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Computational models of beam bending taking into account shear deformation

Firsanov Vic.V.

#### Abstract:

The classical model of beam bending is based on Bernoulli’s hypotheses: there is no transverse linear strain, no shear strain in the plane, where is the longitudinal and is the transverse coordinates of the beam, and no transverse normal stress. At the same time, both the transverse normal and tangent stresses are preserved in the equilibrium equations, since without them the problem of bending the beam has no solution. Implementation of the corresponding physical relations is neglected. For isotropic and orthotropic linear elastic materials, the shear strain is determined by dividing the tangent stress by the shear modulus. The larger the shear modulus, for example, compared to the elastic modulus in tension and bending, the closer we are to the hypothesis of no shear deformations, and Vice versa, the smaller the shear modulus, the more problematic the use of this hypothesis. This is especially true for the problem of bending orthotropic plates that are not reinforced in the transverse direction. Then the shear modules in the transverse direction are mainly determined by the properties of the weak binder and can be significantly less than the physical characteristics of an orthotropic package with planar reinforcement. In a beam, the reinforcement is carried out in the plane, and if the beam cannot be reinforced in the transverse direction due to too small a normal transverse stress, then a small number of layers at angles must be applied, since the bent beam also works for shear. Therefore, the shear modulus is determined not only by the binder, but also by the reinforcing fibers, and can be commensurate with the elastic modulus, and be several times smaller, depending on the number of reinforcing fibers. The aim of the work is to assess the effect of shear deformation on the stress-strain state of the beam.

#### Keywords:

BEAM,

BEND,

BOUNDARY CONDITIONS,

DEFLECTION,

DISPLACEMENT,

equations,

hypotheses,

MODEL,

STRAIN,

STRESS,

балка,

гипотезы,

граничные условия,

деформация,

изгиб,

напряжение,

перемещение модель,

прогиб,

уравнения
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