**Strain energy under axial load:**

Consider a member of constant cross sectional area A, subjected to axial force Pas shown in Fig. 2.8. Let E be the Young’s modulus of the material. Let the member be under equilibrium under the action of this force, which is applied through the centroid of the cross section. Now, the applied force P is resisted by uniformly distributed internal stresses given by average stress σ =P/A as shown by the free body diagram (vide Fig. 2.8). Under the action of axial load P applied at one end gradually, the beam gets elongated by (say) . This may be calculated as follows. The incremental elongation of du small element of length dx of beam is given by,

Now the work done by external loads W= 1/2XPu (3.0)

In a conservative system, the external work is stored as the internal strain energy. Hence, the strain energy stored in the bar in axial deformation is,

Substituting equation (2.0) in (4.0) we get,

**Strain Energy in Bending :**

**Fig .6**

Consider a beam AB subjected to a given loading as shown in figure.

Let,

M = The value of bending Moment at a distance x from end A.

From the simple bending theory, the normal stress due to bending alone is expressed as.

**Castigliano’s theorem:**

**Numerical:**

**Maxwell’s Theorem**

The displacement of a point B on a structure due to a unit load acting at point A is equal to the displacement of point A when the unit load is acting at point B, that is, fBA = fAB.

The rotation of a point B on a structure due to a unit couple moment acting at point A is equal to the rotation of point A when the unit couple moment is acting at point B, that is, αBA = αAB.

These are useful for 2nd-degree-indeterminate and higher structures.

**Unit load method:**