Strain energy

Strain energy

Strain Energy

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


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.

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)

Links to Spring Design
  1. Strain Energy Methods ...A powerpoint presentation of the application of strain energy methods
  2. Elasticity Theory ...Very detailed notes including reference to strain energy
  3. -Strain energy ...Very readable set of relevant notes