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FLEXURAL ANALYSIS OF SYMMETRIC LAMINATED COMPOSITE TIMOSHENKO BEAMS UNDER HARMONIC FORCES – II. FINITE ELEMENT FORMULATION
Abstract
A super-convergent finite element is formulated for the dynamic flexural response of symmetric laminated composite beams subjected to transverse harmonic forces. Based on the assumptions of Timoshenko beam theory, a one-dimensional finite beam element with two-nodes and four degrees of freedom per element is developed. The new beam element is applicable to symmetric laminated composite beams and accounts for the effects of shear deformation, rotary inertia, and Poison's ratio. The analytical closed-form solution for flexural displacement functions developed in a companion paper is employed to develop exact shape functions. The present formulation can be used for quasi-static, steady state flexural analysis of symmetric laminated composite beams. It is also used to extract the natural frequencies and mode shapes for flexural response. The accuracy and efficiency of the present finite element are shown through comparisons with other established exact and Abaqus finite element solutions. The new element is demonstrated to be free from shear locking and discretization errors occurring in conventional finite element solutions and illustrates an excellent agreement with those based on finite element solutions at a fraction of the computational and modeling cost.
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The flexural dynamic response of symmetric laminated composite beams subjected to general transverse harmonic forces is investigated. The dynamic equations of motion and associated boundary conditions based on the first order shear deformation are derived through the use of Hamilton’s principle. The influences of shear deformation, rotary inertia, Poisson’s ratio and fibre orientation are incorporated in the present formulation. The resulting governing flexural equations for symmetric composite Timoshenko beams are solved exactly and the closed form solutions for steady state flexural response are then obtained for cantilever and simply supported boundary conditions. The applicability of the analytical closed-form solution is demonstrated via several examples with various transverse harmonic loads and symmetric cross-ply and angle-ply laminates. Results based on the present solution are assessed and validated against other well established finite element and exact solutions available in the literature.
Vibration and buckling analysis of cross-ply composite beams using refined shear deformation theory is presented. The theory accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from the Hamilton’s principle. The resulting coupling is referred to as triply coupled vibration and buckling. A two-noded C1 beam element with five degree-of-freedom per node is developed to solve the problem. Numerical results are obtained for composite beams to investigate modulus ratio on the natural frequencies, critical buckling loads and load-frequency interaction curves.
Vibrations of angle ply laminated beams are studied using the higher order theory and isoparametric 1d finite element formulations
through proper constitution of elasticity matrix. Subsequent to the validation of the formulation, deep sandwich and composite
beams are critically analyzed for various boundary conditions. Frequencies classified based on their spectrum are presented
along with those of first order theories for comparison.
A two-noded C1 finite element of 8 d.o.f. per node, based on a higher-order shear deformation theory, has been developed for flexural analysis of symmetric laminated composite beams in this note. The element assumes a parabolic variation of shear stress across the thickness of beams to avoid layer-dependent shear correction factors. Numerical results are given for simply supported, symmetric laminated composite beams, loaded equally at the top and bottom surfaces of the beams and are compared with the exact elasticity solution of Pagano. The special feature of the element is that transverse normal stress across the thickness of the beams, which is playing a crucial role in delamination, is reasonably accurately calculated using constitutive law even though unrealistic discontinuities at the interfaces of laminates are shown by the element.
This paper presents a new finite element formulation for the free vibration analysis of composite beams based on the third-order beam theory. This work also studies the influence of the mass components resulting from higher-order displacements on the frequencies of flexural vibration. By using Hamilton's principle, the variational consistent equation of motion in matrix form corresponding to the third-order shear deformation theory is derived. The resulting mass matrices are decomposed into three parts, i.e., the usual part, including the rotary inertia, corresponding to first-order theory, the part resulting from higher-order displacement, and the part resulting from the coupling between the different components of the axial displacement. The numerical examples show that the higher-order and coupling mass matrices have a significant influence on the frequencies of high mode flexural vibration. The present element formulation for composite beams can be easily extended to composite plates and shells.
A refined 2-node, 4 DOF/node beam element is derived based on higher order shear deformation theory for axial–flexural-shear coupled deformation in asymmetrically stacked laminated composite beams. The element has an exact shape function matrix, which is derived by satisfying the static part of the governing equations of motion, where a general ply-stacking is considered. This makes the element super-convergent in static analysis. The numerical results are validated by considering uniformly distributed load with various boundary conditions. Subsequently, the efficiency of the element for free vibration analysis is studied. Numerical examples showing the nature of interlaminar stresses for various ply-stacking configurations and boundary conditions are demonstrated. Also, parametric studies are performed to study the effect of coupling and stiffness on the natural frequencies. Based on these studies, number of observations are drawn.
A 2-noded curved composite beam element with three degrees of freedom per node is proposed for the analysis of laminated beam structures. The formulation accounts for flexural, extensional and transverse shear loadings in the plane of the curved beam. The transverse shear flexibility based on first-order shear deformation theory is incorporated. A cubic polynomial is assumed for the transverse displacement w. The field interpolations for the longitudinal displacement u and section rotation θ are derived using the elemental equilibrium equations. The procedure leads to field interpolations that are coupled by means of coefficients, which are functions of geometrical and material properties of the element. The efficacy of these coupled polynomial fields in improving the accuracy and convergence characteristics of the proposed element has been demonstrated by a series of numerical examples. The lay-up sequence does not affect the accuracy of the element, unlike the conventional 2-noded elements, which make use of independent field interpolations. The element does not exhibit membrane and shear locking. The test problems prove the versatility of the element for the analysis of curved and straight laminated beams.
The work presented in this paper is the theoretical and experimental investigation of the dynamic behavior for laminated composite beams (LCB). A finite element model is used in order to obtain the natural frequencies and mode shapes of the LCB. The model includes separate rotational degrees of freedom for each lamina but does not require additional axial or transverse degrees of freedom beyond those necessary for a single lamina. The shape functions ensure compatibility between laminas. Shear deformation is included but not interfacial slip or delamination. The effect of the fiber orientation is considered. Numerical results are compared with the experimental ones. The experimental tests are carried out by using a hammer test and the frequency response function is displayed on FFT analyzer. The theoretical model gives good results compared with the experimental ones. It is found that the envelope layers have the main effect on the natural frequencies.
A finite element model based on a higher-order shear deformation theory is developed to study the free vibration characteristics of laminated composite beams. The Poisson effect, which is often neglected in one-dimensional laminated beam analysis, is incorporated in the formulation of the beam constitutive equation. Also, the effects of in-plane inertia and rotary inertia are considered in the formulation of the mass matrix. Numerical results for symmetrically laminated composite beams are obtained as special cases and are compared with other exact solutions available in the literature. A variety of parametric studies are conducted to demonstrate the influence of beam geometry, Poisson effect, ply orientation, number of layers and boundary conditions on the frequencies and mode shapes of generally layered composite beams.
A general finite element based on a first-order deformation theory is developed to study the free vibration characteristics of laminated composite beams. The formulation accounts for bi-axial bending as well as torsion. The required elastic constants are derived from a two-dimensional elasticity matrix. As many results are not available for bi-axial bending vibration of beams, the obtained numerical results are compared with those existing in the literature for the case of uni-axial bending. The predominating modes for the first eight frequencies for different fibre orientations are presented. The numerical results given explain the effects of shear deformation on various vibrational frequencies of angle ply laminates. The parametric study conducted brings out the influence of beam geometry and boundary conditions on natural frequencies.
A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.