Mechanical Engineering Department

SOM-2 (UNIT-1) CHAPTER-1

Complementary Strain Energy :

Consider the stress strain diagram as shown Fig 1.1. 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.

Fig 1.1

Let us consider elastic non linear primatic bar subjected to an axial load. The resulting stress strain plot is as shown.

Fig 1. 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

Complementary Energy

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 :

Consider a loaded beam as shown in figure

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 above equation is castigation's theorem:

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

M_i = applied moment

q_i = resulting rotation

Castigliano's First Theorem :

In similar fashion as discussed in previous section suppose the displacement of the structure are changed by a small amount dd_i. While all other displacements are held constant the increase in strain energy can be expressed as

Where

¶U / d_i ® is the rate of change of the starin energy w.r.t d_i.

It may be seen that, when the displacement d_i 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 P_idd_i 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 d_i is equal to the corresponding force P_i provided that the strain is expressed as a function of the displacements.