Lakhasly

The hypothesis of Navier Bernoulli consists in assuming that the sections normal to the average fiber remain flat during the deformation of the beam to the plates.- The principle of Navier Bernoulli amounts to neglecting shear and warping of the cross sections in the study of displacement and deformation of a beam element to plate.Similarly, when we study the torsion, we see that a non-circular section, with two symmetric axes, holds under the effect of a torsional torque a radial warp.It can therefore only be applied to very thin structures.

The hypothesis of Navier Bernoulli consists in assuming that the sections normal to the average fiber remain flat during the deformation of the beam to the plates. This hypothesis, which makes it possible to calculate the normal stresses due to bending moment, is well verified in the case of pure bending where the shear force is zero. On the other hand, in the case of simple bending with shear force, the sections do not remain Planes, but warp in the form of letter S very compressed. Similarly, when we study the torsion, we see that a non-circular section, with two symmetric axes, holds under the effect of a torsional torque a radial warp. The principle of Navier Bernoulli is based on the following observations :

• The warping of a section is always very small with respect to the dimensions of the section.


• The variation of the warpage, when passing from a section to an infinitely neighboring section, is always very small, not only with respect to the distance of the two infinitely neighboring sections.


• The principle of Navier Bernoulli amounts to neglecting shear and warping of the cross sections in the study of displacement and deformation of a beam element to plate.
  It is rare to find a theory that would be applicable to all possible cases (composite, anisotropic, isotropic, large number of layers, sandwich stratification etc.) and to the different domains (static, dynamic and buckling), and Simple and easy and does not cost expensive in computing time. The oldest theory is that of Kirch=off Dhatt which neglects the transverse shear effect. It can therefore only be applied to very thin structures. The first-order theory commonly associated with Mindlin and Reissner, which was one of the first to state its bases, takes into account the effects of transverse shear across the thickness. It leads, by the hypothesis of "straight sections remain straight" to a vector of constant transverse shear stresses in the thickness, in contradiction with a quadratic representation conventionally obtained for the beams (Timoshenko theory) or the plates in bending. To correct this deficiency, so-called transverse shear correction factors are introduced. Finite elements formulated in displacement based on first order theory generally give good results for isotropic and orthotropic structures. They become less precise when applied to composite materials containing several layers with very different anisotropy from one layer to another Topdar et al, in which case it would be necessary to impose conditions of continuity on the interfaces. Indeed, transverse shear correction factors, once introduced in the 1st order models in displacement, have solved problems of multilayer structures but their evaluation unfortunately depends on the number of stratifications. To rule out this type of problem forever, high order theories were introduced in the early 1970s. The first theory was proposed in 1969 by Whitney, who assumed a high order displacement field of 3. It Gave precise results but was abandoned because of its theoretical complexity ; It requires a large number of Whitney parameters. Other theories have appeared later, each of which has advantages and disadvantages, with different formalisms depending on the field of application [78].