Plate theory

The articles plate theory and plate (technical mechanics) overlap thematically. Help me to better differentiate or merge the articles (→ instructions ) . To do this, take part in the relevant redundancy discussion . Please remove this module only after the redundancy has been completely processed and do not forget to include the relevant entry on the redundancy discussion page{{ Done | 1 = ~~~~}}to mark. Acky69 ( discussion ) 23:36, Feb. 23, 2018 (CET)

Plate theories describe the properties of plates in technical mechanics . To make the calculations manageable, it makes use of some simplifications and defines the conditions under which it is valid.

A generally three-dimensional dynamic problem is transformed into a two-dimensional problem that is dynamic, quasi-static or static by neglecting small quantities.

Plate theories may be used if it can be assumed

that the plate as a flat surface structure is only stressed normal to its center plane ( ) (different from the disc ) and ${\ displaystyle z = 0}$

that the normal stress can be normally neglected the central area: .

{\ displaystyle \ sigma _ {Z}}

{\ displaystyle \ sigma _ {Z} \ approx 0}

Theories

Depending on the magnitude of the external loads and accelerations as well as the boundary conditions of the original 3D problem, a suitable plate theory must be selected that approximates the original problem sufficiently well.

According to the geometry of the plate

Depending on the ratio of the plate thickness to the other geometric dimensions, e.g. B. to the characteristic length , are to be distinguished: ${\ displaystyle d}$ ${\ displaystyle l}$

If this ratio is negligibly small: a thin plate theory can be used. In the case of thin plates, straight line sections that originally stood orthogonally on the central surface remain , to a good approximation, straight and orthogonal to the deformed central surface even in the deformed state (normal hypothesis, Kirchhoff-Love hypothesis; cf. Bernoulli bar ). ${\ displaystyle {\ frac {d} {l}} \ ll 1}$

Is only the 3rd power of the above Ratio very small :, it is a question of moderately thick panels. In the case of moderately thick panels, the warping can no longer be neglected and theories must be used in which undeformed straight vertical lines remain straight in the deformed state, but are no longer perpendicular to the middle surface (theory for flexible panels; see Timoshenko beams ) . ${\ displaystyle \ left ({\ frac {d} {l}} \ right) ^ {3} \ ll 1}$

According to the size of the deformation

Depending on the ratio ( amount ) of deformations of the plate to its thickness , a distinction must be made: ${\ displaystyle u}$ ${\ displaystyle d}$

If this ratio is small: a linear plate theory can be used (plate theory according to Kirchhoff ). ${\ displaystyle {\ frac {u} {d}} \ ll 1}$

Is the deflection of the order of the sheet thickness: , so must the non-linear be used theory for moderate deflection, wherein in spite of small distortions the plate and the disc problem is not decoupled (plate theory of Kármán ). ${\ displaystyle {\ frac {u} {d}} \ approx 1}$

If the deflections are in the order of magnitude of the plate dimensions (characteristic length): then a plate theory for large deformations must be used, whereby, depending on the boundary conditions and external loads, either a pure bending problem or a pure membrane or shell problem can act. ${\ displaystyle {\ frac {u} {d}} \ gg 1}$

Overview

large deflections u d or u ≈ l ${\ displaystyle \ gg}$	Bend or cup problem	spatial state of tension
moderate deflections u ≈ d	nonlinear plate theory (Disk problem coupled with disk problem)	plane stress state; Distortions parallel to the median plane are proportional to the distance from the median plane		spatial tension Status
small deflections u d ${\ displaystyle \ ll}$	linear plate theory (Disk problem decoupled from disk problem)			spatial tension Status
		*straight, orthogonal	*straight, not orthogonal	*not really, not orthogonal
		thin plate d / l 1 ${\ displaystyle \ ll}$	moderately thick plate (d / l) ³ 1 ${\ displaystyle \ ll}$	thick plate

*) Lines that are straight and orthogonal to the center plane in the undeformed state are in the deformed state ...

Scientists involved

literature

Karl-Eugen Kurrer : The History of the Theory of Structures. Searching for Equilibrium . Ernst & Sohn , Berlin 2018, pp. 702–716, ISBN 978-3-433-03229-9 .

Web links

Script Prof. Dr. FU Mathiak: Plate Theory (PDF file; 2.87 MB)
Script Dr.-Ing. I. Raecke Uni Magdeburg: Planar structures I / II (PDF file; 1.94 MB)

Individual evidence

↑ English transcription. Born in Saint Petersburg in 1916.

[1] English transcription. Born in Saint Petersburg in 1916.