Strain Energy

     Consider an elastic spring as shown in the Fig.1.0 When the spring is slowly pulled, it deflects by a small amount u1. When the load is removed from the spring, it goes back to the original position. When the spring is pulled by a force, it does some work and this can be calculated once the load-displacement relationship is known. It may be noted that, the spring is a mathematical idealization of the rod being pulled by a force P axially. It is assumed here that the force is applied gradually so that it slowly increases from zero to a maximum value P. Such a load is called static loading, as there are no inertial effects due to motion. Let the load-displacement relationship be as shown in Fig. 2.0. Now, work done by the external force may be calculated as,
 strain energy
linear spring
Force displacement relation
      The area enclosed by force-displacement curve gives the total work done by the externally applied load. Here it is assumed that the energy is conserved i.e. the work done by gradually applied loads is equal to energy stored in the structure. This internal energy is known as strain energy. Now strain energy stored in a spring is,
Work and energy are expressed in the same units. In SI system, the unit of work and energy is the joule (J), which is equal to one Newton metre (N.m). The strain energy may also be defined as the internal work done by the stress resultants in moving through the corresponding deformations. Consider an infinitesimal element within a three dimensional homogeneous and isotropic material. In the most general case, the state of stress acting on such an element may be as shown in Fig. 3.0. There are normal stresses (σ12 and σ3), and and shear stresses (τ1,τ2 and τ3), and acting on the element. Corresponding to normal and shear stresses we have normal and shear strains. Now strain energy may be written as,formula
in which σT is the transpose of the stress column vector i.e.,
    The strain energy may be further classified as elastic strain energy and inelastic strain energy as shown in Fig. 4.0 If the force P is removed then the spring shortens. When the elastic limit of the spring is not exceeded, then on removal of load, the spring regains its original shape. If the elastic limit of the material is exceeded, a permanent set will remain on removal of load. In the present case, load the spring beyond its elastic limit. Then we obtain the load-displacement curve OABCDO as shown in Fig. 4.0. Now if at B, the load is removed, the spring gradually shortens. However, a permanent set of OD is till retained. The shaded area BCD is known as the elastic strain energy. This can be recovered upon removing the load. The area OABDO represents the inelastic portion of strain energy.
     The area corresponds to strain energy stored in the structure. The area is defined as the complementary strain energy. For the linearly elastic structure it may be seen that OABCDOOABEO
Area OBC = Area OBE
i.e. Strain energy = Complementary strain energy
    This is not the case always as observed from Fig. 4.0. The complementary energy has no physical meaning. The definition is being used for its convenience in structural analysis as will be clear from the subsequent chapters.
     Usually structural member is subjected to any one or the combination of bending moment; shear force, axial force and twisting moment. The member resists these external actions by internal stresses. In this section, the internal stresses induced in the structure due to external forces and the associated displacements are calculated for different actions. Knowing internal stresses due to individual forces, one could calculate the resulting stress distribution due to combination of external forces by the method of superposition. After knowing internal stresses and deformations, one could easily evaluate strain energy stored in a simple beam due to axial, bending, shear and torsional deformations.
Strain energy can be described in 4 types of cases.

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