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Strain Energy

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Strain Energy. Variational Theorems. Concept of Minimum Potential Energy. The Ritz Method.
Strain Energy
energy
–
capacity to do work
potential
energy
–
stored energy as a result of work previously done e.g. wind-up toys, spinning tops
strain energy
is a form of potential energy that is stored in materials that have undergone/ been subjected to strain
when the strained material returns to its srcinal dimensions, it does work
strain energy
–
energy stored within a material when work has been done on the material
assumption: material remains within elastic region so that all the energy is recovered i.e. no permanent deformations
thus
with reference to the diagram above: work done by (a gradually applied load) in straining material = area under the load-extension curve
the area above the curve is known as the
complementary energy
Strain Energy due to Normal (direct) Stress
consider the elemental length of bar,
, with coss-sectional area
Young’s modulus,
From which we get,
Substituting this into
gives
For a bar of length
, total strain energy is
If the bar has constant c/s area
Normal stress,
so
Strain Energy due to Shear
Putting the above expressions together gives shear strain energy,
Strain energy
Shear modulus
From which we get
Total energy from shear
Now
Strain Energy due to Bending
Strain energy becomes
Total strain energy from bending
For constant bending moment
Bending theory equation:
Strain energy =work done= now So
Strain Energy due to Torsion
From simple torsion theory
Total strain energy from torsion
Since T is constant in most practical applications
Also from simple torsion theory
So we have
Now the polar moment of inertia for a shaft
Also the volume for a bar is
Substituting these into the above expression for strain energy gives
Torsion theory
Angle of twist
is in radians
Strain energy = work done =

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