ME21203 - Mechanics of Solids
Tough hai !!
Website - Mechanics of Solids (ME21203),
Autumn 2022 (iitkgp.ac.in)
Syllabus Review of concept of stress and strain, Transformation of stress and
strain: Transformation of stress and strain, principal stresses, principal
strains, Mohr s circle for stress and strain.
Torsion: Torsion of cylindrical bars, torsional stress, modulus of rigidity
and deformation.
Flexural loading: Shear and moment in beams;
Up-till this in test
load, shear and moment relationship;
flexure formula;
shear stress in beams;
differential equation of the elastic curve, deflection of beams.
Combined loading: Axial and torsional; axial and bending;
axial, torsional and bending.
Column: Buckling of slender columns, Euler bucking load for different
end conditions.
Bending of Beams: Straight and asymmetrical bending, Shear center.
Thick-walled cylinders.
Energy methods: Castigliano s theorem.
MOS Tutorial solns. + other questions
Notes stress and strain Normal Stress Normal Stress
P
σaverage =
A
Defined in the gauge length while testing the material. P is the
force applied perpendicular to the area.
While pulling the sample, the area will reduce eventually, but we
take only the original area for our stress calculations, not the
current area or the changed area.
Shear Stress
τaverage =
V
A
axial Strain
ϵ =
l − l0
l0
Shear Strain
γ = tan θ ≈ θ
when a piece is tested the stress at the ends in infinite, so it is wider at
the ends, elongated hourglass shape. So we only take into
consideration the linear cylindrical central part of the piece.
The curve based on the original cross-section and gauge length is
called the engineering stress–strain curve, while the curve based on
the instantaneous cross-section area and length is called the true
stress–strain curve. Unless stated otherwise, engineering stress–strain
is generally used.
**This the "Engineering stress strain diagram".
1 -proportional limit
2 - linearly elastic
3- upper yield point
4- lower yield point
5- ultimate tensile strength
6- rupture
The topmost point of this diagram represents the strength of any
material, the higher it is the stronger the material.
Work Hardening/ Strain hardeningto increase strength, load the material physically in its plastic
phase and then reloading it.
[c] => Compression
[T] => elongation
The two have their own separate stress strain diagrams.
Determining Yield Point for metals with no clear
graph spike or drop-
In this case, as there is no specific spike in the graph at the yield point.
What we do is :
We agree on a particular offset value, which is generally 0.2% or 0.4%
so, the offset of 0.002 in epsilon and then make a line parallel to the
original straight line. The place where this line cuts the original curve is
taken as the yield point.
Analysis of StressesStress can be resolved into any directions, for sake of simplicity we use
Augustin Cauchy.
Normal stress - Perpendicular to surface inspected.
Shear Stress - parallel to surface inspected.
Strain -
The normal strain is represented as
Normal strain is defined only with respect to a splitting plane and is
prependicular to that plane
ϵxx
The shear strain on the other hand is represented by
γxy
It is represented in the form that from any point we can take two
perpendicular axes and then after applying loading, they will not remain
perpendicular,
γshearstrain = 90 − θnew
Torsion -
Related to the twisting of rods or beams.
τ = Gγ
gamma is the shear strain, tao is the shear tress.
also,
τxy = Gγxy
τyz = Gγyz
τxz = Gγxz
Also,
γ = tanθ ≈ θ
Where, theta is the rotation angle (the angle by which the shaft is twisted.)
Here,
L = lengthR = Outerradius
ρ = innerradiusof theelement
γ = anglesubtendedbythebentpoint
See how the opposite torques have been used for balancing
Shear Force and Bending moment diagrams -
(413) Understanding Shear Force and Bending Moment Diagrams YouTube
Watch for sign convention and stuff.