A plate is a body bounded by two parallel planes, the distance between which, called the plate thickness h = const, is small in comparison with its other sizes [1] . There is also the following refined definition of a plate: a plate is a body bounded by a cylindrical surface and two planes perpendicular to it, the distance between which is small compared to its other dimensions. [2] In the same meaning as the term “plate”, the term “plate” is also used.
Content
Plate Types
Plate - a term used in structural mechanics to describe the design scheme , taking into account the peculiarities of the geometry of the body. All bodies have three dimensions. In the case when one of the body sizes is significantly different from the other two, to simplify the calculation of strength , rigidity and stability, the real three-dimensional structure can be replaced by its design scheme. For plates, such a calculation scheme is a two-dimensional flat body, the displacements of which are determined by the displacements of the plane, which bisects the thickness of the plate. This plane is called the `` median plane. '' When the plate is bent, the median plane turns into a curved surface. The line of intersection of the lateral surface of the plate with the median plane is called the contour of the plate.
The term “significantly different” used in the definition of a plate is not completely defined. Depending on the characteristics of loading the plate, different limit ratios between the thickness and other dimensions of the plate are adopted. The most reliable condition that a construction object can be considered as a plate is to compare the calculation results with two methods: as a plate and as a flat three-dimensional body. Roughly accepted is the condition that for a plate its thickness is less than other sizes by at least 5 times. A thin plate in which the maximum deflection under the influence of a transverse load exceeds a quarter of its thickness is called a flexible plate [1]
A plate bending from its own plane is called a plate . When calculating a slab, two assumptions are usually used: the first - it is assumed that the rectilinear elements normal to the median plane remain straight after the deformation, normal to the deformed median surface (hypothesis of direct normal); second, it is believed that the plate is not compressible in thickness. These assumptions make it possible to express the displacements of all points of the plate through the transverse displacements of the median plane. Calculation of plates using these assumptions forms the basis of the technical theory of plate bending. The deformed state of the plate, in which the median plane passes into a cylindrical surface, is called a cylindrical bend, and such a plate is called a beam plate.
A vertically located plate in a plane stress state is called a wall or a beam-wall. Thin walls under the action of external loads parallel to the median surface can lose local stability. When checking the stability of thin walls, as in the calculation of plates, the hypothesis of direct normals is used.
By design, the plates can be single-layer and multi-layer (of two or more layers). Lamellas having ribs located at a constant pitch in one or two directions are called ribbed plate. If there are five or more ribs in each direction, the plate can be calculated as an anisotropic structure. A ribbed rectangular plate whose edges are parallel to its sides is called an orthotropic plate.
The history of the creation of the technical theory of plates
The founder of the theory of bending and vibration of plates is Jacob Bernoulli Jr. (1759-1789), who in 1789 received the differential equation of bending of a plate, considering it as a system of strings stretched in two mutually perpendicular directions. In 1828, Augustin Cauchy (1789-1857), and then in 1829 Simeon Poisson (1781-1840) used equations of elasticity theory to bend plates. [3]
Gustav Robert Kirchhoff (1824–1887), the famous German physicist, known for his work on the theory of calculating electric circuits and the deformation of solids, in 1850 developed the theory of plate bending. The theory he proposed is based on two assumptions that simplify the calculation: the hypothesis of direct normals and the assumption of the incompressibility of the plate material over its thickness.
I. G. Bubnov proposed a method for integrating differential equations for solving boundary value problems. This method was used by I. G. Bubnov in 1902 to calculate the plates working in the ship's hull system. B. G. Galerkin , apparently independently of I. G. Bubnov, proposed a similar method for integrating differential equations, which was widely used to calculate rectangular plates for various loading and fixing schemes of plates. The method has received in the technical literature the name of the Bubnov-Galerkin method.
Modern methods for calculating plates are based on the use of the finite element method .
Constructions
The plate can be an independent design or be part of the plate system. Separate plates are used in construction in the form of wall panels, wall beams, slabs and floor and cover panels, foundation slabs, etc.
Horizontal and vertical plates, interconnected by bonds, form a supporting system, which in relation to buildings is called a wall system.
Inclined plates can form span bearing structures. A system of rectangular inclined plates, the middle surface of which unfolds on a plane, is called a fold. A system of equilateral triangular or trapezoidal plates connected by the sides of the same length is called a tent covering or tent.
Literature
- Feodosiev V.I. Resistance of materials: Textbook for high schools. - 10th ed., Revised. and add. - M .: Publishing House of MSTU. N.E. Bauman, 1999 .-- T. 2 .-- 592 p. - (Mechanics at a technical university). - ISBN 5-7038-1340-9 ; UDC 539.3 / 6 (075.8); LBC 30.121 F42.
Notes
- ↑ 1 2 Handbook of the designer of industrial, residential and public buildings and structures. Settlement and theoretical. Book 2. M., Stroyizdat, 1973.
- ↑ Collection of recommended terms. Issue 82. Structural mechanics. M., ed. Science, 1970.
- ↑ Grigoryan A.G. Mechanics from antiquity to the present day. M., ed. Science, 1974.