Strain Energy

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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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