Transformations of displacement and force quantities between systems of axes and generic points of rigid open cross-sections

摘要

The displacement and force quantities of Vlasov-beams with a rigid in plane open cross-section are defined in a system of axes and generic points, depending on six system parameters, expressed in an "absolute" system of axes: two coordinates of the origin, two coordinates of the sectorial pole, the direction angle of the axes, and the constant term (or zero point) of the sectorial area. For a non-guided vlasov beam the stiffness matrix is formed in the fundamental system characteristic of each cross section, in which the origin is placed at the centroid and the pole at the shear center, the direction of axes is that of the principle axes, and the sectorial area is an equilibrium system. The force quantities, defined in this system, are mutually orthogonal in the sense that they do not do virtual work in one another's reference deformations. For a guided beam the deformations are restricted by longitudinally continuous restraints preventing one or several lateral or axial degrees of freedom. Then the active, the effective degrees of freedom are often mutually orthogoal in some other, transformed, non-fundamental system of axes and special points, here called the solving system, in which the stiffness (sub)matrix for effective degrees of freedom is now formed. (Special points have physical meaning, generic points are arbitrary.) In order that the force quantities from elements with different cross-sections can be added and the displacement quantities connected like vectors at nodes to form the basic equations of FEM of Vlasov beams, the force and displacement quantities from elements with different cross-sections must be transformed to one (nodal) system of axes and generic points. The state of deformation of a Vlasov beam can be regarded as a mathematically objective, tensor-like quantity, able to be expressed in the form of displacement and force "quantity vectors" in arbitrary Systems of axes and generic points. The transformation matrices from one system of axes and generic points to any other within one prismatic Vlasov element are derived in this article.