c = distance from neutral axis to outer fibre(m)

E = Young's Modulus (N/m^{2})

F = Axial Force (N)

G = Modulus of Rigidity (N/m^{2})(m)

I = Moment of Inertia (m^{4})(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/m^{2})

τ _{max} = Max shear stress (N/m^{2})

θ = Deflection (radians)

**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**

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

The total increase in axial force over slice dx for the section of the beam from z

Solving for τ

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

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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