The document discusses concepts of stress and strain in materials. It defines stress as an internal force per unit area within a material. Stress can be normal (perpendicular to the surface) or shear (parallel to the surface). Normal stress can be tensile or compressive. Strain is a measure of deformation in response to stress. Hooke's law states that stress is proportional to strain in the elastic region. Poisson's ratio describes the contraction that occurs perpendicular to an applied tensile load. Stress-strain diagrams are used to analyze a material's behavior under different loads. The document also discusses volumetric strain, shear stress and strain, bearing stress, and provides examples of stress and strain calculations.

1. Chapter-1
Concepts of stress and strain
 Introduction to strength of materials:
 Strength of materials is a branch of applied mechanics that deals with the
behavior of solid bodies (structural and machine members) under the action
of external loads, taking into account the internal forces created and resulting
deformation.
 It study about the load carrying capacity of various members such as beams
and columns.
 The objectives of this course is to determine stress, strain, displacement,
deformations etc. and to understand mechanical strength of materials which
is essential for safe design of all structures.
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2. Cont…
 One of our principal concerns in this course is material behavior
(Strength). But strength models are often intimately related to stress
and strain.
In statics, the concept of static equilibrium was defined and the
difference between internal and external forces introduced. However, it is
not possible solely on basis of force calculation to assess the
structural integrity of a component or structure.
 Hence: Strength of materials is obviously a core subject for
mechanical and materials engineers, since it enables us to determine
by calculation, if the components we design will function as intended
or fail. 2

3. In order to do so, we define the term stress as a measure for internal
force per area acting inside a structure.
Stresses, however, cannot be directly measured, but stain is
measurable and can be directly related to stress.
Cont…
3

4.  Normal Stress and Strain:
When a material is subjected to an external force (load), a resisting
force is set up within the component, this internal resistance force per
unit area is called stress.
A normal stress (σ), results when a member is subjected to an axial load
applied through acts in a direction perpendicular to the cut surface.
Thus, normal stressed may be either tensile (+ve) or compressive (-ve).
Normal Load (Axial load): Load is perpendicular to the supporting material.
- Tension Load: As the ends of material
are pulled apart to make the material
longer, the load is called a tension load.
- Compression Load: As the ends of
material are pushed in to make the material
smaller, the load is called a compression load. 4

5.  The simplest type of load (P) is a direct pull or push, known technically
as tension or compression.
The Force transmitted across any section, divided by the area of that
section, is called intensity of stress or stress.
Cont…
5

6. Cont…
In engineering applications, we use the original cross section area of the
specimen and it is known as conventional stress or Engineering stress.
We generally assume the normal stress distribution in an axially loaded
member is uniform, except near the vicinity of the applied load know as
Saint Venant’s Principle.

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7. True stress and true strain
Engineering stress was calculated by dividing the load “P” by the initial
cross section of the specimen.
But it is clear that as the specimen elongates its diameter decreases and
the decrease in cross section is apparent during necking phase.
Hence, the True (actual) stress which is obtained by dividing the load by
the actual cross sectional area.
Some engineering applications like metal forming process involve large
deformations and they require actual or true strains that are obtained
using the successive recorded lengths to calculate the strain.
Cont…
The difference between the true
stress and the engineering stress
is negligible for smaller loads,
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8. Strain is the concept used to compare the elongation of a material to
its original, undeformed length.
Or
Cont…
Hydraulic Machine
for Tension &
Compression test.
8

9.  Stress-Strain Diagram:
The mechanical properties of materials used in engineering
are determined by tests performed on small specimen of
materials.
Stress and strain are calculated from easily measurable
quantities (normal load, diameter, elongation, original
length) and can be plotted against one another as in Figure
below.
Such Stress-Strain diagrams are used to study the behavior of
a material from the point it is loaded until it breaks. 9

10. Cont…
10

11. Cont…
Plastic Region (Point 2 –3
 If the material is loaded beyond
the yield strength, the material
will not return to its original shape
after unloading.
11

12.  Hooke’s Law and Poisson's Ratio:
Initial part of the loading indicates a linear relationship between
stress and strain, and the deformation is completely recoverable in
this region for both ductile and brittle materials.
This linear relationship, i.e., stress is directly proportional to
strain, is popularly known as Hooke's law.
A material is said to be elastic if all the deformations are
proportional to the load.
Within the limits for which Hooke’s law is obeyed, the ratio of the
direct stress to the strain produced is called young’s modules or the
modules of Elasticity, i.e. 12

13. Most of the engineering structures are designed to function
within their linear elastic region only.
Poisson’s ratio:
Consider a rod under an axial tensile load P as shown in figure
above such that the material is within the elastic limit.
The normal stress on x plane is σxx = P/A and the associated
longitudinal strain in the x-direction can be found from εx =σxx/E.
Cont…
13

14. As the material elongates in the x-direction due to the load P, it also
contracts in the other two mutually perpendicular directions, i.e., y and
z directions.
Hence, despite the absence of normal stresses in y and z directions,
strains do exist in those directions and they are called lateral strains.
The ratio between the lateral strain and the axial/longitudinal strain
for a given material is always a constant within the elastic limit and this
constant is referred to as Poisson's ratio. It is denoted by ν or μ.
Cont…
14

15. Since the axial and lateral strains are opposite in sign, a negative sign is
introduced in equation.
Using the above equation, the lateral strain in the material can be obtained
by:
Poisson's ratio can be as low as 0.1 for concrete and as high as 0.5 for rubber.
In general, it varies from 0.25 to 0.35 and for steel it is about 0.3.
Cont…
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16. Cont…
Moreover:
16

17. Example 1: A Steel rod (E=200 GPa) has a circular cross section
and is 10m long. Determine the minimum diameter if the rod
must hold a 30 kN tensile force without deforming more than
5mm. Assume the steel stays in the elastic region. Note, 1 GPa = 10^9 Pa.
Cont…
17

18.  Volumetric Strain (εv) and Bulk Modulus (K):
When a body is subjected to three mutually perpendicular stresses of
equal intensity the ratio of direct stress to the corresponding volumetric
strain is known as bulk modules (K) i.e. K = σ/εv.
Then, the body undergoes uniform changes in three directions without
the distortion of the shape.
The ratio of change in volume to
original volume has been defined as
Volumetric strain (εv).
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Relation between K and E…???

19.  Shear Stress and Strain:
A shear stress (τ) can be defined as the shear force (V) (which acts
parallel or tangential to the surface) divided the cross sectional area (A)
in which it acts.
 Shear stress can be explained using rectangular block subjected to a
shear force “P” tangential at top surface as shown below.
19

20. Cont…
20
 Consider a bolted connection in which two plates are connected by a
bolt with cross section “A” as shown in figure below.
 The tensile loads applied on the plates will tend to shear the bolt at the
section AA. Hence, it can be easily concluded from the FBD of the bolt
that the internal resistance force V must act in the plane of the section
AA and it should be equal to the external load P.
 The internal forces are called shear forces and when they are divided
by the corresponding section area is called shear stress.

21. In general, the shear stress is found to be maximum at the center and
zero at certain locations on the edge.
 Consider a bolt subjected under double shear, as shown below:
The bolt experiences shear on two sections AA and BB. Hence, the bolt is
said to be under double shear and the shear stress (τ)on each section is
given as:
Cont…
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22. The shear forces V are the resultants of the shear stresses, thus the average
shear stress:
 The shear stress is reduced by half in double shear when compared to a
single shear.
Shear stresses are generally found in bolts, pins and rivets that are used to
connect various structural members and machine components.
Cont…
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23.  If large shear forces act on a component and the component does not fail,
these will naturally lead to deformations in the plane of the section.
While normal forces actually changes the length of faces in a considered
section, shear forces will however only distort the shape of the section,
see Figure below.
Therefore simply apply the distortion angle “γ” as measure for shear
strain.
Shear strain:
23

24. Cont…
Figure: Relation between stress and strain respectively for (A) normal and (B) shear
deformations at elastic region.
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 Modules of rigidity (G): For elastic material shear strain is proportional to the
shear stress.

25.  Bearing Stress in Connections:
A bearing stress (σb), is a compressive normal stress that occurs on the surface of
contact between two interacting members.
The average bearing stress in the member is obtained by dividing the magnitude of the
bearing force “F” by the area of interest.
Bolts, pins and rivets create bearing stresses along the surface of contact.
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26. Example 2: A circular hollow tube made of steel is used to support a
compressive load of 500kN. The inner and outer diameters of the tube are
90mm and 130mm respectively and its length is 1000mm. Due to
compressive load, the contraction of the rod is 0.5mm. Determine the
compressive stress and strain in the post.
Example 3: A steel bar of length 2.5 m with a square cross section 100 mm on
each side is subjected to an axial tensile force of 1300 kN (see figure). Assume that E
= 200 GPa and v = 0.3. Determine the increase in volume of the bar.
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Cont…

27. Example 4: A punch for making holes in steel plates is shown in Fig.
below. Assume that a punch having diameter d 20 mm is used to punch a
hole in an 8-mm plate, as shown in the cross-sectional view (Fig. b). If a force
P 110 kN is required to create the hole, what is the average shear stress in
the plate and the average compressive stress in the punch?
Cont…
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END OF CH-1 !!