Bending of a rod under concentrated load

At the free end of the rod it is not acted upon by any reaction force. Say at any cross-section at a length from wall, the reaction force is , then from force balance,

=>

(14.1)

and

From equation 13.1, . Then from , we have the following:

For a circular rod , so that .

But from equation 13.3, , so that

(14.2)

Now from the following expression of the tangent to any point on the rod and its derivatives,

(14.3a)

(14.3b)

From equation 14.3a and b, we derive,

(14.4)

Putting and , we finally obtain,

(14.5)

Equation 8.36 is solved with the boundary conditions:

at , and at , and

(14.6)

Integrating equation 14.5 and using b.c. 14.6,

(14.7)

Integrating the above expression we obtain the following integral, known as the elliptic integral

(14.8)

and, is obtained in terms of elliptic functions.

The shape of the rod is finally obtained in terms of dimensionless lengths as: