Relations of the theory of thin timoshenko-type shells in terms of the curvilinear coordinates of the reference surface: PMM vol. 42, no. 4, 1978, pp. 753–758

Abstract

A method is given for solving the problem of parametrization and determination of the metric of the middle surface of a shell of complex configuration. The method is based on introducing, into the space, a surface σ 0 with simple geometry, and using this as a reference surface on which the middle surface a is mapped. The position of a point on σ is defined in terms of the Gaussian coordinates α 1, α 2 of the point on σ 0 and the distance H ( α 1, α 2) between σ and σ 0 measured along the normal to σ 0. Expansion of the shell displacement vector in terms of the basis vectors on σ, the basis representing a mapping of the basis on σ 0, and use of a Timoshenko-type theory, yield a formulation of a nonlinear boundary value problem of computing shells of complex configuration. A method is given of reducing the problem of investigating the open “non-classical” shells in terms of the coordinates of their middle surfaces to a “conditionally classical” problems in terms of the coordinates of the reference surface. A theory of shells shallow relative to the reference surface is proposed, which generalizes the classical theory of shallow shells the middle surface of which is shallow relative to a plane.