A generalized element for the free vibration analysis of composite beams
TL;DR: In this article, a general finite element based on a first-order deformation theory is developed to study the free vibration characteristics of laminated composite beams, which accounts for bi-axial bending as well as torsion.
About: This article is published in Computers & Structures.The article was published on 1994-06-03. It has received 49 citations till now. The article focuses on the topics: Finite element method & Vibration.
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13 Jan 2021
TL;DR: In this article, the effect of CNT orientation and gradation distribution on static and free vibration analysis of Functionally Graded CNT-Reinforced Composite (FG-CNTRC) beams is addressed.
Abstract: Structural tailoring can provide a promising performance for Functionally Graded (FG) components in engineering. Moreover, utilizing advanced Carbon Nanotube (CNT) as embedded reinforcement in nanocomposite structures, excellent mechanical properties can be tailored and designed to meet requirements. This research addressed the issue of a particular effect for CNT orientation and gradation distribution on static and free vibration analysis of Functionally Graded CNT-Reinforced Composite (FG-CNTRC) beams. First, an efficient finite beam element capable of controlling both parameters was derived based on the Timoshenko beam theory. Single-Walled CNT (SWCNT) was used as primary reinforcement and graded through-thickness. Then, an extensive parametric study was done for model convergence, static, and dynamic analysis. The proposed model offers unique shape function depends on material properties and cross-section geometry, high-accuracy, and expanded to cover both orientations and grading exponents. This expansion allows passive-control of the beam stiffness and strength without any increment in structural weight. Wherein constituent materials quantities and volume fractions were not changed. Finally, obtained findings concerned about orientation angle and power-law exponent, which showed that they significantly affect the structural response, and therefore offer a practical approach of structure tailoring for applied loads, required response, and specific weight limitations.
1 citations
TL;DR: In this article, a beam dynamic finite element analysis was performed by using the displacement element construction principle to obtain the general solution of displacement equation to the mode expressed by beam end displacements, and taking the mode as displacement trial function, element stiffness matrix and element mass matrix for beam flexural vibration and axial vibration were established, respectively, by applying principle of minimum potential energy.
Abstract: For beam dynamic finite element analysis, according to differential equation of motion of beam with distributed mass, general analytical solution of displacement equation for the beam vibration is obtained. By applying displacement element construction principle, the general solution of displacement equation is conversed to the mode expressed by beam end displacements. And taking the mode as displacement trial function, element stiffness matrix and element mass matrix for beam flexural vibration and axial vibration are established, respectively, by applying principle of minimum potential energy. After accurate integral, explicit form of element matrix is obtained. The comparison results show that the series of relative error between the solution of analytical trial function element and theoretical solution is about and the accuracy and efficiency are superior to that of interpolation trial function element. The reason is that interpolation trial function cannot accurately simulate the displacement mode of vibrating beam. The accuracy of dynamic stiffness matrix method is almost identical with that of analytical trial function. But the application of dynamic stiffness matrix method in engineering is limited. The beam dynamic element obtained in this paper is analytical and accurate and can be applied in practice.
1 citations
Cites methods from "A generalized element for the free ..."
...Nabi and Ganesan [22] put forward a finite element method based on free vibration analysis theory of composite beam with the first order shear deformation....
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...[22] S. M. Nabi and N. Ganesan, “A generalized element for the free vibration analysis of composite beams,” Computers and Structures, vol. 51, no. 5, pp. 607–610, 1994....
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TL;DR: In this article, the effect of the longitudinal to transverse moduli ratio on the first three purely out-of-plane resonance frequencies of 0°/90/90°/0° laminated straight beams with square section by considering different slenderness ratios and boundary conditions is considered.
Abstract: This work deals with the effect of the longitudinal to transverse moduli ratio on the first three purely outof-plane resonance frequencies of 0°/90/90°/0° laminated straight beams with square section by considering different slenderness ratios and boundary conditions. Numerical solution of the free vibration problem is performed with the help of the transfer matrix method based on the distributed parameter mathematical model. The rotary inertia and shear deformation effects are considered based on the first order shear deformation theory in the analysis. The overall transfer matrix of the beam is obtained making use of the series solution of a set of differential equations. Comparison of the exact natural frequencies obtained in this study with the existing results is quite good. INTRODUCTION As is well known, fiber-reinforced composite materials are being successfully used in many engineering applications due to their outstanding engineering properties. This has led to an increase in the number of studies relevant to applications of composites in structures. However, most of the work done in the field of vibration of composites has been theoretical in nature. Abarcar and Cunniff /1 / used the transfer matrix method based on the discrete mathematical model to determine the natural frequencies of clamped-free beams of general orthotropy. Ritchie et al. Ill calculated 211 Vol. 12, No. 4, 2001 Longitudinal to Transverse Moduli Ratio: Resonance Frequencies the spectrum of torsional and flexural resonant frequencies for orthotropic bars with free-free ends by the transfer matrix method. Miller and Adams /3/ presented the vibration characteristics of orthotropic fixed-free beams without including the effect of the shear deformation. Using the energy approach, Teoh and Huang /4/ worked out the free vibration of generally orthotropic beams with fixed-free ends. Teh and Huang 151 offered two finite element models based on a first order shear deformation theory for the free vibration analysis of fixed-free beams of general orthotropy. Teh and Huang 161 further investigated the influence of fiber orientation on normal mode shapes. Using the finite element technique utilizing transfer matrix analysis, Wallace and Bert 111 calculated the flexural and torsional natural frequencies of bars made of planar anisotropic materials. Free vibration analysis of symmetric laminated beams based on the first order shear deformation theory was reported by Chen and Yang /8/. Vinson and Sierakowski 191 presented the exact solution of a simply supported composite beam based on the classical lamination theory, which neglects the effects of the rotary inertia and shearing deformation. Kapania and Raciti /10/ examined the nonlinear vibrations of laminated beams. Kapania and Raciti II1/ also presented a survey in the vibration analysis of laminated composite beams. Chandrasekhara et al. /12/ derived the equation of motion of composite beams using a first order shear deformation theory and obtained exact frequencies and mode shapes of composite beams with several boundary conditions. Suresh et al. /13/ worked out the effect of warping on the freevibration characteristics of torsionally and flexurally coupled composite beams. Hodges et al. /14/ solved the equations of motion by both an exact integration method and a mixed finite element method. Exact solutions based on the Timoshenko type equations for symmetrically laminated composite beams with ten different boundary conditions are given by Abramovich /15/. The rotary inertia and shear deformation effects were investigated for simply supported beams with square section for the axial and out-of-plane bending vibration in Abramovich's study /15/. Chandrashekhara and Bangera /16/ studied the free vibration characteristics of laminated composite beams by the finite element analysis using a higher order shear deformation theory. Analytical solutions of the free vibration problem of laminated composite beams were obtained by Krishnaswamy et al. /17/. Singh and Abdelnassar /18/ worked out the forced vibration problem based on the third order shear deformation theory. Abramovich and Livshits /19/ studied the free vibration of symmetric laminated composite beams. Khdeir and Reddy /20/ employed the transfer matrix method in the free vibration analysis of cross-ply laminated beams based on the higher order shear deformation theory. The finite element method based on the first order shear deformation theory was presented by Nabi and Ganesan /21/. Eisenberger et al. 1221 utilized the dynamic stiffness analysis to study the free vibration of laminated beams using a first order shear deformation theory. Abramovich et al. /23/ studied vibrations of multi-span non-symmetric laminated composite beams. Khdeir /24/ examined the dynamic response of anti-symmetric cross-ply laminated composite beams for different boundary conditions. Harmonic response of tapered composite beams was examined by the finite element analysis based on the higher order shear deformation theory by Rao and Ganesan 1251. Zappe and Lesieutre 126/ presented an iterative smeared model for the vibration analysis of laminated beams. Song and Waas 1211 studied both buckling and free vibration analyses of laminated composite beams. They 1211 also investigated the shear deformation effects. Kant et al. /28/
Cites background from "A generalized element for the free ..."
...The ratio of the longitudinal modulus to transverse shear modulus E,/G,2 is also high (16-40) for fiber reinforced materials....
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TL;DR: In this article, a modeling method for the modal analysis of a composite trapezoidal plate undergoing in-plane translational acceleration is presented, where the equations of motion for the plate are derived and transformed into a dimensionless form.
Abstract: A modeling method for the modal analysis of a composite trapezoidal plate undergoing in-plane translational acceleration is presented in this paper. The equations of motion for the plate are derived and transformed into a dimensionless form. The effects of the inclination angles, fiber orientation angle and the acceleration on the modal characteristics of the plate are investigated.
References
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Book•
28 Feb 1986TL;DR: In this paper, the authors introduce the concept of anisotropic elasticity and composite Laminate Theory for composite materials, and present a test standard for polymer matrix composites.
Abstract: Preface to the Second Edition. Preface to the First Edition. 1. Introduction to Composite Materials. 2. Anisotropic Elasticity and Composite Laminate Theory. 3. Plates and Panels of Composite Materials. 4. Beams, Columns and Rods of Composite Materials. 5. Composite Material Shells. 6. Energy Methods For Composite Material Structures. 7. Strength and Failure Theories. 8. Joining of Composite Material Structures. 9. Introduction to Composite Design. Appendices: A-1. Micromechanics. A-2. Test Standards for Polymer Matrix Composites. A-3. Properties of Various Polymer Composites. Author Index. Subject Index.
1,144 citations
TL;DR: A review of the recent developments in the analysis of laminated beams and plates with an emphasis on vibrations and wave propagations is presented in this paper, where a significant effort has been spent on developing appropriate continuum theories for modeling the composite materials.
Abstract: A summary of the recent developments in the analysis of laminated beams and plates with an emphasis on vibrations and wave propagations in presented. First, a review of the recent studies on the free-vibration analysis of symmetrically laminated plates is given. These studies have been conducted for various geometric shapes and edge conditions. Both analytical (closed-form, Galerkin, Rayleigh-Ritz) and numerical methods have been used. Because of the importance of unsymmetrically laminated structural components in many applications, a detailed review of the various developments in the analysis of unsymmetrical ly laminated beams and plates also is given. A survey of the nonlinear vibrations of the perfect and geometrically laminated plates is presented next. It is seen that due to the bending-stretching coupling, the nonlinear behavior of the unsymmetrically laminated perfect and imperfect plates, depending upon the boundary conditions, may be hardening or softening type. Similar behavior also is observed for imperfect isotropic and laminated plates. Lastly, the developments in studying the wave propagation in laminated materials are reviewed. It is seen that a significant effort has been spent on developing appropriate continuum theories for modeling the composite materials. Some recent studies on the linear and nonlinear transient response of laminated materials also are described.
288 citations
TL;DR: In this paper, exact solutions for the free vibration of symmetrically laminated composite beams are presented for the first-order shear deformation and rotary inertia have been included in the analysis.
216 citations
TL;DR: In this article, a finite element model based on a higher-order shear deformation theory is developed to study the free vibration characteristics of laminated composite beams and the effects of in-plane inertia and rotary inertia are considered in the formulation of the mass matrix.
130 citations