R. Bantsuri and N. Shavlakadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

THE CONTACT PROBLEM OF INTERACTION OF ELASTIC BEAM OF VARIABLE RIGIDITY WITH ELASTIC BASIS

A contact problem for anisotropic plate in the shape of angle (−Q < arg z < 0) is investigated when a semi-infinite beam of variable bending rigidity dx^{α} (d > 0, α ≥ 0) leans on one of the edges and the other edge of the angle is free. A distribution of strains in the plate and flexure in the beam must be determined in case when a normal load of intensity P_{0}(x) effects on the beam.

Applied problem is reduced to the Karleman boundary-value problem for a strip, an exact solution for any value α is obtained.

It is proved that an unknown contact normal strain in the vertex of angle for any Q and in case of α ≥ 3 admits the estimate: P (x)−P_{0}(x) = O(x^{α−3}), x → 0 and for large x : P (x) − P_{0}(x) = O(x^{−(1+τ_{0}^{+})}), τ_{0}^{+} > 0.

In case of 2 < α < 3, in the vertex of the angle we get: P (x) − P_{0}(x) = O(x^{−(1+μ_{0}^{−})}), −1 < μ_{0}^{−} < 0.

For α < 2 a normal contact strain can be bounded or unbounded near the point x = 0 and for large x : P (x) − P_{0}(x) = O(x^{−(3−α)}).

Concrete results for different values of Q and also the results in case of an orthotropic plate are obtained.