Bending of a rod under concentrated vertical load

At any cross section, the internal stress is constant and is equal to . Similar to earlier examples, the expression of the tangent to the rod at any location and its derivatives can be written in terms of the angle ,

(14.9a,b)

From equation (14.9)a and b,

or

(14.10)

Boundary conditions are:

At

(14.11)

Solution of equation 14.10:

Multiplying both left and write hand side of equation 14.10

Integrating,

(14.12)

Using boundary condition , we have,

From equation 14.12 we have,

(14.13)

Then, for small , we have

(14.14)

The above expression yields a critical force for buckling of the rod under vertical load

(14.15)