strictly as per AKTU syllabus for BTech first yr 2021 subject- fundamentals of mechanical and mechatronics subject code - KME 101T

1. Subject / Code: RME – 101 T
Faculty: Mr. Vivek singh chauhan,
Branch : Mechanical Engineering
Section Number : 09
(Assistant Professor, ME Department)

30. Example - 2

33. Fx Fy

38. This law is used to determine the resultant of two forces acting at a point of a rigid
body in a plane and is inclined to each other at an angle of Ɵ. It state that
“If two forces acting simultaneously on a particle, be represented in magnitude and
direction by two adjacent sides of a parallelogram then their resultant may be
represented in magnitude and direction by the diagonal of the parallelogram, which
passes through their point of intersection.”
Derivation -
Let two forces P and Q act at a point ‘O’
as shown in fig (a).The force P is
represented in magnitude and direction by
vector OA, Where as the force Q is
represented in magnitude and direction by
vector OB, Angle between two force is
‘Ɵ’.The resultant is denoted by vector OC
in fig (b). Drop perpendicular from C on
OA.
α Ɵ
Ɵ

39. Let, P,Q = Forces whose resultant is required to be found out.
α = Angle which the resultant forces makes with one of the forces
Ɵ = Angle between the forces P and Q
Now ∠CAD = θ { because OB II CA and OA is common base }
In OCD applying Pythagoras theorem
𝑂𝐶2
= 𝑂𝐷2
+ 𝐶𝐷2
𝑅2
= (𝑂𝐴 + 𝐴𝐷)2
+ (𝑄𝑠𝑖𝑛𝜃)2
𝑅2 = (𝑃 + 𝑄𝑐𝑜𝑠𝜃)2 + (𝑄𝑠𝑖𝑛𝜃)2
𝑅2 = 𝑃2 + 𝑄2𝑐𝑜𝑠2𝜃+ 2PQcosθ + 𝑄2𝑠𝑖𝑛2𝜃
𝑅2
= 𝑃2
+ 𝑄2
(𝑐𝑜𝑠2
𝜃+ 𝑠𝑖𝑛2
𝜃) + 2PQcosθ
R = √(𝑃2
+ 𝑄2
+ 2PQcosθ) for magnitude
In ∆ OCD tanα =
𝐶𝐷
𝑂𝐷
=
𝑄𝑠𝑖𝑛𝜃
𝑃+𝑄𝑐𝑜𝑠𝜃
α = tan−1 𝑄𝑠𝑖𝑛𝜃
𝑃+𝑄𝑐𝑜𝑠𝜃
for resultant direction

40. WHAT IS “Stress” ?

41. Introduction to Stress

42. Shear Stress
 Forces parallel to the area resisting the force cause shearing stress.
 It differs to tensile and compressive stresses, which are caused by forces perpendicular to
the area on which they act.
 Shearing stress is also known as tangential stress.
τ= F/A
Combined Stress
In combined stress there are two types of stress
 Shear stress
 Tortional stress

43. Introduction to Stress

44. Introduction to Stress
Tortional stress
• The stresses and deformations induced in a circular shaft by a twisting moment.
Strain
Also known as unit deformation, strain is the ratio of the change in dimension caused by
the applied force, to the original dimension. where δ is the deformation and L is the
original length, thus ε is dimensionless.

45. Types of strain:
Tensile strain
Compressive strain
Shear strain
Volumetric strain
Tensile strain
It is the ratio of the increase in length to its original length.
Tensile strain = increase in length,(l-l0)/original length,(l0)

46. Introduction to Stress
• Compressive strain
It is ratio of the decrease in length to its original length.
compressive strain = decrease in length,(l0-l)/original length,(l0)
Shear strain
We can define shear strain exactly the way we do longitudinal
strain: the ratio of deformation to original dimensions.

47. Introduction to Stress
• Volumetric strain
Volumetric strain of a deformed body is defined as the ratio of the change in volume of the
body to the deformation to its original volume.
volumetric strain = change in volume/original volume

48. Stress strain diagram for ductile material

49. • The curve starts from the origin ‘O’ showing thereby that there is no initial stress
or strain in the test specimen.
• Up to point ‘A’ Hooke’s law is obeyed and stress is proportional to strain
therefore ‘OA’ is straight line and point ‘A’ is called the proportionality limit stress.
• The portion between ‘AB’ is not a straight line, but up to point ‘B’, the material
remains elastic.
• The point ‘B’ is called the elastic limit point and the stress corresponding to
that is called the elastic limit stress.
• Beyond the point ‘B’, the material goes to plastic stage until the upper yield point
‘C’ is reached.
• At this point the cross-sectional area of the material starts decreasing and the
stress decreases to a lower value to a point ‘D’, called the lower yield point.
• Corresponding to point ‘C’, the stress is known as upper yield point stress.
Stress strain diagram for ductile material

50. • At point ‘D’ the specimen elongates by a considerable amount without any
increase in stress and up to point ‘E’.
• The portion ‘DE’ is called the yielding of the material at constant stress.
• From point ‘E’ onwards , the strain hardening phenomena becomes pre-
dominant and the strength of the material increases thereby requiring more
stress for deformation, until point ‘F’ is reached.
• Point ‘F’ is called the ultimate point and the stress corresponding to this point
is called the ultimate stress.
• It is the maximum stress to which the material can be subjected in a simple
tensile test.
• At point ‘F’ the necking of the material begins and the cross sectional area
starts decreasing at a rapid rate.
Stress strain diagram for ductile material

51. • Due to this local necking the stress in the material goes on
decreasing inspite of the fact that actual stress intensity goes on
increasing.
• Ultimately the specimen breaks at point ‘G’, known as the breaking
point, and the corresponding stress is called the normal breaking
stress bared up to original area of cross section.
Stress strain diagram for ductile material

52. = 1/m
(µ)
Poisson’s ratio

53. Elastic Limit:
When an external force acts on a body, the body tends to undergo some deformation. If the
external force is removed and the body comes back to its origin shape and size, the body is
known as elastic body. This property, by virtue of which certain materials return back to their
original position after the removal of the external force, is called elasticity.
The body will regain its previous shape and size only when the deformation caused by the
external force, is within a certain limit. Thus there is a limiting value of force up to and
within which, the deformation completely disappears on the removal of the force. The value
of stress corresponding to this limiting force is known as the elastic limit of the material.
Hooke’s law:
It states that when a material is loaded within elastic limit, the stress is proportional to the
strain produced by the stress. This means the ratio of the stress to the corresponding strain is a
constant within the elastic limit.
Stress /Strain = Constant
This constant is known as elastic constant.

54. Types of Elastic Constants:
There are three elastic constants;
Normal stress/ Normal strain = Young’smodulus or Modulus of elasticity (E)
Shear stress/ Shear strain = Shear modulus or Modulus of Rigidity (G)
Direct stress/ Volumetric strain = Bulk modulus (K)
Young’sModulus or Modulus of elasticity (E):
It is defined as the ratio of normal stress (σ) to the longitudinal strain (e).
E = (σ) / (e)
Modulus of Rigidity or Shear Modulus (G or C):
It is the ratio between shear stress (τ) and shear strain (es). It is denoted by G or C.
G= τ/ϕ

55. Bulk Modulus or Volume Modulus of Elasticity (K):
It may be defined as the ratio of normal stress (on each face of a solid cube) to volumetric strain.
It is denoted by K. Bulk modulus is a measure of the resistance of a material to change of volume
without change of shape or form.
K = Direct Stress / Volumetric strain
= σ/ev
Relation between E, K and Poisson’s Ratio (μ or 1/m)
Consider a cubical element subjected to volumetric stress σ
which acts simultaneously along the mutually perpendicular x,
y and z-direction.
The resultant strains along the three directions can be worked
out by taking the effect of individual stresses.

66. load
actual
the
to
due
component
in the
Stress
)
(
component
the
of
Strength
FoS
y
u,S
S


68. | BY VIVEK SINGH CHAUHAN
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