Modeling of the beam discontinuity with two analyses in strong and weak forms using a torsional spring model

In this paper, a discontinuity in beams whose intensity is adjusted by the spring stiffness factor is modeled using a torsional spring. Adapting two analyses in strong and weak forms for discontinuous beams, the improved governing differential equations and the modified stiffness matrix are derived respectively. In the strong form, two different solution methods have been presented to make an analogy between the formulation of the Euler-Bernoulli and Timoshenko theories that indicates the influence of the shear deformation in discontinuous beams. The flexural stiffness of discontinuous beams is corrected by using the Dirac’s delta function. In the weak form, the reduced stiffness matrix is derived from the strain energy equation established by the continuity, kinematics and constitutive equations. The linearity assumption of the geometry and material is considered to construct the kinematics and constitutive equations respectively. The continuity conditions mathematically connect two divided parts of the Euler-Bernoulli beam for which an improved Hermitian shape function is employed to interpolate displacement field. An application shows the comparison and validation of the results of the strong and weak forms, and also the quasi-static behavior of discontinuous beams.