Application of linear conjugation method for mixed flexural problem in transtropic plates
Abstract
Formulas and dependences derived in works [1,2] for plates made of orthotropic material were used for solving the flexural problem of a hinged edge supported semi-infinite plate. These formulas are written through the analytic functions of a complex variable. The formulas are given for the case of a transversally isotropic material, when certain parameters must satisfy equality.
A new refined model formulas representation of transtropic plates bending in the complex form is proposed in the article. This form is convenient for application of the linear conjugation of analytic functions method by M.I. Mushelishvili. The linear conjugation method of analytic functions is convenient for solving problems with mixed boundary conditions at the edges, or by a piecewise distributed moment load along the support line in particular.
At the same time, it is assumed that torque and displacement of the median plate surface on this boundary are equal to zero. It means that in this case we assume that condition of zero equality at the boundary of generalized rotation angle (instead of the torque) does not significantly affect the general solution of the problem.
The solution method for plane problem of the elasticity theory and bending problems of Kirchhoff’s thin plates is elaborated in detail in the works of M.I. Muskhelishvili and I.O. Prusov. In this paper, the equations obtained by authors differ from the well-known representations of S. Timoshenko’s theories by members, which consider the transverse compression and nonlinearity of the normal stresses distribution in thickness of the plate. This non-linearity is provided by dependencies consideration of transverse forces. Ratios of displacement modules are the multipliers in these dependencies, which can be quite significant.
In the considered case for transtropic material, the complex potentials of M.I. Muskhelishvili can be expressed through one complex variable . Formulas for displacements, generalized rotation angles, bending moments and transverse forces in a transtropic plate could also be expressed in a compact form through the complex potentials.
