The exact axial, bending, and torsion stiffness matrices have been developed for an arbitrary nonuniform beam element. The coefficients of the bending stiffness matrix require the evaluation of three integrals, while the axial and torsion stiffness matrices require only one integral. These coefficients are evaluated for a uniform beam (verification) and a nonuniform beam with either linearly- or parabolically-varying cross-section dimensions. Two sets of numerical results are presented to provide a comparison of the current exact approach with a commonly used displacement-based approach and an approximate approach found in most commercial finite element programs. The two existing approaches produced acceptable results for an extremely small range of tapers. As more elements are used with the two existing approaches, their solutions will converge to the current exact solution which requires only one element. © 1992.
Friedman, Z., & Kosmatka, J. B. (1992). Exact stiffness matrix of a nonuniform beam-I. Extension, torsion, and bending of a bernoulli-euler beam. Computers and Structures, 42(5), 671–682. https://doi.org/10.1016/0045-7949(92)90179-4