^{1}

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.

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 replacement of traditionally metallic structural components with laminated composites has resulted in new and unique design challenges. Metallic structures exhibit mainly isotropic material properties and failure modes. By contrast, composite materials are anisotropic, which can result in more complex failure modes. Delamination is a common failure mode in layered structures. It may arise from loss of adhesion between two layers of the structure, from interlaminar stresses arising from geometric or material discontinuities, or from mechanical loadings. The presence of delamination may significantly reduce the stiffness and strength of the structures. A reduction in the stiffness will affect the vibration characteristics of the structures, such as the natural frequencies and mode shapes. Changes in the natural frequency, as a direct result of the reduction of stiffness, may lead to resonance if the reduced frequency is close to an excitation frequency.

The dynamic modeling of flexible delaminated multi-layer beams has been a topic of interest for many researchers. With the increase in use of laminated composite structures, the requirement for accurate delamination models has also grown. The earliest delamination models, formulated in the 1980s [

The accuracy of dynamic analysis, and forced response calculation, of a flexible structure depends greatly on the reliability of the modal analysis method used and the resulting natural frequencies and modes. There are various numerical, semianalytical, and analytical methods to extract the natural frequencies of a system. The conventional Finite Element Method (FEM) has a long well-established history and is the most commonly used method for modal analysis. The FEM is a general systematic approach to formulate the constant element mass and stiffness matrices for a given system and is easily adaptable to complex systems containing variations in geometry or loading. Nonuniform geometry, for example, is often modeled as a stepped, piecewise-uniform configuration. The conventional FEM formulation, based on polynomial shape functions, leads to constant mass and stiffness matrices and results in a linear eigenvalue problem from which the natural frequencies and modes of the system can be readily extracted. Lee [

Alternatively, one can use semianalytical formulations, such as the so-called Dynamic Finite Element (DFE) method [

Analytical methods, namely the dynamic stiffness matrix (DSM), have also been used for the vibrational analysis of isotropic [

The aim of this paper is to present a DSM formulation for the free vibration analysis of a delaminated two-layer beam, using the free mode delamination model. The delamination is represented by two intact beam segments; one for each of the top and bottom sections of the delamination. The delaminated region is bounded on either side by intact, full-height beams. The beams transverse displacements are governed by the Euler-Bernoulli slender beam bending theory. Shear deformation and rotary inertia, commonly associated with Timoshenko beam theory, are neglected. For harmonic oscillation, the governing equations are developed and used as the basis for the DSM development. Continuities of forces, moments, displacements, and slopes at the delamination tips are enforced, leading to the DSM of the system. Assembly of element DSM matrices and the application of boundary conditions results in the nonlinear eigenvalue problem of the defective system. In addition, two computer codes, based on the conventional Finite Element Method (FEM) and the analytical solutions reported in the literature [

Figure

The coordinate system and notation for a delaminated composite beam.

Consider a delamination tip after deformation. According to the numbering scheme in Figure

The faces of the delamination remain planar after deformation.

at stations

Likewise, using (

Through continuity conditions, a coupling relationship can be found within the delamination region to reduce the total number of unknowns from eight (

Numerical checks were performed to confirm the predictability, accuracy, and practical applicability of the proposed DSM method. DSM and FEM formulations, as well as the Coefficient Method (CM), were programmed in Matlab codes. To solve the nonlinear eigenproblem (

In what follows, an illustrative example of fixed-fixed, homogeneous, 2-layer delaminated beam is examined. The natural frequencies of the system with a central split, about the midsection (

Table

Natural frequencies

Delamination Length | Present DSM | Wang et al., as reported in [ | Della and Shu [ | FEM [ | ||||

Mode 1 | Mode 2 | Mode 1 | Mode 2 | Mode 1 | Mode 2 | Mode 1 | Mode 2 | |

Intact | 22.39 | 61.67 | 22.39 | 61.67 | 22.37 | 61.67 | 22.36 | 61.61 |

0.1 | 22.37 | 60.80 | 22.37 | 60.76 | 22.37 | 60.76 | 22.36 | 60.74 |

0.2 | 22.36 | 55.99 | 22.35 | 55.97 | 22.36 | 55.97 | 22.35 | 55.95 |

0.3 | 22.24 | 49.00 | 22.23 | 49.00 | 22.24 | 49.00 | 22.23 | 48.97 |

0.4 | 21.83 | 43.89 | 21.83 | 43.87 | 21.83 | 43.87 | 21.82 | 43.86 |

0.5 | 20.89 | 41.52 | 20.88 | 41.45 | 20.89 | 41.45 | 20.88 | 41.50 |

0.6 | 19.30 | 41.03 | 19.29 | 40.93 | 19.30 | 40.93 | 19.28 | 41.01 |

A split beam FEM, exploiting cubic Hermite [

Natural frequencies

Delamination Length | Present DSM | FEM; 6 elements | FEM; 10 elements | Layerwise FEM [ | ||||

Mode 1 | Mode 2 | Mode 1 | Mode 2 | Mode 1 | Mode 2 | Mode 1 | Mode 2 | |

Intact | 22.39 | 61.67 | — | — | — | — | 22.36 | 61.61 |

0.5 | 20.89 | 41.52 | 20.89 | 41.57 | 20.89 | 41.55 | 20.88 | 41.50 |

0.6 | 19.30 | 41.03 | 19.29 | 41.08 | 19.29 | 41.04 | 19.28 | 41.01 |

Figure

The first two natural modes for a 2-layered beam with centrally located midplane delamination, compared with those of the intact configuration. (Element totals are based on 1 element each for the outer intact segments, and equal element divisions for the top and bottom delaminated beam segments. Delamination tips and beam endpoints are visualized.), (a): 1st mode shapes, (b): 2nd mode shapes.

The inadmissible mode: interpenetration of equi-thickness top and bottom beams. While mathematically possible, this situation would not be encountered in practical applications; 60% of span, midplane delamination.

The first opening mode for a delaminated beam with top beam thickness equal to 40% the height of the intact beam; 60% of span, off-midplane delamination.

Based on the “exact” dynamic stiffness matrix (DSM) formulation, a new element for the free vibration analysis of a delaminated layered beam has been developed using the free mode delamination model. The DSM element exploits the closed form solution to the governing equation of the system and is “exact” within the limitations of the theory. For homogeneous beams with a central, midplane delamination, a 6-element model of the delaminated system provides excellent agreement with those models presented in the literature. A conventional finite element model was also briefly discussed. System natural modes, pole behaviour and opening modes for both midplane and off-midplane delaminations were also examined and illustrated.

The authors wish to acknowledge the support provided by NSERC, Ontario Graduate Scholarship (OGS), and Ryerson University, as well as the reviewers for their helpful comments.