.3 is devoted to the use ofsingularity functionsfor the determination of beamdeflections and slopes. It is optional and assumes that Section 5.4 has been covered previously. Itis recommended that both Sections 5.4 and 9.3 be included in the course, since singularityfunctions provide the students with an effective and versatile method for the determination ofdeflections and slopes under the most diverse loading conditions. In addition, and as indicatedearlier, singularity functions are well suited to the use of computers.Section 9.4A discusses themethod of superpositionfor the determination of beam deflectionsand slopes. It shows how the expressions given in Appendix D for various simple loadings canbe used to obtain the deflection and slope of a beam supporting a more complex loading. InSection 9.4B, the method of superposition is applied to the determination of the reactions at thesupports ofstatically indeterminate beams.Sections 9.5 through 9.6 are optional. They deal with the application of the moment-areamethods to the determination of the deflection of beams and may be omitted in courses that placea greater emphasis on analytical methods and make use of singularity functions. It should benoted, however, that these methods provide a very practical means for the determination of thedeflection and slope of beams ofvariable cross section.The two moment-area theorems are derived in Section 9.5A and are immediately applied inSection 9.5B to the computation of the slope and deflection ofcantilever beamsandbeams withsymmetric loadings(simply supported or overhanging beams). Section 9.5C shows how to drawabending-moment diagram by parts. This approach greatly facilitates the determination ofmoment areas in all but the simplest loading situations.

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xivSection 9.6 deals with simply supported and overhanging beams withunsymmetric loadings.The analysis of such beams hinges on the use of a reference tangent drawn through one of thesupports after the tangential deviation of the second support has been computed from thebending-moment diagram. Section 9.6B describes how to locate the point ofmaximum deflectionand how to compute that deflection.Section 9.6C deals with the analysis ofstatically indeterminate beamsand the determination ofthe reactions at their supports.Chapter 10ColumnsSection 10.1 introduces the concept ofstability of a structure. The example considered in thissection consists of a block supported by two spring-connected rigid rods. It is shown that theposition of equilibrium in which both rods are aligned is stable if this position is the onlypossible position of equilibrium of the system. The same criterion is applied to an elasticpin-ended column in Section 10.1A in order to deriveEuler’s formula. Section 10.1B shows howEuler’s formula for pin-ended columns can be used to determine the critical load of columnswith other end conditions.Section 10.2 is optional; it deals with the eccentric loading of a column and gives the derivationof thesecant formula.Section 10.3 discusses thedesign of columns under a centric load. Empirical formulas developedby various engineering associations for the design of steel columns, aluminum columns, and woodcolumns are presented in Section 10.3A. Section 10.3B is devoted to an optional discussion ofthe application ofLoad and Resistance Factor Designto steel columns. As noted at the end ofthis section, the design formulas presented in this section are intended to provide introductoryexamples of different design approaches. These formulas do not provide all the requirements thatare needed for more comprehensive designs often encountered in engineering practice.Section 10.4 discusses thedesign of columns under an eccentric loadand presents two of themost frequently used methods: theallowable-stress methodand theinteraction method.Chapter 11Energy MethodsSection 11.1A introduces the concept ofstrain energyby considering the work required tostretch a rod of uniform cross section. This work, which is equal to the area under the load-deformation curve, represents the strain energy of the rod. Thestrain-energy densityis defined inSection. 11.1B, as well as themodulus of toughnessand themodulus of resilienceof a givenmaterial. The formula for the elastic strain energy associated with normal stresses is derived inSection 11.2A, as well as the expressions for the strain energy corresponding to an axial loadingand to pure bending. The formula for the strain energy associated with shearing stresses isderived in Section 11.2B, as well as the expressions corresponding to torsion and transverseloading.

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xvSection 11.3, which is optional, covers the strain energy for a general state of stress and derivesan expression for thedistortion energy per unit volume, both in the general case of three-dimensional stress and in the particular case of plane stress.Section 11.4A discussesimpact loadingsand Section 11.4B the design of a structure for animpact load. To facilitate the solution of impact-loading problems, it is shown in Section 11.5Athat the strain energy of a structure subjected to a single concentrated loadPcan be obtained byequating the strain energy to the work ofP.(Appendix D is used to express the deflection interms ofP). As shown in Section 11.5B, the reverse procedure can be used to determine thedeflection of a structure at the point of application of a single loadPor a single coupleM; thestrain energy of the structure is computed from one of the formulas derived in Section 11.2, andthe work ofPorMis equated to the expression obtained for the strain energy.Sections 11.6 through 11.9 are optional. In Section 11.6 an expression for the strain energy of astructure subjected to several loadsis obtained by computing the work of the loads as they aresuccessively applied. Reversing the order in which the loads are applied, one provesMaxwell’sreciprocal theorem. The expression obtained for the strain energy of the structure is used inSection 11.7 to proveCastigliano’s theorem. Section 11.8 is devoted to the application ofCastigliano’s theorem to the determination of the deflection and slope of a beam and to thedeflection of a point in a truss. Finally, Section 11.9 deals with the application of Castigliano’stheorem to the determination of the reactions at the supports of statically indeterminate structuressuch as beams and trusses.

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TABLE I: LIST OF TOPICS COVERED INMECHANICS OF MATERIALS,Seventh EditionSuggested Number of PeriodsCoreAdditionalAdvancedSectionsTopicsTopicsTopicsTopicsChapter 1: Introduction – Concept of Stress1.1-2Stress Under Axial Loading1-21.3-5Components of Stress; Factor of Safety1Chapter 2: Stress and Strain – Axial Loading2.1Stress-Strain Diagrams; Deformations Under1-2Axial Loading2.2-3Statically Indeterminate Problems12.4-5Poisson’s Ratio; Generalized Hooke’s Law1*2.6Dilatation; Bulk Modulus0.25-0.52.7-8Shearing Strain0.5*2.9Stress-Strain Relationships for Fiber-Reinforced0.5-1Composite Materials2.10-12Stress Concentrations; Plastic Deformations0.5-1*2.13Residual Stresses0.5Chapter 3: Torsion3.1Stresses in Elastic Range13.2-3Angle of Twist; Statically Indeterminate Shafts1-23.4-5Design of Transmission Shafts; Stress1Concentrations*3.6-8Plastic Deformations; Residual Stresses1-2*3.9-10Noncircular Members;Thin-Walled Hollow Shafts1-2Chapter 4: Pure Bending4.1-3Stresses in Elastic Range1-24.4-5Members Made of Several Materials; Stress1-2Concentrations*4.6Plastic Deformations; Residual Stresses1-24.7Eccentric Axial Loading1-24.8-9Unsymmetric Bending; General Eccentric1-2Axial Loading*4.10Bending of Curved Members1-2Chapter 5: Analysis and Design of Beams for Bending5.1Shear and Bending-Moment Diagrams1-1.55.2Using Relations Betweenw, V,andM1-1.55.3Design of Prismatic Beams in Bending1-2*5.4Use of Singularity Functions to DetermineVandM1-2*5.5Nonprismatic Beams1-2Chapter 6: Shearing Stresses in Beams and Thin-WalledMembers6.1Shearing Stresses in Beams1-2*6.2Shearing Stresses in Narrow Rectangular Beam0.256.3-4Shearing Stresses in Thin-Walled Members1-2*6.5Plastic Deformations0.25*6.6Unsymmetric Loading; Shear Center1-2

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TABLE I: LIST OF TOPICS COVERED INMECHANICS OF MATERIALS,Seventh Edition (CONTINUED)Suggested Number of PeriodsCoreAdditionalAdvancedSectionsTopicsTopicsTopicsTopicsChapter 7: Transformation of Stress and Strain7.1Transformation of Plane Stress1-27.2Mohr’s Circle for Plane Stress1-27.3-4Three-Dimensional Analysis of Stress0.5-1*7.5Yield and Fracture Criteria0.5-17.6Thin-Walled Pressure Vessels0.5-1*7.7-8Analysis of Strain; Mohr’s Circle1-1.5*7.9Strain Rosette0.5Chapter 8: Principal Stresses under a Given Loading8.1Principal Stresses in a Beam0.5-18.2Design of Transmission Shafts0.5-18.3Stresses under Combined Loadings1-3Chapter 9: Deflection of Beams9.1-1AEquation of Elastic Curve0.5-1*9.1BDirect Determination of Elastic Curve from0.5Load Distribution9.2Statically Indeterminate Beams0.5 - 1*9.3Use of Singularity Functions1-29.4Method of Superposition1-2Application of Moment-Area Theorems to:*9.5Cantilever Beams and Beams with1-2Symmetric Loadings*9.6A-6BBeams with Unsymmetric Loadings; Maximum1-1.5Deflection*9.6CStatically Indeterminate Beams0.5Chapter 10: Columns10.1Euler’s Column Formula1-2*10.2Eccentric Loading; Secant Formula110.3Design of Columns under a Centric Load1-210.4Design of Columns under an Eccentric Load1-2Chapter 11: Energy Methods11.1-2Strain Energy1-211.3Strain Energy for General State of Stress0.511.4Impact Loading0.5-111.5Deflections by Work-Energy Method0.5-1*11.6-8Castigliano’s Theorem1-2*11.9Statically Indeterminate Structures1-2_____________________Total Number of Periods24-41½21-38½3-6

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