The incompressibility condition for an isotropic linearly elastic material seriously limits the application of the classical hypotheses of the beam bending theory formulated by Bernoulli for small deformations and displacements. At the same time, it is assumed that such a strong kinematic condition as the volume immutability condition, which is valid for elastic components of linear deformations, must be unconditionally fulfilled. When an incompressible beam is bent by a force load, the incompressibility condition, which characterizes the absence of deformation of the volume change, is obviously homogeneous, that is, the volume of the beam at the micro and macro levels does not change during deformation, and under the action of a bending thermal load, the deformation of the volume change is proportional to the operating temperature, and the elastic component of the total deformation of the volume change is zero. Consequently, mechanical incompressibility manifests itself when a force load acts on the beam, but in the case of thermal action, the deformation of the volume change is a function of temperature. Of course, this is a serious difference between the two conditions, however, even in the case of a temperature load, the condition of partial or mechanical incompressibility may be conflicting with respect to the classical hypotheses of beam bending, which may lead to the degeneration of the problem. It would be a mistake to completely reject the classical hypotheses of beam bending, but some should be abandoned and other hypotheses should be introduced that will not lead to a serious complication of the tasks being solved If we accept the absence of linear deformation in the direction transverse to the neutral axis of the beam, the absence of shear deformation and at the same time fulfill the condition of proportionality of the deformation of the volume change to the operating temperature load, then the two desired displacements, one transverse, the other longitudinal, can be determined from these ratios, and the solutions obtained will not correspond to the temperature problem being solved. Because of this, it is advisable to abandon the hypothesis of the absence of shear deformation, then the two kinematic desired functions will be related only by the dependence of the deformation of the volume change on the temperature load. In this case, the physical connection between shear deformations and stresses is restored. Taking into account these shear deformations is especially important for materials with low shear stiffness in transverse directions. The hypothesis about the non-compressibility of the beam fibers in the transverse direction will remain valid for physical relations, but these stresses are preserved in the equilibrium equations.

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