c = distance from neutral axis to outer fibre(m)
Strain Energy Pure Tension and compression
Strain Energy Pure Torsion
Strain Energy Direct Shear
Alternatively allowing z to be a variable:..
Strain Energy Beam in bending
Illustrating the case when M is fixed and note related to x
Illustrating the case when M is related, very simply to x
Strain Energy due to tranverse shear stress
There is a linear distribution of axial stress σx 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 z1 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 z1 = 0 and the minimum shear stress is at the outer fibre when z1 = 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 [l2 /h2]/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.
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)