Transverse Displacement Field Research Articles

SPECTRAL-ELEMENT-BASED SOLUTIONS FOR WAVE PROPAGATION ANALYSIS OF MULTIPLY CONNECTED UNSYMMETRIC LAMINATED COMPOSITE BEAMS

A spectrally formulated element that has three degrees of freedom (axial motion, transverse motion and rotation) per node for analysis of slender multiply connected composite beams under high-frequency impact loading is presented. The model represents a linear distributed parameter system. The element has an exact dynamic stiffness matrix as it is derived from exact solution to the governing wave equation in frequency domain. The present element can handle any kind of cross-sectional unsymmetry due to ply orientation. The results from the present formulation are compared with the results from a time domain finite element model based on linear axial and cubic transverse displacement field. It is shown that the formulated element is able to predict the reflected and transmitted response accurately through a rigid angle joint for varying joint angles and axial–flexural coupling. Also, the effect of ply-stacking sequence on dynamic response is illustrated. With this approach, a very large composite 2-D beam network can be analyzed under high-frequency impact loading with much smaller system size and lower computational cost compared to time domain finite element method.

First-order zig-zag sublaminate plate theory and finite element model for laminated composite and sandwich panels

A refined laminated plate theory and three-dimensional finite element based on first-order zig-zag sublaminate approximations has been developed. The in-plane displacement fields in each sublaminate are assumed to be piecewise linear functions and vary in a zig-zag fashion through-the-thickness of the sublaminate. The zig-zag functions are evaluated by enforcing the continuity of transverse shear stresses at layer interfaces. This in-plane displacement field assumption accounts for discrete layer effects without increasing the number of degrees of freedom as the number of layers is increased. The transverse displacement field is assumed to vary linearly through-the-thickness. The transverse normal strain predictions are improved by assuming a constant variation of transverse normal stress in each sublaminate. In the computational model, each finite element represents one sublaminate. The finite element is developed with the topology of an eight-noded brick, allowing the thickness of the plate to be discretized into several elements, or sublaminates, where each sublaminate can contain more than one physical layer. Each node has five engineering degrees of freedom, three translations and two rotations. Thus, this element can be conveniently implemented into general purpose finite element codes. The element stiffness coefficients are integrated exactly, yet the element exhibits no shear locking due to the use of an interdependent interpolation scheme and consistent shear strain fields. Numerical performance of the current element is investigated for a composite armored vehicle panel and a sandwich panel. These tests demonstrate that the element is very accurate and robust.

A discrete layer beam finite element for the dynamic analysis of composite sandwich beams with integral damping layers

A discrete layer finite element is presented for the dynamic analysis of laminated beams. The element uses C 0 continuous linear and quadratic polynominals to interpolate the in-plane and transverse displacement field, respectively, and is free from the effects of shear locking. Modal frequencies and damping are estimated using both the modal strain energy method and the complex modulus method. A forced response version of the model is also presented. The model predictions are compared with experimental data for composite sandwich beams with integral damping layers. Four damping configurations are considered, a constrained layer treatment, a segmented constrained layer treatment and two internal treatments.

A wavelet approach to the active structural acoustic control

This paper presents a new cost function to be used in active control strategies, approximating the radiated acoustic power from a planar structure by a wavelet transform. This cost function is defined to be global, i.e., in terms of energy, and in the wave number domain to exploit the fact that only wave number components located in the supersonic region contribute to sound power radiation. It is shown that the wavelet transform of the transverse displacement field can provide an energy localization both in the supersonic region and spatially on the structure. Thus, the new cost function is defined as being representative of the radiated acoustic power and is obtained from a wavelet transform which depends solely on a scale parameter which is directly related to the frequency so that this representation extends over a broadband frequency range without requiring more than a single step, i.e., the wavelet transformation. The wavelet approach is compared to the radiated acoustic power from simply supported beams of finite and infinite width and very good agreement is obtained. Optimal control simulations are performed to validate the use of the wavelet transform as the cost function in an active control strategy. The performance obtained using the wavelet approach as the cost function compares advantageously with the performance obtained using the radiated acoustic power.

Nonlinear Vibrations of Elastically-Constrained Rotating Disks

An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

The two-level FETI method for static and dynamic plate problems Part I: An optimal iterative solver for biharmonic systems

We present a Lagrange multiplier based substructuring method for solving iteratively large-scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems. The proposed method is essentially an extension of the FETI domain decomposition algorithm to fourth-order problems. The main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations. This results in a two-level FETI substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures, and grows at most polylogarithmically with the number of elements per substructure. These theoretically proven optimal convergence properties of the new FETI method are numerically demonstrated for several finite element plate bending static and transient problems. The two-level iterative solver presented in this paper is applicable to a large family of biharmonic time-independent as well as time-dependent systems. It is also extendible to shell problems.

Experimental results on active structural acoustic control using fiber-optic strain sensors

A new strategy for the active control of the sound power radiated from a flexural vibrating beam is experimentally investigated. This strategy involves sensing the discrete structural strain using white-light interferometric multimode fiber-optic strain sensors. Two approaches are used to relate the radiated sound power to the structural strain. The first approach is based on the reconstruction of the transverse structural displacement field from the structural strain field using a finite-difference scheme, and a wave-number transformation of the structural displacement to obtain the radiated sound field. The second approach is based on an exact expression directly relating the radiated sound field to the wave-number transform of the structural strain. In both approaches, the wave-number transform is performed over the supersonic (radiating) components of the structural information. The dynamic strain measurements are first compared with polyvinylidene film strain sensors. As a second step, a straightforward control scheme is implemented into a multichannel filtered-x LMS algorithm to control the strain amplitudes on the beam actively. Finally, both strategies for using the strain information are implemented to perform the active control of the sound power radiated from the beam. Preliminary performance results of both strategies using four strain sensors are then presented.

FrameView: Object-Oriented Visualization System for Frame Analysis

Rapidly developing desktop computing capabilities, which include high resolution graphics and interactive graphical user interfaces, are leading to a new generation of engineering software. One of the challenges in engineering software development is the effective use of computer graphics for visualization of data. Object-oriented methodologies hold the greatest promise to support reuse and rapid prototyping for large-scale software development. This paper outlines some of the essential characteristics of FrameView, an object-oriented software system for visualization of responses from any type of frame analysis model. FrameView provides a graphical user interface that is based on the X-Window system and Motif. The graphics library used is PEX, which is a three-dimensional (3D) extension to the X-Window system. The programming language used is C++. Important attributes of PEX, and tedious programming details required when using PEX, have been encapsulated in a number of classes. General classes are developed to abstract the tools needed for viewing frame-element responses and two-dimensional (2D) and 3D graphics modeling. Viewing of results from 2D frame elements based on an assumed cubic transverse displacement field for drawing the deflected shape, and an assumed linear function for any other general response quantities along the element length, have been incorporated in FrameView. An example of how FrameView has been extended to handle other graphical representations of frame-element responses is discussed.

Structural deformation in oxidized and reduced YBa 2(Cu 1− yCo y) 3O 6+ x compounds (0

We have investigated in detail the structural properties of the system YBa 2 CuPin1− y Co y ) 3O 6+ x with different values of x and y. X-ray diffraction shows that the Co doping induces an orthorhombic-to-tetragonal transition at y≈0.025. Electron diffraction and electron microscopy indicate that for y<0.025, displacement fields oriented along the 〈110〉 directions exist throughout the crystallite. For y0.025 electron diffraction shows the presence of intense streaks of diffuse scattering located at the tails of the Bragg reflections in the (001) plane of the reciprocal space, and oriented along the 〈110〉 directions. For the same oxidation treatment the size of the diffuse streaks increases with increasing dopant concentration. They disappear when the sample undergoes a reduction. Two-beam images show a contrasted cross-hatched pattern whose hatchings are also oriented along the 〈110〉 directions. These features are displacive in origin and correspond to the superposition of transverse displacement fields whose propagation vectors are oriented along the 〈110〉 directions. X-ray diffraction experiments with a conventional source as well as with synchroton radiation at LURE were carried out on a well oxidized single crystal ( y=0.04). The diffuse scattering is of the same type as that seen with electron diffractiion. These X-ray diffraction experiments confirms that the diffuse scattering is displacive in origin and shows that the displacements concern all atoms. The displacements are linked to the deformations created by the specific arrangement of the oxygen atoms forming the -Cu1-O4 microchains. One way to relax the internal constrains due to the microchains is to form microtwins. The lattice is distorted at each twin wall and this distortion induces transverse zig-zag displacements of all atoms, perpendicular to the twin walls.

Axisymmetric buckling of moderately thick polar orthotropic annular plates

The problems of axisymmetric elastic buckling of moderately thick polar orthotropic annular plates that are simply-supported or clamped at outer edges but free at inner edges and subjected to external and/or internal pressure are addressed in this work. By assuming the Mindlin-type first-order transverse shear deformable displacement field, governing differential equations are derived through application of the variational principle. Numerical solutions are obtained with the aid of the finite difference method. Comparisons with classical plate theory with Kirchhoff-type displacement fields are made. Effects of parameters, such as ratios of radii, elastic moduli, thickness to external radius are studied. An important finding is that the effect of transverse shear deformation has an extremely large influence on the buckling pressures of the thick and highly polar orthotropic plates. Such an effect, as demonstrated by the numerical examples, cannot be neglected in practice.

A High Precision Higher Order Triangular Element For Free Vibration Of General Laminated Plates

A 72 degree of freedom (d.o.f.) high precision triangular plate element based on a simplified higher order shear deformation plate theory is developed for the free vibration analysis of composite laminated plates. The in-plane and transverse displacement fields of the plate are assumed, respectively, to vary cubically and constantly through the plate thickness, so that zero transverse shear stress conditions on the free boundary planes can be satisfied and the shear correction factor is not required. The total displacement of the plate is expressed as the sum of the displacement due to bending and that due to shear deformation. Thus, the independent variables involved in the formulations can be reduced and the element formulation process can also be simplified. The element presents no shear locking problem and the results obtained for plates with various material properties and boundary conditions are compared with those obtained by using the three-dimensional (3-D) elasticity solution and other available alternative solutions.

Buckling and free vibration of cross-ply laminated circular cylindrical shells subjected to axial thrust and lateral pressure loading according to a higher order displacement field

An analytical solution is presented for the stability and free vibration of the cross-ply laminated thin circular cylindrical shells which are simply supported at both ends and subject to axial and lateral pressure. Based on the principles of virtual work and a nine-term higher order transverse shear and transverse normal deformable displacement field, a set of governing equations for thermal buckling of thin circular cylindrical shells together with the associated boundary conditions are derived. Navier double series method is then applied to find the closed-form solutions. Effects of important parameters such as degree of orthotropy, ratio of length to radius, ratio of radius to thickness, etc., are studied.

Vibration analysis of plates using the multivariable spline element method

The vibration analysis of plates using the multivariable spline element method is presented in this paper. The spline functions are applied to construct bending moments, twisting moments and transverse displacement field functions. The spline equations of eigenvalue problems with multiple variables of vibration of plates are derived based on the Hellinger-Reissner mixed variational principle. For simplicity, the boundary conditions which consist of three local spline points are amended to fit any specified boundary conditions. Several numerical solutions of plate vibration analysis are presented which illustrate the accuracy and convergence of the method.

Cylindrical bending of arbitrary cross-ply laminated plates

A refined shear deformation theory for cylindrical bending of arbitrary cross-ply laminated plates is presented. A piecewise linear in-plane displacement and constant transverse displacement field is employed. The transverse shear strains across any two different layers are assumed to be proportional to each other. Governing equations and consistent boundary conditions are deduced based upon the Principle of Minimum Potential Energy. The accuracy of the present theory is examined by applying it to a cylindrical bending problem of laminated plates which has been solved exactly. The results indicate that the present theory can accurately predict transverse displacement and in-plane displacement and normal stress at low span-to-thickness ratios.

A [formula omitted] three-node shallow shell element

A highly desirable three-node shallowly curved shell element is proposed for general shell analysis. Constrained-type C 0- anisoparametric interpolations are derived for the membrane and transverse displacement fields by the use of explicit edge constraints. Pertinent issues on shell theory and finite element approximations are addressed. Several numerical studies are carried out which demonstrate the effectiveness of this new triangular element in the regime of thin and moderately thick shells.

Shear Effect in Beam Stiffness Matrix

Formulation of stiffness matrix, consistent geometric stiffness matrix and consistent mass matrix for a beam element, including shear effect by coupling transverse and shear angular displacement field is presented.

Acoustic impedance of rectangular panels

The modal radiation impedance of a rectangular panel simply supported in an infinite baffle in the presence of an inviscid, uniform, subsonic flow is determined. The analysis is based on an expansion in normal modes of the transverse vibration displacement field of the panel and the use of wavenumber-frequency transforms. Integral expressions for the modal radiation impedance and the cross-modal coupling impedance are formulated. A Gaussian quadrature with reusable abscissas is developed to evaluate these modal radiation impedances. Changes with respect to flow speed in modal radiation damping of a rectangular membrane are measured. For the first observable low order mode the change is significant and agrees with the theory. For higher order modes non-significant changes are predicted and measured. The computed modal radiation reactances of the membrane are compared with the best available experiments. Significant discrepancies exist. Their causes are discussed.

Stress Analysis of Anisotropic Laminated Plates

Membrane and interlaminar stresses in statically loaded and vibrating composite structures are obtained by determining optically and numerically the respective partial derivatives of the holographically recorded transverse displacement field. A clamped, circular, orthotropic boron-epoxy plate under a uniformly applied static pressure and vibration are considered separately. Under uniform static pressure, yr0 = 0, while the anisotropy of the laminated composite results in ir9 ^ 0, which is unlike the corresponding isotropic case. The vibrating plate has complicated, unsymmetrical displacements producing both in-plane shearing strains and stresses. The higher order derivatives are obtained optically by holographic-moire and numerically by employing cubic-spline and discretequadratic differentiate functions.