The explicit form of the expressions makes this method applicable either in exact mathematical calculations or in standard displacement based finite element programs.

The relationships presented in the technical literature for the solution of non-prismatic beams are usually based on a numerical solution in which the beam is divided into smaller elements having constant mechanical properties in each element. The error encountered in these methods is reduced by increasing the number of the elements.

The bending of variable thickness elements by the displacement based formulations (stiffness method) includes the inherent approximations due to the fundamental assumptions of the displacement fields in those methods. These assumptions usually lead to the violation of one of the three fundamental equations as the necessary and sufficient conditions for the problem solution, namely equilibrium equations, consistency equations, constitutive equations of material behaviour.

One of the most convenient methods for overcoming this problem is to use a flexibility based formulation (force method) in the analysis. In this paper, a new formulation is presented. for the analysis of non-prismatic beams based on the implicit derivation of exact shape functions The flexibility basis of the method ensures the exact fulfilment of equilibrium at any interior point of the element. The explicit form of the expressions is well suited to be calculated by standard numerical procedures. The results obtained show the success of the proposed formulation in both exactness and economy in comparison with the methods present in the technical literature.

The innovative method for derivation of implicit shape functions through basic movement functions is presented. Utilizing this method, the stiffness, geometric stiffness and consistent mass matrices as well equivalent nodal loads of these beams have been obtained in an exact fashion, and the results could be extended to provide new elements for plate and shell elements of variable depths.

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