LECTURE 39 Complementary Strain Energy : Consider the stress strain diagram as shown Fig 39.1. The area enclosed by the inclined line and the vertical axis is called the complementary strain energy. For a linearly elastic materials the complementary strain energy and elastic strain energy are the same.
Let us consider elastic non linear primatic bar subjected to an axial load. The resulting stress strain plot is as shown.
The new term complementary work is defined as follows So In geometric sense the work W* is the complement of the work ‘W' because it completes rectangle as shown in the above figure
Likewise the complementary energy density u* is obtained by considering a volume element subjected to the stress s The complementary energy density is equal to the area between the stress strain curve and the stress axis. The total complementary energy of the bar may be obtained from u* by integration Sometimes the complementary energy is also called the stress energy. Complementary Energy is expressed in terms of the load and that the strain energy is expressed in terms of the displacement.
Let the two Loads P Let the Load P Let DP Now the increase in strain energy Suppose the increment in load is applied first followed by P Since the resultant strain energy is independent of order loading, Combing equation 1, 2 and 3. One can obtain or upon taking the limit as DP For a general case there may be number of loads, therefore, the equation (6) can be written as
The statement of this theorem can be put forth as follows; if the strain energy of a linearly elastic structure is expressed in terms of the system of external loads. The partial derivative of strain energy with respect to a concentrated external load is the deflection of the structure at the point of application and in the direction of that load. In a similar fashion, castigliano's theorem can also be valid for applied moments and resulting rotations of the structure Where
In similar fashion as discussed in previous section suppose the displacement of the structure are changed by a small amount dd Where ¶U / d It may be seen that, when the displacement d The work which is equal to P By rearranging the above expression, the Castigliano's first theorem becomes The above relation states that the partial derivative of strain energy w.r.t. any displacement d |