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.
Fig 39. 2
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 s1 and Î1, in a manner analogous to that used in defining the strain energy density. Thus
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.
Castigliano's Theorem : Strain energy techniques are frequently used to analyze the deflection of beam and structures. Castigliano's theorem were developed by the Italian engineer Alberto castigliano in the year 1873, these theorems are applicable to any structure for which the force deformation relations are linear
Castigliano's Therom :
Let the two Loads P1 and P2 produce deflections Y1 and Y2 respectively strain energy in the beam is equal to the work done by the forces.
Let the Load P1 be increased by an amount DP1.
Let DP1 and DP2 be the corresponding changes in deflection due to change in load to DP1.
Now the increase in strain energy
Suppose the increment in load is applied first followed by P1 and P2 then the resulting strain energy is
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 DP1 approaches zero [ Partial derivative are used because the starin energy is a function of both P1 and P2 ]
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
Castigliano's First Theorem :
In similar fashion as discussed in previous section suppose the displacement of the structure are changed by a small amount ddi. While all other displacements are held constant the increase in strain energy can be expressed as
¶U / di ® is the rate of change of the starin energy w.r.t di.
It may be seen that, when the displacement di is increased by the small amount dd ; workdone by the corresponding force only since other displacements are not changed.
The work which is equal to Piddi is equal to increase in strain energy stored in the structure
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 di is equal to the corresponding force Pi provided that the strain is expressed as a function of the displacements.