On the restricted torsion of narrow rectangular cross section by Kirchhoff's thin plate theory

1982 ◽  
Vol 3 (4) ◽  
pp. 507-520
Author(s):  
Chang Fo-van
Keyword(s):  
Cross Section ◽  
Thin Plate ◽  
Plate Theory ◽  
Rectangular Cross Section ◽  
Thin Plate Theory

Asymptotic Solutions for Warping and Distortion of Thin-Walled Box Beams

1987 ◽  
Vol 54 (1) ◽  
pp. 165-173 ◽  
Author(s):  
C. D. Balch ◽  
C. R. Steele
Keyword(s):  
Cross Section ◽  
Plate Theory ◽  
Rectangular Cross Section ◽  
Beam Width ◽  
Asymptotic Solutions ◽  
Perturbation Procedure ◽  
Unperturbed Problem ◽  
Thin Plate Theory ◽  
Box Beams ◽  
Thin Walled

The equations of conventional thin plate theory are used to formulate an eigenvalue problem for effects of self-equilibrating end loads in thin-walled rectangular cross section tubes, or box beams. The problem is analyzed by a perturbation procedure, which is based on a small parameter proportional to the square root of the ratio of wall thickness to cross section width. Solution of the unperturbed problem yields a family of membrane and inextensional end-effect eigenfunctions which are found to have decay distances on the order of the beam width or shorter. The perturbation procedure is carried out to obtain closed form asymptotically valid solutions for warping and distortional effects which decay much more slowly. These asymptotic solutions compare favorably with previous analytical and experimental results.

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.

scholarly journals A Method for Determining the Thicknesses of the Reserved Protective Layers in Complex Goafs Using a Joint-Modeling Method of GTS and Flac3D

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Xin Zhao ◽  
Dianshu Liu ◽  
Shenglin Li ◽  
Meng Wang ◽  
Shuaikang Tian ◽  
...  
Keyword(s):  
Thin Plate ◽  
Plate Theory ◽  
Three Dimensional ◽  
Joint Modeling ◽  
Modeling Method ◽  
Underground Cavity ◽  
Protective Layers ◽  
Safe Production ◽  
Thin Plate Theory ◽  
The Stability

In this study, a C-ALS underground cavity scanner was used to detect the shapes of mining goafs. In addition, GTS software was adopted to establish a three-dimensional geological model based on the status of the stopes, geological data, and mechanical parameters of each rock mass and to analyze the roof areas of the goafs. In regard to the morphology of the study area, based on a thin plate theory and the obtained field sampling data, a formula was established for the thicknesses of the reserved protective layers in the goafs. In addition, a formula for the thicknesses of the protective layers in the curved gobs was obtained. The thickness formula of the protective layers was then successfully verified. The detection results showed that the roof shapes of the goafs in the Yuanjiacun Iron Mine were mainly arc-shaped, and the spans of the goafs were generally less than 20 m. The stability of the arc-shaped roofs was found to be greater than that of the plate-shaped roofs. Therefore, by reducing the thicknesses of the protective layers in mining goafs, the ore recovery rates can be increased on the basis of safe production conditions. The formula of the thickness of the security layers obtained through the thin plate theory was revised based on the statistical results of the roof shapes of the goafs and then combined using GTS and FLAC3D. The modeling method successfully verified the stability of the mined-out areas. It was found that the verification results were good, and the revised formula was able to improve the recovery rate of the ore under the conditions of meeting safe production standards. Also, it was found that the revised formula could be used in the present situation. At the same time, it was also determined that the complexity of the rock masses obstructed the full identification of the joints and fissures in the present orebodies. Therefore, it is necessary to incorporate C-ALS underground cavity scanners to regularly observe the shapes of the goafs in order to ensure that stability and safety standards are maintained.

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