In this work, the Picard’s successive iteration method was used to solve the elastic buckling problem of Euler columns with pinned ends. The problem was represented by a second order ordinary differential equation in the deflection function subject to the boundary conditions at the pinned ends. The boundary value problem was expressed in integral form, and the Picard’s iteration scheme developed from the integral form. A suitable buckling shape function was used to obtain an initial approximation to the deflection, and the Picard’s iteration scheme used to obtain first, second and third iterations for the modal buckling functions through the use of corresponding boundary conditions. Corresponding Picard’s approximations for the first, second and third iterations of the critical buckling load were obtained as 1 2 ( ) 9 6 0 / . , cr P E I l  ( ) 2 2 9 8361 / . cr EI l  and ( ) 3 2 9 8657 / . . cr EI l  The errors in the first, second and third iterates of the critical buckling load were -2.732%, -0.339% and -0.040% respectively. The use of the exact buckling shape function in the Picard’s iteration scheme was found to result in the exact closed form solution for the critical buckling load.