c = distance from neutral axis to outer fibre(m)
E = Young's Modulus (N/m2)
F = Axial Force (N)
G = Modulus of Rigidity (N/m2)(m)
I = Moment of Inertia (m4)(m)
l = length (m)
M = moment (Nm)
V = Traverse Shear force Force (N)
x = distance from along beam (m)
z = distance from neutral (m)
γ = Angular strain = δ/l
δ = deflection (m)
τ = shear stress (N/m2)
τ max = Max shear stress (N/m2)
θ = Deflection (radians)
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
Consider a beam subject to traverse shear loading as shown. The beam is subject to stresses as a result of bending moments. It is also subject to stresses as a result of traverse shear load. These notes only relate to the stresses due to the traverse shear load.
Consider the beam as shown and specifically a slice dx wide.
The beam width is b
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.
b is width: For a rectangle b = constant: For other section b may be a function of x
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.
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 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.
Note :The constant K for the traverse shear option is shown in the section on traverse shear above. For a Structural section (K = 1)