A comparative study of the Galerkin approximation utilized in the Timoshenko beam theory

✍ Scribed by Mohsen Dadfarnia; Nader Jalili; Ebrahim Esmailzadeh

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 277 KB
• Volume: 280
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1016/j.jsv.2004.02.028

✦ Synopsis

The time-function selection utilized in Galerkin approximation is investigated for the Timoshenko beam theory. Both identical and different time functions for transverse deflection and beam's cross-sectional rotation are considered. In order to explain the underlying concept here, we consider a general Timoshenko beam system with uniform cross-section, which is moving in the horizontal plane. To characterize the elastic deformations, we associate to each point on the undeformed neutral axis of the beam two quantities nðx; tÞ and cðx; tÞ: The transverse elastic deflection is represented by n, while c denotes the orientation of the beam cross-section. x is the reference variable along the beam measured from one end of the support, and t is the time. Fig. 1 shows the kinematics of deformation of a beam element that undergoes a shear deformation in addition to a pure bending.

The equations of motion can be derived by applying Hamilton's principle. However, to facilitate the analysis, typically an assumed mode expansion (i.e., Galerkin approximation) is