Order Plate Theory Research Articles

Functionally graded plates behave like homogeneous plates

In recent years many articles concerned with the mechanics of functionally graded plates have been published. Usually new analysis methods are developed to handle the continuous variation in material properties through the thickness of the plate and extensive results are presented. This article shows that no special tools are required because functionally graded plates behave like homogeneous plates. This simple result is developed using the classical plate theory and is shown to hold true when higher order plate theories or the three dimensional elasticity theory is used. The variation in material properties through the thickness of the plate introduces a coupling between the inplane and bending deformations which complicates the analysis. Here we show that by a proper choice of the reference surface, this coupling can be eliminated so that the bending of the plate is governed by the same equation of motion as that of homogeneous plates.

A HIGHER ORDER FINITE ELEMENT MODELING OF PIEZOLAMINATED SMART COMPOSITE PLATES AND ITS APPLICATION TO ACTIVE VIBRATION CONTROL

This paper presents a higher order — field consistent — piezolaminated 8-noded plate finite element with 36 elastic degrees-of-freedom per element and two electric degrees-of-freedom per element, one each for the piezoelectric sensor and actuator. The higher order plate theory used satisfies the stress and displacement continuity at the interface of the composite laminates and has zero shear stress on the top and bottom surfaces. The transverse shear deformation is of a higher order represented by the trigonometric functions allowing us to avoid the shear correction factors. In order to maintain the field consistency, the inplane displacements, u and v are interpolated using linear shape functions, the transverse displacement w is interpolated using hermite cubic interpolation function, while rotations θx and θy are interpolated using quadratic interpolation function. The element is developed to include stiffness and the electromechanical coupling of the piezoelectric sensor/actuator layers. The active vibration control performance of the piezolaminated smart composite plates has been studied by modeling them with the above element and applying various control strategies.

Cross-Laminated Timber Plates: Evaluation and Verification of Homogenized Elastic Properties

Cross-laminated solid wood panels are used in timber structures as load bearing plates and shear panels. Since timber is a relatively soft construction material, the design of such structures is driven by serviceability criteria. Therefore, accurate elastic properties are required. In this paper a fully automated procedure to determine global elastic properties of full-scale cross-laminated wood panels is developed. Experimental modal analysis is used to determine resonance frequencies and mode shapes of rectangular wooden specimens. An analytical model based on Reddy's higher order plate theory is applied to calculate natural frequencies and mode shapes numerically. Corresponding frequencies are allocated using the modal assurance criterion. All three shear moduli and the two in-plane stiffness moduli are identified successfully by minimizing the difference between measured and estimated resonance frequencies in a total least squares sense. By comparing resonance frequencies and additionally by a static bending experiment, it is shown that the global mechanical behavior of the specimen is accurately described using an orthotropic, homogenized, linear elastic material behavior.

On stress singularities at angular corners of plates of arbitrary thickness under tension

Stress singularities at angular corners of plates of arbitrary thickness with free boundaries subjected to in-plane loading are studied within the first order plate theory. By adapting a stress resultant function approach a characteristic equation for determining the structure and orders of the singularities near the vertex is developed. This equation differs from that obtained within the classical plane theory of elasticity (Williams’ solution) and includes singularities of the out-off-plane shear stress resultants.

Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order plate theory). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.

A Higher Order Asymptotic Analysis for Orthotropic Plates in Stress Edge Conditions

A higher order asymptotic analysis for orthotropic plates is presented with the leading order interior solution reduced to the well-known Kirchhoff plate theory. The boundary-layer solutions are decoupled into the plane strain and torsional deformations of a boundary-layer plane strip, which is analyzed by the Laplace transform method to formulate Saint-Venant’s principle in plate studies. The necessary and sufficient conditions for stress edge-data generating exponentially decaying solutions are established to derive the stress edge conditions for the higher order plate theory.

Shape control of smart composite plate with non-rectangular piezoelectric actuators

Quasi-static shape control of a smart structure may be achieved through optimizing the applied electric fields, loci, shapes and sizes of piezoelectric actuators attached to the structure. In this paper, a finite element analysis (FEA) software has been developed for analyzing static deformation of smart composite plate structures with non-rectangular shaped PZT patches as actuators. The mechanical deformation of the smart composite plate is modeled using a 3rd order plate theory, while the electric field is simulated based on a layer-wise theory. The finite element formulation is verified by comparing with experimentally measured deformation. Numerical results are obtained for the optimum values of the electric field in the PZT actuators to achieve the desired shape using the linear least square (LLS) method. The numerical results demonstrate the influence of the shapes of actuators.

Finite element vibration analysis of composite box structures using the high order plate theory

This study deals with free vibration of folded structures and box beams made of composite materials using a four-noded Lagrangian and Hermite finite element that incorporates high order transverse shear deformation and rotary inertia. An 8×8 matrix is assembled to transform the system element matrices from the local to global co-ordinates, in which an eighth drilling degree of freedom (d.o.f.) per node is appended to the existing 7-d.o.f. system. The results obtained are in good agreement with the semi-analytical solutions and numerical results reported by other investigators. Sample studies are carried out for various layup configurations and boundary conditions. The significance of the high order plate theory in analyzing folded structures is enunciated in this paper.

Dynamic behavior of folded composite plates analyzed by the third order plate theory

This study investigates the dynamic behavior of multiply-folded composite laminates analyzed the high order plate theory. Using the third order finite element program developed for this study, the effects of folding angles and ply orientations on the transient responses for various loading and boundary conditions are studied. The numerical results obtained are in good agreement with those reported by other investigators. Furthermore, the new results reported in this paper show the interactions between folding angels and layup sequences. Key observation points are discussed and a brief design guideline is given.

A new unconstrained third-order plate theory for Navier solutions of symmetrically laminated plates

Among the many higher order plate theories, the third-order shear deformation plate theory (TSDT) of Reddy is, perhaps, the most widely adopted model in the study of laminated plates. It, however, imposes a restriction that transverse shear stress vanishes on the top and bottom surfaces of the plate. Such requirement, although reasonable in many engineering applications, is not valid when the plate is subject to shear traction parallel to the surface. To account for such problems, the present plate model releases the constraints of vanishing transverse shear stress on the top and bottom plate surfaces. This unconstrained third-order shear deformation plate theory (UTSDT) is particularly useful for the study of a plate with contact friction or present in a flow field where the boundary layer shear stress is significant. The governing differential equations of UTSDT are of similar complexity as that of TSDT but it yields more accurate solutions. In addition, it is more flexible as it can be degenerated to the first-order shear deformation plate theory (FSDT) of Reissner and Mindlin if the higher-order rotation coefficients are neglected and a shear correction factor is considered, or to the TSDT if the relevant rotation coefficients are constrained. The present study further develops the unconstrained theory in composite laminates. Navier solutions for bending and stress analysis of multilayered and symmetrically laminated composite plates are presented. It is concluded that the present plate model provides more accurate solutions than that of TSDT, with similar level of analytical complexity, when compared with the 3D elasticity exact solutions.

Wave reflection and transmission in beams containing delamination and inhomogeneity

This paper presents an analytical approach using higher order plate theories to determine wave reflections from and transmissions through a damaged region in a beam. The damaged region is either treated as two split beams or as an inhomogeneity. The reflection ratios and transmission ratios are found to depend strongly on the frequency of the incident flexural waves, as well as the size of the damage, which gives rise to strong stop/pass band behaviour. Using the spectral analysis method, the transient wave propagation in a beam with a part-through delamination is predicted and compared with experimental results, indicating a good agreement in the phases and amplitudes of both the reflected and transmitted waves.

Lamb wave scattering from a circular partly through-thickness hole in a plate

The scattering problem of an incident plane S 0 Lamb wave in a plate with a circular partly through-thickness hole is considered in this paper. A 3D approach is used where the wave fields in the outer part outside the hole and in the inner part beneath the hole are expanded in the possible Lamb modes and the horizontally polarized shear modes. Both propagating modes and evanescent modes are included in the expansions. The expansion coefficients are obtained by utilizing the boundary conditions at the hole boundary and the continuity conditions below the hole. A linear system of equations for the expansion coefficients is obtained by projecting the different boundary conditions onto an orthogonal set of projection functions. A second approach is also presented, mainly for validation of the 3D model. This approach is based on the lowest order plate theories for extensional and flexural motion. Consequently, this model is only valid for very low frequencies. The two plate theories are combined and the wave fields in the different regions of the plate are obtained from the boundary conditions at the hole. Comparisons are made between the two different models for low frequencies and they show good agreement. A few examples are also given from the 3D model for higher frequencies where the plate theory model is not valid.

PLANE WAVE SOLUTIONS AND MODAL ANALYSIS IN HIGHER ORDER SHEAR AND NORMAL DEFORMABLE PLATE THEORIES

We use the three-dimensional Hellinger–Reissner mixed variational principle to derive a K th order (K = 0,1,2,...) shear and normal deformable plate theory. The balance laws, the constitutive relations and the boundary conditions for the plate theory are deduced. The constitutive relations incorporate the shear and the normal tractions applied on the top and the bottom surfaces of the plate. For a K th order plate theory with displacements expressed as a power series in the thickness co-ordinate z with terms up to z K , the transverse shear and the transverse normal stresses involve terms upto z K +2 while in-plane stress components have terms up to z K . The equations for the plate theory are expressed in a compact form by taking Legendre polynomials as the basis functions. The plate theory is used to study plane travelling waves and in particular the lengths of decay of the displacement components; this allows for a rigorous ordering of the importance of the displacement descriptors in terms of decaying properties.Finally, we study the free vibrations of a simply supported rectangular orthotropic thick plate; results from the present theory are compared with an exact three-dimensional solution and with other plate theories. To this end, a K th order compatible plate theory is also deduced; the term “compatible” alludes to the fact that the reduction map for the stress fields is induced by the kinematical reduction map, whilst in the “mixed” models it is postulated independently. It is found that the frequencies up to the fifth mode of vibration computed with the fifth order theory and without introducing any shear correction factors match very well with the corresponding analytical solution. Also, through-the-thickness distribution of all of the stress components is found to agree well with the three-dimensional elasticitysolution, while the stress distribution obtained from the compatible plate theories deviatesconsiderably, especially for the higher modes.

Systematic closed form modeling of hybrid parameter multiple body systems

Presented in this paper is a systematic approach to modeling non-holonomic hybrid parameter multiple body systems. The continuum bodies are represented with the postulates usually associated to the non-linear theories, the Timoshenko (like) beam theories, the higher order plate and shell theories, and the rational theories (e.g. rods) with intrinsic rotary inertia properties. The methodology is an extension of previous work. It is founded in variational principles, but uses vector algebra to eliminate tedium. The variational nature of the methodology allows rigorous equation formulation providing not only the complete non-linear hybrid differential equations, but also the boundary conditions. The methodology is formulated satisfying general non-holonomic constraints; it produces a minimal realization. The spatial dimensions of the continuua are not restricted and the inter-body connections are completely general. To demonstrate the application of the technique, a two-link elastic pendulum or manipulator is modeled. The algorithmic modeling steps are demonstrated. Numerical simulations are presented.

Nonlinear vibration of an initially stressed plate based on a modified plate theory

Abstract Nonlinear partial differential equations for the vibrating motion of a plate based on a modified higher order plate theory with seven kinematic variables are derived. The present seven-variable modified higher order plate theory satisfies the stress-free boundary conditions. Using these derived governing equations, the large amplitude vibrations of a simply supported thick plate subjected to initial stresses are studied. The Galerkin method is used to transform the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge–Kutta method is used to obtain the ratio of linear to nonlinear frequencies. Frequency ratios obtained by the present theory are compared with the Mindlin plate theory results and Lo's 11-variable higher order plate theory results. It can be concluded that present modified plate theory predicts frequency ratios very accurately. Also, the benefit of significant simplification can be observed as compared with the Lo's higher order plate theory. The effects of initial stress and other factors on frequency ratio are investigated.

Analysis of thick sandwich construction by a {3,2}-order theory

This study is an extension of the {1,2}-order plate theory to a higher order {3,2} theory. Based on the equivalent single-layer assumptions, the in-plane and transverse displacement components are expressed as cubic and quadratic expansions through the thickness of the sandwich construction. Also, the transverse stress component is assumed to vary as a cubic function through the thickness. Utilizing Reissner’s definitions for kinematics of thick plates, the displacement components at any point on the plate are approximated in terms of weighted-average quantities (displacements and rotations) that are functions of the in-plane coordinates. The undetermined coefficients defining the in-plane and transverse displacement fields are then expressed in terms of the weighted-average displacements and rotations and their derivatives by directly employing Reissner’s definitions and enforcing the zero transverse-shear-stress conditions on the upper and lower surfaces of the sandwich panel. The coefficients defining the transverse stress component are obtained by requiring the transverse strain component, which is expressed in terms of the unknown coefficients of the transverse stress component from a mixed constitutive relation, to be the least-squares equivalent of the kinematic definition of the transverse strain component. The resulting expressions for the unknown coefficients of the transverse stress component are related to resultant strains and curvatures defined from kinematic relations. The equations of equilibrium and boundary conditions of the sandwich plate based on the {3,2}-higher-order theory are derived by employing the principles of virtual displacements. The robustness and accuracy of this {3,2}-order plate theory are established through comparisons with exact solutions available in the literature. The finite element implementation of the present {3,2}-order plate theory is also discussed.

A {1,2}-order plate theory accounting for three-dimensional thermoelastic deformations in thick composite and sandwich laminates

A {1,2}-order theory for laminated composite and sandwich plates is extended to include thermoelastic effects. The theory incorporates all three-dimensional strains and stresses. Mixed-field assumptions are introduced which include linear in-plane displacements, parabolic transverse displacement and shear strains, and a cubic distribution of the transverse normal stress. Least squares strain compatibility conditions and exact traction boundary conditions are enforced to yield higher polynomial degree distributions for the transverse shear strains and transverse normal stress through the plate thickness. The principle of virtual work is used to derive a 10th-order system of equilibrium equations and associated Poisson boundary conditions. The predictive capability of the theory is demonstrated using a closed-form analytic solution for a simply-supported rectangular plate subjected to a linearly varying temperature field across the thickness. Several thin and moderately thick laminated composite and sandwich plates are analyzed. Numerical comparisons are made with corresponding solutions of the first-order shear deformation theory and three-dimensional elasticity theory. These results, which closely approximate the three-dimensional elasticity solutions, demonstrate that through-the-thickness deformations even in relatively thin and, especially in thick, composite and sandwich laminates can be significant under severe thermal gradients. The {1,2}-order kinematic assumptions insure an overall accurate theory that is in general superior and, in some cases, equivalent to the first-order theory.

A higher order plate theory for dynamic stability analysis of delaminated composite plates

A higher order shear deformation theory is used to investigate the instability associated with delaminated composite plates subject to dynamic loads. Both transverse shear and rotary inertia effects are taken into account. The procedure is implemented using the finite element method. Delamination is modeled using the penalty parameter approach. The natural frequencies are computed and compared with NASTRAN 3D results and available experimental data. The effect of delamination on the critical buckling load and the first two instability regions is investigated for various loading conditions, plate thickness and boundary conditions. As expected the natural frequencies and the critical buckling load of the delaminated plate are lower than those of the nondelaminated plate. They decrease with increase in delamination length. Increase in delamination length causes instability regions to be shifted to lower parametric resonance frequencies and the normalized width of the instability regions to increase.

PHASE VELOCITY CURVES ON NEW ARBITRARILY HIGHER ORDER PLATE THEORY

PHASE VELOCITY CURVES ON NEW ARBITRARILY HIGHER ORDER PLATE THEORY

EXACT CORRESPONDENCE BETWEEN EIGENVALUES OF MEMBRANES AND FUNCTIONALLY GRADED SIMPLY SUPPORTED POLYGONAL PLATES

We use Reddy's third order plate theory to study buckling and steady state vibrations of a simply supported functionally gradient isotropic polygonal plate resting on a Winkler–Pasternak elastic foundation and subjected to uniform in-plane hydrostatic loads. Young's modulus and the Poisson ratio for the material of the plate are assumed to vary only in the thickness direction. Effects of rotary inertia are considered. The problem of determining the critical buckling load or the vibration frequency of the plate is found to be analogous to that of ascertaining the frequency of a membrane clamped at the edges and whose shape coincides with that of the plate. The critical buckling load and the vibration frequency are shown to be positive. Some available results for plates symmetric about the mid-plane can be retrieved from the present analysis.