Stress
where,
σ=normal stress, or tensile stress, pa
P=force
applied, N
A=cross-sectional area of the bar, m2
=shearing stress, Pa
As=total area in shear, m2
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Strain
where,
=tensile or compressive strain, m/m
=total elongation in a bar, m
=original length of the bar, m
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Hooke's Law
Stress is proportional to strain
where,
E=proportionality constant
called the elastic modulus or modulus of elasticity or
Young’s modulus, Pa
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Piosson's
Ratio
where,
v=Poisson’s ratio
=lateral strain
=axial strain
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Unit Volume
Change
where,
=change in volume
=original volume
=strain
=Poisson’s ratio
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Elongation
due to its weight
where,
=total elongation in a material which
hangs vertically under its own weight
W=weight of the material
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Thin Rings
where,
=Circumferential or hoop Stress
S=Circumferential or hoop tension
A=Cross-sectional area
=Circumferential strain
E=Young’s modulus
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Strain
Energy
where,
U=total energy stored in the bar or strain
energy
P=tensile load
=total elongation in the bar
L=original length of the bar
A=cross-sectional area of the bar
E=Young’s
modulus
U=strain energy per unit volume
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Thin Walled
Pressure vessels
where,
=normal or circumferential or hoop
stress in cylindrical vessel, Pa
=normal or circumferential or hoop
stress in spherical vessel, Pa and longitudinal stress around the circumference
P=internal pressure of cylinder, Pa
r=internal radius, m
t=thickness of wall, m
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Mohr's
Circle for Biaxial Stress
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Pure Shear
where,
=Shearing Stress, Pa
=Shearing Strain or angular deformation
G=Shear modulus, Pa
E=Young’s modulus, Pa
V=Poisson’s ratio
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Torsion
formula for Thin walled tubes
where,
=maximum shearing stress, Pa
=Shearing stress at any point a distance
x from the centre of a section
r=radius of the section, m
d=diameter of a solid circular shaft, m
=polar moment of inertia of a
cross-sectional area, m4
T=resisting torque, N-m
N= rpm of shaft
P=power, kW
=angle of twist, radian
L=length of shaft, m
G=shear modulus, Pa
do=outer diameter of hollow
shaft, m
di=inner diameter of hollow shaft, m
and
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Torsion
formula for Circular Shafts
where,
=Ip, polar moment of inertia
for thin-walled tubes
r=mean radius
t=wall thickness
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Flexure
Formula
where,
=Stress on any point of cross-section at
distance y from the neutral axis
=stress at outer fibre of the beam
c=distance measured from the neutral axis
to the most remote fibre of the beam
I=moment of inertia of the
cross-sectional area about the centroidal axis
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Shear
Stress In Bending
where,
F=Shear force
Q=statistical moment about the neutral axis
of the cross-section
b=width
I=moment of inertia of the cross-sectional
area about the Centroidal axis.
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Thin-Walled
Hollow Members (Tubes)
where,
=shearing stress at any point of a blue
t=thickness of tube
q=shear flow
T=applied torque
R=distance between a reference point and
segment ds
Π=angle of twist of a hollow tube
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Stress
Concentration
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Curved Beam
in Pure Bending
where,
=normal stress
M=bending moment
dA=cross-sectional area of an element
r=distance of curved surface from the
centre of curvature
A=cross-sectional area of beam
R=distance of neutral axis from the
centre of curvature
R1=distance of centroidal axis
from the centre of curvature
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Bending of
a Beam
(a) Bending
of a Beam Supported at Both Ends
(b)
Bending of a Beam Fixed at one end
where, d= bending
displacement, m
F=force applied, N
I=length of the beam, m
a=width of beam, m
b=thickness of beam, m
Y=Young’s modulus, N/m2
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