Plate (engineering mechanics)
In technical mechanics or in structural engineering, a plate is a component that is spread out on the plane and consists of stiff material (flat surface structure ) and is loaded by forces acting perpendicularly on it and by moments about axes that lie in the plane of the plate.
In contrast to a plate, a disk is loaded by forces in its plane .
In linear plate theory , any load can be broken down into a plate problem and a disk problem that is decoupled from it . The two problems cannot be decoupled in nonlinear plate theories.
A surface component that is not flat but curved in the unloaded state is called a shell .
In deviation from the above definition and the general understanding that a disk is a spread in the plane member in is track-laying into a screw used component with a U-shaped profile also as a plate referred to (see "clamp" in rail fastening system ).
requirements
- There are loads that are perpendicular to the median plane.
- A distinction is made between plate theories for thin plates, moderately thick plates and thick plates, depending on whether the plate thickness is much smaller than the plate width, moderately smaller or smaller.
- A distinction is made between plate theories in which the deflections are considerably smaller than the plate thickness (plate theory according to Kirchhoff ), in the order of magnitude of the plate thickness (plate theory according to von Kármán ) or in the order of magnitude of the plate width and thus considerably greater than the plate thickness.
- In the case of thin and moderately thick plates, the deflections of which are small or moderately large, a plane stress state is a good approximation . In addition, here the distortions parallel to the median plane are proportional to the distance from the median plane; perpendicular to the median plane they are zero.
- Straight and orthogonal line segments that are perpendicular to the center of the panel remain straight in the bent state for thin and moderately thick panels and even orthogonal for thin panels. This leads to a linear distribution of the bending stresses over the plate thickness.
- Linear plate theories (for small deflections) assume linear material behavior .
Overview
|
large deflections
u d or u ≈ l |
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 |
linear plate theory
(Disk problem decoupled from disk problem) |
|||
| *straight,
orthogonal |
*straight,
not orthogonal |
*not really,
not orthogonal |
||
|
thin plate
d / l 1 |
moderately thick plate
(d / l) ³ 1 |
thick plate | ||
*) Lines that are straight and orthogonal to the center plane in the undeformed state are in the deformed state ...
application
Plates are used mainly as applied floor cover , as foundation plates and bridges . They are usually made of concrete or steel , also in shipbuilding , vehicle construction , offshore structures and in semiconductor electronics , e.g. B. electronic wafers .
storage
Slabs are clamped in one or two axes between supports . The supports are linear (walls) or point-like ( columns ). They can be arranged on the edge or at any point under the plate. The simplest panels are rectangular and supported at the edge, but they can also have any floor plan shape and holes (recesses).
Calculation and dimensioning
Plate theory is used to calculate plates that are not too complicated. For simple concrete slabs there are tables for dimensioning with reinforcement , for very complicated shapes you need computer programs based on the finite element method .
literature
- J. Bluhm: planar structures . Institute for Mechanics - Faculty of Engineering - Department of Building Sciences, University of Duisburg-Essen
- Philippe Ciarlet : Mathematical Elasticity, North Holland 1991 (Volume 2 Theory of Plates, Volume 3 Theory of Shells)