Boundary Layer Phenomena in Large Deflection, Small Strain Plate Theory

1993 ◽  
Vol 60 (1) ◽  
pp. 229-232
Author(s):  
L. J. Berg
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Large Deflection ◽  
Large Deformations ◽  
Plate Theory ◽  
Small Strain ◽  
Thin Plates ◽  
Large Deflections ◽  
Layer Solution

Boundary layers exist at the edges of thin plates undergoing large deformations because the interior of the plate must assume a developable shape. The developable shape is sometimes incompatible with the force and moment resultants prescribed at the plate’s boundary, in particular when the edge of the plate is stress free. A boundary layer solution is presented which describes the shape of a boundary layer in a plate undergoing large deflections. The boundary layer is a slight perturbation of the interior shape which allows the appropriate boundary conditions to be satisfied. Since developable shells are applicable to a plane, the boundary layer is also appropriate for arbitrary developable shells.

Very Large Deformation but Small Strain Analysis of Plates and Folded Plates by Finite Strip Method

2005 ◽  
Vol 8 (6) ◽  
pp. 547-560 ◽  
Author(s):  
A. R. Shahidi ◽  
M. Mahzoon ◽  
M. M. Saadatpour ◽  
M. Azhari
Keyword(s):  
Large Deformations ◽  
Virtual Work ◽  
Strain Analysis ◽  
Small Strain ◽  
Linear Mapping ◽  
Thin Plates ◽  
Computational Domain ◽  
Finite Strip ◽  
Strip Method ◽  
Finite Strip Method

In this paper a Finite strip method is developed to analyze very large deformations of thin plates and folded plates by use of the elastic Cosserat theory. The principle of virtual work is exploited to present the weak form of the governing differential equations. Through a linear mapping, a rectangular strip is transformed into a standard square computational domain in which the deformation and director fields are developed together with the general forms of the uncoupled nonlinear equations. The geometric and material tangential stiffness matrices are formed through linearization, and a step by step procedure is presented to complete the scheme. The validity and the accuracy of the method are illustrated through certain numerical examples and comparison of the results with other researches. The method is shown to be capable of handling numerical analysis of plates experiencing very large deformations.

Proposing a Pretwisted Bending Actuator

Aerospace ◽  
2006 ◽  
Author(s):  
Frank Dienerowitz ◽  
Nicole Gaus ◽  
Wolfgang Seemann
Keyword(s):  
Magnetic Fields ◽  
Cross Section ◽  
Large Deflection ◽  
Mechanical Energy ◽  
Electric Energy ◽  
Small Strain ◽  
Large Deflections ◽  
Direct Transformation ◽  
Active Elements ◽  
Bending Actuator

Piezoelectric bending actuators take advantage of both piezoelectricity and kinematics of beams, i.e. (1) direct transformation of electric energy into mechanical energy without causing significant magnetic fields and (2) to be capable of turning small strain modifications into large deflections, provided the cross-section is rather flat. Unfortunately the latter usually implies that bending actuators provide only one axis of large deflection. Herein a pretwisted bending actuator is investigated, similar to a helicoid. The active elements along the beam axis are subdivided and controlled separately, hence allowing independent control of the curvature of each section. Due to the pretwist, this bending actuator can provide not only one but two axes of deflection. For a slender pretwisted bending actuator the problems emerging are presented and discussed, covering the work space of the actuator, optimization of electrode connecting patterns and experimental results.

scholarly journals Rigid plastic plates at large deformations

1985 ◽  
Vol 7 (1) ◽  
pp. 20-23
Author(s):  
Vu Van The
Keyword(s):  
Reinforced Concrete ◽  
Boundary Conditions ◽  
Large Deformations ◽  
Plate Thickness ◽  
Large Deflections ◽  
Rigid Plastic ◽  
Collapse Pressure ◽  
Yield Load ◽  
Yield Behaviour ◽  
Relationship Of

By being statically loaded plastic plates can support loads exceeding the bending collapse pressure, the behaviour of perfectly rigid plastic plates beyond the yield load depends on changes in geometry to the plastic flow. Therefore in post yield behaviour the deflection can not be considered small in comparison with the plate thickness. In this paper we employ the equations of plates at moderately large deflections and the approximate live behaviour of plates introduced in/2,3/ by dividing plates into a number of rigid regions which have been separated by line hinges situated at locations where their discontinue ties in w,i occur, an estimative method of the toad - deflection relationship of arbitrarily shaped plates having arbitrarily boundary conditions is developed. This method is directly extended to anisotropic and reinforced concrete plates.

Finite amplitude convection cells

1967 ◽  
Vol 30 (3) ◽  
pp. 577-600 ◽  
Author(s):  
J. L. Robinson
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Cross Section ◽  
Circular Cross Section ◽  
Maximum Heat ◽  
Mean Field Equations

In this paper we consider two-dimensional steady cellular motion in a fluid heated from below at large Rayleigh number and Prandtl number of order unity. This is a boundary-layer problem and has been considered by Weinbaum (1964) for the case of rigid boundaries and circular cross-section. Here we consider cells of rectangular cross-section with three sets of velocity boundary conditions: all boundaries free, rigid horizontal boundaries and free vertical boundaries (referred to here as periodic rigid boundary conditions), and all boundaries rigid; the vertical boundaries of the cells are insulated. It is shown that the geometry of the cell cross-section is important, such steady motion being not possible in the case of free boundaries and circular cross-section; also that the dependence of the variables of the problem on the Rayleigh number is determined by the balances in the vertical boundary layers.We assume only those boundary layers necessary to satisfy the boundary conditions and obtain a Nusselt number dependence $N \sim R^{\frac{1}{3}}$ for free vertical boundaries. For the periodic rigid case, Pillow (1952) has assumed that the buoyancy torque is balanced by the shear stress on the horizontal boundaries; this is equivalent to assuming velocity boundary layers beside the vertical boundaries (rather than the vorticity boundary layers demanded by the boundary conditions) and leads to a Nusselt number dependence N ∼ R¼. If it is assumed that the flow will adjust itself to give the maximum heat flux possible the two models are found to be appropriate for different ranges of the Rayleigh number and there is good agreement with experiment.An error in the application of Rayleigh's method in this paper is noted and the correct method for carrying the boundary-layer solutions round the corners is given. Estimates of the Nusselt numbers for the various boundary conditions are obtained, and these are compared with the computed results of Fromm (1965). The relevance of the present work to the theory of turbulent convection is discussed and it is suggested that neglect of the momentum convection term, as in the mean field equations, leads to a decrease in the heat flux at very high Rayleigh numbers. A physical argument is given to derive Gill's model for convection in a vertical slot from the Batchelor model, which is appropriate in the present work.

Nonlinear Asymptotic Solution of the Reissner Plate Equations

1985 ◽  
Vol 52 (4) ◽  
pp. 907-912 ◽  
Author(s):  
L. A. Taber
Keyword(s):  
Plate Theory ◽  
Small Strain ◽  
Circular Plates ◽  
Exponential Form ◽  
Plate Edge ◽  
Reissner Plate ◽  
Plate Equations ◽  
Special Case ◽  
Clamped Edges ◽  
Layer Solution

Based on the Reissner plate equations for large displacements and rotations within the limits of small strain, asymptotic solutions are developed for circular plates under uniform pressure. With the boundary layer solution assumed in exponential form, the boundary conditions are applied directly at the plate edge without the need for matched asymptotic expansions. Results are presented for plates with clamped edges. When compared to the solution for the special case of von Ka´rma´n plate theory, stresses generally deviate by less than 10 percent for rotation angles up to about 30 deg.

Periodic Solutions with a Boundary Layer of Reaction–Diffusion Equations with Singularly Perturbed Neumann Boundary Conditions

2014 ◽  
Vol 24 (08) ◽  
pp. 1440019 ◽  
Author(s):  
Valentin F. Butuzov ◽  
Nikolay N. Nefedov ◽  
Lutz Recke ◽  
Klaus R. Schneider
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Diffusion Equations ◽  
Time Periodic

We consider singularly perturbed reaction–diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, ε) with boundary layers and derive conditions for their asymptotic stability. The boundary layer part of u(x, t, ε) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order ε. Another peculiarity of our problem is that — in contrast to the case of Dirichlet boundary conditions — it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the description of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions.

A CURVATURE-CONSTRUCTED METHOD FOR BENDING ANALYSIS OF THIN PLATES USING THREE-NODE TRIANGULAR CELLS

2012 ◽  
Vol 09 (01) ◽  
pp. 1240021 ◽  
Author(s):  
X. Y. CUI ◽  
G. R. LIU ◽  
G. Y. LI
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Weak Form ◽  
Field Variable ◽  
Thin Plates ◽  
Bending Analysis ◽  
Thin Plate Theory ◽  
Edge Points ◽  
System Equations ◽  
Gradient Smoothing

This paper presents a curvature-constructed method (CCM) for bending analysis of thin plates using three-node triangular cells and assumed piecewisely linear deflection field. In the present CCM, the formulation is based on the classic thin plate theory, and only deflection field is treated as the field variable that is assumed piecewisely linear using a set of three-node triangular background cells. The slopes at nodes and/or the mid-edge points of the triangular cells are first obtained using the gradient smoothing techniques (GST) over different smoothing domains. Three schemes are devised to construct the curvature in each cell using these slopes at nodes and/or the mid-edge points. The generalized smoothed Galerkin weak form is then used to create the discretized system equations. The essential rotational boundary conditions are imposed in the process of constructing the curvature field, and the translational boundary conditions are imposed in the same way as in the standard FEM. A number of numerical examples, including both static and free vibration analyses, are studied using the present CCM and the numerical results are compared with the analytical ones and those in the published literatures. The results show that outstanding schemes can obtain very accurate solutions.

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