Local effects in a thermoelastic plate stiffened by a rod/ring

Abstract

539.377On the basis of the revised theory of plates, which makes it possible to determine the components of the stress tensor, we study the thermostressed state of an unbounded plate stiffened by a rod-ring. In the stiffening and in the contact zone of the plate the stresses differ significantly, both quantitatively and qualitatively, from those predicted by the classical theories of elasticity.A considerable number of papers have been devoted to the problem of determining the stress state of stiffened shells and plates. The main results and a rather complete bibliography can be found in the monographs [1,3,4,8]. (We note that the majority of these studies were carried out in L'vov.) However all of these studies were based on a theory of plates and shells that does not describe the three-dimensional effects that can occur in the contact zones. An analysis of the thermal stresses in plates with an inhomogeneous inclusion [2,7] based on the revised theory [6] gives reason to expect a significant contribution from these effects in the case of stiffening.We consider a uniformly heated infinite plate of thickness 2h stiffened along the curve r = rl by a rod-ring of the same thickness and made of a different material. Determining the stress-strain state of such a system using the revised theory, which makes it possible to determine all the components of the stress tensor [6], reduces to solving the following system of singularly perturbed differential equations for the functions w and ~P:Here t is the temperature, r is the radial coordinate, =45-r~rr r , ai=(1-2-)at is the coefficient of linear thermal expansion, and L, is the Poisson coefficient. We introduce the following notation for the Kelvin functions [5]:fl(x) = berx, f2(x) = beix, f3(x) = kerx, f4(x) = keix,We then write the general solution of the system (1) in the formi=l where T/= ?~E -1 and (..The boundary conditions (the absence of stresses at infinity and on the surface r = r0, where ro is the interior radius of the rod-ring) and the coupling conditions (continuity of the stresses ar and ~r~ and