scholarly journals Moment and Stress Analysis Solutions of Clamped Rectangular Thick Plate

2020 ◽  
Vol 5 (4) ◽  
pp. 531-534 ◽  
Author(s):  
O. M. Ibearugbulem ◽  
Festus Chukwudi Onyeka
Keyword(s):  
Thick Plate ◽  
Plate Theory ◽  
Analytic Solutions ◽  
Numerical Comparison ◽  
Theory Analysis ◽  
Third Order ◽  
Total Potential ◽  
Out Of Plane ◽  
Plane Displacement

The bending solutions of rectangular thick plate with all four edges clamped (CCCC) were investigated in this study. The basic governing equations used for analysis are based on third-order shear deformation plate theory analysis under uniformly distributed load. Using a formulated total potential energy equation, the three coupled general governing differential equations for the determination of the out of plane displacement and shear deformations rotation along the direction of x and y coordinates were obtained. These equations as obtained are solved simultaneously after minimization to determine the coefficients of displacements of the plate and other the mentioned functions. By solving these equations, the analytic solutions of rectangular thick plate with all four edges clamped were derived. From the formulated expression, the formula for calculation of the maximum deflection, moment, stress and in-plane displacements were deduced. The proposed method obviates the need of shear correction factors, which is associated with Mindlin’s theory (FSDT) for the solution to the problem. Moreover, numerical comparison shows the correctness and accuracy of the results.

scholarly journals Analysis Bending Solutions of Clamped Rectangular Thick Plate

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Yang Zhong ◽  
Qian Xu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Shear Deformation ◽  
Thick Plate ◽  
Plate Theory ◽  
Analytic Solutions ◽  
Higher Order ◽  
Numerical Comparison ◽  
Governing Equations ◽  
Partial Differential

The bending solutions of rectangular thick plate with all edges clamped and supported were investigated in this study. The basic governing equations used for analysis are based on Mindlin’s higher-order shear deformation plate theory. Using a new function, the three coupled governing equations have been modified to independent partial differential equations that can be solved separately. These equations are coded in terms of deflection of the plate and the mentioned functions. By solving these decoupled equations, the analytic solutions of rectangular thick plate with all edges clamped and supported have been derived. The proposed method eliminates the complicated derivation for calculating coefficients and addresses the solution to problems directly. Moreover, numerical comparison shows the correctness and accuracy of the results.

Warpage of Flip Chip BGA Package

2006 ◽  
Author(s):  
Milena Vujosevic
Keyword(s):  
Flip Chip ◽  
Plate Theory ◽  
Contour Plot ◽  
Thermally Induced ◽  
Ball Grid Arrays ◽  
Fundamental Understanding ◽  
Out Of Plane ◽  
Bga Package ◽  
Plane Displacement ◽  
Analytical Expressions

The work focuses on the thermally induced out of plane displacement of Flip Chip Ball Grid Arrays (FCBGA). Analytical expressions for substrate displacements are derived based on the Plate Theory and Suhir's solution for stresses in tri-material assembly. The validity of the model is established by comparing the analytical solution to the finite element results as well as to the experimental data. The benefits of the model are twofold: 1) it provides a tool for fundamental understanding of the parameters that influence warpage, and 2) has a predictive capability. With respect to 1) an analysis is presented on the nature and degree of influence that different geometric and material parameters have on the FCBGA warpage. With respect to 2) the "Warpage Contour Plot" is proposed as a tool for warpage prediction that can be easily utilized in the early stages of the design process.

Analysis of Stress Singularities at Bi-Material Corners in Reddy's Theory of Plate Bending

2006 ◽  
Vol 22 (1) ◽  
pp. 67-75 ◽  
Author(s):  
C. S. Huang
Keyword(s):  
Thick Plate ◽  
Isotropic Plate ◽  
Plate Theory ◽  
Stress Singularity ◽  
Numerical Approach ◽  
Sharp Corner ◽  
Equilibrium Equations ◽  
Third Order ◽  
Isotropic Plates ◽  
Expansion Technique

AbstractThe order of stress singularity at a sharp corner of a plate needs to be known before a numerical approach can be taken to determine accurately the stress distribution of a plate with irregular geometry (such as a V-notch) under loading. This work analyzes the order of the stress singularity at a bi-material corner of a thick plate under bending, based on Reddy's third-order shear deformation plate theory. An eigenfunction expansion technique is used to derive the asymptotic displacement field in the vicinity of the sharp corner by solving the equilibrium equations in terms of displacement functions. This paper explicitly shows the first known characteristic equations for determining the order of the stress singularity at the interface corner of a bonded dissimilar isotropic plate. Moreover, the numerical results are given in graphic form for the order of stress singularity at the interface corner in bonded dissimilar isotropic plates and at the vertex of a bi-material wedge with free radial edges. The results presented herein fill some of the gaps in the literature

Vibration of Thick Circular Plate--Determination of Elastic Properties Through the Resonant Behaviour

1983 ◽  
Vol 7 (2) ◽  
pp. 58-64
Author(s):  
P. Priolo ◽  
C. Sitzia
Keyword(s):  
Finite Elements ◽  
Elastic Properties ◽  
Thick Plate ◽  
Plate Theory ◽  
Resonant Condition ◽  
High Frequencies ◽  
Thick Circular Plate ◽  
Properties Of Materials ◽  
Points Of View

The authors examine, from two complementary points of view, the main problem deriving from the necessity of deducing elastic properties of materials by considering the resonant condition of transversely vibrating discs, that is the determination of the efficiency at high frequencies of finite elements formulated with the assumptions of the thick plate theory. The first approach consists, having standardized the basic relations for various thick annular semi-analytical finite elements, in testing convergence and correspondence to known analytical solutions. The second consists in the experimental evaluation of the influence of thickness in deducing the Young’s modulus of a series of polycarbonate resin discs at frequencies corresponding to modes with up to eight nodal circles.

Buckling Analysis of Mindlin Plates Under the Green–Lagrange Strain Hypothesis

2015 ◽  
Vol 15 (06) ◽  
pp. 1450079 ◽  
Author(s):  
Eugenio Ruocco ◽  
Vincenzo Minutolo
Keyword(s):  
Solution Method ◽  
Plate Theory ◽  
Mode Shapes ◽  
Governing Equations ◽  
Nonlinear Strain ◽  
Out Of Plane ◽  
Plates And Shells ◽  
Moderately Thick Plates ◽  
Lagrange Strain ◽  
Plane Displacement

In the present paper, the influence of Green–Lagrange nonlinear strain-displacement terms, usually considered negligible under the von Kármán hypothesis, on the buckling of isotropic, moderately thick plates and shells, is investigated. The first order shear deformation plate theory is applied and the governing equations, containing nonlinear terms related to both in-plane displacement and out-of-plane rotations usually ignored in the literature, are derived using the principle of minimum of the strain energy. The general Levy type solution method is employed, and exact buckling loads and mode shapes are derived. To verify the accuracy of the solution obtained, comparisons with existing data are first made. Then, through graphics and tables, the effect of the nonlinear strain-displacement terms for a range of boundary and load conditions, variations of aspect ratio, thickness ratio and changes in geometry is presented. The results obtained show that the von Kármán's model can sensibly overestimate the critical load for structures characterized by the modes involving comparable in-plane and out-of-plane displacements.

On the Singularity Induced by Boundary Conditions in a Third-Order Thick Plate Theory

2002 ◽  
Vol 69 (6) ◽  
pp. 800-810 ◽  
Author(s):  
C. S. Huang
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Thick Plate ◽  
Plate Theory ◽  
Vertex Angle ◽  
Singular Behavior ◽  
Third Order ◽  
Characteristic Equations ◽  
Shear Deformation Plate Theory ◽  
Comparison Results

This paper thoroughly examines the singularity of stress resultants of the form r−ξFθ for 0<ξ⩽1 as r→0 (Williams-type singularity) at the vertex of an isotropic thick plate; the singularity is caused by homogeneous boundary conditions around the vertex. An eigenfunction expansion is applied to derive the first known asymptotic solution for displacement components, from the equilibrium equations of Reddy’s third-order shear deformation plate theory. The characteristic equations for determining the singularities of stress resultants are presented for ten sets of boundary conditions. These characteristic equations are independent of the thickness of the plate, Young’s modulus, and shear modulus, but some do depend on Poisson’s ratio. The singularity orders of stress resultants for various boundary conditions are expressed in graphic form as a function of the vertex angle. The characteristic equations obtained herein are compared with those from classic plate theory and first-order shear deformation plate theory. Comparison results indicate that different plate theories yield different singular behavior for stress resultants. Only the vertex with simply supported radial edges (S(I)_S(I) boundary condition) exhibits the same singular behavior according to all these three plate theories.

5,10,15,20-Tetrakis(diphenylmethyl)porphyrin - A Nonplanar Porphyrin with Intermediate Degree of Ruffling

1999 ◽  
Vol 54 (5) ◽  
pp. 662-666 ◽  
Author(s):  
Steffen Runge ◽  
Mathias O. Senge ◽  
Karin Ruhlandt-Senge
Keyword(s):  
Crystal Structure ◽  
Structure Determination ◽  
Title Compound ◽  
Spectroscopic Data ◽  
Crystal Structure Determination ◽  
Free Base ◽  
Out Of Plane ◽  
Intermediate Degree ◽  
Plane Displacement

The title compound was prepared by condensation of pyrrole and diphenylacetaldehyde in 16% yield. Based on spectroscopic data, the free base porphyrin exhibits a nonplanar macrocycle in solution with a degree of distortion between that of 5,10,15,20-tetracyclohexylporphyrin and 5,10,15,20-tetra(t-butyl)porphyrin. A crystal structure determination of the cobalt(II) complex reveals a ruffled macrocycle conformation with an average out-of-plane displacement of the meso carbon atoms by 0.55 Å.

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