- The Governing Equations of Thin-Plate Theory

2012 ◽  
pp. 527-564
Keyword(s):  
Thin Plate ◽  
Plate Theory ◽  
Governing Equations ◽  
Thin Plate Theory

scholarly journals Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment

2021 ◽  
Vol 43 (5) ◽  
Author(s):  
Giovanni Tocci Monaco ◽  
Nicholas Fantuzzi ◽  
Francesco Fabbrocino ◽  
Raimondo Luciano
Keyword(s):  
Thin Plate ◽  
Strain Gradient ◽  
Plate Theory ◽  
Laminated Composite ◽  
Equilibrium Equations ◽  
Thin Plate Theory ◽  
Bending Behavior ◽  
Hygrothermal Environment ◽  
Load Types ◽  
Novel Applications

AbstractIn this work, the bending behavior of nanoplates subjected to both sinusoidal and uniform loads in hygrothermal environment is investigated. The present plate theory is based on the classical laminated thin plate theory with strain gradient effect to take into account the nonlocality present in the nanostructures. The equilibrium equations have been carried out by using the principle of virtual works and a system of partial differential equations of the sixth order has been carried out, in contrast to the classical thin plate theory system of the fourth order. The solution has been obtained using a trigonometric expansion (e.g., Navier method) which is applicable to simply supported boundary conditions and limited lamination schemes. The solution is exact for sinusoidal loads; nevertheless, convergence has to be proved for other load types such as the uniform one. Both the effect of the hygrothermal loads and lamination schemes (cross-ply and angle-ply nanoplates) on the bending behavior of thin nanoplates are studied. Results are reported in dimensionless form and validity of the present methodology has been proven, when possible, by comparing the results to the ones from the literature (available only for cross-ply laminates). Novel applications are shown both for cross- and angle-ply laminated which can be considered for further developments in the same topic.

Normal Loading on a Wedge-Shaped Plate

1955 ◽  
Vol 6 (3) ◽  
pp. 196-204 ◽  
Author(s):  
D. E. R. Godfrey
Keyword(s):  
Thin Plate ◽  
Mellin Transform ◽  
Plate Theory ◽  
Normal Loading ◽  
Thin Plate Theory

SummaryThe equations of thin plate theory are expressed in polar co-ordinates and transformed using the Mellin transform. Problems involving discontinuous and isolated normal loadings may then be solved in the case of the built-in or freely supported wedge-shaped boundary.

NODAL INTEGRATION THIN PLATE FORMULATION USING LINEAR INTERPOLATION AND TRIANGULAR CELLS

2011 ◽  
Vol 08 (04) ◽  
pp. 813-824 ◽  
Author(s):  
X. Y. CUI ◽  
S. LIN ◽  
G. Y. LI
Keyword(s):  
Boundary Conditions ◽  
Thin Plate ◽  
Plate Theory ◽  
Weak Form ◽  
Linear Interpolation ◽  
Integration Domain ◽  
Nodal Integration ◽  
Thin Plate Theory ◽  
System Equations ◽  
Gradient Smoothing

This paper presents a thin plate formulation with nodal integration for bending analysis using three-node triangular cells and linear interpolation functions. The formulation was based on the classic thin plate theory, in which only deflection field was required and dealt with as the field variables. They were assumed to be piecewisely linear and expressed using a set of three-node triangular cells. Based on each node, the integration domain has been further derived, where the curvature in the domain was computed using a gradient smoothing technique (GST). As a result, the curvature in each integration domain is a constant whereby the deflection is compatible in the whole problem domain. The generalized smoothed Galerkin weak form is then used to create the discretized system equations where the system stiffness is obtained using simple summation operation. The essential rotational boundary conditions are imposed in the process of constructing the curvature field in conjunction with imposing the translational boundary conditions in the same way as undertaken in the standard FEM. A number of numerical examples were studied using the present formulation, including both static and free vibration analyses. The numerical results were compared with the reference ones together with those shown in the state-of-art literatures published. Very good accuracy has been achieved using the present method.

A Thick/Thin Plate Element Based on the Area Coordinate Analytical Trial Functions

2013 ◽  
Vol 790 ◽  
pp. 341-346
Author(s):  
Li Wang ◽  
Yu Lin Lu ◽  
Hao Ran Lou ◽  
Jia Wei ◽  
Min Zhu
Keyword(s):  
Coordinate System ◽  
Thin Plate ◽  
Analytical Solutions ◽  
Plate Theory ◽  
Good Precision ◽  
Shear Locking ◽  
Plate Element ◽  
Governing Equations ◽  
Numerical Examples ◽  
Fundamental Analytical Solutions

In this paper, a generalized conforming thick/thin plate element based on the quadrilateral area coordinate system called AATF-PQ4 is developed. Based on the governing equations of Mindlin-Reissner plate theory, the fundamental analytical solutions are first derived, then using this trial functions to formulate element AATF-PQ4. In the case of thin plate, this thick/thin plate element is degraded into corresponding thin plate element automatically and is free from shear locking. Numerical examples show that the proposed element, AATF-PQ4, has a good precision for thick and thin plate.

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