c = distance from neutral axis to outer fibre(m) Illustrating the case when M is related, very simply to x
_{x} at a section at a distance x along the
beam =
Along the slice dx the axial stress
increases to (M + Vdx)z/ I . Thus along the slice dx there is a increase
in axial stress of [(Vdx)z] / I.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 xSolving 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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