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Notes On Strain energy
Strain Energy
c = distance from neutral axis to outer fibre(m) Strain Energy Pure Tension and compression Strain Energy Pure Torsion Strain Energy Direct Shear Strain Energy Beam in bending Illustrating the case when M is related, very simply to x
Strain Energy due to tranverse shear stress The total increase in axial force over slice dx for the section of the beam from z_{1} to the outer fibre of the beam is balanced by a shear force = τ _{xz} w dx as shown below. b is width: For a rectangle b = constant: For other section b may be a function of x Solving for τ _{xz} The maximum shear stress is at the neutral axis when z_{1} = 0 and the minimum shear stress is at the outer fibre when z_{1} = c. The equation for shear stress at any distance z from the neutral axis for a rectangular suction, with constant width b,subject to a traverse shear force V is as shown below. To obtain the strain energy substitute this equation into that derived for direct shear For the solid rectangle ( c = h/2, width = b, height = h, and length = x )subject to a traverse force V load along its length the strain energy = ... Using similar principles the strain energy for different sections subject to traverse shear can be identified as shown below Comparing the strain energy due to direct shear in a beam and that due to bending: For the simply supported rectangular section beam with a central traverse force of 2V of length l the strain energy due to bending and due to traverse shear as shown below. For a simply supported rectangular beam loaded, with single central load, The strain energy resulting from the bending moments is [l^{2} /h^{2}]/3 times that due to traverse shear loading. For a typical beam of l/h ratio = 10 the bending shear energy is 33 times the traverse force shear energy. The traverse force shear energy can be neglected for most beams of significant length. Summary The strain energy in a member or component for each type is loading is shown below: Note :The constant K for the traverse shear option is shown in the section on traverse shear above. For a Structural section (K = 1) 
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Last Updated 25/01/2013