Analytical Construction of Shape Functions and Stiffness Matrix of Finite Element in Method of Displacements

An element of a circular ring in a form of a quarter of a circle is considered without loss of generality. For the analytical construction of shape functions of parameters of stress and strain state, excluding problems connected with neglecting the displacement as a whole rigid body, methods of forces and displacements are jointly used, which allow one to exclude a differential equation while constructing the stiffness matrix of the element but while constructing the shape functions of displacements to restrict oneself to the simplest equation of elastic line that permits the simple derivation of Green function. Boundary elements may be considered using analogous relationships.