in the name of god
For NMAA and Kardan University
Faculty of Engineering
Lecture by :
LT “Sayed Dawod karimi”
dawodkarimi2007@gmail.com
0799560376
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Lesson #05
Strain
• 1. Lesson Objectives
• At the conclusion of this lesson, you should be
able to do the following:
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• Define the terms displacement, deformation,
strain, modulus of elasticity, and Poisson’s Ratio.
• Apply Hooke's Law to calculate stress or strain in
the elastic region.
• Apply Poisson's Ratio to calculate lateral strain
and lateral deformation.
2. Orientation
Why is this topic important?
• Deformation is an important measure of the
performance of a structural element or an
entire structure. Strain, a non-dimensional
measure of the intensity of deformation, is
used extensively in the characterization of
materials. Because it is easily measured and
can be mathematically related to stress, strain
can be used to indirectly measure the stress in
a member.
How does this topic relate to prior learning?
• In Lesson 15, we were introduced to normal
stress in an axially loaded member. In Lesson
17, we will learn how stresses and strains are
related to each other.
3. Reading Assignment
• Riley, Sturges, and Morris: Study Sections 4-4
and 4-5
4. Key Definitions
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Displacement – A movement of a point with respect to a reference axis system. In a body or
system of bodies subjected to forces, displacements may be caused by the translation or
rotation of a body as a whole, or they may result from a change in the dimensions or shape
of a body.
Deformation – A change in the dimensions or shape of a body. Normal (axial) deformation
(dn) is calculated as dn = Lf - Lo, where Lf is the final length and Lo is the original length.
Strain – A measure of the intensity of deformation. Normal strain (e) is calculated as
e = dn / Lo, where dn is the normal (axial) deformation and Lo is the original length. Strain is
a non-dimensional quantity; however, normal strain is often expressed in terms of in/in or
mm/mm. Tensile normal strain is considered to be positive (+), while compressive normal
strain is considered to be negative (-).
Poisson's Ratio – A material property representing the ratio of lateral to longitudinal strain.
Poisson's Ratio (n) is calculated as n = -elat /elong, where elat is the lateral strain, measured
perpendicular to the direction of the applied load, and elong is the longitudinal strain,
measured parallel to the direction of the applied load.
Modulus of Elasticity - A material property representing the slope of the normal stress-strain
(s-e) curve in the elastic region. The modulus of elasticity is also called Young's Modulus.
Hooke's Law - The linear relationship between normal stress (s) and normal strain (e).
Hooke's Law is written mathematically as s = Ee, where E is the modulus of elasticity.
5. Lesson Notes
Why are Deformations Important?
• they did not have to allow for as much uncertainty in their
design, and so they were able to use considerably less steel
in the building. Less steel made the structure lighter and
more economical than any previous skyscraper had been.
However, the building was also much more flexible than it
should have been. It was susceptible to much larger
deformations than older, heavier buildings had been, and
so even in light winds, the tower flexed and twisted far
more than anyone had anticipated. Because window glass
is itself very stiff, the windows were unable to flex along
with the building frame, and they simply popped out of
their mountings. It is important to note that the structural
design was, in fact, safe. The stresses in the structural
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members were within allowable limits. Nonetheless, the building failed to fulfill
its intended function because of excessive deformations.
A deformation is simply a change in the dimensions or shape of a body.
Deformations can be caused by applied loads or by changes in temperature.
Deformations are important to engineers for four principal reasons:
Deformation is a measure of performance. As the case of the John Hancock
tower illustrates, when structural elements deform excessively, an entire
structure may fail to function as intended.
The tendency to deform under load is one of several methods commonly used to
characterize materials.
Unlike stresses, deformations can be directly measured. And because there is a
clear mathematical relationship between stress and deformation, measured
deformations can be used to indirectly determine the stresses in a body. Thus
deformations are very important to our understanding about how structures
work.
Deformations are particularly valuable for analyzing statically indeterminate
structures. A statically indeterminate structure is one for which the equations of
equilibrium alone are not sufficient to solve for all unknown reactions and
internal forces. In such problems, deformations are used as the basis for
formulating the additional equations necessary to analyze the structure. We will
not analyze statically indeterminate structures in CE-301, but civil engineering
majors will do so in a later course.
Deformation and Strain
• where dh and dw are the changes in height and width dimensions,
respectively. It is important to note that, while there are two
different expressions for lateral strain, a given axially loaded
member with a given applied load has just one value of lateral
strain. The lateral deformations might be different in the two
lateral directions—but the lateral strain is always the same in both
directions.
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• In all cases, the general expression for normal strain is the
deformation divided by the original length. Because this calculation
is always a length divided by a length, strain is actually a
dimensionless quantity. However, as a standard convention, strain
is usually expressed as inches per inch (in/in) or millimeters per
millimeter (mm/mm), to emphasize how it is calculated.
• It is important to recognize, however, that this equation is rarely
applied in the form shown above. Because Poisson’s Ratio is a
material property, it is usually obtained from reference tables like
Table A-17 and A-18 in your textbook. The equation above is
typically used to calculate lateral or longitudinal strain for a given
material.
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• Poisson’s Ratio is a scalar quantity, usually between 0.1 and 0.4 for
most common materials. The minus sign in the equation above
reflects that fact that longitudinal elongation (positive elong) is
always accompanied by a decrease in the lateral dimensions
(negative elat), and longitudinal shortening (negative elong) is always
accompanied by an increase in the lateral dimensions (positive elat).
The Stress-Strain Curve and Hooke’s Law
• At any given magnitude of load, this steel specimen experiences some
normal stress, s, which is equal to the load divided by the cross-sectional
area of the specimen. At the same time, this load causes the specimen
to elongate. If we measure the elongation, we can calculate the
corresponding longitudinal strain, e, as the measured deformation
divided by the original length.
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• As we increase the load applied to the test specimen, both the stress and
strain will increase simultaneously as well. When the resulting
relationship between stress and strain is plotted as a graph, the result is
a stress-strain curve. Figure 4-28 in your textbook shows stress-strain
curves for three different materials—steel, magnesium, and cast iron.
For each of these examples, it is important to recognize that the stressstrain curve is a function solely of the material being tested. The size
and shape of the test specimens have no effect on the curve. For a given
material, the stress-strain curve will always be essentially the same. And
note each of the three stress-strain curves has a characteristic shape and
characteristic magnitudes of the key points on the curve—illustrating the
important point that the stress-strain curve is used to characterize a
material.
• We will examine stress-strain curves in more detail next
lesson. For now, it is only necessary to note two key
characteristics of the example curves shown in Figure 4-28.
First, a stress-strain curve always starts at the origin. This is
expected, because zero load results in zero deformation,
and so zero stress should result in zero strain.
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• Second, note that stress and strain tend to increase in
direct proportion to each other within the lower portion of
the curve, especially for steel and magnesium. This linear
portion of the stress-strain is called the elastic region. The
term elastic means that, if the load is removed from the
material, it will return to its original shape with no
permanent deformation. Since it is highly desirable that
structural components and machine parts not experience
permanent deformation under normal loading conditions,
engineers almost always design such components to
function safely within the elastic region.
• The linear relationship between stress and strain in the elastic
region of the stress-strain curve is one of the most important
concepts underlying all of structural mechanics. This linear
relationship is called Hooke’s Law, named after the great British
scientist Robert Hook, and can be expressed mathematically as
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• where E is the modulus of elasticity, the slope of the elastic portion
of the stress-strain curve. The modulus of elasticity is a material
property and is also sometimes called Young’s Modulus, after the
British mathematician and scientist, Thomas Young. Because strain,
e, is a dimensionless quantity, the units for the modulus of elasticity
are the same as for stress—psi, ksi, N/m2, Pa, MPa, etc.
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• Textbook Example Problems 4-8 and 4-10 provide good examples of
the calculation of stress, strain, modulus of elasticity, and Poisson’s
ratio. In Example Problem 4-10, ignore the reference to the
modulus of rigidity. We will work with this property in a future
lesson.
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In-Class Problem 17-1
End of lesson
Question
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