Mixed Edge Conditions Research Articles

Analysis of layered panels with mixed edge boundary conditions using state space differential quadrature method

The state space differential quadrature method (SSDQM) has been extensively used to obtain accurate semi-analytical solutions for static and dynamic response of layered composite structures. The analysis of layered panels with mixed boundary conditions through-the-thickness of edges has received very little attention. In this work, a general and flexible method to extend SSDQM for analysis of panels with mixed edge boundary conditions is introduced. A simplified mathematical condition is derived from the constraints inherent in SSDQM, to implement the mixed edge boundary conditions in a systematic manner. Considering the limited application of SSDQM to analyze panels with large thickness-to-length and interlayer stiffness ratios, the proposed approach is used to study bending and free vibration response of thick, soft-core sandwich panels with mixed edge conditions. Convergence studies and comparison with finite element solutions show that SSDQM based on the proposed approach provides rapid convergence and excellent accuracy.

Application of the DSC-Element method to flexural vibration of skew plates with continuous and discontinuous boundaries

This paper generalizes the newly developed DSC-Element method for free vibration analysis of skew plates using the first-order shear deformable plate theory. Basically, the DSC-Element method not only embraces the discrete singular convolution (DSC) delta type wavelet kernel as a trial function with the Ritz principle, but also incorporates the concept of the finite element method. The current approach is novel and flexible as contrast to the global numerical methods in treating the complex kinematic supporting edges. The objective of this paper is to examine the efficiency and validity of the DSC-Element method for oblique plates having large skew angles. Parametric studies for the vibration analysis of skew plates with various skew angles, thickness ratios, aspect ratios and continuous or discontinuous periphery supports are presented as well. The natural frequencies are directly compared and discussed with those reported in the open literature. Some frequency solutions for skew plates with mixed edge conditions are also presented.

Differential cubature method for buckling analysis of arbitrary quadrilateral thick plates

In this paper, a novel numerical solution technique, the differential cubature method is employed to study the buckling problems of thick plates with arbitrary quadrilateral planforms and non-uniform boundary constraints based on the first order shear deformation theory. By using this method, the governing differential equations at each discrete point are transformed into sets of linear homogeneous algebraic equations. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Detailed formulation and implementation of this method are presented. The buckling parameters are calculated through solving a standard eigenvalue problem by subspace iterative method. Convergence and comparison studies are carried out to verify the reliability and accuracy of the numerical solutions. The applicability, efficiency, and simplicity of the present method are demonstrated through solving several sample plate buckling problems with various mixed boundary constraints. It is shown that the differential cubature method yields comparable numerical solutions with 2.77-times less degrees of freedom than the differential quadrature element method and 2-times less degrees of freedom than the energy method. Due to the lack of published solutions for buckling of thick rectangular plates with mixed edge conditions, the present solutions may serve as benchmark values for further studies in the future.

Elastic analysis of the transfer of shearing stress from partially electroded piezoelectric actuators to composite plates in cylindrical bending

An analysis of laminated composite plates forced into cylindrical bending by partially electroded piezoelectric actuators attached to the surfaces is performed using the equations of linear elasticity. The partially electroded actuator is treated as a thin surface film and mixed edge conditions are employed to simulate simple supports. The calculations reveal that earlier works using approximate plate equations overestimate the maximum shearing stress transferred by an order of magnitude and underestimate the length over which the transfer occurs. Nevertheless, these works indicate that ending the electrodes a short distance from the edge of the actuator eliminates the shearing stress singularity that occurs when the electrodes extend to the edge. Since this work is accurate, a reliable determination of the short distance required for the shearing stress at the edge of the actuator to be an order of magnitude smaller than the maximum is made.

The vibration of a simply supported rectangular elastic plate due to piezoelectric actuators

An analysis of a simply supported rectangular elastic plate forced into bending vibrations by the application of time harmonic voltages to piezoelectric actuators attached to its bottom and top surfaces is performed by using the equations of linear elasticity. The actuators have been modeled as thin surface films and mixed edge conditions are employed to simulate simple supports.

The cylindrical bending vibration of a laminated elastic plate due to piezoelectric actuators

We analyze the forced cylindrical bending vibrations of a laminated elastic plate with actuators at the top and bottom surfaces and forced into vibrations by applying time harmonic voltages to the actuators. The actuators are modeled as thin films and mixed edge conditions are employed to simulate simple supports. The analysis is performed by using the method of Fourier series and the solution is exact within the assumptions of linear elasticity and plane strain deformations. Numerical results are computed for an aluminum plate with actuators affixed to its two major surfaces.

An elastic analysis of laminated composite plates in cylindrical bending due to piezoelectric actuators

An analysis of laminated composite plates forced into cylindrical bending by the application of voltages to piezoelectric actuators attached to the top and bottom surfaces is performed using the equations of linear elasticity. In the treatment the actuator is modeled as a thin surface film and mixed edge conditions are employed to simulate simple supports. The results obtained from the linear elastic solution are compared with those obtained using the elementary equations of the extension and flexure of thin composite plates, which are the ones usually employed in practice. When the properties of adjacent laminates are very different the comparison reveals that for large span-length-to-thickness ratios the agreement is quite good but for small ones it is not good at all. The existence of a shearing stress singularity under the edge of the fully electroded actuator is exhibited.

Correct asymptotic theories for the axisymmetric deformation of thin and moderately thick cylindrical shells

A refined shell theory is formulated for the elastostatics of long, moderately thick cylindrical shells in axisymmetric deformation. This theory corresponds to a two-term outer asymptotic expansion of the exact solution for small values of the dimensionless shell thickness parameter. The complexity of the known exact solution for the three-dimensional elasticity problem has stimulated an interest in thin and thick shell theories to provide accurate approximate solutions in the shell interior without any reference (or matching) to the inner asymptotic solution. The principal difficulty in developing a shell theory lies in the determination of an appropriate set of twodimensional boundary conditions for the shell solution from the prescribed edge data of the threedimensional theory. The derivations of boundary conditions for the thin shell theory and the refined shell theory constitute a main contribution of this paper. Correct boundary conditions obtained for the first time include (i) displacement edge conditions for thin shell theory, and (ii) stress and two types of mixed edge conditions for the refined shell theory. Several applications of the refined theory are given to show that corrections to the thin shell solution can be important.

Effective Breadth Problems by an Eigenfunction Approach

The problem of effective breadth of a stiffened panel with two different edge or support conditions, namely (i) stres -free ed es and (ii) displacement restrained edges, has been solved for the first time, in this paper, by the e.• enfunctoon approach o the end problem of semi-infinite strips. It is shown that these two edge condollens lead to complex eogenvalue problems and thus the plane-stress problem of the plating cannot be solved by the classical methods which have been hitherto successfully used to obtain the estimates of effective breadths for the two other possible mixed edge conditions, namely, (i) vanishing direct stress and tangential displacement and (ii) vanishing normal displacement and shear stress. It is brought out that this end eigenlunetion formulation, using a generalized orthogonality of Papkovich, provides a general analytical framewerk for the stress field problems of stiffened panels in bending. Numerical results are presented for the first problem nly. The two classical problems have been revisited from this viewpoint, and certain well-known effectove breadth results have been rederived and compared. An alternative form of the generalized orthogonality relation, which offers a certain analytical advantage, has been used and the exostence of such a form is proved for all four edge conditions.

Eigenvalue Problems of Rectangular Plates With Mixed Edge Conditions

The vibration and stability of plates with mixed edge conditions are considered in this paper. A simply supported rectangular plate which is clamped along the central portion on two opposite edges and a plate with partial clamping along one edge are analyzed. The problems are formulated as dual series equations and reduced to homogeneous Fredholm integral equations of the second kind. Comparisons with numerical results obtained by other investigators are made. Vibration and buckling mode shapes are illustrated. A vibration analysis of a plate simply supported adjacent to the corners is also made. This case is formulated as a coupled system of dual series equations which is reduced to a system of homogeneous integral equations. In all of the solutions given, the singularity is isolated and treated analytically.