Trapezoidal Plates Research Articles

Free vibration and buckling analyses of composite plates with straight-sided quadrilateral domain based on DSC approach

Discrete singular convolution (DSC) method has been proposed to obtain the frequencies and buckling loads of composite plates. By using geometric transformation, the straight-sided quadrilateral domain is mapped into a square domain in the computational space using a four-node element. Plates having different geometries such as rectangular, skew, trapezoidal and rhombic plates are presented. The obtained results are compared with those of other numerical methods. Numerical results indicate that the DSC is a simple, accurate and reliable algorithm for vibration and buckling analyses of composite plates.

Plate osteosynthesis of the mandibular condyle

The aim of this study is to evaluate and compare the biomechanical stability of various osteosynthesis materials for mandible condylar-process fractures. On 160 porcine mandibles, four different monocortical plating techniques (40 per group) were investigated. Condyles were fractured at a defined location from the incisure to the posterior border. After correct anatomical reduction the fractures were plated, using four different techniques. Osteosynthesis materials used were the delta plate, the trapezoid plate, the dynamic compression plate and double mini-plates. Each group was subjected to linear loading in lateral to medial, medial to lateral, anterior to posterior and posterior to anterior directions by a universal mechanical testing machine TIRAtest 2720. Yield load, yield displacement were measured for the different plates. Statistically significant differences were noted between the fixation groups in all four directions. Rigid internal fixation with double mini plates showed the best stability in all directions except posterior to anterior. In this direction, the delta-plate resisted the highest loads. In the three other directions, the delta plate was second best with data similar to double miniplates but lower in magnitude.

Development and biomechanical testing of a new osteosynthesis plate (TCP ®) designed to stabilize mandibular condyle fractures

In this paper, the different steps of development and experimental validation of a new type of three-dimensional (3-D) trapezoidal osteosynthesis plate (Modus TCP 2.0, Medartis, Basel, Switzerland) is described. These plates have been designed to stabilize sub-condylar and condylar neck fractures of the mandible. In order to apply the principles of functionally stable osteosynthesis to the mandibular condyle, i.e. to put the plate as close as possible to the tensile strain lines occurring during function, two new 4- and 9-hole 3-D trapezoidal plates were designed. Tests were conducted on fresh human mandibles before and after osteosynthesis of a standardised unilateral sub-condylar 'fracture', and a static biting exercise between the ipsilateral first molars was reproduced on a test bench. The resulting condylar fragment displacement in the sagittal plane was measured and the alterations of the condylar tensile strain lines induced by the osteosynthesis were investigated by using photoelastic strain tests. None of the plates broke. No macroscopic condylar displacement was noted when assessing the quality of the primary stabilization. Strain analysis showed the ability of these 3-D plates to transmit physiological strains across the fracture line and the absence of potentially damaging strains around the plate. These results were accredited to the 3-D and trapezoidal features of the plates. The Modus TCP plates experimentally fulfil the principles of functionally stable osteosynthesis in the condylar region and are able to resist physiological strains.

Experimental and numerical investigations for the free vibration of cantilever trapezoidal plates

Based on the advantages of non‐contact and full field measurement, the optical technique called amplitude‐fluctuation electronic speckle pattern interferometry (AFESPI) with an out‐of‐plane setup is employed to investigate the free vibration of cantilever trapezoidal plates with various taper ratios and sweep‐back angles. Twenty different plate configurations are analyzed, including triangular and trapezoidal plates, and the first seven vibration modes of each plate are measured. The AF‐ESPI method is very convenient for measuring vibrating objects because no contact is required in contrast to classical modal analysis using accelerometers. Based on the fact that clear fringe patterns will appear only in resonance, both resonant frequencies and corresponding mode shapes can be obtained experimentally using the present technique. Numerical calculations by finite element method are also performed and the results are compared with the experimental measurements. Excellent agreements are obtained for both results of resonant frequencies and mode shapes. The influences of taper ratios and sweep‐back angles on the vibration behavior of cantilever trapezoidal plates are also demonstrated in terms of the dimensionless frequency parameter.

Needs and trends for biomaterials and biodevices in oral and maxillofacial surgery

AbstractThe evolution of the devices used for the fracture fixation is now oriented toward the bioresorbable materials (PLA, PGA, PDX) which should gradually replace the metallic materials (titanium, SS) in the next 10‐15 years. The osteosynthesis devices will have to fulfil more precise biomechanical criteria, in certain anatomical zones (i.e. trapezoidal plates for the mandibular condyle).The most known TMJ prostheses are either definitive prostheses (no more available on the European market, only in the American market) or temporary condylar prostheses. In this field the lag compared with the orthopaedics is certainly over 20 years. Very few biomechanical studies have been performed and there is few interest from the industrials in Europe for this type of prostheses because of the small number of cases and the scarcity of the indications.Numerous materials of all classes have been tried as bone substitutes but all are lacking of osteoinductivity and therefore combined materials are largely studied. Actually, in maxillofacial surgery, the most used materials are HA, TCP, Coral eventually enriched with growth factors. Concerning tissue engineering, nice models are elaborated in various laboratories in the world but huge difficulties remain regarding the optimal molecular environment and the biomechanical functionality.

Transverse Vibration of Mindlin Plates on Two-Parameter Foundations by Analytical Trapezoidal p-Elements

An analytical trapezoidal hierarchical element for the transverse vibration of Mindlin plates resting on two-parameter foundations is presented. Legendre orthogonal polynomials are used as enriching shape functions to avoid the shear-locking problem and to improve considerably the computational efficiency. Element matrices are integrated in closed form eliminating the numerical integration errors conventionally found. With the C0 continuity requirement, the element can be used to analyze any triangular and polygonal plates without difficulty, while the Kirchhoff p -version elements requiring C1 continuity are not as versatile. The computed natural frequencies for rectangular, skew, trapezoidal, triangular, annular, and polygonal plates on two-parameter foundations show that the convergence of the proposed element is very fast compared to the conventional linear finite elements with respect to the number of degrees of freedom used. Many numerical examples are given.

Using Spline Functions to Solve Boundary-Value Problems for Laminated Orthotropic Trapezoidal Plates of Variable Thickness

The bending problem for laminated orthotropic trapezoidal plates of variable thickness is solved by parametrizing the domain of interest and reducing the two-dimensional boundary-value problem to one-dimensional using the spline-collocation method. The resulting problem is solved by the stable discrete-orthogonalization method. The influence of orthotropy and the variability of the thickness on the displacement and stress fields is analyzed

A monopolar patch antenna with very wide impedance bandwidth

A monopolar patch antenna that utilizes a bandwidth broadening technique is investigated experimentally and numerically. The proposed antenna consists of a monopole that is formed by connecting four trapezoidal plates orthogonally. A circular patch, which is shorted to the circular ground plane by four shorting wires, is placed at the top of the monopole. The antenna possesses 10-and 14-dB return loss bandwidths of 138% and 107.2% respectively, which is about 49 times that of the corresponding monopolar wire-patch antenna. Also, it provides monopole-like radiation patterns over such wide operating bandwidth. As the proposed antenna can achieve such wide impedance bandwidth with relatively low profile, it is very suitable for multi-band mobile communication systems.

Vibrations of skewed cantilevered triangular, trapezoidal and parallelogram Mindlin plates with considering corner stress singularities

Based on the Mindlin shear deformation plate theory, a method is presented for determining natural frequencies of skewed cantilevered triangular, trapezoidal and parallelogram plates using the Ritz method, considering the effects of stress singularities at the clamped re-entrant corner. The admissible displacement functions include polynomials and corner functions. The admissible polynomials form a mathematically complete set and guarantee the solution convergent to the exact frequencies when sufficient terms are used. The corner functions properly account for the singularities of moments and shear forces at the re-entrant corner and accelerate the convergence of the solution. Detailed convergence studies are carried out for plates of various shapes to elucidate the positive effects of corner functions on the accuracy of the solution. The results obtained herein are compared with those obtained by other investigators to demonstrate the validity and accuracy of the solution. Copyright © 2005 John Wiley & Sons, Ltd.

Multiband PIFA for WLAN mobile terminals

A novel compact planar inverted-F antenna (PIFA) to be mounted on laptop computers is presented. The proposed multiband single-feed PIFA simultaneously operates in the IEEE 802.11a, IEEE 802.11b/g, and HIPERLAN2 frequency bands. The multiband behavior is obtained by combining a trapezoidal feed plate with two different resonance paths in the radiating structure. The overall size of the antenna renders it suitable to be installed at the border of the display of a notebook. Simulation results are compared with measurements performed on an antenna prototype.

Buckling analysis of trapezoidal composite sandwich plate subjected to in-plane compression

The results of the stability analysis of simply supported layered isosceles trapezoidal plate subjected to axial in-plane compression are presented. The layup configuration is confined to symmetric laminates. The solution has been obtained by means of the Galerkin orthogonalisation method combined with the proposed method of the coordinate system transformation. The results of the analytical solution are compared with the verifying FEM calculation. The computed results in the form of graphs of the buckling force as a function of material and geometrical parameters of the panel are included.

Geometric nonlinear vibration of clamped Mindlin plates by analytically integrated trapezoidal p-element

A trapezoidal hierarchical finite element for nonlinear free and forced vibration analyses of skew and trapezoidal Mindlin plates is introduced. Legendre orthogonal polynomials are used as enriching shape functions to avoid the shear-locking problem. The element matrices are analytically integrated in closed form. With the enriching degrees of freedom, the accuracy of the computed results and the computational efficiency are greatly improved. The arc-length iterative method is used to solve the nonlinear motion equation. The results of linear and nonlinear vibration show that the convergence of the proposed element is very fast with respect to the number of degrees of freedom. Clamped condition on the boundaries and weakly nonlinearity without bifurcation will be considered. The skew angles of plates hardly influence the shape of backbone curves, but the decreasing of chord ratios increases the hardening spring effect of the backbone curves.

Boundary element method in dynamic analysis of structures with cracks

A general characteristics of the boundary element approaches used in the analysis of structures with cracks: the subregion method, the displacement discontinuity method and the dual method are presented. The dual method uses simultaneously the displacement and the traction boundary integral equations. The method is combined with the time domain method, the integral transform method and the dual reciprocity method. The boundary integral equations for a two-dimensional analysis and their numerical implementations are described. The dynamic stress intensity factors, which characterize the stress field at the crack tip are computed using the crack opening displacements and the path independent integrals. The crack growth with constant and variable velocity, which depends on the fracture criterion is modelled. The methods are used to compute displacements, dynamic stress intensity factors and crack paths for dynamically loaded machine parts: a rectangular plate, a support, a crank, a rotating disc, a trapezoidal plate and to model a dynamic crack experiment.

Post Local Buckling of Skew and Trapezoidal Plates

A numerical procedure is developed for the post-buckling analysis of skew and trapezoidal plates. The procedure uses virtual work method in conjunction with the natural coordinates. The post-buckling equilibrium path is controlled by the Hookean strains and the total potential energy is formulated as a function of the compressive strain rather than stresses. The method is programmed, and several numerical examples are presented to illustrate the scope and ability of the procedure. Initial local buckling results obtained for skew plates are compared with known solutions in the literature. Post-buckling behavior of skew and trapezoidal plates is studied for different geometries. The non-dimensional loads carried in the plate and dimensionless lateral displacements of skew and trapezoidal plate in the post buckling region are given in several tables.

Transverse vibration of thick polygonal plates using analytically integrated trapezoidal Fourier p-element

A trapezoidal Fourier p-element for the transverse vibration analyses of thick plates is introduced. Trigonometric functions are used as enriching shape functions to avoid ill-conditioning problems associated with high order polynomials in a conventional p-element. The element matrices are analytically integrated in closed form. With the enriching Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly improved. A simply supported square thick plate is analyzed by a rectangular Fourier p-element, two trapezoidal Fourier p-elements and the conventional linear finite elements, respectively. The results show that the convergence of the present element is very fast with respect to the number of trigonometric terms and it produces higher accurate modes than the linear finite elements for the same number of degrees of freedom. The proposed element is applied to find the natural modes of the rectangular, skew and trapezoidal plates with different boundary conditions. Since one can always break a triangle into three trapezoids, the range of application of the present element is as good as the triangular element. A triangular, an irregular quadrilateral and two polygonal thick plates are analyzed by the present element. The solutions of the trapezoidal Fourier p-element are in excellent agreement with those of available published methods.

Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns

The main purpose of this study is to compare the methods of differential quadrature (DQ) and harmonic differential quadrature (HDQ). For this purpose, DQ and HDQ methods are presented for buckling, bending, and free vibration analysis of thin isotropic plates and columns. Plates of different shapes such as rectangular, circular, square, skew, trapezoidal, annular, and sectorial plate subjected to different boundary conditions are selected to demonstrate the accuracy of the method. Four different support conditions are taken into consideration for columns. Numerical results are presented to illustrate the method and demonstrate its efficiency. It is emphasized that the HDQ method gives more accurate results and needs less grid points than the DQ method.

Spatial system identification of a simply supported beam and a trapezoidal cantilever plate

Dynamic models of structural and acoustic systems are usually obtained by means of either modal analysis or finite element modeling. Detrimentally, both techniques rely on a comprehensive knowledge of the system's physical properties. As a consequence, experimental data and a nonlinear optimization are required to refine the model. For the purpose of control, system identification is often employed to estimate the dynamics from disturbance and command inputs to set of outputs. Such discretization of a spatially distributed system places unknown weightings on the control objective, in many cases, contradicting the original goal of optimal control. This paper introduces a frequency domain system identification technique aimed at obtaining spatially continuous models for a class of distributed parameter systems. The technique is demonstrated by identifying a simply supported beam and a trapezoidal cantilever plate, both with bonded piezoelectric transducers. The plate's dimensions are based on the scaled side elevation of a McDonnell Douglas FA-18 vertical stabilizer.

면내 병진 가속을 받는 복합재 사다리꼴 평판의 진동 해석

A modeling method for the modal analysis of a composite trapezoidal plate undergoing in-plane translational acceleration is presented in this paper. The equations of motion for the plate are derived and transformed into a dimensionless form. The effects of the inclination angles, fiber orientation angle and the acceleration on the modal characteristics of the plate are investigated.

A high precision shear deformable element for free vibration of thick/thin composite trapezoidal plates

A high precision shear deformable triangular element has been proposed for free vibration analysis of composite trapezoidal plates. The element has twelve nodes at the three sides and four nodes inside the element. Initially the element has fifty-five degrees of freedom, which has been reduced to forty-eight by eliminating the degrees of freedom of the internal nodes through static condensation. Plates having different side ratios (b/a), boundary conditions, thickness ratios (h/a=0.01, 0.1 and 0.2), number of layers and fibre angle orientations have been analyzed by the proposed shear locking free element. Trapezoidal laminate with concentrated mass at the centre has also been analyzed. An efficient mass lumping scheme has been recommended, where the effect of rotary inertia has been included. For validation of the present element and formulation few results of isotropic trapezoidal plate and square composite laminate have been compared with those obtained from open literatures. The numerical results for composite trapezoidal laminate have been given as new results.

Plate on layered foundation analyzed by a semi-analytical and semi-numerical method

A semi-analytical and semi-numerical method is developed for the analysis of plate-layered soil systems. Applying a Hankel transform, an expression relating the surface settlement and the reaction of the layered soil is derived. Such a reaction can be treated as a load acting on the plate in addition to the applied external load. Having the plate modeled by eight-noded isoparametric elements, the governing equations of the plate can be formed and solved. Numerical examples, including square, trapezoidal and circular plates resting on elastic layered soil, are given to demonstrate the advantages, accuracy and versatility of this method.