The least squares weighted residual method was used in this work to solve the boundary value problem (BVP) of an Euler column of length l fixed at x = 0, and pinned at x = l. Polynomial shape (spline) functions for Euler columns with fixedpinned ends were used to obtain one – and two parameter buckling shape functions in terms of unknown generalised parameters. The one and two parameter buckling shape functions were used to construct least squares weighted residual integral statements of the boundary value problem. The least squares weighted residual statements simplified the boundary value problem (BVP) to algebraic eigenvalue – eigenvector problems. The solution for non trivial cases yielded characteristic buckling equations which were solved to obtain the buckling loads. One parameter coordinate shape function yielded the critical load as Qcr = 21EI/l2, while the two parameter buckling shape function yielded Qcr = 20.34614EI/l2. One parameter least squares weighted residual solution yielded a relative error of 4 % while the two parameter least squares weighted residual solution yielded a relative error of 0.77% compared to the exact solution.