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Strain Energy

The external work done on an elastic member in causing it to distort from its unstressed state is transformedinto strain energy which is a form of potential energy. The strain energy in the form of elastic deformation ismostly recoverable in the form of mechanical work.

Nomenclature

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)

= Angular strain = /l = deflection (m) = shear stress (N/m 2 )

max = Max shear stress (N/m 2) = Deflection (radians)

Strain Energy Pure Tension and compression

Strain Energy Pure Torsion

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Strain Energy Direct Shear

Alternatively allowing z to be a variable:..

Strain Energy Beam in bending

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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 tothe 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 x at a section at a distance x along the beam =

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Along the slice dx the axial stress increases to (M + Vdx)z/ I . Thus along the slice dx there is a increase inaxial stress of [(Vdx)z] / I.

The total increase in axial force over slice dx for the section o f the beam from z 1 to the outer fibre of the beamis balanced by a shear force = 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 xz

The maximum shear stress is at the neutral axis when z 1 = 0 and the minimum shear stress is at the outerfibre when z 1 = c.

The equation for shear stress a t any distance z from the n eutral axis for a rectangular suction, withconstant 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 loadalong its length the strain energy = ...

Using similar principles the strain energy for different sections subject to traverse shear can be identified asshown below

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