Analysis of elastic systems with discontinuous parameters
Constructions in the form of rectangular planar plates and shallow shells reinforced by a cross system of ribs of great rigidity and loaded with an arbitrary lateral load are considered. The stress-strain state of such structures is described by technical theories of plates and shallow shells. The effect of the stiffeners is taken into account in the equations as additional terms containing factors with the Dirac δ-function and its derivatives. The plate and the shell may have a stepwise variable thickness. The ribs are arbitrary thin-walled beams of variable cross-section that play an essential role in providing the load-carrying capacity of the structure as a whole. The stress-strain state of not only the plate and the shell but also the stiffeners themselves is determined. The questions of the rate of convergence of solutions in the form of trigonometric series describing the stress-strain state in the stiffeners are discussed. The results of analyses of reinforced plates are presented.
Calculation of the ribbed plate in the mixed form “deflection - function of efforts”
Analytical solution of a boundary-value problem for a system of differential equations of deformation is obtained for the rectangular plate supported with ribs of rigidity in the “function of deflection - function of efforts” mixed form. The solution is presented in the form of a series over combinations of regular and special discontinuous functions. The solution leads to fast converging series. A simple algorithm for computations is provided.
Equilibrium of elastic rectangle: Mathieu-Inglis-Pickett solution revisited
This paper addresses a general analytical method for investigating the two-dimensional distributions of stresses set up in a rectangular plate by a load applied along its sides in any arbitrary manner. Proposed independently by Mathieu (1890), Inglis (1921) and Pickett (1944), and later named the ‘superposition method’, it has been applied with success to the study of distribution of stresses inside a rectangle. The object of this paper is to prove the advantages of that approach when studying a stress field near the boundaries, including specific cases of discontinuous and concentrated normal and shear loadings. The method is illustrated by several numerical examples, the rapidity of convergence and the accuracy of results are investigated. The distribution of stresses along some typical lines in the plate are computed and shown graphically.