Shear Flexible Finite Element Research Articles

Structural–Acoustic Response Analysis of Variable Stiffness Laminates with Inherent Material Damping

Sound radiation and transmission loss characteristics of variable stiffness composite plate reinforced with the curvilinear fibers are investigated numerically. The formulation is developed using higher-order shear flexible finite element model combined with Helmholtz wave equation. The governing equations obtained using Hamilton’s approach are further solved through the modal super position method to analyze the vibration response under steady state excitation. The inherent material damping of the laminate is accounted through the modal damping calculated using the modal strain energy approach. The acoustic pressure of the variable stiffness laminates is estimated using the Raleigh integral. Subsequently, acoustic response characteristics such as acoustic power level, radiation efficiency, directivity pattern, and transmission loss from the laminates are predicted using the estimated sound pressure for various forcing frequencies. A parametric study covering a wide range of design variables including center and edge fiber angles, lamination scheme, thickness ratio, and boundary conditions on the acoustic sound behavior arising from the vibration of curvilinear fiber composite plate is detailed. This study reveals that the acoustic response of the curvilinear fiber composite plate is significantly influenced by the curvilinear fiber angles at the center/edge fiber angle of the layers. It is hoped that the results obtained here will be useful for designers in developing structures with desired acoustic response characteristics.

Nonlinear bending of porous curved beams reinforced by functionally graded nanocomposite graphene platelets applying an efficient shear flexible finite element approach

In the present study, the nonlinear flexural bending of thick and thin porous curved composite beams reinforced and functionally graded by graphene platelets is carried out using a three-noded C1 continuous curved beam finite element developed introducing an efficient shear deformation theory based on trigonometric function. The nonlinearity through the strain–displacement relationship by introducing von Karman’s assumptions is considered. The nonlinear equilibrium equations resulting from minimum potential energy principle are numerically solved based on the direct iteration technique. The bending nonlinear features through the load–deflection relationship are presented by selecting various design parameters like long and short beams, shallow and deep curved cases, support conditions, the variation of graphene platelets in the metal foam and the existence of porosity. This investigation reveals that the level of nonlinearity gets noticeably affected by the depth coupled with the curved beam slenderness ratio.

Free vibration characteristics of thick doubly curved variable stiffness composite laminated shells using higher-order shear deformation theory

In the present work, the natural frequencies of cylindrical and spherical laminated shells with variable stiffness are numerically studied using a shear flexible isogeometric finite element. The kinematics relies on cubic shear deformation theory in which cubic variation is assumed for the surface displacements and a quadratic variation for the traverse displacement along the thickness. A zig-zag function, used for the in-plane displacements, accounts for the abrupt discontinuity at the boundaries of the laminae. The Lagrangian equations of motion is deployed to solve the frequencies of curved panels. A detailed parametric analysis examines the influence of fibre centre/edge angles, shell geometric variables, material anisotropy and edge conditions on frequencies and mode shapes.

A finite element study on the large amplitude flexural vibration characteristics of FGM plates under aerodynamic load

The large amplitude flexural vibration characteristics of functionally graded material (FGM) plates are investigated here using a shear flexible finite element approach. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of the constituents. The effective material properties are then evaluated based on the rule of mixture. The FGM plate is modeled using the first-order shear deformation theory based on exact neutral surface position and von Kármán’s assumptions for large displacement. The third-order piston theory is employed to evaluate the aerodynamic pressure. The governing equations of motion are solved by harmonic balance method to study the vibration amplitude of FGM plates under supersonic air flow. Thereafter, the non-linear equations of motion are solved using Newmark’s time integration technique to understand the flexural vibration behavior of FGM plates in time domain (simple harmonic or periodic or quasi-periodic). This work is new in the sense that it deals with the non-linear flutter characteristics of FGM plates under high supersonic airflow accounting for both the geometric and aerodynamic non-linearities. Some parametric study is conducted to understand the influence of these non-linearities on the flutter characteristics of FGM plates.

Optimum design of laminated composite under axial compressive load

In the present study optimal design of composite laminates, with and without rectangular cut-out, is carried out for maximizing the buckling load. Optimization study is carried out for obtaining the maximum buckling load with design variables as ply thickness, cut-out size and orientation of cut-out with respect to laminate. Buckling load is evaluated using a ‘simple higher order shear deformation theory’ based on four unknown displacements u, v, w b and w s . A C1 continuous shear flexible finite element based on HSDT model is developed using Hermite cubic polynomial. It is observed that for thick anti-symmetric laminates, the non-dimensional buckling load decreases with increase in aspect ratio and increase in fibre orientation angle. There is a decrease in the non-dimensional buckling load of symmetric laminate in the presence of cut-out.

English

The influence of localised in-plane load on the flexural vibration characteristics of isotropic and composite plates have been studied using a four-noded shear flexible high precision plate bending finite element. First, the critical buckling loads of such plates subjected to partial or concentrated compressive loads were calculated, then the linear and nonlinear flexural vibration frequencies were obtained. Limited parametric study was carried out to study the influences of location and distribution of tensile or compressive in-plane load on the vibration frequencies of such plates. Defence Science Journal, 2010, 60(1), pp.106-111 , DOI:http://dx.doi.org/10.14429/dsj.60.117

Shear-flexible thin-walled element for composite I-beams

A shear-flexible finite element based on an orthogonal Cartesian coordinate system is developed for the flexural and buckling analyses of thin-walled composite I-beams with both doubly and mono-symmetrical cross-sections. Using the first-order shear deformable beam theory, the derived element includes both the transverse shear and the restrained warping induced shear deformations. Governing equations are derived from the principle of minimum total potential energy. Three different types of finite elements, namely, linear, quadratic and cubic elements are developed to solve the governing equations. The geometric stiffness for the buckling analysis of axially loaded, thin-walled composite beams is developed. The resulting linearized buckling problem is solved using a shifted inverse iteration algorithm. A parametric study of the effects of the aspect ratio and the fibre orientation on the tip displacement is presented. The convergence of the elements is also investigated. The elastic buckling loads for mono- and doubly-symmetric I-beam cross-sections are compared with other results available in the literature and with solutions using shell elements in a commercially available finite element program.

FREQUENCY RESPONSE OF A MODERATELY THICK ANTISYMMETRIC CROSS-PLY CYLINDRICAL PANEL USING MIXED TYPE OF FOURIER SOLUTION FUNCTIONS

A hitherto unavailable analytical solution to the boundary-value problem of the free vibration response of shear-flexible antisymmetric cross-ply laminated cylindrical panels is presented. The laminated shell theory formulation is based on the first order shear deformation theory (FSDT) including rotatory and surface-parallel inertias. The governing equations of the panel are defined by five highly coupled partial differential equations in five unknowns—three displacements, and two rotations. The assumed solution functions for the eigen/boundary-value problem are selected in terms of mixed-type double Fourier series. Numerical results presented for parametric effects, such as length-to-thickness ratio and radius-to-thickness ratio, should serve as a bench mark for future comparison. A four-node shear-flexible finite element is selected to compare the results with the present solution.

Free vibration response of a three-node shear flexible finite element for moderately deep shells

A free vibration response of a shear-locking free isoparametric shear flexible three-node triangular finite element suitable for moderately deep shell configurations is presented. The moderately deep shell configuration based on the Sanders’ shell theory is incorporated into the element formulation. A shear correction term is introduced in transverse shear strain components to alleviate the shear-locking phenomenon. The element is developed with a full integration scheme. Natural frequencies and mode shapes are obtained and compared for doubly curved shells with the available analytical and finite element solutions.

FREE VIBRATION RESPONSE OF SHEAR-DEFORMABLE ANTISYMMETRIC CROSS-PLY CYLINDRICAL PANELS

A hitherto unavailable analytical solution to the boundary value problem of free vibration response of shear-flexible antisymmetric cross-ply laminated cylindrical panels is presented. The equivalent single layer approach based on a first order shear deformation theory including rotary and in-plane inertias is incorporated into the shell formulation. The characteristic equations of the panel are defined by five highly coupled second and third order partial differential equations in five unknowns, i.e., three displacements, and two rotations. A recently developed solution methodology, based on a boundary-continuous double Fourier series approach, is utilized to solve the eigenvalue problem. Numerical results presented for various parametric effects such as length-to-thickness ratio, radius-to-thickness ratio, aspect ratio, and major-to-minor modulus ratio, etc., should serve as a bench mark for future comparison. A four-node shear-flexible finite element is selected to compare the results with the present solution.

A study on thermal postbuckling behaviour of laminated composite plates using a shear-flexible finite element

This paper deals with a nonlinear finite element formulation for studying the postbuckling behaviour of laminated plates induced by a uniform/nonuniform temperature field. The thermal stress analysis is carried out for the nonuniformly distributed temperature loading over the plate before solving the buckling problem. The formulation is based on the Reissner-Mindlin first order shear deformation theory. The nonlinearity due to moderately large deformation of the plate is included in the formulation. Deflection-load relationships are obtained through an iterative method. An element based on C° continuity developed recently is used to investigate the postbuckling/secondary instability behaviour of the laminates. Detailed numerical results based on various parameters are presented for cross-ply and angle-ply laminated composite plates with different boundary conditions.

Coupled rigid-elastic motion of filament-wound composite robotic arms

This paper presents a methodology for the analysis of robotic arms consisting of composite box-beams and rigid bodies undergoing large rotations. A two-node hybrid-stress shear-flexible finite element is used in modeling the links. The mass properties of the finite element which are required for multibody dynamic analysis are developed. A recursive formulation based on Kane's equations is used in the analysis of the multi-link system. Floating body reference frames allow the imposition of different boundary conditions and the dynamical equations contain all the nonlinear interactions between the rigid and elastic motions. Isotropic and composite spatial robotic arms of the same weight are simulated to illustrate the improvement obtained by using composite materials.

Filament-wound composite links in multibody systems

Dynamics of multi-link systems consisting of filament-wound composite beams and rigid bodies undergoing large rotations and translations is formulated. A two-node hybrid-stress shear-flexible finite element is used in modeling the links. The mass properties of the finite element which are required for multibody dynamic analysis are developed. A recursive formulation based on Kane's equations is used in the analysis of the multi-link system. Closed loops and prescribed motions are allowed in the system and the corresponding constraint equations are imposed. The constraint violations due to numerical errors are stabilized by using Baumgarte's method. A slider-crank mechanism is simulated to illustrate the differences between the responses of isotropic and composite mechanisms of the same weight.

Thermal stress analysis of laminated doubly curved shells using a shear flexible finite element

The thermal stress analysis oflaminated doubly curved shallow shells is presented using a shear flexible finite element model. The basic equations of the laminated shell theory are the extensions of Sanders' shell theory to include shear deformation and the thermal strains. The present finite element solution for isotropic and composite plates is compared with the closed form solutions available in the literature. A wide variety of laminated shell problems subjected to different temperature fields are studied. The influence of temperature dependent material properties, panel sizes, and boundary conditions on the thermal deformations are demonstrated.

Non-linear static and dynamic analysis of heated laminated plates: A finite element approach

The static and dynamic thermoelastic responses of laminated plates have been investigated by the use of a shear flexible finite element model. The model accounts for large deflections of the plate and non-uniform distributions of temperature. A nine-noded isoparametric element is implemented to obtain the numerical solutions. The validity of the model is demonstrated by comparison with solutions available in the literature. A wide variety of results are presented for the non-linear response of rectangular and circular plates under thermal and thermo-mechanical loads. The influences of anisotropy, the boundary conditions, aspect ratio, rotary inertia and stacking sequence on the thermally induced response are studied.

Nonlinear vibration of laminated plates using a refined shear flexible finite element

A finite element model based on a higher order shear deformation theory is developed to study the large amplitude-free vibration of laminated composite plates. The large deformation is accounted for by using von Karman strain-displacement relations. The higher order theory accounts for the parabolic variation of transverse shear strains through the thickness of the laminate. A C° nine noded isoparametric element with seven degrees of freedom at each node is implemented. The direct iteration method is used to solve the nonlinear equations which are evaluated at the point of reversal of motion. A wide variety of case studies are performed to examine the nonlinear vibration characteristics of laminated plates.

Thermal buckling of laminated plates using a shear flexible finite element

The buckling behavior of laminated composite plates subjected to a uniform temperature field is studied. Transverse shear flexibility is accounted for in the analysis using the thermo-elastic version of the first-order shear deformation theory. The prebuckling behavior of the plate under a uniform temperature field is studied prior to predicting the initial thermal buckling load. Numerical results are presented for thermal buckling problems using a nine-noded isoparametric quadrilateral finite element. The influence of boundary conditions, ply orientation, and plate geometries on the critical buckling temperature is examined.

A shear flexible finite element for nonuniform, laminated composite beams

A shear flexible finite element is formulated for linearly tapered, symmetrically laminated composite beams. The element has three nodes and 18 degrees of freedom. The three-node configuration is obtained from a five-node parent element by constraining the shear angle variation to be linear. The bending in two planes, twisting and stretching are considered. The performance of the element is tested with isotropic and composite materials, constant and variable cross-sections, and straight and curved geometries. The element proves to be accurate and versatile. The compatibility with plate and shell elements as a stiffener is assured through the use of simple nodal variables of C 0-type.

Stability and natural vibration analysis of laminated plates by using a mixed element based on a refined plate theory

A mixed shear flexible finite element, with relaxed continuity, is developed for the geometrically linear and non-linear analysis of layered anisotropic plates. The element formulation is based on a refined higher order theory which satisfies the zero transverse shear stress boundary conditions on the top and bottom faces of the plate and requires no shear correction coefficients. The mixed finite element developed herein consists of eleven degrees of freedom per node which include three displacements, two rotations and six moment resultants. The element is evaluated for its accuracy in the analysis of the stability and vibration of anisotropic rectangular plates with different lamination schemes and boudary conditions. The mixed finite element described here for the higher order theory gives very accurate results for buckling loads and natural frequencies.

A refined mixed shear flexible finite element for the nonlinear analysis of laminated plates

The present study is concerned with the development of a mixed shear flexible finite element with relaxed continuity for the geometrically linear and nonlinear analysis of laminated anisotropic plates. The formulation of the element is based on a refined higher-order theory. This theory satisfies the zero transverse shear stress boundary conditions on the top and bottom faces of the plate. Shear correction coefficients are not needed. The developed element consists of 11 degrees-of-freedom per node, taking into account three displacements, two rotations, and six moment resultants. An evaluation of the element is conducted with respect to the accuracy obtained in the bending of laminated anistropic rectangular plates with different lamination schemes, loadings, and boundary conditions.