STRENGTH OF MATERIALS
I3681VN
STRESSES AND STRAINS IN ONE DIMENSIONS
LECTURER
MS. EMILIA MUPWEDI
DEPARTMENT OF CIVIL AND MINING ENGINEERING
UNIVERSITY OF NAMIBIA
STRESS AND STRAIN IN ONE-DIMENSION
On completion of this lecture, students should be able to:
✓Define Stress
✓Components of Stress
✓One Dimensional Stress Systems
✓General State of Stress
✓Strain
STRESS AND STRAIN IN ONE-DIMENSION
Revision from Engineering Mechanics:
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•
•
•
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Scalar and Vector Quantities
Forces and Resultant Forces
Parallelogram Law of Forces
Triangle Law of Forces
Moment of a Force
STRESS AND STRAIN IN ONE-DIMENSION
Branches of Engineering
Mechanics
STRESS AND STRAIN IN ONE-DIMENSION
Definition:
The strength of a material may be defined as ability, to resist its
failure and behaviour, under the action of external forces. It has been
observed that, under the action of these forces, the material is first
deformed and then its failure takes place. A detailed study of forces
and their effects, along with some suitable protective measures for
the safe working conditions, is known as Strength of Materials.
STRESS AND STRAIN IN ONE DIMENSION
Therefore, strength of material is defined by:
• Internal stresses or intensity of force in it.
STRESS AND STRAIN IN ONE-DIMENSION
Let us look at the effects produced by the application of loads on the
material:
• Elasticity
• Stress
✓ Normal Stress
✓ Shear Stress
• Strain
STRESS AND STRAIN IN ONE-DIMENSION
Let us look at the effects produced by the application of loads on the
materials:
Elasticity
The property of certain materials of returning back to their original
position, after removing the external force, is known as elasticity.
STRESS AND STRAIN IN ONE-DIMENSION
A body is said to be perfectly elastic, if it returns back completely to
its original shape and size, after the removal of external forces.
If the body does not return back completely to its original shape and
size, after the removal of the external force, it is said to be partially
elastic.
STRESS AND STRAIN IN ONE-DIMENSION
Stress
Every material is elastic in nature. That is why, whenever some
external system of forces acts on a body, it undergoes some
deformation. As the body undergoes deformation, its molecules set
up some resistance to deformation. This resistance per unit area to
deformation, is known as stress.
STRESS AND STRAIN IN ONE-DIMENSION
Mathematically stress may be defined as the force per unit area:
𝑃
𝜎=
𝐴
Where,
𝑃 – Load or force acting on the body
𝐴 – Cross-sectional area of the body
STRESS AND STRAIN IN ONE-DIMSION
➢ Stress has the same units as pressure, and in fact pressure is one
special variety of stress. However, stress is a much more complex
quantity than pressure because it varies both with direction and
with the surface it acts on.
➢ Compression: Stress that acts to shorten an object.
➢ Tension: Stress that acts to lengthen an object.
STRESS AND STRAIN IN ONE-DIMENSION
Normal Stress
Stress that acts perpendicular to a surface.
Can be either compression or tension.
Cross-section: Section perpendicular to
longitudinal axis of the bar
STRESS AND STRAIN IN ONE-DIMENSION
Tensile Stress is induced in a body
when it is subjected to two equal and
opposite pulls.
STRESS AND STRAIN IN ONE-DIMENSION
Compressive Stress is induced in a
body when two equal and opposite
pushes.
STRESS AND STRAIN IN ONE-DIMENSION
Direct Stress: Stress that varies with direction.
➢ Stress under a stone slab is directed; there is a force in one
direction but no counteracting forces perpendicular to it.
➢ This is why a person under a thick slab gets squashed but a scuba
diver under the same pressure doesn’t. The scuba diver feels the
same force in all directions
STRESS AND STRAIN IN ONE-DIMENSION
Stress is denoted by 𝝈.
➢ 𝝈𝒙 , 𝝈𝒚 , 𝝈𝒛 : represents the components of normal stress in the x,
y and z direction.
STRESS AND STRAIN IN ONE-DIMENSION
Stress (𝝈) sign convention:
𝝈 > 𝟎, Positive
𝝈 < 𝟎, Negative
STRESS AND STRAIN IN ONE-DIMENSION
Shear Stress (𝝉) acts parallel to a surface. It can cause one object to
slide over another. It also tends to deform originally rectangular
objects into parallelograms.
STRESS AND STRAIN IN ONE-DIMENSION
Shear Stress (𝝉) is denoted by a double suffix notation, 𝝉𝒙𝒚 .
➢ The first suffix gives the direction of the normal to the plane on
which the stress is acting.
➢ The second suffix indicates the direction of shear stress
component.
STRESS AND STRAIN IN ONE-DIMENSION
Shear Stress (𝝉) sign convention:
𝝉 > 𝟎, Positive
𝝉 < 𝟎, Negative
STRESS AND STRAIN IN ONE-DIMENSION
Strain
As already mentioned, whenever a single force (or a system of
forces) acts on a body, it undergoes some deformation. This
deformation per unit length is known as strain.
Therefore, strain is defined as the amount of deformation an object
experiences compared to its original size and shape.
Note that strain is dimensionless.
STRESSES AND STRAIN
Mathematically strain may be defined as the deformation per unit
area:
𝛿𝑙
𝜀=
𝑙
Where,
𝛿𝑙 – Change of length of the body
𝑙 – Original length of the body
STRESSES AND STRAIN
Longitudinal or Linear Strain
Strain that changes the length of a line without changing its direction.
Can be either compressional or tensional.
➢ Compression Longitudinal strain that shortens an object.
➢ Tension Longitudinal strain that lengthens an object.
STRESSES AND STRAIN
Shear strain: Strain that changes the angles of an object. Shear
causes lines to rotate.
Infinitesimal Strain : Strain that is tiny, a few percent or less.
Allows a number of useful mathematical simplifications and
approximations.
Finite Strain Strain larger than a few percent. Requires a more
complicated mathematical treatment than infinitesimal strain.
STRESSES AND STRAIN
Homogeneous Strain Uniform strain. Straight lines in the original
object remain straight. Parallel lines remain parallel. Circles deform
to ellipses. Note that this definition rules out folding, since an
originally straight layer has to remain straight.
Inhomogeneous Strain How real geology behaves. Deformation
varies from place to place. Lines may bend and do not necessarily
remain parallel.
STRESS AND STRAIN IN ONE-DIMENSION
STRESS-STRAIN RELATIONSHIP
➢ Hooke’s law
States that when a material is loaded, within its elastic limit, the
stress is proportional to the strain.
𝑆𝑡𝑟𝑒𝑠𝑠
= 𝐸 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑆𝑡𝑟𝑎𝑖𝑛
STRESS AND STRAIN IN ONE-DIMENSION
STRESS-STRAIN
RELATIONSHIP
➢ From OA the material
obeys Hooke’s law
• Material is able to
regain shape
after load is removed
• Strain would return to
zero
STRESS AND STRAIN IN ONE-DIMENSION
STRESS-STRAIN
RELATIONSHIP
➢ At point B, the point is
called elastic limit
• If the material is stressed
beyond this
point, some plastic
deformation will
occur i.e. strain is not
recoverable if
load is removed
STRESS AND STRAIN IN ONE-DIMENSION
STRESS-STRAIN
RELATIONSHIP
➢ Point C is yield point; there
is an appreciable strain
even without further
increases in load
➢ At point D, the bar begins
to form a local “neck”
➢ At point E, fracture takes
place
STRESS AND STRAIN IN ONE-DIMENSION
STRESS-STRAIN
RELATIONSHIP
➢ Maximum or ultimate
tensile stress is calculated
by dividing the load at D
by the original crosssection area
➢ Note: in design the
material will only be used
in the range OA
STRESS AND STRAIN IN ONE-DIMENSION
Ductility
The capacity of a material to be drawn plastically before breaking is
called ductility and is measured using the two quantities.
Hooke’s law states that strain is proportional to the stress producing
it
• This law is obeyed within certain limits of stress by most ferrous
alloys, it can be assumed to apply with sufficient accuracy to other
engineering materials such as timber, concrete and non-ferrous
alloys.
STRESS AND STRAIN IN ONE DIMENSION
STRESS AND STRAIN
RELATIONSHIP
The result of the tensile test of a typical
ductile material such as mild steel is
shown on the right. The result is in the
form of stress strain relationship.
Stress-Strain diagram for a ductile material
STRESS AND STRAIN IN ONE-DIMENSION
Modulus of Elasticity
Within the limits for which Hooke’s law is obeyed, the ratio of the
direct stress to the strain produced is called Young’s modulus (E)
For a bar of uniform cross-section area A and length 𝑙
STRESS AND STRAIN IN ONE-DIMENSION
The principle of superposition
• When a number of loads are acting together on an elastic material,
the principle of superposition states:
✓ that the resultant strain will be the sum of the individual strains
caused by each load acting separately.
STRESS AND STRAIN IN ONE-DIMENSION
The relation for the resulting deformation may be modified as:
𝑃𝑙
1
𝛿𝑙 =
=
= (𝑃1 𝑙1 + 𝑃2 𝑙2 + 𝑃3 𝑙3 + … )
𝐴𝐸 𝐴𝐸
Where,
𝑃1 – Force acting on section 1
𝑙1
– Length of section 1
𝑃2 , 𝑙2 – Corresponding values of section 2, and so on.
STRESS AND STRAIN IN ONE-DIMENSION
Class Example:
A steel bar of cross-sectional area 200 𝑚𝑚2 is loaded as shown in the
Figure below. Find the change in length of the bar. Take E as 200
GPa.
STRESS AND STRAIN IN ONE-DIMENSION
Class Example:
A brass bar, having cross-sectional area of 500 mm2 is subjected to
axial forces as shown in the Figure shown below. Find the total
elongation of the bar. Take E = 80 GPa.
STRESS AND STRAIN IN ONE-DIMENSION
Poisson’s ratio
• If a bar is subjected to a longitudinal stress, there will be a strain in
𝜎
the longitudinal direction equal to
𝐸
• Other strain will be in all directions at right angles to 𝜎
• For an elastic material the lateral strain is proportional to the
longitudinal strain and is of opposite type
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
= −𝑣
𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
STRESS AND STRAIN IN ONE-DIMENSION
Poisson’s ratio
• This ratio is called Poisson’s ratio named after Simeon Poisson
𝜎
𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 = −𝑣
𝐸
• The negative sign is included here since longitudinal elongation
(positive strain) causes lateral contraction (negative strain), and
vice versa.
STRESS AND STRAIN IN ONE-DIMENSION
Class Example:
𝑣𝑠𝑡 = 0.32
𝐸 = 200 𝐺𝑃𝑎
STRESS AND STRAIN IN ONE-DIMENSION
Shear Modulus or Modulus of rigidity: Self Study
It has been experimentally found that within the elastic limit, the shear
stress is proportional to the shear strain.
𝜏 ∝ φ or 𝜏=C × φ
Where,
𝜏 – Shear stress
Φ– Shear strain
C – A constant, known as shear modulus or modulus of rigidity.
It is also denoted G or N.
STRESS AND STRAIN IN ONE-DIMENSION
Thermal Stress and Strain
• A change in temperature can cause a body to change its
dimensions.
• Generally, if the temperature increases, the body will expand,
whereas if the temperature decreases, it will contract.
• Ordinarily this expansion or contraction is linearly related to the
temperature increase or decrease that occurs.
STRESS AND STRAIN IN ONE-DIMENSION
Thermal Stress and Strain
• If this is the case, and the material is homogeneous and isotropic,
it has been found from experiment that the displacement of the end
of a member having a length L can be calculated using the
formula.
𝛿𝑇 = ∝ ∆𝑇𝐿
STRESS AND STRAIN IN ONE-DIMENSION
Thermal Stress and Strain
𝛿𝑇 =∝ ∆𝑇𝐿
Where,
𝛿𝑇 - the algebraic change in the length of the member
∝ - a property of the material, referred to as the
linear coefficient of thermal expansion
∆𝑇- the algebraic change in temperature of the member
𝐿 - the algebraic change in the length of the member
STRESS AND STRAIN IN ONE-DIMENSION
Thermal strain
𝛿𝑇
𝜀𝑇ℎ𝑒𝑟𝑚𝑎𝑙 =
=∝ ∆𝑇
𝐿
Coupled strain
If we consider both mechanical strain and thermal strain in the
structure, then the total strains in the X direction would be computed
as:
𝜎𝑥
𝜀 𝑥 = 𝜀𝑇 + 𝜀 =∝ ∆𝑇 +
𝐸
STRESS AND STRAIN IN ONE-DIMENSION
Axially Loaded Bars
✓Introduction
✓Stresses in Bars of Different Sections
✓Stresses in the Bars of Uniformly Tapering Sections
✓Stresses in the Bars of Composite Structures
✓Stresses in Simple Statically Indeterminate Structures
STRESS AND STRAIN IN ONE-DIMENSION
Introduction
In the previous lectures, we developed the concept of stress as a
means of measuring the force distribution within a body and strain as
a means of measuring a body’s deformation.
✓ We have also shown that the mathematical relationship between
stress and strain depends on the type of material from which the
body is made.
✓ In particular, if the material behaves in a linear elastic manner, then
Hooke’s law applies, and there is a proportional relationship
between stress and strain.
STRESS AND STRAIN IN ONE-DIMENSION
Saint-Venants Principle
States that the stress and strain produced at points in a body
sufficiently removed from the region of external load application will
be the same as the stress and strain produced by any other applied
external loading that has the same statically equivalent resultant and
is applied to the body within the same region.
STRESS AND STRAIN IN ONE-DIMENSION
Using Hooke’s law and the definitions of stress and strain, we will
now develop an equation that can be used to determine the elastic
displacement of a member subjected to axial loads.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in Bars of Different Sections
Sometimes a bar is made up of different lengths having different
cross-sectional areas as shown below.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in Bars of Different Sections
In such cases, the stresses, strains and changes in lengths for each
section is worked out separately as usual. The total changes in length
is equal to the sum of the changes of all the individual lengths.
✓ It may be noted that each section is subjected to the same external
axial pull or push.
𝛿𝑙 = 𝛿𝑙1 + 𝛿𝑙2 + 𝛿𝑙3 +….
STRESS AND STRAIN IN ONE-DIMENSION
Class Example
An automobile component shown in the figure below is subjected to
a tensile load 160 kN. Determine the total elongation of the
component, if its modulus of elasticity is 200 GPa.
STRESS AND STRAIN IN ONE-DIMENSION
Class Example
A member formed by connecting a steel bar to an aluminium bar is
shown below.
Assuming that the bars are prevented from buckling sidewise,
calculate the magnitude of force P, that will cause the total length of
the member to decrease by 0.25 mm. The values of elastic modulus
for steel and aluminium are 210 GPa and 70 GPa respectively.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in Bars of Uniformly Tapering Sections
So far, we have discussed the stresses in the bars of different sections
or stepped sections. But now we shall discuss the stresses in the bars
of uniformly tapering sections. Two types of uniformly tapering
sections:
✓ Bars of uniformly tapering circular sections.
✓ Bars of uniformly tapering rectangular sections.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in the Bars of Uniformly Tapering Circular Sections
Let us consider a circular bar AB uniformly tapering section shown
below:
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in the Bars of Uniformly Tapering Circular Sections
Derive Equation…
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in the Bars of Uniformly Tapering Rectangular Sections
Sometimes, the uniformly tapering section varies from square section
at one end to another square section at the other. Or it may also vary
from rectangular section at one end to another rectangular section at
the other. In such cases, the stresses should be found out from the
fundamentals.
STRESS AND STRAIN IN ONE-DIMENSION
Class Example
An alloy bar of 1 m length has square section throughout, which
tapers from one end of 10 mm × 10 mm to the other end of 20 mm ×
20 mm. Find the change in its length due to an axial tensile load of 30
kN. Take E for the alloy as 120 GPa.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in the Bars of Composite Structures
A bar made up of two or more different materials, joined together is
called a composite bar. The bars are joined in such a manner, that
the system extends or contracts as one unit, equally, when subjected
to tension or compression. Following two points should always be
kept in view, while solving example on composite bars:
✓ Extension or contraction of the bar is equal. Therefore, strain is
also equal.
✓ The total external load, on the bar, is equal to the sum of the loads
carried by the different material.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in the Bars of Composite Structures
Consider a composite bar made up of two different materials as
shown below.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in the Bars of Composite Structures
1. For the sake of simplicity, we have considered the composite bar
made up of two different materials only. But this principle may be
extended for a bar made up of more than two different materials
also.
2. If the lengths of the two bars are different, then elongations should
be separately calculated and equated.
𝐸1
3. The ratio is known as modulas ratio of the two materials and is
𝐸2
denoted by the letter m.
STRESS AND STRAIN IN ONE-DIMENSION
Class Example
A reinforced concrete circular section of 50,000 𝑚𝑚2 cross-sectional
area carries 6 reinforcing bars whose total area is 500 𝑚𝑚2 . Find the
safe load, the column can carry, if the concrete is not to be stressed
more than 3.5 MPa. Take modular ratio for steel and concrete as 18.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in Simple Statically Indeterminate Structures
• In the previous sections, we have been discussing the cases, where
simple equations of statics were sufficient to solve the examples.
• Sometimes, the simple equations are not sufficient to solve such
problems. Such problems are called statically indeterminate
problems, and the structures are called statically indeterminate
structures.
STRESS AND STRAIN IN ONE-DIMENSION
Stresses in Simple Statically Indeterminate Structures
The structures in which the stresses can be obtained by forming two
or more equations are called simple statically indeterminate
structures. The stresses in such structures may be found out with the
help of two or three compatible equations.
STRESS AND STRAIN IN ONE-DIMENSION
Class Example
A square bar of 20 mm side is held between two rigid plates and
loaded by an axial force P equal to 450 kN as shown in below. Find
the reaction at the ends A and C and extension of the portion AB.
Take E = 200 GPa.
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