Exact dynamic stiffness matrix for a thin‐walled beam of doubly asymmetric cross‐section filled with shear sensitive material

Abstract

SUMMARYAn exact dynamic stiffness matrix is developed for the flexural motion of a three-dimensional, bi-material beam of doubly asymmetric cross-section. The beam comprises a thin walled outer layer that encloses and works compositely with its shear sensitive core material. The outer layer may have the form of an open or closed section and provides flexural, warping and Saint-Venant rigidity, while the core material provides Saint-Venant and shear rigidity. The uniform distribution of mass in the member is accounted for exactly and thus necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick-Williams algorithm, which enables the required natural frequencies to be converged upon to any required accuracy with the certain knowledge that none have been missed. Such a formulation enables the powerful modelling features associated with the finite element technique to be utilized when establishing structural models. Three examples are included to validate and illustrate the method. The work also holds considerable potential in its application to the approximate analysis of asymmetric, multi-storey, three-dimensional wall-frame structures.

Coupled flexural–torsional dynamic stiffness matrix of an elastically supported axially loaded beam with shear resistant in‐fill

“…The results of third example compared with the results obtained using finite element abaqus software. Example Figure shows the continuous, thin‐walled beam with doubly asymmetric cross‐section that was originally considered by Rafezy and Howson (). The beam encloses a shear core between support points A and B, for which the following material and geometric properties apply: The shear core is then omitted over the remainder of the beam, where the original properties remain unchanged except that GA x = GA y = GJ c = 0 and the small change in m has been ignored, as it was in the original problem.…”

“…Example (1) Figure 3 shows the continuous, thin-walled beam with doubly asymmetric cross-section that was originally considered by Rafezy and Howson (2007). The beam encloses a shear core between support points A and B, for which the following material and geometric properties apply:…”

“…Ghandi et al . () extended Rafezy and Howson's work () by replacing the Euler–Bernoulli theory with Timoshenko theory when modeling the beam's thin‐walled outer layer. Such sandwich constructions is now finding progressively more application across many areas of mechanical, aeronautical and structural engineering (Zenkert (), Vinson ()).…”

Free Vibration Analysis of a Thin-Walled Beam with Shear Sensitive Material

“…Figure 1 shows a uniform, three dimensional beam of length . It has a doubly asymmetric cross-section comprising a thin-walled outer layer that encloses shear sensitive material [24]. The outer layer may have the form of an open or closed section that is assumed to provide warping and Saint-Venant rigidity, while the core materials provide Saint-Venant and shear rigidity.…”

“…In the beams, the elastic center and the center of mass are not coincident, so the translational and torsional modes are inherently coupled as a result of this offset. Rafezy and Howson [24] proposed an exact dynamic stiffness matrix approach for the three-dimensional, bimaterial beam of doubly asymmetric cross-section. The beam comprises a thin-walled outer layer that encloses and works compositely with its shear sensitive core material.…”