LECTURE 38 Energy Methods Strain Energy Strain Energy of the member is defined as the internal work done in defoming the body by the action of externally applied forces. This energy in elastic bodies is known as elastic strain energy : Strain Energy in uniaxial Loading Fig .1 Let as consider an infinitesimal element of dimensions as shown in Fig .1. Let the element be subjected to normal stress s_{x}.
Assuming the element material to be as linearly elastic the stress is directly proportional to strain as shown in Fig . 2. Fig .2 \ From Fig .2 the force that acts on the element increases linearly from zero until it attains its full value.
\ Therefore the workdone by the above force
For a perfectly elastic body the above work done is the internal strain energy “du”. where dv = dxdydz
By rearranging the above equation we can write The equation (4) represents the strain energy in elastic body per unit volume of the material its strain energy – density ‘u_{o}' . From Hook's Law for elastic bodies, it may be recalled that In the case of a rod of uniform cross – section subjected at its ends an equal and opposite forces of magnitude P as shown in the Fig .3. Fig .3 Modulus of resilience : Fig .4 Suppose ‘ s_{x}‘ in strain energy equation is put equal to s_{y} i.e. the stress at proportional limit or yield point. The resulting strain energy gives an index of the materials ability to store or absorb energy without permanent deformation So The quantity resulting from the above equation is called the Modulus of resilience The modulus of resilience is equal to the area under the straight line portion ‘OY' of the stress – strain diagram as shown in Fig .4 and represents the energy per unit volume that the material can absorb without yielding. Hence this is used to differentiate materials for applications where energy must be absorbed by members. Modulus of Toughness : Fig .5 Suppose ‘Î' [strain] in strain energy expression is replaced by Î_{R} strain at rupture, the resulting strain energy density is called modulus of toughness From the stress – strain diagram, the area under the complete curve gives the measure of modules of toughness. It is the materials. Ability to absorb energy upto fracture. It is clear that the toughness of a material is related to its ductility as well as to its ultimate strength and that the capacity of a structure to withstand an impact Load depends upon the toughness of the material used. ILLUSTRATIVE PROBLEMS
Solution : From the above results it may be observed that the strain energy decreases as the volume of the bar increases.
Solution :
Therefore, the strain energy of the rod should be u = 5 [13.6] = 68 N.m Strain Energy density
Yield Strength : As we know that the modulus of resilience is equal to the strain energy density when maximum stress is equal to s_{x} . It is important to note that, since energy loads are not linearly related to the stress they produce, factor of safety associated with energy loads should be applied to the energy loads and not to the stresses. Strain Energy in Bending : Fig .6 Consider a beam AB subjected to a given loading as shown in figure. Let
ILLUSTRATIVE PROBLEMS
Solution : The bending moment at a distance x from end Substituting the above value of M in the expression of strain energy we may write Problem 2 :
Solution: a.
For Portion AD of the beam, the bending moment is For Portion DB, the bending moment at a distance v from end B is Strain Energy : Since strain energy is a scalar quantity, we may add the strain energy of portion AD to that of DB to obtain the total strain energy of the beam. b. Substituting the values of P, a, b, E, I, and L in the expression above. Problem 3) Determine the modulus of resilience for each of the following materials.
4) For the given Loading arrangement on the rod ABC determine (a). The strain energy of the steel rod ABC when
(b). The corresponding strain energy density in portions AB and BC of the rod. |