A Dynamic Stiffness Element for Free Vibration Analysis of Delaminated Layered Beams

Nicholas H Erdelyi,Seyed M Hashemi

doi:10.1155/2012/492415

Abstract

A dynamic stiffness element for flexural vibration analysis of delaminated multilayer beams is developed and subsequently used to investigate the natural frequencies and modes of two-layer beam configurations. Using the Euler-Bernoulli bending beam theory, the governing differential equations are exploited and representative, frequency-dependent, field variables are chosen based on the closed form solution to these equations. The boundary conditions are then imposed to formulate the dynamic stiffness matrix (DSM), which relates harmonically varying loads to harmonically varying displacements at the beam ends. The bending vibration of an illustrative example problem, characterized by delamination zone of variable length, is investigated. Two computer codes, based on the conventional Finite Element Method (FEM) and the analytical solutions reported in the literature, are also developed and used for comparison. The intact and defective beam natural frequencies and modes obtained from the proposed DSM method are presented along with the FEM and analytical results and those available in the literature.

Highlights

Layered structures have seen greatly increased use in civil, shipbuilding, mechanical, and aerospace structural applications in recent decades, primarily due to their many attractive features, such as high specific stiffness, high specific strength, good buckling resistance, and formability into complex shapes, to name a few
The aim of this paper is to present a dynamic stiffness matrix (DSM) formulation for the free vibration analysis of a delaminated two-layer beam, using the free mode delamination model
Two computer codes, based on the conventional Finite Element Method (FEM) and the analytical solutions reported in the literature [7, 8], taking into account the same continuity conditions, are developed and used as a benchmark for comparison

Summary

Introduction

Layered structures have seen greatly increased use in civil, shipbuilding, mechanical, and aerospace structural applications in recent decades, primarily due to their many attractive features, such as high specific stiffness, high specific strength, good buckling resistance, and formability into complex shapes, to name a few. Equations of motion were derived from Hamilton’s principle, a Finite Element Method (FEM) was developed to formulate the problem, and the effects of location, size, and number of delaminations on vibration frequencies of delaminated beams were investigated [6]. The DSM describes the free vibration of the system and exhibits both inertia and stiffness properties of the syetem Based on this exact member theory, the DSM produces exact results for simple structural elements, such as uniform beams, using only one element [10, 11]. These expressions were used to develop a cracked DSM formulation, and the free vibration of doubly coupled cracked composite beams was investigated Given these considerations, the DSM method for a single beam can be modified to accurately model delaminated multilayer beams.

Mathematical Model

Numerical Tests

Findings

Conclusion