Showing papers on "Plate theory published in 1999"

The Behavior of Sandwich Structures of Isotropic and Composite Materials

Abstract: SANDWICH STRUCTURES: ORIGINS, ADVANTAGES, AND USES Description of Various Sandwich Constructions Advantages of Sandwich Construction over Construction Monocoque Thin Walled Construction Origins of Sandwich Construction Uses of Sandwich Construction Present Approach to Analysis Problems References ANISTROPIC ELASTICITY AND COMPOSITE LAMINATE THEORY Introduction Derivation of the Anisotropic Elastic Stiffness and Compliance Matrices The Physical Meaning of the Components of the Orthotropic Elasticity Tensor Methods to Obtain Composite Elastic Properties from Fiber and Matrix Properties Thermal and Hygrothermal Considerations Time-Temperature Effects on Composite Materials High Strain Rate Effects on Material Properties Laminae of Composite Materials Laminate Analysis [A], [B], and [D] Stiffness Matrices for a Mid-Plane Symmetric Sandwich Structure Piezoelectric Effects Problems References DERIVATION OF THE GOVERNING EQUATIONS FOR SANDWICH PLATES (PANELS) Introduction Plate Equilibrium Equations The Bending of Composite Material Laminated and/or Sandwich Plates: Classical Theory Classical Plate Theory Boundary Conditions Analysis of Composite Materials Laminated and/or Sandwich Panels Including Transverse Shear Deformation Effects Boundary Conditions for a Plate Using the Refined Plate Theory Laminated or Sandwich Plate on an Elastic Foundation Laminated or Sandwich Plates Subjected to Dynamic Loads Problems References BEAMS, COLUMNS, AND RODS OF COMPOSITE MATERIALS Development of Classical Beam Theory Some Simplified Sandwich-Beam Solutions Eigenvalue Problems of Sandwich Beams: Natural Vibrations and Elastic Stability Other Considerations Problems References ENERGY METHODS FOR SANDWICH STRUCTURES Introduction Theorem of Minimum Potential Energy Analysis of a Beam in Bending Using the Theorem of Minimum Potential Energy Reissner's Variational Theorem and Its Applications Static Deformation of Moderately Thick Beams Flexural Vibrations of Moderately Thick Beams Flexural Natural Frequencies of a Simply Supported Beam Including Transverse Shear Deformation and Rotatory Inertia Effects Minimum Potential Energy for Rectangular Plates A Rectangular Composite Material Plate Subjected to Lateral and Hygrothermal Loads In-Plane Shear Strength Determination of Composite Materials in Laminated and Sandwich Panels Problems References SOLUTIONS FOR RECTANGULAR SANDWICH PLATES Introduction Navier Solutions for Rectangular Sandwich Plates Levy Solutions for Plates of Composite Materials Perturbation Solutions for the Bending of a Composite Material Sandwich Plate, with Mid-Plane Symmetry and No Bending-Twisting Coupling Isotropic Sandwich Panels Subjected to a Uniform Lateral Load Minimum Weight Optimization for a Sandwich Panel Subjected to a Distributed Lateral Load Analysis of an Isotropic Sandwich Plate on an Elastic Foundation Subjected to a Uniform Lateral Load Static Analysis of Sandwich Plates of Composite Materials Including Transverse Shear Deformation Effects Exact Solution Other Considerations Problems References DYNAMIC EFFECTS ON SANDWICH PANELS Introduction Natural Flexural Vibrations of Sandwich Plates: Classical Theory Natural Flexural Vibrations of Sandwich Plates Including Transverse Shear Deformation Effects Forced-Vibration Response of a Sandwich Plate Subjected to a Dynamic Lateral Load Dynamic Response of Sandwich Plates to Localized Loads Large Amplitude Nonlinear Oscillations of Sandwich Plates Simply Supported on All Edges Linear and Nonlinear Oscillations of Specially Orthotropic Sandwich Panels with Various Boundary Conditions Vibration Damping Problems References THERMAL AND MOISTURE EFFECTS ON SANDWICH STRUCTURES General Considerations Derivation of the Governing Equations for a Thermoplastic Isotropic Plate Boundary Conditions General Treatment of Plate Nonhomogeneous Boundary Conditions Thermoelastic Effects on Beams Self-Equilibrium of Thermal Stress Rectangular Composite Material Plate Subjected to Lateral and Hygrothermal Loads References ELASTIC INSTABILITY (BUCKLING) OF SANDWICH PANELS General Considerations The Buckling of an Orthotropic Sandwich Plate Subjected to In-Plane Loads Classical Theory Elastic Stability of a Composite Sandwich Panel Including Transverse Shear Deformation and Hygrothermal Effects The Buckling of an Isotropic Plate on an Elastic Foundation Subjected to Biaxial In-Plane Compressive Loads The Buckling of Honeycomb Core Sandwich Panels Subjected to In-Plane Compressive Loads The Buckling of Solid- or Foam-Core Sandwich Panels Subjected to In-Plane Compressive Loads Buckling of a Truss-Core Sandwich Panel Subjected to Uniaxial Compression Elastic Stability of a Web-Core Sandwich Panel Subjected to a Uniaxial Compressive In-Plane Load Buckling of Honeycomb-Core Sandwich Panels Subjected to In-Plane Shear Loads Buckling of Solid-Core or Foam-Sandwich Panel Subjected to In-Plane Shear Loads Buckling of a Truss-Core Sandwich Panel Subjected to In-Plane Shear Loads Buckling of a Web-Core Sandwich Panel Subjected to an In-Plane Shear Load Other Considerations Problems References STRUCTURAL OPTIMIZATION TO OBTAIN MINIMUM-WEIGHT SANDWICH PANELS Introduction Minimum Weight Optimization of Honeycomb-Core Sandwich Panels Subjected to a Unidirectional Compressive Load Minimum Weight Optimization of Foam-Core Sandwich Panels Subjected to a Unidirectional Compressive Load Minimum Weight Optimization of Truss-Core Sandwich Panels Subjected to a Unidirectional Compressive Load Minimum Weight Optimization of Web-Core Sandwich Panels Subjected to a Unidirectional Compressive Load Minimum Weight Optimization of Honeycomb-Core Sandwich Panels Subjected to In-Plane Shear Loads Minimum Weight Optimization of Solid- and Foam-Core Sandwich Panels Subjected to In-Plane Shear Loads Minimum Weight Optimization of Truss-Core Sandwich Panels Subjected to In-Plane Shear Loads Minimum Weight Optimization of Web-Core Sandwich Panels Subjected to In-Plane Shear Loads Optimal Stacking Sequences for Composite Material Laminate Faces for Various Sandwich Panels Subjected to Various Loads Problems References SANDWICH SHELLS Introduction Analysis of Sandwich Cylindrical Shells under Axially Symmetric Loads A General Solution for Orthotropic-Sandwich Cylindrical Shells under Axially Symmetric Loads Shells with Mid-Plane Asymmetry Other Considerations Problems References BUCKLING OF SANDWICH CYLINDRICAL SHELLS Buckling of a Solid- or Foam-Core Sandwich Cylindrical Shell with Isotropic Faces Subjected to an Axially Symmetric Compressive End Load Buckling of a Solid- or Foam-Core Sandwich Cylindrical Shell with Orthotropic Composite Faces Subjected to an Axially Symmetric Compressive Load Buckling of a Honeycomb-Core Sandwich Cylindrical Shell with Composite Faces Subjected to an Axially Symmetric Compressive End Load Overall Buckling of Sandwich Cylindrical Shells Subjected to an Overall Bending Moment Buckling of a Sandwich Cylindrical Shell Due to External Pressure Buckling of a Sandwich Cylindrical Shell Due to Torsion Dynamic Buckling Problems References MINIMUM WEIGHT OPTIMIZATION OF SANDWICH CYLINDRICAL SHELLS General Discussion Minimum Weight Optimization of a Solid Foam-Core Sandwich Cylindrical Shell with Isotropic Faces Subjected to an Axially Compressive Load Minimum Weight Optimization of a Solid- or Foam-Core Sandwich Cylindrical Shell with Orthotropic Composite Material Faces Subjected to an Axially Compressive Load Minimum Weight Optimization of a Honeycomb-Core Sandwich Cylindrical Shell with Composite Material Faces Subjected to an Axially Symmetric Compressive Load Problems References APPENDIX 1: Core Materials APPENDIX 2: Face Materials APPENDIX 3: American Society for Testing Materials (ASTM) Standards for Sandwich Structures and Materials INDEX

Theory and analysis of elastic plates

Abstract: Preface 1.General Introduction and Preliminaries 2.Virtual Work Principles and Solution Methods 3.Theory and Analysis of Euler-Bernoulli Beams 4.Theory and Analysis of Timoshenko Beams 5.The Classical Plate Theory 6.The First Order Shear Deformation Theory 7.Variational Solutions of Plates 8.Finite Element Analysis of Beams and Plates 9.Refined Theories of Plates 10.Nonlinear Analysis of Beams and Plates Subject Index

Axisymmetric bending of functionally graded circular and annular plates

Abstract: Axisymmetric bending and stretching of functionally graded solid and annular circular plates is studied using the first-order shear deformation Mindlin plate theory. The solutions for deflections, force and moment resultants of the first-order theory are presented in terms of the corresponding quantities of isotropic plates based on the classical Kirchhoff plate theory. This gives the Mindlin solution of functionally graded circular plates whenever the Kirchhoff solution to the problem is known. Numerical results for displacements and stresses are presented for various percentages of ceramic-metal volume fractions.

Multilayered Shell Theories Accounting for Layerwise Mixed Description, Part 2: Numerical Evaluations

Abstract: The mixed layerwise shell theories that are presented in the companion article (E. Carrera, Multilayered Shell Theories Accounting for Layerwise Mixed Description, Part 1: Governing Equations' AIAA Journal, Vol. 37, No. 9, 1999, pp. 1107-1116) are evaluated here by solving several problems related to orthotropic cross-ply laminated, circular, cylindrical, and spherical shells subjected to static loadings for which closed-form solutions are given. Particular cases related to layerwise and equivalent single-layer models, based on classical displacement formulations, are evaluated for comparison purpose. A further comparison with three-dimensional elasticity exact solutions and to other higher-order shear deformations studies have been made. Results are given in the form of tables and diagrams. Approximations introduced by Donnell's shallow shell theories are evaluated for most of the problems. It has been concluded that the proposed mixed layerwise theories leads to a better description than the related analyses, which are based on displacement formulations. An excellent agreement, with respect to the exact solution, has been found for displacement and transverse stress components. These stresses have been herein calculated a priori. The importance of an adequate description of curvature terms related to the shell thickness to radii ratio h/R is also underlined. These effects have been contrasted by extensive use of fictitious interfaces in the conduced layerwise investigations.

Water Entry of a Wedge by Hydroelastic Orthotropic Plate Theory

Abstract: Water entry of a hull with wedge-shaped cross sections is analyzed. The stiffened platings between two transverse girders on each side of the keel are separately modeled. Orthotropic plate theory is used. The effect of structural vibrations on the fluid flow is incorporated by solving the two-dimensional Laplace equation in the cross-sectional fluid domain by a generalized Wagner's theory. The coupling with the plate theory provides three-dimensional flow effects. The theory is validated by comparison with full-scale experiments and drop tests. The importance of global ship accelerations is pointed out. Hydrodynamic and structural error sources are discussed. Systematic studies on the importance of hydroelasticity as a function of deadrise angle and impact velocity are presented. This can be related to the ratio between the wetting time of the structure and the greatest wet natural period of the stiffened plating. This ratio is proportional to the deadrise angle and inversely proportional to the impact velocity. A small ratio means that hydroelasticity is important and a large ratio means that hydroelasticity is not important.

Analytical Solution for Rectangular Thick Laminated Plates Subjected to Arbitrary Boundary Conditions

Abstract: Three-dimensional deformations of a multilayered, linear elastic, anisotropic rectangular plate subjected to arbitrary boundary conditions at its edges are analyzed by the generalized Eshelby-Stroh formalism. The rectangular laminate consists of anisotropic and homogeneous laminae of arbitrary thicknesses. Perfect bonding is assumed between the adjoining laminae in the sense that both surface tractions and displacements are assumed to be continuous across their interfaces. The analytical solution is in terms of infinite series, and the effect of truncating the series on the accuracy of the solution is scrutinized. The method is also applicable to rectangular laminated plates, with edges of each lamina subjected to different boundary conditions. Results are presented for thick plates with different sets of edge boundary conditions, e.g., two opposite edges simply supported and the other two subjected to eight different conditions or all four edges clamped.

Finite Element and Plate Theory Modeling of Acoustic Emission Waveforms

Abstract: A comparison was made between two approaches to predict acoustic emission waveforms in thin plates. A normal mode solution method for Mindlin plate theory was used to predict the response of the flexural plate mode to a point source, step-function load, applied on the plate surface. The second approach used a dynamic finite element method to model the problem using equations of motion based on exact linear elasticity. Calculations were made using properties for both isotropic (aluminum) and anisotropic (unidirectional graphite/epoxy composite) materials. For simulations of anisotropic plates, propagation along multiple directions was evaluated. In general, agreement between the two theoretical approaches was good. Discrepancies in the waveforms at longer times were caused by differences in reflections from the lateral plate boundaries. These differences resulted from the fact that the two methods used different boundary conditions. At shorter times in the signals, before reflections, the slight discrepancies in the waveforms were attributed to limitations of Mindlin plate theory, which is an approximate plate theory. The advantages of the finite element method are that it used the exact linear elasticity solutions, and that it can be used to model real source conditions and complicated, finite specimen geometries as well as thick plates. These advantages come at a cost of increased computational difficulty, requiring lengthy calculations on workstations or supercomputers. The Mindlin plat theory solutions, meanwhile, can be quickly generated on personal computers. Specimens with finite geometry can also be modeled. However, only limited simple geometries such as circular or rectangular plates can easily be accommodated with the normal mode solution technique. Likewise, very limited source configurations can be modeled and plate theory is applicable only to thin plates.

Nonlinear Flutter of Composite Panels Under Yawed Supersonic Flow Using Finite Elements

Abstract: A finite element formulation is presented for the effects of arbitrary flow direction on the large-amplitude supersonic flutter of composite panels. The von Karman large-deflection plate theory is used to account for large-amplitude limit-cyde oscillations, quasisteady first-order piston theory aerodynamics is employed for aerodynamic loading, and first-order shear deformation theory is used for laminated composite panels. An efficient solution procedure is presented by using the modal transformation to reduce the number of nonlinear panel flutter equations and then applying the linearized updated mode with nonlinear time function approximation to the reduced nonlinear panel flutter modal equations. A modal participation is defined and the minimum number of linear modes for accurate and converged limit-cycle response can be ensured. Examples are given for isotropic and composite panels at yawed supersonic flow.

On the accuracy of Mindlin plate predictions for the frequency-temperature behavior of resonant modes in AT- and SC-cut quartz plates

Abstract: The frequency spectra of resonant modes in AT- and SC-cut quartz plates and their frequency-temperature behavior were studied using Mindlin first- and third-order plate equations. Both straight-crested wave solutions and two-dimensional plate solutions were studied. The first-order Mindlin plate theory with shear correction factors was previously found to yield inaccurate frequency spectra of the modes in the vicinity of the fundamental thickness-shear frequency. The third-order Mindlin plate equations without correction factors, on the other hand, predict well the frequency spectrum in the same vicinity. In general, the frequency-temperature curves of the fundamental thickness-shear obtained from the first-order Mindlin plate theory are sufficiently different from those of the third-order Mindlin plate theory that they raise concerns. The least accurately predicted mode of vibration is the flexure mode, which results in discrepancies in its frequency-temperature behavior. The accuracy of other modes of vibrations depends on the degree of couplings with the flexure mode. Mindlin first-order plate theory with only the shear correction factors is not sufficiently accurate for high frequency crystal vibrations at the fundamental thickness-shear frequency. Comparison with measured resonant frequencies and frequency-temperature results on an AT-cut quartz plate shows that the third-order plate theory is more accurate than the first-order plate theory; this is especially true for the technically important fundamental thickness shear mode in the AT-cut quartz plate.

Active control of composite plates with integrated piezoelectric sensors and actuators under various dynamic loading conditions

Abstract: Theoretical formulations based on the classical laminated plate theory (CLPT) and Navier solutions are presented for the analysis of laminated composite plates with integrated sensors and actuators and subjected to both mechanical and electrical loadings. A negative force-cum-moment feedback control algorithm coupling the direct and inverse piezoelectric effects is presented and used as an active control of the dynamic response of the integrated plate structures through closed loop control. Emphasis in this study is also given to different types of dynamic loading condition. Three types of loading condition are considered, namely, an initial displacement being applied to the plate, harmonic surface loading and the plate being subjected to a moving point load.

Thermoelastic analysis of periodic thin lines deposited on a substrate

Abstract: Thermoelastic stresses and curvatures arising from patterned thin lines on initially flat isotropic substrates are analyzed. A connection is made between substrates with patterned lines and laminated anisotropic composites containing transverse matrix cracks. Using this analogy along with anisotropic plate theories, approximate analytical expressions are derived for volume-averaged stresses as well as curvatures along and normal to the lines, for any thickness, width and spacing of the lines. The predictions of the analysis are shown to compare favorably with finite element simulations of stresses and curvatures for Si substrates with Al, Cu or SiO 2 lines. The predictions also match prior experimental measurements of curvatures along and normal to patterned SiO 2 lines on Si wafers, and further capture the general experimental trends reported previously for curvature evolutions in Si wafers with Al lines. The model presented here thus provides a very convenient and simple analytical tool for extracting stresses in thin lines on substrates from a knowledge of experimentally determined film stress, thereby circumventing the need for detailed computations for a wide range of unpassivated line geometries of interest in microelectronic applications.

Characterization of Fatigue Behavior of Bonded Composite Repairs

Abstract: Composite patches are bonded to a cracked metallic surface either symmetrically (double sided ) or unsymmetrically (single sided ) to extend service life. The stresses in the metallic panel are greatly affected by the repair symmetry. Unsymmetric repairs present the greatest challenge because of the presence of out-of-plane bending. Thermal residual stresses are present because of the thermal coefe cient mismatch of the patch and the aluminum plate. Debonding along an adhesive ‐adherend interface can reduce the patch effectiveness. A simple analysis with Mindlin plate theory is investigated to model the host and the repair plate. The two plates are connected by an adhesive layer modeled by effective springs. Large dee ection theory is used in the case of unsymmetric repairs. The springs are ineffective in the debond zone and are removed. Both the aluminum and the debond cracks are characterizedbyfracturemechanicsby useofthestressintensity factorand strain-energyreleaserate,respectively. Experiments on aluminum 2024-T3 plate, AS4/3501-6 carbon/epoxy composite patch and FM73 adhesive include determining the thermal residual stresses in the aluminum plate and observation of debond development by use of an ultrasonic C-scan. Tests are conducted to examine the metallic and debond crack growth interaction on unsymmetric repairs.

Effects of interfacial damage on the global and local static response of cross-ply laminates

Abstract: The problem of the implications of interfacial damage on the local and global static response of cross-ply laminated flat structures is addressed in this paper. As a basic prerequisite, a third-order generalized zig-zag nonlinear plate theory incorporating the effect of interfacial damage is used. The theory rests upon a representation of the displacement field which: (i) fulfills a priori the shear traction continuity conditions at each interface of the laminate and the free shear traction conditions on the top and bottom external planes of the plate, (ii) satisfies the requirement of continuous displacements at the perfectly bonded interfaces, and (iii) incorporates the interfacial tangential displacement jump condition when slip-type interlayer imperfections are present. The theory also incorporates the geometrical nonlinearities and the thermal effects. Numerical results highlighting the effect of interfacial damage on the static response of laminated composite plates in cylindrical bending are displayed and conclusions on their implications upon their load-carrying capacity are outlined.

Modeling and Validation of Induced Strain Actuation of Composite Coupled Plates

Abstract: A consistent plate finite element model is formulated for coupled composite plates with induced strain actuation and validated with test data obtained from cantilevered isotropic and anisotropic plates. Actuators are modeled as additional plies fully integrated into substrate laminae, and the formulation is based on modified thin classical laminated-plate theory. The analysis is formulated for a generic anisotropic plate with a number of piezoactuators of arbitrary size, surface-bonded or embedded at arbitrary locations. Composite plates with extension-twist and bending-twist couplings were built and tested. Two rows of piezoceramic elements are surface mounted on both top and bottom surfaces near the root. Static tests are carried out using induced strain actuation, and mechanical loading and measured data are correlated with predictions for bending and twist distributions. For an extension-twist coupled plate, the agreement between predicted and measured induced twist due to extensional strain with piezoactuation is excellent. For the strongly bending-twist coupled composite plate, the predicted induced twist due to bending strain with piezoactuation agreed well in trends, but magnitudes were underpredicted by a maximum of 20% from measured values. For the weakly bending-twist coupled composite plate, the predicted induced-twist angle agreed extremely well with measured data. The modeling and validation results show the usability of the piezoactuation in the field of plate shape control.

Spline FSM postbuckling analysis of shear-deformable rectangular laminates

Abstract: A spline finite strip method is developed for the prediction of the geometrically non-linear response of rectangular, composite laminated plates to progressive in-plane loading. The development takes place within the context of the use of the first-order shear deformation plate theory and the non-linearity is introduced in the strain-displacement equations in the manner of the von Karman assumption. A number of applications of the new capability is described, involving laminates subjected to progressive uniform end shortening and to progressive in-plane shearing. In all the applications a close comparison of the finite strip results with independent finite element results is demonstrated.