A novel stiffness/flexibility-based method for Euler-Bernoulli/Timoshenko beams with multiple discontinuities and singularities

Nonprismatic beams have vast and various applications in mechanical and structural systems; thus, much research is dedicated to develop well-performed stiffness matrices for beams with different forms of section changes and singularities. This paper introduces stiffness matrices by the use of flexibility and stiffness methods. For this purpose, the spring model of the beam element is introduced. This model provides an innovative physical interpretation for the beam element. It can also be used to develop finite elements for both Euler-Bernoulli and Timoshenko beams with different combinations of tapering, singularity and discontinuity. In this model, beam sections are represented by some appropriate virtual springs; their connection in series/parallel gives the flexibility/stiffness matrix of the beam element. The obtained matrices are in general forms and applicable to different beam conditions.