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

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.

A Finite Element Formulation for Bending‐Torsion Coupled Vibration Analysis of Delaminated Beams under Combined Axial Load and End Moment

“…e systems are assumed to have a central split, about the midsection (L 1 � L 4 ) and of various lengths up to 60% of the span (0 ≤ L 2 /L ≤ 0.6). e split is occurring symmetrically along the midplane of the beam and surrounded by intact beam segments, as has also been presented and studied in [4,18]. e defective FEM models are then created and used to evaluate the natural frequencies and mode shapes of various prestressed delaminated beam configurations.…”

“…e un-prestressed defective beam, in this case, is assumed to be clamped-clamped and isotropic, with P 0 � 0, M zz � 0, H 2 /H � 0.3, and L 2 /L � 0.4. To validate the present FE solution method, the first two nondimensional natural frequencies, λ 2 , of a defective clamped-clamped beam with a through-the-width delamination occurring symmetrically about the midsection (L 1 � L 4 ) on the midplane (H 2 � H 3 ) for various delamination lengths are compared with the analytical results reported by Wang et al [4], as well as the DSM [18] and FEM data [19] by Erdelyi and Hashemi, where…”

“…Later, they developed an analytical model using coupled linear layerwise laminate theory and a FE model to investigate both static and dynamic response of flexible defective composite beams, including delaminations and active piezoelectric sensors [15]. Della and Shu [16,17] reported an analytical solution method, and more recently, Erdelyi and Hashemi used Dynamic Stiffness Matrix (DSM) [18] and presented a FEM-based novel assembly technique [19] to investigate the behaviour of a delaminated slender beam and compared their results with those reported by Wang et al [4] and Lee [20]. In a recent work, Szekrényes [21] performed stability analysis on delaminated composite beams undergoing coupled flexurallongitudinal vibration.…”

“…In two recent publications, the authors exploited the classical [24] and frequency-dependent (dynamic) FEM [25] formulations for the free vibration of intact prestressed beams. e free vibration/buckling of intact and delaminated axially-loaded beams has also been investigated before (e.g., [17][18][19][20][21][22][23]). However, to the authors' best knowledge, the vibration of prestressed delaminated beams, subjected to combined axial load and end moment, has not been reported in the open literature.…”

A dynamic model of a cantilever beam with a closed, embedded horizontal crack including local flexibilities at crack tips

“…While some studies [33,34,41,43,46] are focused on detection of delamination in a laminated material, some [4,16,31,32,[35][36][37][38]40,42,44,45] are focused on modeling delaminations in beam structures based on analytical methods, 4 the finite element (FE) method, and experimental methods. The compatibility conditions at the junctions are formulated as changes in the axial forces and bending moments there [16,31,32,[35][36][37]39,40,42,44], which cannot describe local flexibilities at crack tips due to the presence of a crack. Wang and Qiao [38] used a shear compliance coefficient at a crack tip to describe the crack tip deformation for a simply-supported end-notched beam specimen.…”