Delamination Effect on Flutter of Homogeneous Laminated Plates

The effect of delamination on the flutter boundary of two‐dimensional laminated plates are investigated theoretically. Linear‐plate theory and qusai‐steady aerodynamic theory are employed. A simple beam‐plate‐theory model is developed to predict the flutter boundaries of delaminated homogeneous plates with simply supported ends. The effects of delamination position, size, and thickness on the flutter boundary are studied in detail. The results reveal that the presence of a delamination degraded the stiffness and the natural frequencies of the plate and thereby decreases the flutter boundary of the plate. However, for certain geometries the flutter boundaries were raised due to the flutter coalescence modes of the plate altered by the presence of a delamination in the plate.     

Analysis of Thick Circular Plates Undergoing Large Deflections

This investigation considers the effect of transverse shear deformation on bending of the axisymmetrically loaded isotropic and orthotropic circular and annular plates undergoing large deflection. The analysis treats the nonlinear terms of lateral displacement as fictitious loads acting on the plate. The solution of a von Kármán‐type plate is, therefore, reduced to a plane problem in elasticity and a linear plate‐bending problem. Results are presented for simply supported and clamped plates and are in good agreement with the available solutions. For plates considered in this study, the influence of shear deformation on lateral displacement becomes more significant as the orthotropic parameter increases. The linear and nonlinear solutions for orthotropic plates deviate at a low value of the maximum deflection‐to‐thickness ratio (w/h). Consequently, the extent of w/h within which the small‐deflection theory is applicable to orthotropic plates is much lower than the value of about 0.4 typically used for isotropic plates, and it depends, in general, on the degree of orthotropy. The technique employed in this study is well suited for the analysis of nonlinear plate problems.     

A Higher Order Shear Deformation Model for Bending Analysis of Functionally Graded Plates

In this paper, the static response of simply supported functionally graded plates subjected to a transverse uniform load and resting on an elastic foundation is examined by using a new higher order displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The 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 material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the new higher order shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.     

Thermal Postbuckling Behavior of Composite Sandwich Plates

By considering the total transverse displacement of a sandwich plate as the sum of the displacement due to bending of the plate and that due to shear deformation of the core, a 72 degrees of freedom high precision high order triangular-plate element is developed for the thermal postbuckling analysis of rectangular composite sandwich plates. Due to an uneven thermal expansion coefficient in the two local material directions, the buckling mode of the plate can be changed from one mode to another as the fiber orientation or aspect ratio of the plate is varied. By examining the local minimum of total potential energy of each mode, a clear picture of buckle pattern change is presented. Numerical results show that for a sandwich plate with cross-ply laminated faces, buckle pattern change may occur when the plate has a long narrow shape. However, for sandwich plates with angle-ply laminated faces, the buckling mode is dependent on the fiber orientation and aspect ratio of the plate. The effect of temperature gradient on the postbuckling behavior of the sandwich plate is limited except for angle-ply laminated sandwich plates with fiber angle greater than 70° or less than 20°.     

Plastic-Buckling of Rectangular Plates under Combined Uniaxial and Shear Stresses

This paper is concerned with the plastic-buckling of rectangular plates under uniaxial compressive and shear stresses. In the prediction of the plastic-buckling stresses, we have adopted the incremental theory of plasticity for capturing the inelastic behavior, the Mindlin plate theory for the effect of transverse shear deformation, the Ramberg-Osgood stress–strain relation for the plate material, and the Ritz method for the bifurcation buckling analysis. The interaction curves of the plastic uniaxial buckling stress and the plastic shear buckling stress for thin and thick rectangular plates are presented for various aspect ratios. The effect of transverse shear deformation is examined by comparing the interaction curves obtained based on the Mindlin plate theory and the classical thin plate theory.     

Observations on Higher‐Order Beam Theory

A parabolic shear‐deformation beam theory assuming a higher‐order variation for axial displacement has been recently presented. In this theory, the axial displacement variation can be selected so that it results in a suitable admissible transverse shear‐strain variation across the depth of the beam. This paper examines several transverse shear‐strain variations that can go with the aforementioned higher‐order theory. Apart from the usual simple parabolic variation, six other shear‐strain variations are considered: the sinusoidal variation, cubic, quartic, quintic, and sixth‐order polynomials. All these variations for transverse shear‐strain satisfy the requirement that the shear strain be zero at the extreme fibers (z? = ?±h/2) and nonzero elsewhere along the depth of the beam. Comparison of the results from this paper with results from others show that the simple parabolic distribution for transverse shear strain gives most accurate results. Also, Timoshenko's theory (with a shear factor of five‐sixths) and the current formulation which uses the parabolic shear‐strain distribution, give identical values for deflections.     

Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory

This paper is concerned with the bending problem of micro- and nanobeams based on the Eringen nonlocal elasticity theory and Timoshenko beam theory. In the former theory, the small-scale effect is taken into consideration while the effect of transverse shear deformation is accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. General solutions for the deflection, rotation, and stress resultants are presented for transversely loaded beams. In addition, specialized bending solutions are given for beams with various end conditions. These solutions account for a better representation of the bending behavior of short, stubby, micro- and nanobeams where the small-scale effect and transverse shear deformation are significant. Considering particular loading and boundary conditions, the effects of small-scale and shear deformation on the bending results may be observed because of the analytical forms of the solutions.     

Postbuckling of Moderately Thick Imperfect Rings under External Pressure

A fully nonlinear finite-element analysis for postbuckling response of a moderately thick imperfect ring under applied hydrostatic pressure is presented. The fully nonlinear theory employed here, in contrast to the von Karman approximation generally prevalent in the existing literature, for a moderately thick ring does not, on employment of the conventional Love–Kirchhoff hypothesis (originally developed for the small deflection regime), automatically guarantee vanishing of the transverse normal and shear strains in the large deflection regime. A curved six-node element, based on an assumed quadratic displacement field (in the circumferential coordinate), employs a two-dimensional hypothesis, known as linear displacement distribution through thickness theory, to capture the effect of the transverse shear/normal (especially, shear) deformation behavior. Numerical results show that even for a sufficiently thin ring, the conventional nonlinear theory, based on von Karman approximation, produces an error on the order of 10%.     

Vibration and Stability of Thick Plates on Elastic Foundations

Natural frequencies and buckling stresses of a thick isotropic plate on two-parameter elastic foundations are analyzed by taking into account the effect of shear deformation, thickness change, and rotatory inertia. Using the method of power series expansion of the displacement components, a set of fundamental dynamic equations of a two-dimensional, higher-order theory for thick rectangular plates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are used to solve the eigenvalue problems of a simply supported thick elastic plate. To assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling stress are examined in detail. The distribution of modal transverse stresses are obtained by integrating the three-dimensional equations of motion in the thickness direction. The present approximate theories can accurately predict the natural frequencies and buckling stresses of thick plates on elastic foundations as compared with Mindlin plate theory and classical plate theory.     

Flutter Control of Smart Composite Structures in Hygrothermal Environment

Flutter control of smart composite plates under subsonic airflow is investigated in hygrothermal environment. The active fiber composite (AFC) which is more effective and adaptive than the conventional monolithic piezoelectric material is used in the present analysis to control the undesirable response due to hygrothermal effect. The velocity and displacement feedback control algorithm are subsequently established to reduce actively the response of the plate. Numerical examples of isotropic and laminated composite plates with or without hygrothermal effect are presented. It is observed that the structures become weak in the presence of hygrothermal load in the form of reduced flutter boundary. The flutter boundary can be enhanced with the help of AFC. The parametric study is performed and it is observed that the flutter boundary can be enhanced with the help of a feedback-control system activating the AFC. Therefore, one can say that the AFC is effective to enhance the flutter boundary of the present aeroelastic structure in the hygrothermal environment.     

Thermal Postbuckling of Preloaded Shear Deformable Laminated Plates

Postbuckling analysis is presented for a simply supported, shear deformable laminated plate subjected to a uniform lateral pressure and thermal loading, and resting on an elastic foundation. The temperature fields considered are associated with a nonuniform tentlike and parabolic distribution over the plate surface. The material properties are assumed to be independent of temperature. The lateral pressure is first converted into an initial deflection, and the initial geometric imperfection of the plate also is taken into account. The formulations are based on Reddy's higher-order shear deformation plate theory and include the plate-foundation interaction and thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine the thermal postbuckling equilibrium paths. The numerical illustrations concern the thermal postbuckling behavior of preloaded antisymmetric angle-ply laminated plates under a tentlike temperature field and symmetric cross-ply laminated plates under a parabolic temperature field resting on Pasternak-type or softening nonlinear elastic foundations from which the results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, fiber orientation, transverse shear deformation, the plate aspect ratio, thermal load ratio, and initial geometric imperfection as well as initial lateral pressure are studied.     

Deducing Buckling Loads of Sectorial Mindlin Plates from Kirchhoff Plates

This study presents a relationship between the buckling loads of sectorial plates based on the Kirchhoff (or classical thin) plate theory and the Mindlin plate theory. Whereas the former plate theory neglects the effect of transverse shear deformation, the latter plate theory allows for it. This effect becomes significant when dealing with moderately thick plates and sandwich plates. The relationship allows easy and accurate deduction of the buckling loads of the Mindlin plates from their corresponding Kirchhoff solutions.     

Localized Buckling of a Bilinear Elastic Ring under External Pressure

A fully nonlinear finite element analysis for prediction of localization in moderately thick imperfect rings under applied hydrostatic pressure is presented. The present nonlinear finite element solution methodology includes all the nonlinear terms in the kinematic equations and utilizes the total Lagrangian formulation in the constitutive equations and incremental equilibrium equations. A curved six-node element, based on an assumed quadratic displacement field (in the circumferential coordinate), employs a two-dimensional hypothesis, known as linear displacement distribution through thickness theory, to capture the effect of the transverse shear/normal (especially, shear) deformation behavior. The driving factor behind this analysis is to determine the onset of localization arising out of the bilinear material behavior of the ring with modal imperfection. Numerical results suggest that material bilinearity is primarily responsible for the appearance of a limit or localization (peak pressure) point on the postbuckling equilibrium path of an imperfect ring.     

Stochastic Free Vibration Response of Soft Core Sandwich Plates Using an Improved Higher-Order Zigzag Theory

In this paper, an improved higher-order zigzag theory for vibration of soft core sandwich plates with random material properties is proposed. The theory satisfies the condition of continuity in transverse shear stresses at all the layer interfaces and transverse shear stress free condition at the top and bottom of the plate, including the transverse flexibility effect of the core. The variation of in-plane displacements through thickness is assumed to be cubic while transverse displacement varies quadratically within the core and constant throughout the faces. The core is modeled as a 3D elastic continuum. An efficient C0 finite element in conjunction with a first-order perturbation approach is developed for the implementation of the proposed plate theory in a random environment and is employed to evaluate the second-order statistics of the eigensolutions by modeling lamina material properties as basic random variables. The mean and standard deviations of natural frequencies and their mode shapes are computed and validated with Monte Carlo simulation.     

Nonlinear Analysis of Angle‐Ply Composite and Sandwich Laminates

A refined higher order shear deformation theory for linear and geometrically nonlinear behavior of fiber‐reinforced angle‐ply composite and sandwich laminates is established. Laminae material is assumed to be linearly elastic, homogeneous and isotropic/orthotropic. The theory accounts for nonlinear quadratic variation of transverse shear strains through the thickness of the laminate and higher order terms in Green's strain vector in the sense of von Karman. A simple C0 finite‐element formulation of this theory is then presented with a total Lagrangian approach, and a nine node Lagrangian quadrilateral element is chosen with nine degrees of freedom per node. Numerical results are presented for linear and geometrically nonlinear analyses of multilayer angle‐ply composite and sandwich laminates. The theory is shown to predict displacements and stresses more accurately than first‐order shear deformation theory. The results are compared with available closed‐form and numerical solutions of plate theories and three‐dimensional finite‐element solutions. New results are also generated for future evaluations.     

Vibration of Laminate-Faced Sandwich Plate by a New Refined Element

An efficient six-noded triangular element based on refined plate theory is developed for the analysis of sandwich plates with stiff laminated face sheets and it is applied to a free vibration problem in this paper. The plate theory represents parabolic through thickness variation of transverse shear stresses with continuity at the layer interfaces, which introduces discontinuity at these interfaces for the shear strains. It is to be noted that the plate theory requires unknowns at the reference plane only. Moreover, it ensures a shear stress-free condition at the top and bottom surfaces of the plate. Thus, the plate theory has all of the features required for an accurate modeling of laminated sandwich plates. The plate theory suffers from a problem in its finite element implementation since it requires C1 continuity of transverse displacement at the element interfaces. As very few elements based on this plate theory exist and they possess certain disadvantages, an attempt has been made to develop this new element. It has been utilized to study some interesting problems of laminated sandwich plate.     

Quantification of Cable Deformation with Time Domain Reflectometry—Implications to Landslide Monitoring

Time domain reflectometry (TDR) technology has become a valuable tool for detecting displacements and locating shear planes in rock or soil slopes. It is based on transmitting an electromagnetic pulse into a coaxial cable grouted in rock or soil mass and watching for reflections of this transmission due to cable deformity induced by the ground deformation. Early detection of localized shear deformation in soft soils and quantifying the shear displacement using TDR remains a challenging work. The TDR response due to localized shear deformation is affected by cable resistance, soil-grout-cable interaction, and shear bandwidth. A comprehensive TDR wave propagation model considering cable resistance is introduced to model TDR response to cable deformity. Effects of the influencing factors on the relationship between the reflection spike and the shear displacement are investigated through laboratory tests and numerical simulations. The implications to enhancing TDR response and quantifying shear displacement are stressed. Practical suggestions are made, including procedure for correcting resistance effect, selection of cable and grout, and how to quantify shear displacement using TDR.     

Nonlinear Zigzag Theory for Buckling of Hybrid Piezoelectric Rectangular Beams under Electrothermomechanical Loads

A new efficient electromechanically coupled geometrically nonlinear (of von Karman type) zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electrothermomechanical loads. The thermal and potential fields are approximated as piecewise linear in sublayers. The deflection is approximated as piecewise quadratic to explicitly account for the transverse normal strain due to thermal and electric fields. The longitudinal displacement is approximated as a combination of third order global variation and a layerwise linear variation. The shear continuity conditions at the layer interfaces and the shear traction-free conditions at the top and bottom are used to formulate the theory in terms of three primary displacement variables. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of symmetrically laminated simply supported beams under electrothermal loads are obtained for comparing the results with the available exact two-dimensional (2D) piezothermoelasticity solution. The comparison establishes that the present results are in excellent agreement with the 2D solution which neglects the prebuckling transverse strain effect.     

Transverse Shear Stiffness of Composite Honeycomb Core with General Configuration

Based on the zeroth-order approximation of a two-scale asymptotic expansion, equivalent elastic shear coefficients of periodic structures can be evaluated via the solution of a local function τklij(y), and the homogenization process reduces to solving the local function τklij(y) by invoking local periodic boundary conditions. Then, effective transverse shear stiffness properties can be analytically predicted by reducing a local problem of a given unit cell into a 2D problem. In this paper, an analytical approach with a two-scale asymptotic homogenization technique is developed for evaluation of effective transverse shear stiffness of thin-walled honeycomb core structures with general configurations, and the governing 3D partial differential equations are solved with the assumptions of free warping constraints and constant variables through the core wall thickness. The explicit formulas for the effective transverse shear stiffness are presented for a general configuration of honeycomb core. A detailed study is given for three typical honeycomb cores consisting of sinusoidal, tubular, and hexagonal configurations, and their solutions are validated with existing equations and numerical analyses. The developed approach with certain modifications can be extended to other sandwich structures, and a summary of explicit solutions for the transverse shear stiffness of common honeycomb core configurations is provided. The lower bound solution provided in this study is a reliable approximation for engineering design and can be efficiently used for quick evaluation and optimization of general core configurations. The upper bound formula, based on the assumption of uniform shear deformation, is also given for comparison. Further, it is expected that with appropriate construction in the displacement field, the more accurate transverse stiffness can be analytically attained by taking into account the effect due to the face-sheet constraints.     

Bending Solutions of Axisymmetric Levinson Plates in Terms of Corresponding Kirchhoff Solutions

This paper presents exact axisymmetric bending solutions of circular and annular plates based on the higher-order plate theory of Levinson. The solutions are displayed in terms of the corresponding Kirchhoff (or classical thin) plate solutions. These Kirchhoff-Levinson bending relationships are derived using the mathematical similarity of the governing equations of the two plate theories and the basis of load equivalence. The relationships allow one to readily deduce the more accurate Levinson plate solutions that account for the effect of transverse shear deformation, without having to solve the more complicated Levinson plate equations.