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A variant of refinement of the classical theory of thin plate bending

Firsanov Vic.V.

#### Abstract:

The classical theory of bending of thin plates based on Kirchhoff’s hypotheses about the absence of normal stresses in the transverse to the bases direction, invariability of the length of the normal element to the middle plane of the plate, which means the invariability of the thickness and lack of linear deformation in the transverse direction, the absence of shear strains in planes perpendicular to the bases of the plate. At the same time, in the equilibrium equations, both normal stresses in the transverse direction and tangential stresses associated with shear deformations by physical relations remain, but the physical connections are obviously broken. Refinement of the classical theory is usually associated with the rejection of all Kirchhoff hypotheses, which significantly complicates such a model, or the rejection of one or two kinematic hypotheses. For example, you can set a displacement in the transverse direction as a power series along the transverse coordinate. In this case, if the degrees are even, the linear deformation in the transverse direction is different from zero, but the normal element connecting the bases of the plate does not change its length, which is not in contradiction with the Kirchhoff hypothesis. However, this approach may not lead to significant refinements of the classical model, so for a more or less significant refinement, it is assumed that the most acceptable rejection of the hypothesis of the absence of shear deformations in the planes transverse to the plate bases. In this case, the physical relationship between shear and stress is restored. Accounting for these shear deformations is especially important for materials with low shear stiffness in the transverse directions. Another reason for refining the classical model of plate bending is that some boundary conditions are not satisfied accurately enough, which is due to the introduction of a generalized Kirchhoff shear force into the calculation model, which consists of a purely shear force and an increment along one of the plane coordinates of the torque. With certain refinements, it is possible to solve the problem of three boundary conditions on the free edges of the plate.

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