Bending of a rod under distributed load

Here we will consider large bending of a rod which remains strongly attached to a rigid wall. Let us say that the rod of length and area of cross-section bends under the action of a uniform load, e.g. gravity. If the density of the rod is , then assuming that the rod does not undergo any extension, the reaction force at the rigid wall can be written as:

(13.12)

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,

=>       =>

(13.13)

Integrating the vectorial form of above expression,

and

(13.14)

From equation 13.14,

For a circular rod , we have,

, so that,

(13.15)

Now

	(13.16a)
	(13.16a)

Thus from equation 13.16a and b we can write the following force balance equation for bending of the rod

(13.17)

Putting and , we have

This equation is solved with the following boundary conditions:

, and ,

(13.18)

Solution of equation 13.17 along with the b.c. 13.18 yields the following graph,