Friday, April 5, 2013

Strain energy stored in a member

When an elastic body is subjected to the external loading it may undergo the deformation or strain. If the material is strained withing its elastic limit, then according to the principle of conservation of energy, the work by the external load on the body should be equal to the strain energy that is stored in the body. Strain energy can also be called as the potential energy of the body.
If you remove the loading from the body, it will regain its original shape, so this energy is again converted to the mechanical energy.
Axial Loading on bar:
If a member having elasticity of "E" is subject to an axial loading of  stress "f",
then the amount of the strain energy stored in  body per unit of its length can be calculated to be equal to
U= f^2/2E
There are numbers of methods which uses the strain energy as the basis for the analysis of the given structures which can not be analysed by the general conditions of equilibrium, which are only 3 in numbers. Castigliano's theorems are much famous which use this energy principle also.
Bending loading on an elastic bar:
If the loading on the bar is a bending moment of value 'M', then the flexural stresses are induced in the cross section of the bar, which vary from the center of the cross section to the extreme fibers.
at any distance 'y' from the center, the value of stress is given by,
f = M.y/I
M is the applied moment, and I is the second moment of the area of the section about the central axis of the cross section.
Energy stored in any element of length, 'ds' and cross sectional area of 'da' is given by,
 (f^2.ds.da)/2E = (M.y/I)^2.ds.da/ 2E
Now the energy stored in the total cross section of the bar can be found by adding the energy stored in the infinite nos. of elements coming in the section of length ds.
now in the above equation, da.y^2 is the second moment of area. So summing up all the second moments of areas will give us the moment of Inertia of the section.
So the energy stored in the length of ds = M^2.ds/2EI
the total energy stored in the whole length of the bar can be found out by integrating the above equation with the limits of the length of the bar.


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