Issue 38

N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01 2 of an infinity system of the algebraic equations. The variation method was used for the analogical problem’s solving in [6- 8]. The energetic method was applied to the problem of the semi-strip with the free lateral edges and loaded short edge in [9]. In [10] authors constructed a special system of byorthogonal functions, with the help of which they solved the problem on a semi-strip loading at it’s the short edge. The problem for the semi-strip with the free longitudinal sides was solved with the help of the stress function in [11, 12]. The Laplace’s integral transformation was used for the problem’s solving in [13]. The approach based on the use of Fadle-Papkovich functions was applied in [14-16]. In this paper the method, which was worked out by G. Ya. Popov, was used [17]. Accordingly to it the integral transformations were applied directly to the equilibrium equations and boundary conditions of a problem. It leaded the initial problem to one-dimensional boundary problem in the transformation’s domain. The last one was formulated as the vector boundary valued problem and solved exactly with the apparatuses of the matrix differential calculations and Green’s matrix function [18]. The problem was reduced to the singular integral equation’s solving. Investigation of the signature’s nature of the singular integral equation’s solving was under consideration of many famous scientists. Today the new theories are appeared, which describe the solution’s behavior at the particular points [19]. The investigations of the singularities’ nature for the complex medium are continued [20]. But in most studies the authors did not pay attention to the fixed singularities at the angular points of the semi-strip usually, although these singularities play a main role in the estimation of the stress state. One approach that allows to find and to take such singularities into account was proposed in widely known work [21]. It was used in this paper for the fixed singularities’ consideration. The special generalized method, which was proposed in [22, 23], was applied to obtain the solution of the (SIE) with regarding of the solution’s two fixed singularities at the end of the integration’s interval. Figure 1 : Geometry of the problem. T HE STATEMENT OF A PROBLEM he elastic ( G is a share module,  is a Poison’s coefficient) semi-strip, x a y 0 , 0      is considered. At the edge y x a 0, 0    the semi-strip is loaded   y xy x p x x x a ( , 0) , ( , 0) 0, 0        (1) where p x ( ) is the known function. At the lateral sides x y 0, 0     and x a y , 0     the boundary conditions of the fixed are given         u y v y u a y v a y y 0, 0, 0, 0, , 0, , 0, 0        (2) T