Solution Manual For Mechanics Of Materials, 7th Edition

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Solutions Manual
to accompany
Mechanics of Materials
Seventh Edition
Ferdinand P. Beer
Late of Lehigh University
E. Russell Johnston, Jr.
Late of University of Connecticut
John T. DeWolf
University of Connecticut
David F. Mazurek
United States Coast Guard Academy
Prepared by
Amy Mazurek

v
DESCRIPTION OF THE MATERIAL CONTAINED IN
MECHANICS OF MATERIALS, Seventh Edition
Chapter 1
Introduction–Concept of Stress
The main purpose of this chapter is to introduce the concept of stress. After a short review of
Statics in Section 1.1 emphasizing the use of free-body diagrams, Sections 1.2 through 1.2
discuss normal stresses under an axial loading, shearing stresses—with applications to pins and
bolts in single and double shear—and bearing stresses. This section also introduces the student
to the concepts of analysis and design. Section 1.2A 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 loading be centric.
Section 1.2D is devoted to the application of these concepts to the analysis of a simple structure.
Section 1.2E describes how students should approach the solution of a problem in mechanics of
materials using the SMART methodology: Strategy, Modeling, Analysis and Reflect & Think.
Section 1.2E also discusses the numerical accuracy to be expected in such a solution. Problems
included 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 stresses on oblique planes under
an axial loading, while Section 1.4 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 Chapter 7, after students
have discovered for themselves the need for such techniques.
Section 1.5 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 2
Stress and Strain–Axial Loading
This chapter is devoted to the analysis and design of members under a centric axial loading.
Section 2.1A introduces the concept of normal strain, while Section 2.1B 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 21C, which is optional,
defines true stress and true strain. Section 2.1D 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

vi
whose mechanical properties depend upon that direction. Among the latter are fiber-reinforced
composite materials, that are described in this section.
Section 2.1E discusses the elastic and the plastic behavior of a material and defines its elastic
limit, while Section 2.1F is devoted to fatigue and the behavior of materials under repeated
loadings. The first lesson of Chapter 2 ends with Section 2.1G, 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. Section 2.2 discusses
statically indeterminate problems involving members under an axial load. As indicated in the
preface of the text and in the introduction to this manual, 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.
Section 2.3 discusses the thermal expansion of rods and shows how to determine stresses in
statically indeterminate members subjected to temperature changes.
Section 2.4 introduces the concept of lateral strain for an isotropic material and defines
Poisson’s ratio. Section 2.5 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 Section 2.5, and the
conditions under which it can be used are clearly stated. Section 2.6 is optional. It discusses the
change in volume of a material under a multiaxial loading and defines the dilatation and the bulk
modulus or modulus of compression of a given material.
Section 2.7 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. Section 2.8 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. (2.35), that expresses the relation among these three constants.
Section 2.9, which is optional, extends the stress-strain relationships to fiber-reinforced
composite materials. The relations obtained are expressed by Eqs. (2.37) and (2.39) and involve
three different values of the modulus of elasticity and six different values of Poisson’s ratio.
Section 2.10 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 2.11
discusses stress concentrations near circular holes and fillets in flat bars under axial loading.
Section 2.12 is devoted to the plastic deformation of members under centric axial loads and
introduces the concept of an elastoplastic material. As stated in the preface of the text, the
authors believe that students should be exposed to the concept of plastic deformation in the first

vii
course in mechanics of materials, if only to let them realize the limitations of the assumption of a
linear stress-strain relation in engineering applications. By introducing this concept early in the
course in connection with axial loading, rather than later with torsion or bending, one makes it
easier for the students to understand and accept it. For the same reason, residual stresses are
discussed in Section 2.13 in connection with axial loading. However, since some instructors may
not want to include the concept of residual stresses in an elementary course, this section is
optional and can be omitted without any prejudice to the understanding of the rest of the text.
Chapter 3
Torsion
The Introduction introduces this type of loading, while Section 3.1 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 Section 3.1A, 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 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. 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 3.1C 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 Section 3.2. This
section also contains various applications involving the twisting of single and gear-connected
shafts. Section 3.3 deals with the solution of problems involving statically indeterminate shafts.
Section 3.4 is devoted to the design of transmission shafts and begins with the determination of
the torque required to transmit a given power at a given speed, both in SI and U.S. customary
units. Note that the effect of bending on the design of transmission shafts will be discussed in
Section 8.2, which is optional. Section 3.5 discusses stress concentrations at fillets in circular
shafts.
Sections 3.6 through 3.7 deal with the plastic deformations and residual stresses in circular
shafts and are optional. Since a similar presentation of the plastic deformations and residual
stresses of members in pure bending is given in Chapter 4, the instructor may decide to include
only one of these presentations in the course. Section 3.6 describes the general method for the
determination of the torque corresponding to a given maximum shearing stress in a shaft made of
a material with a nonlinear stress-strain diagram, while Sections 3.7 and 3.8 deal, respectively,
with the deformations and the residual stresses in shafts made of an elastoplastic material.
Sections 3.9 and 3.10 are also optional. They are devoted, respectively, to the torsion of solid
members and thin-walled hollow shafts of noncircular section.

viii
Chapter 4
Pure Bending
The Introduction 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. Sections 4.1 and 4.1A establish 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 Section 4.1B, 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. 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 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 4.3 discusses the
anticlastic curvature of members in pure bending and also states the loading conditions required
for the ends of the member to remain plane.
Section 4.4 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 4.5 deals with the stress concentrations at
fillets and grooves in flat bars under pure bending.
Section 4.6 is optional. This section discusses the plastic deformations and residual stresses in
members subjected to pure bending in much the same way that these were discussed in Sections 3.6
through 3.8 in the case of members in torsion. Section 4.6 describes the general method for the
determination of the bending moment corresponding to a given maximum normal stress in a
member possessing two planes of symmetry and made of a material with a nonlinear stress-strain
diagram. Section 4.6A deals with members made of an elastoplastic material and derives
formulas relating the thickness of the elastic core and the radius of curvature with the applied
bending moment in the case of members with a rectangular cross section. It also defines the
shape factor and the plastic section modulus of members with a nonrectangular section. Section
4.6B deals with the determination of the plastic moment of members made of an elastoplastic
material and possessing a single plane of symmetry, while Section 4.6C discusses residual
stresses.
Section 4.7 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 4.8 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

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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. Sample Problem 4.10 has been included in
this section to provide an opportunity for students to use material on Mohr’s Circle to determine
stresses in non-symmetrical sections. The material is based on what students will have learned in
a study of Statics where the transformation of moments of inertia is covered for problems where
it is necessary to look at the rotation of coordinate axes. Problems 4.141 through 4.143 are also
asterisked 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 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.
Section 4.10 is optional; it deals with the bending of curved members.
Chapter 5
Analysis and Design of Beams
for Bending
In the Introduction 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 determined in this chapter, using the flexure formula (5.1), while the latter
will be discussed in Chapter 6.
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 will be
introduced 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 on
relations among load, shear, and bending moment, is presented in Section 5.2. 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 5.3 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. An optional paragraph on page 372
describes the application of Load and Resistance Factor Design to beams under transverse loading.
Section 5.4 introduces the concept of singularity functions and shows how these functions can
provide an alternative and effective method for the determination of the shear and bending

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moment at any point of a beam under the most general loading condition. While this section is
optional, it should be included in the lesson schedule if singularity functions are to be used later
for the determination of the slope and deflection of a beam (Section 9.3). It is pointed out on
page 387 that singularity functions are particularly well suited to the use of computers, and
several optional problems requiring the use of a computer (Probs. 5.118 through 5.125) have
been included in this assignment.
Section 5.5, which is optional, is devoted to nonprismatic beams, such as forged or cast beams
designed to be of constant strength, and rolled-steel beams reinforced with cover plates.
Chapter 6
Shearing Stresses in Beams and Thin-Walled Members
It is shown in the Introduction 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
Sections 61A that the horizontal shear per unit length q, or shear flow, is equal to VQ/I. This
result is applied in Concept Application 6.1 to the determination of the shear force in the nails
connecting three planks forming a wooden beam, as well as in Probs. 6.1 through 6.4. Probs. 6.5
through 6.8 apply the same concepts to steel beams made of sections bolted together
In Section 6.1B 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
  (6.6)
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. 6.1 and to design a timber beam in Sample Prob. 6.2. Problems 6.9 through 6.12
and 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 and
6.3 the designs obtained on the basis of normal stresses, respectively, for a timber beam in
Sample Prob. 5.7 and for a rolled-steel beam in Sample Prob. 5.8 are checked and found to be
acceptable 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 expression q = VQ/I obtained on Section 6.1A for the shear flow on the
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 Concept Application 6.4 and in Probs. 6.29
through 6.33 to the determination of the shearing forces and shearing stresses in nailed and glued
vertical surfaces.

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Section 6.4 deals with the determination of shearing stresses in thin-walled members and shows
that Eq. (6.6) can be applied to the determination of the average shearing stress in a section of
arbitrary orientation.
Section 6.5, which is optional, describes the formation of plastic zones in beams subjected to
transverse loads.
Section 6.6, which is also optional, deals with the unsymmetric loading of thin-walled members,
the determination of the shear center, and the computation of the shearing stresses caused by a
shearing force exerted at the shear center.
Chapter 7
Transformations of Stress and Strain
After a short introduction, formulas for the transformation of plane stress under a rotation of
axes are derived in Section 7.1A, while the principal planes of stress, principal stresses, and
maximum shearing stress are determined in Section 7.1B.
Section 7.2 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; in other words, this convention is the same as that used in statics for the
transformation of moments and products of inertia. Attention is called to the statement at the
bottom 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 that
three principal axes of stress and three principal stresses exist. Section 7.4 shows how three
different Mohr’s circles can be used to represent the transformations of stress associated with
rotations of the element about the principal axes. The results obtained are used to show that in a
state of plane stress, the maximum shearing stress does not necessarily occur in the plane of
stress.
Section 7.5 is optional. Section 7.5A presents the two criteria most commonly used to predict
whether a ductile material will yield under a given state of plane stress, while Section 7.5B
discusses the two criteria used to predict the fracture of brittle materials.
Section 7.6 deals with stresses in thin-walled pressure vessels; it is limited to the analysis of
cylindrical and spherical pressure vessels.
The second part of the chapter (Sections 7.7 through 7.9) deals with transformations of strain
and is optional. Section 7.7A presents the derivation of the formulas for the transformation of strain
under a rotation of axes. It should be noted that this derivation is based on the consideration of an
oblique triangle (Fig. 7.51) and the use of the law of cosines, and that the determination of the
shearing strain is facilitated by the use of Eq. (7.43), which relates it to the normal strain along
the coordinate axes and their bisector.

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Section 7.7B introduces Mohr’s circle for plane strain, and Section 7.8 discusses the three-
dimensional analysis of strain and its application to the determination of the maximum shearing
strain in states of plane strain and of plane stress. Section 7.9 deals with the use of strain rosettes
for the determination of states of plane strain.
Chapter 8
Principal Stresses under a Given Loading
This chapter is devoted to the determination of the principal stresses and maximum shearing
stress in beams, transmission shafts subjected to transverse loads as well as to torques, and
bodies of arbitrary shape under combined loadings.
In the Introduction it is shown that, while only normal stresses occur on a square element with
horizontal and vertical faces located at the surface of a beam, shearing stresses will occur if the
element is rotated through 45o (Fig. 8.1). The reverse situation is observed for an element with
horizontal 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 shearing
stress at various points.
This is done in Section 8.1 for cantilever beams of various rectangular sections subjected to a
single concentrated load at their free end. It is found that the principal stress σmax does not exceed
the maximum normal stress σm determined by the method of Chapter 5 except very close to the
load. While this result holds for most beams of nonrectangular section, it may not be valid for
rolled-steel W and S beams, and the analysis and design of such beams should include the
determination of the principal stress σmax at 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 of transmission shafts using gears or sprocket
wheels to transmit power to and from the shaft. These shafts are subjected to transverse loads as
well as to torques. The design of such shafts is the subject of Sample Prob. 8.3 and Probs. 8.15
through 8.30.
The determination of the stresses at a given point K of a body due to a combined loading is the
subject of Section 8.3. First, the loading is reduced to an equivalent system of forces and couples
in a section of the body containing K. Next, the normal and shearing stresses are determined at K.
Finally, using one of the methods of transformation of stresses presented in Chapter 7, the
principal planes, principal stresses, and maximum shearing stress may be determined at K. This
procedure is illustrated in Concept Application 8.1 and Sample Probs. 8.4 and 8.5.

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Chapter 9
Deflection of Beams
The relation derived in Chapter 4 between the curvature of a beam and the bending moment is
recalled in Section 9.1 and used to predict the variation of the curvature along the beam. In
Section 9.1A, 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.
Concept Applications 9.1 and 9.2 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. Concept Application 9.3 indicates how to proceed when the bending moment must be
represented by two different functions of x.
Section 9.1B is optional; it shows in the case of a beam supporting a distributed load, 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 9.2 is devoted to the analysis of statically indeterminate beams and to the determination
of the reactions at their supports. It is suggested that a minimum of two lessons be spent on
Sections 9.1 through 9.2 if neither the use of singularity functions (Section 9.3) nor the
moment-area method (Sections 9.5 through 9.6) are to be covered in the course.
Section 9.3 is devoted to the use of singularity functions for the determination of beam
deflections and slopes. It is optional and assumes that Section 5.4 has been covered previously. It
is recommended that both Sections 5.4 and 9.3 be included in the course, since singularity
functions provide the students with an effective and versatile method for the determination of
deflections and slopes under the most diverse loading conditions. In addition, and as indicated
earlier, singularity functions are well suited to the use of computers.
Section 9.4A discusses the method of superposition for the determination of beam deflections
and slopes. It shows how the expressions given in Appendix D for various simple loadings can
be used to obtain the deflection and slope of a beam supporting a more complex loading. In
Section 9.4B, the method of superposition is applied to the determination of the reactions at the
supports of statically indeterminate beams.
Sections 9.5 through 9.6 are optional. They deal with the application of the moment-area
methods to the determination of the deflection of beams and may be omitted in courses that place
a greater emphasis on analytical methods and make use of singularity functions. It should be
noted, however, that these methods provide a very practical means for the determination of the
deflection and slope of beams of variable cross section.
The two moment-area theorems are derived in Section 9.5A and are immediately applied in
Section 9.5B to the computation of the slope and deflection of cantilever beams and beams with
symmetric loadings (simply supported or overhanging beams). Section 9.5C shows how to draw
a bending-moment diagram by parts. This approach greatly facilitates the determination of
moment areas in all but the simplest loading situations.

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Section 9.6 deals with simply supported and overhanging beams with unsymmetric loadings.
The analysis of such beams hinges on the use of a reference tangent drawn through one of the
supports after the tangential deviation of the second support has been computed from the
bending-moment diagram. Section 9.6B describes how to locate the point of maximum deflection
and how to compute that deflection.
Section 9.6C deals with the analysis of statically indeterminate beams and the determination of
the reactions at their supports.
Chapter 10
Columns
Section 10.1 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 Section 10.1A in order to derive Euler’s formula. Section 10.1B shows how
Euler’s formula for pin-ended columns can be used to determine the critical load of columns
with other end conditions.
Section 10.2 is optional; it deals with the eccentric loading of a column and gives the derivation
of the secant formula.
Section 10.3 discusses the design of columns under a centric load. Empirical formulas developed
by various engineering associations for the design of steel columns, aluminum columns, and wood
columns are presented in Section 10.3A. Section 10.3B is devoted to an optional discussion of
the application of Load and Resistance Factor Design to steel columns. As noted at the end of
this section, the design formulas presented in this section are intended to provide introductory
examples of different design approaches. These formulas do not provide all the requirements that
are needed for more comprehensive designs often encountered in engineering practice.
Section 10.4 discusses the design of columns under an eccentric load and presents two of the
most frequently used methods: the allowable-stress method and the interaction method.
Chapter 11
Energy Methods
Section 11.1A introduces the concept of strain energy by considering the work required to
stretch 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. The strain-energy density is defined in
Section. 11.1B, as well as the modulus of toughness and the modulus of resilience of a given
material. The formula for the elastic strain energy associated with normal stresses is derived in
Section 11.2A, as well as the expressions for the strain energy corresponding to an axial loading
and to pure bending. The formula for the strain energy associated with shearing stresses is
derived in Section 11.2B, as well as the expressions corresponding to torsion and transverse
loading.

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Section 11.3, which is optional, covers the strain energy for a general state of stress and derives
an expression for the distortion energy per unit volume, both in the general case of three-
dimensional stress and in the particular case of plane stress.
Section 11.4A discusses impact loadings and Section 11.4B the design of a structure for an
impact load. To facilitate the solution of impact-loading problems, it is shown in Section 11.5A
that the strain energy of a structure subjected to a single concentrated load P can be obtained by
equating the strain energy to the work of P. (Appendix D is used to express the deflection in
terms of P ). As shown in Section 11.5B, the reverse procedure can be used to determine the
deflection of a structure at the point of application of a single load P or a single couple M; the
strain energy of the structure is computed from one of the formulas derived in Section 11.2, and
the work of P or M is 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 a
structure subjected to several loads is obtained by computing the work of the loads as they are
successively applied. Reversing the order in which the loads are applied, one proves Maxwell’s
reciprocal theorem. The expression obtained for the strain energy of the structure is used in
Section 11.7 to prove Castigliano’s theorem. Section 11.8 is devoted to the application of
Castigliano’s theorem to the determination of the deflection and slope of a beam and to the
deflection of a point in a truss. Finally, Section 11.9 deals with the application of Castigliano’s
theorem to the determination of the reactions at the supports of statically indeterminate structures
such as beams and trusses.

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TABLE I: LIST OF TOPICS COVERED IN MECHANICS OF MATERIALS, Seventh Edition
Suggested Number of Periods
Core Additional Advanced
Sections Topics Topics Topics Topics
Chapter 1: Introduction – Concept of Stress
1.1-2 Stress Under Axial Loading 1-2
1.3-5 Components of Stress; Factor of Safety 1
Chapter 2: Stress and Strain – Axial Loading
2.1 Stress-Strain Diagrams; Deformations Under 1-2
Axial Loading
2.2-3 Statically Indeterminate Problems 1
2.4-5 Poisson’s Ratio; Generalized Hooke’s Law 1
*2.6 Dilatation; Bulk Modulus 0.25-0.5
2.7-8 Shearing Strain 0.5
*2.9 Stress-Strain Relationships for Fiber-Reinforced 0.5-1
Composite Materials
2.10-12 Stress Concentrations; Plastic Deformations 0.5-1
*2.13 Residual Stresses 0.5
Chapter 3: Torsion
3.1 Stresses in Elastic Range 1
3.2-3 Angle of Twist; Statically Indeterminate Shafts 1-2
3.4-5 Design of Transmission Shafts; Stress 1
Concentrations
*3.6-8 Plastic Deformations; Residual Stresses 1-2
*3.9-10 Noncircular Members;
Thin-Walled Hollow Shafts 1-2
Chapter 4: Pure Bending
4.1-3 Stresses in Elastic Range 1-2
4.4-5 Members Made of Several Materials; Stress 1-2
Concentrations
*4.6 Plastic Deformations; Residual Stresses 1-2
4.7 Eccentric Axial Loading 1-2
4.8-9 Unsymmetric Bending; General Eccentric 1-2
Axial Loading
*4.10 Bending of Curved Members 1-2
Chapter 5: Analysis and Design of Beams for Bending
5.1 Shear and Bending-Moment Diagrams 1-1.5
5.2 Using Relations Between w, V, and M 1-1.5
5.3 Design of Prismatic Beams in Bending 1-2
*5.4 Use of Singularity Functions to Determine
V and M 1-2
*5.5 Nonprismatic Beams 1-2
Chapter 6: Shearing Stresses in Beams and Thin-Walled Members
6.1 Shearing Stresses in Beams 1-2
*6.2 Shearing Stresses in Narrow Rectangular Beam 0.25
6.3-4 Shearing Stresses in Thin-Walled Members 1-2
*6.5 Plastic Deformations 0.25
*6.6 Unsymmetric Loading; Shear Center 1-2

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TABLE I: LIST OF TOPICS COVERED IN MECHANICS OF MATERIALS,
Seventh Edition (CONTINUED)
Suggested Number of Periods
Core Additional Advanced
Sections Topics Topics Topics Topics
Chapter 7: Transformation of Stress and Strain
7.1 Transformation of Plane Stress 1-2
7.2 Mohr’s Circle for Plane Stress 1-2
7.3-4 Three-Dimensional Analysis of Stress 0.5-1
*7.5 Yield and Fracture Criteria 0.5-1
7.6 Thin-Walled Pressure Vessels 0.5-1
*7.7-8 Analysis of Strain; Mohr’s Circle 1-1.5
*7.9 Strain Rosette 0.5
Chapter 8: Principal Stresses under a Given Loading
8.1 Principal Stresses in a Beam 0.5-1
8.2 Design of Transmission Shafts 0.5-1
8.3 Stresses under Combined Loadings 1-3
Chapter 9: Deflection of Beams
9.1-1A Equation of Elastic Curve 0.5-1
*9.1B Direct Determination of Elastic Curve from 0.5
Load Distribution
9.2 Statically Indeterminate Beams 0.5 - 1
*9.3 Use of Singularity Functions 1-2
9.4 Method of Superposition 1-2
Application of Moment-Area Theorems to:
*9.5 Cantilever Beams and Beams with 1-2
Symmetric Loadings
*9.6A-6B Beams with Unsymmetric Loadings; Maximum 1-1.5
Deflection
*9.6C Statically Indeterminate Beams 0.5
Chapter 10: Columns
10.1 Euler’s Column Formula 1-2
*10.2 Eccentric Loading; Secant Formula 1
10.3 Design of Columns under a Centric Load 1-2
10.4 Design of Columns under an Eccentric Load 1-2
Chapter 11: Energy Methods
11.1-2 Strain Energy 1-2
11.3 Strain Energy for General State of Stress 0.5
11.4 Impact Loading 0.5-1
11.5 Deflections by Work-Energy Method 0.5-1
*11.6-8 Castigliano’s Theorem 1-2
*11.9 Statically Indeterminate Structures 1-2
_______ _______ _______
Total Number of Periods 24-41½ 21-38½ 3-6

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