Table Of ContentCopyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
МИНИCTEPCTBO ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ
ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ АВТОНОМНОЕ
ОБРАЗОВАТЕЛЬНОЕ УЧРЕЖДЕНИЕ ВЫСШЕГО ОБРАЗОВАНИЯ
«СЕВЕРО-КАВКАЗСКИЙ ФЕДЕРАЛЬНЫЙ УНИВЕРСИТЕТ»
Е. А. Чеботарев, Х. Р. Сугаров
СОПРОТИВЛЕНИЕ МАТЕРИАЛОВ
УЧЕБНОЕ ПОСОБИЕ
на английском языке
Направление подготовки 08.03.01 – Строительство
Профиль подготовки «Промышленное
и гражданское строительство»
Бакалавриат
E. Chebotarev, Kh. Sugarov
STRENGTH OF MATERIALS
COURSE OF LECTURES AND ASSIGNMENTS
For students major in Civil Engineering
Ставрополь
2017
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
УДК 539. 3/.6 (075.8) Печатается по решению
ББК 30.121 я 73 редакционно-издательского совета
Ч 34 Северо-Кавказского федерального
университета
Чеботарев Е. А., Сугаров Х. Р.
Ч 34 Сопротивление материалов: учебное пособие на англ. яз. –
Ставрополь: Изд-во СКФУ, 2017. – 205 с.
Пособие создано в соответствии с федеральными государственными
образовательными стандартами в технических областях. Изложены ос-
новные темы курса "Сопротивление материалов", необходимых для фор-
мирования общего машиностроения (профессиональных) компетенций.
Приведены примеры расчетов практических задач, а также задания для
самостоятельной работы студентов.
Предназначено для студентов, обучающихся по направлению подго-
товки 08.03.01 – Строительство, профилю «Промышленное и гражданское
строительство».
УДК 539. 3/.6 (075.8)
ББК 30.121 я 73
E. Chebotarev, Kh. Sugarov
Strength of materials: course of lectures and assignments. –
Stavropol: Publisher NCFU, 2017. – 205 p.
The textbook was created in line with the Federal State Educational
Standards in technical areas. The textbook describes the main topics of the
course "Strength of Materials", necessary for the formation of general engineer-
ing (professional) competencies. The examples of calculations of practical
problems are given as well as the tasks for students self-work.
The textbook is intended for students major in Civil Engineering.
Reviewers:
Doctor of Technical Sciences А. Bratsikhin
Candidate of Technical Sciences, Associate Professor
A. Malsugenov (Don State Technical University)
© ФГАОУ ВО «Северо-Кавказский
федеральный университет», 2017
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▪ CONTENTS ▪
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1. Fundamental concepts of Strength of Materials . . . . . . . . . . . . 5
2. The method of sections and classification of loadings . . . . . . . . 11
3. Engineering methods of strength calculations . . . . . . . . . . . . . . 14
4. Axial tension or compression . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5. Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6. Geometrical properties of cross-sections . . . . . . . . . . . . . . . . . 46
7. Direct (simple) bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8. Oblique (biaxial, unsymmetrical) bending
(bending in two planes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9. Simple shearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10. Bending with tension (compression) . . . . . . . . . . . . . . . . . . . . 104
11. Eccentric tension (compression) . . . . . . . . . . . . . . . . . . . . . . . 107
12. Strength and fracture theories . . . . . . . . . . . . . . . . . . . . . . . . . 115
13. Torsion with bending of round shaft . . . . . . . . . . . . . . . . . . . . . 137
14. Displacements of framed structure . . . . . . . . . . . . . . . . . . . . . 141
15. Stability of compressed rods (Structural Stability) . . . . . . . . . . 161
16. Strength under cyclic loadings (Fatigue Strength) . . . . . . . . . . 167
Tests to verify acquired knowledge . . . . . . . . . . . . . . . . . . . . . . . . 173
The recommended literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
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▪ INTRODUCTION ▪
Strength of materials is an engineering science about methods of
calculation of the most widespread elements of designs in terms of
strength, rigidity (stiffness) and stability at simultaneous satisfaction of
the reliability and profitability requirements.
Practical confirmation of the main provisions of the discipline
"Strength of Materials" has important methodological significance in the
formation of an engineering student's thinking.
During the study of the discipline "Strength of materials" student
major in Civil Engineering acquire following competencies:
GPC-1 – Ability for data searching, storing, processing and analy-
sis using different data sources and database; ability to represent data
in required format using information, computer and network technolo-
gies;
GPC-2 – Ability to apply basic foundations of Science to his pro-
fessional activity; ability to apply methods of mathematical analysis and
modeling, theoretical and experimental studies.
The main objective of a basic mechanics course should be to de-
velop in the engineering student the ability to analyze a given problem
in a simple and logical manner and to apply to its solution a few funda-
mental and well-understood principles. This textbook aims to help stu-
dents taking course taught in English at North-Caucasus Federal Uni-
versity, Civil Engineering department, in their studies of one of the most
important and, at the same time, most difficult engineering topics.
This course will be taken not only by foreign students (speaking
good English and knowing technical and mathematical terms in Eng-
lish), but also by Russian students intending to improve their English
while studying a professional subject.
The authors would like to thank А. Bratsikhin and A. Malsugenov
who have assisted in preparing of this book by the reading a manu-
script, commentaries and suggestions.
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1. FUNDAMENTAL CONCEPTS
OF STRENGTH OF MATERIALS
(Lectures – 2 hours)
Strength of materials is an engineering science about methods of
calculation of the most widespread elements of designs in terms of
strength, rigidity (stiffness) and stability at simultaneous satisfaction of
the reliability and profitability requirements.
Strength is a property of a design or construction to withstand ex-
ternal impact without failure or development of excessive deformations.
Rigidity is an ability of a design or construction to resist the defor-
mation or to resist the external influences without considerable defor-
mations.
Stability is an ability of a design and its elements to keep the initial
equilibrium form unchanged under loading.
Reliable design is a design that keeps the operational ability during
the provided time interval. The increase in sizes raises the reliability of
a design but reduces its profitability (i.e. increases its cost), and con-
versely. The methods of strength of materials allow optimizing these
parameters.
The vast variety of designs can be divided into elements, which
can be attributed to one of three groups by their form.
The first group is a variety of elements with two dimensions much
smaller in comparison to the third dimensional size. Such elements are
called as bars (rods, beams).
Geometrically the
form of a bar can be
formed by the movement
of a flat figure (section)
along some line (bar ax-
is) so that the plane of a
figure would be perpen-
dicular to the axis of a
bar and would cross it in
the center of gravity (Fig- Figure 1.1
ure 1.1).
Each instant position of this flat figure is called as the cross-
section of a bar.
Depending on a form of an axis rectilinear and curvilinear bars are
distinguished. Their cross-sections can be constant along of an axis or
change smoothly or in steps.
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The rectilinear bar stretched or compressed by forces along its ax-
is is also called as a rod.
The design consisting of rods is called as a rod system. The rod
system consisting of rectilinear rods connected by hinges on their edg-
es is called as a truss. The rod system with rigidly connected rods is
called as a frame. Vertical rods of a frame are called frame legs, hori-
zontal – crossbars.
Round bar that withstands to torsion is called a shaft.
If the rectilinear bar is loaded so that its axis is bent at defor-
mation, it is called as a beam. The beam is called as a simply-
supported beam if both edges are supported – one by pinned support
and another by roller (Figure 1.2a) or cantilever beam, or console, if
only one edge is rigidly fixed (Figure 1.2b).
a b
Figure 1.2. Types of beams
Bodies with one of the dimensional sizes much smaller in compari-
son with two others belong to the second group and are called as
shells. Flat shells are called as thin plates.
The third group is formed by bodies with all three dimensional siz-
es about the same and is called as massifs or a solid blocks.
The subject "Strength of materials" generally studies the first group
of bodies.
There is vast variety of classification of forces. In the course of
mechanics, particularly in the course of strength of materials, following
types of external forces are considered.
1) Surface forces, i.e. forces exerted on the body surface:
i) point (concentrated) forces, which we consider as spot loadings alt-
hough they are actually distributed over a certain surface (which is very
small in comparison with the whole surface), e.g. concentrated loads on
beams, forces in hangers, reactions in supports, etc. (Figure 1.3);
ii) distributed loads, which are applied across the length or area in-
stead of at one point. Strictly speaking, all the forces are distributed
over some finite area (even a needle eventually is acting upon some
area, but obviously can be considered as concentrated force on macro-
scopic level).
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Figure 1.3
Surface forces can be subdivided into:
i) active, which are caused by some external resources (no matter
what is the nature of this resources) and often called as external load-
ings or simply loadings (loads);
ii) reactive, which are caused by the supports of the body and are
called as support reactions.
2) Volume forces (i.e. forces generated by a field of force), loading
the whole body mass, e.g. the dead weight of a body, the force of iner-
tia, centrifugal force, etc.
Types of forces can be also divided with respect to:
1) changing of the force over time:
i) static loading, which can be either constant or slowly increasing
from a certain value (usually from zero) up to its nominal magnitude,
and then remains unchanged. Slow force application is necessary so
that deformation time changes can develop fully and the forces of iner-
tia can be neglected;
2) dynamic loading, which can be for instance:
i) impact load, characterized by great instantaneous load accelera-
tion;
ii) cyclic load (periodically changing), which can lead to body fa-
tigue fracture;
3) point of the force application:
i) fixed load, i.e. the load point does not change in time;
ii) moving load, i.e. the load point changes in time, e.g. crane crab,
a train loading a bridge, etc.
A distributed load can be equated with a concentrated load applied
at a specific point along the bar. The magnitude of the resultant force is
equivalent to the area under the curve of the distributed load. The loca-
tion of the resultant force is at the center of mass of the distributed load.
A special case of loading is so called force couple (we will call it as
concentrated moment). A couple consists of two parallel forces that are
equal in magnitude, opposite in direction and do not share a line of ac-
tion. It does not produce any translation, only rotation. It can be repre-
sented this way: imagine a tap rigidly fixed to a beam and that we are
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trying to rotate tap as if we are turning on the water. The resultant force
of a couple is zero. However, the resultant of a couple is not zero; it is a
pure moment.
The following principles and hypothesis are used to solve prob-
lems of strength of materials:
1. Hypothesis of continuity, which assumes that materials, of which
the bodies are made, are continuous.
2. Hypothesis of uniformity, which assumes that properties of ma-
terial in all points of a body are identical.
3. Principle of small deformations, which assumes that we can
write down the balance equations ignoring the deformations (according
to their small values) and using unstrained sizes and forms of a design
and its parts.
4. Principle of independency of action of forces (principle of super-
position). According to this principle the deformation of a design that is
working under a group of loadings does not depend on the sequence of
the appliance of each force and it is equal to the sum of the defor-
mations for each of loading separately.
The property of bodies to return their initial form after unloading is
called as elasticity and the deformations that disappear after unloading –
elastic. The deformations that remain after release of applied loading
are called as plastic deformations (or residual).
Under the action of external forces a body deforms and the forces
of interaction between particles of the body appear. These forces are
called as internal forces.
The intensity P of internal forces for some of the cross-sectional
area A is characterized by the magnitude of the stresses
P дР
р lim , (1.1)
F0A дA
where ΔР – resultant of internal forces on the elementary (indefinitely
small) area ΔA of cross-section A.
The stress is a vector. The vector of a full stress can be divided to
to components (Figure 1.4):
– on a normal (perpendicular) to section, this component is called
normal stress (σ);
– in the cross-section plane, this component is called tangential
stress (τ). Stress which is tangential to the surface is often called shear-
ing stress.
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Figure 1.4
The units of stress measure are usually pascals 1 Pa = 1 N/m2
(more often 1 MPa = 106Pa). But there are also other units used, such
as kg/m2, kN/mm2 and so on. They can be easily converted one to an-
other.
Deformations arising under the loadings are subdivided to linear
and angular. Linear deformations are named lengthening, assuming
that it is positive if the size of the body increases and conversely – neg-
ative if size of the body decreases (Figure 1.5).
Figure 1.5
Strain (or reduced deformation or relative lengthening) is a math-
ematical term that expresses the trend of the deformation change within
the material field. Strain is the deformation per unit of length:
l l l
x ; y ; z , (1.2)
х l y l z l
x y z
where l , l , l – absolute lengthening’s (deformations) in the
x y z
l l l
direction of coordinate axes; , , – the initial size of a deforma-
x y z
ble body on the corresponding axes of coordinates.
In the case of uniaxial loading the displacements of a specimen (for
example a bar element) lead to a calculation of strain expressed as the
quotient of the displacement and the original length of the specimen.
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Angular deformations are called shifts (Figure 1.6) and are consid-
ered positive if the right angle at the deformation decreases.
Figure 1.6
Values of γ , γ and γ are called as angles of the shift.
xy yz zx
Body deformations are called ideally elastic if after removal of
loading the body restores at once to the initial size and form. For the
majority of rigid materials, there is a certain limit of deformations below
which deformations are elastic and practically proportional to the ap-
plied loading. Such deformations are called linearly-elastic.
A creep is the process of deformations change under the constant
loading.
The phenomenon of the reduction of a pressure at constant de-
formation is called relaxation.
▪ Control questions
1. What does the term «strength of materials» mean?
2. What does the term «durability» mean?
3. What does the term «rigidity» mean?
4. What does the term «stability» mean?
5. Name and describe three groups of bodies. What particular
types of bars there are?
6. Classification of external loads, their description.
7. What are the hypothesis and principles used to solve problems
of strength of materials?
8. What's the main idea of Hypothesis of continuity?
9. What's the main idea of Hypothesis of uniformity?
10. What is the principle of small deformations?
11. What is the principle of forces action independence?
12. Describe Saint-Venant’s principle.
13. What is called as internal forces, stresses? What types of
stresses there are? What are the units to measure forces and stresses?
14. What does the term strain mean? What types of deformations
there are? What is called as lengthening, shift?
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