1. SUBJECT: Marine Engineering Automation and Control
Dr. S. Janakiraman
Assistant Professor ( Mechanical Engineering Department)
2. Unit I - Strength of Materials
Introduction to stress & strain
Types of stress & strain
Hook’s law
Poisson’s ratio
Elastic constant & their relationship
Stress & strain diagram for ductile material
Stress & strain diagram for brittle material
Factor of safety
Numerical based on stress & strain
11. Now, stress is defined as the force
intensity or force per unit area.
Here we use a symbol to represent
the stress
Where A is the area of the
section X-X
The basic units of stress in S.I units
i.e. (International system) are
N/m2
(or Pa)
MPa = N/mm2
= 106
Pa, N /mm2
is
equivalent to MPa
GPa = 103
N/mm2
= 109
N/m2
=
109
Pa
KPa = 10-3
N /mm2
= 103
Pa.
12. Now, stress is defined as the force
intensity or force per unit area.
Here we use a symbol to represent
the stress
Where A is the area of the
section X-X
The basic units of stress in S.I units
i.e. (International system) are
N/m2
(or Pa)
MPa = N/mm2
= 106
Pa, N /mm2
is
equivalent to MPa
GPa = 103
N/mm2
= 109
N/m2
=
109
Pa
KPa = 10-3
N /mm2
= 103
Pa.
15. when some external system of
load acts on a body, the
internal forces (equal &
opposite) are set up at various
section of the body, which
resist the external force. This
force per unit area at any
section is known as stress.
STRESS:
Let us consider a rectangular
bar of some cross–sectional
area and subjected to some
load or force (in Newton’s)
16. Normal Stress (σ)
When a force is applied to an elastic body, the body
deforms. The way in which the body deforms depends
upon the type of force applied to it.
Normal stress is the stress which acts in the direction
perpendicular to the area , it is represented by symbol ‘σ ’
( a ) Tensile Stress due to tensile
force
A tensile stress induced in a
body, when subjected to two
equal & opposite pull which
have a tendency to makes the
body longer or elongate
17. TYPES OF DIRECT STRESS:
Only two basic stresses exist:
(1)Normal stress
(2) Shear stress.
Other stresses either are similar
to these basic stresses or are a
combination of this e.g. bending
stress is a combination tensile,
compressive and shear stresses.
Torsional stress, as encountered
in twisting of a shaft is a
shearing stress.
18. (b) Compressive Stress due to
compressive force
A compressive stress induced in a
body, when subjected to two equal
& opposite push which have a
tendency to makes the body shorter
or shorten
19. SHEAR STRESS
The stress induced in a
body when subjected to
two equal & opposite
forces which are acting
tengentially across the
resisting section. As a
result of which body
tend to shear off across
the section. It is known
as shear stress
20. Strain: Strain is the deformation of
a material from stress. It is simply a
ratio of the change in dimension to the
original dimension. Deformations that
are applied perpendicular to the cross
section are normal strains.
Stain is a measure of the measure of
the deformation produced in the
member by the load.
If a rod of length L is in tension and
the elongation produced is dL,
then the direct strain= Elongation /
Original length
Tensile strain will be positive
compressive strain will be negative.
21. TYPES OF STRAIN ACCORDING TO FORCE
Tensile Strain: if there is some increase in length of a body due to
external force then the ratio of increase of length to the original length
of body is known as tensile strain.
Compressive strain : if there is some decrease in length of body . Then
the ratio of decrease of length of the body to the original length is
known as compressive strain
22. Volumetric Strain is defined as the ratio of change in volume to the initial
volume.
23. Shear strain is defined as the strain accompanying a shearing action. It is
the angle in radian measure through which the body gets distorted when
subjected to an external shearing action. It is denoted by Φ
Consider a cube ABCD subjected to equal and opposite forces Q across
the top and bottom forces AB and CD. If the bottom face is taken fixed,
the cube gets distorted through angle Φ to the shape ABC’D’. Now
strain or deformation per unit length is
Shear strain of cube = CC’ / CD = CC’ / BC = Φ radian
25. Lateral Strain:
Lateral strain of a deformed
body is defined as the ratio
of the change in length
(breadth of a rectangular bar
or diameter of a circular bar)
of the body due to the
deformation to its original
length (breadth of a
rectangular bar or diameter
of a circular bar) in the
direction perpendicular of
the force
26. When an external force is applied on a body and it undergoes
some deformation. If the body returns back to its original
shape and size on complete removal of the load, the body is
called elastic body.
Elasticity: The property of a material by which it returns back
to its original position (i.e. shape and size) on the removal of
external force or load, is called elasticity.
Elastic Limit: It is defined as the value of stress upto and
within which the material return back to their original position
(i.e. shape and size) on the removal of external force.
If the value of external force is such that it exceeds the elastic
limit, than the body will not completely regain its original
position. The body loses its property of elasticity to some
extent. And if the external force acting on the body is
removed, in that condition the body will not return to its
original shape and size and there will be a residual
deformation in the material
Elasticity & Elastic limit
28. Hooke’s law stated that when a
material is loaded within elastic limit,
stress is directly proportional to strain
In which
“σ” is the axial stress
“ε” is the axial strain
“E” is a constant of proportionality
known as the modulus of elasticity for
the material.
The modulus of elasticity is the slope
of the stress-strain diagram in the
linearly elastic region.
The equation σ = Eε is commonly
known as Hooke’s law
Hook’s law:
29. When an elastic body
is subjected to stress,
a proportionate
amount of strain is
produced. The ratio of
the applied stresses
to the strains
generated will always
be constant and is
known as elastic
constant. Elastic
constant represents
the elastic behaviour
of objects.
ELASTIC CONSTANT & THEIR RELATION
SHIP
Elastic Constants
Different elastic constants are as
follows :
1.Young’s modulus or Modulus of
elasticity, E
2.Bulk modulus
3.Modulus of rigidity or shear
modulus
4.Poisson’s ratio
30. Young’s Modulus
According to Hooke’s law,
when a body is subjected to
tensile stress or compressive
stress, the stress applied is
directly proportional to the
strain within the elastic limits
of that body. The ratio of
applied stress to the strain is
constant and is known as
Young’s modulus or modulus
of elasticity.
Young’s modulus is denoted
by letter “E”. The unit of
modulus of elasticity is the
31. Bulk Modulus (B) or
K
When a body is subjected
to mutually perpendicular
direct stresses which are
alike and equal, within its
elastic limits, the ratio of
direct stress to the
corresponding volumetric
strain is found to be
constant. This ratio is called
bulk modulus and is
represented by letter “K”.
Unit of Bulk modulus is
Mpa.
32. Rigidity Modulus(G)
When a body is subjected to
shear stress the shape of the
body gets changed, the ratio
of shear stress to the
corresponding shear strain is
called rigidity modulus or
modulus of rigidity. It is
denoted by the letters “G” or
“C” or “N”. Unit of rigidity
modulus is Mpa
33. Poisson’s ratio
With in the elastic limit, the ratio of lateral strain to longitudinal strain or
linear strain is called Poisson's ratio
Poisson’s ratio is maximum for an ideal elastic incompressible material
and its value is 0.5. For most of the engineering materials, Poisson’s
ratio lies between 0.25 and 0.33. It has no units.
34. Relationship between Elastic Constants
1. The relationship between Young’s modulus (E), rigidity
modulus (G) and Poisson’s ratio (µ) is expressed as :
2.The relationship between Young’s modulus (E), bulk
modulus (K) and Poisson’s ratio (µ) is expressed as :
35. 3. Young’s modulus can be expressed in terms of bulk
modulus (K) and rigidity modulus (G) as :
4. Poisson’s ratio can be expressed in terms of bulk
modulus (K) and rigidity modulus (G) as :
36. Definition of factor of safety
It is the ratio of the ultimate strength of a member or piece of
material (as in an airplane) to the actual working stress or the
maximum permissible stress when in use
37. STRESS STRAIN DIAGRAM
The relation between stress and strain is generally shown by
plotting a stress-strain (σ-ϵ) diagram. Stress is plotted on
ordinate (vertical axis) and strain on abscissa (horizontal
axis). Such diagrams are most common in strength of
materials for understanding the behaviour of materials.
Stress-strain diagrams are drawn for different loadings.
Stress-Strain Curves (Tension)
When a bar or specimen is subjected to a gradually increasing
axial tensile load, the stresses and strains can be found out
for number of loading conditions and a curve is plotted up to
the point at which the specimen fails. giving what is known as
stress-strain curve.
Stress-strain carves for ductile materials : A material is said
to be ductile in nature, if it elongates appreciably before
fracture. One such material is mild steel. The shape of stress-
strain diagram for the mild steel is shown in Fig.(a)
39. STRESS STRAIN DIAGRAM
Portion OA: This portion is absolutely
straight, where the stress is
proportional to strain and the material
obeys Hooke’s law (σ =E ). The value
ϵ
of stress at point A is called
proportional limit.
Portion AB: In this portion, Hook’s
law is not obeyed, although the
material may still be elastic. The point
B indicates the elastic limit.
40. STRESS STRAIN DIAGRAM
Portion BC: In this portion, the metal
shows a strain even without increase in
stress and the strain is not fully return
when load is removed.
Portion CD: Yielding start in this
portion and there is a drop of stress at
the point D directly after yielding
begins at C. The point D is termed as
lower yield point and C is called upper
yield point.
41. STRESS STRAIN DIAGRAM
Portion DE: After yielding has taken
place at D, further straining takes place
at this portion by increasing the stress
and the stress–strain curve continues to
rise up to the point E. Strain in this
portion is about 100 times that of
portion O-A. At the point E, the bar
begins to form a local neck. The point
E is termed as ultimate tensile stress
point.
42. STRESS STRAIN DIAGRAM
Portion EF: In this portion, the load is
falling off from the maximum and
fracture at F takes place. The point F is
termed as fracture or breaking point
and the identical stress is called
breaking stress.
43. STRESS STRAIN DIAGRAM
Stress Strain Curves for Brittle Materials
Materials which show very small elongation
before they fracture are called brittle
materials. The shape of curve for high
carbon steel, concrete and high strength
light alloys or any brittle materials is shown
in fig. For most brittle materials the
permanent elongation (i.e. increase in
length) is less than 10%. Principal
mechanical properties The characteristics of
the materials which describe their behaviour
44. PRINCIPLE MECHANICAL
PROPERTIES
1 - Elasticity
Elasticity of a material is power of coming back to original position
when the stress or load is removed. The greatest stress that a material can
withstand without permanent distortion is called elastic limit.
2- Plasticity
The plasticity of a material is ability to undergo some permanent
deformation without failure. Plastic deformation will take place only after
the elastic range has been exceeded, beyond (point c).
Plasticity is an important property and widely used in several mechanical
processes like forming, shaping, extruding and many other hot and cold
working processes. In general, plasticity increases with increasing
temperature.
Due to this property various metals can be transformed into different
products of required shape and size. This conversion into desired shape
and size is effected either by the application of pressure or heat or both.
45. PRINCIPLE MECHANICAL
PROPERTIES
3- Ductility
Ductility of a material their enables to draw out into thin wire with
application the load. Ductile material such as mild steel, wires of gold,
silver, copper, aluminium, etc. are drawn by extrusion or by pulling
through a hole in a die due to the ductile property. The ductility decreases
with increase of temperature. The percent elongation and the reduction in
area in tension are often used as empirical measures of ductility.
4-Strength
It is the resistance offered by a material when subjected to external
loading, so stronger the material can be withstand with greater the load.
Depending upon the type of load applied the strength can be tensile,
compressive, shear or torsional strength. The maximum stress that any
material will withstand before destruction is called its ultimate strength
(point E as shown in Fig. 1).
46. PRINCIPLE MECHANICAL
PROPERTIES
5- Brittleness
The brittleness of a material is the property of breaking without much
permanent distortion. There are many materials, which break or fail
before much deformation take place, such as glass, cast iron, etc.
Therefore, a non-ductile material is said to be a brittle material. A brittle
material should not be considered as lacking in strength, it is only shows
the lack of elasticity. On stress-strain diagram, these materials don’t have
yield point and value of E is small.
6- Toughness
The toughness of a material is ability to withstand both plastic and elastic
deformations. It is a highly desirable quality for structural and machine
parts to withstand to shock and vibration. Manganese steel, mild steels are
tough materials. For Ex: If a load is suddenly applied to a piece of mild
steel and then to a piece of glass the mild steel will absorb much more
energy before failure occurs. Thus, mild steel is said to be much tougher
than a glass.
47. PRINCIPLE MECHANICAL
PROPERTIES
7- Hardness
Hardness is closely related to strength. It is the ability of a material to
resist scratching, abrasion, penetration with apply external load.
8- Stiffness (Rigidity)
The resistance of a material to deflection is called stiffness or rigidity.
Steel is stiffer or more rigid than aluminium. Stiffness is measured by
Young’s modulus E. The higher value of the Young’s modulus this mean
stiffer the material. E is the ratio of stress over strain and is given by the
slope of line O–A.
