Force method application for static calculation of closed curved bar
Journal Title: Вісник Тернопільського національного технічного університету - Year 2015, Vol 79, Issue 3
Abstract
Within geometrical linear theory of deformations using of the hypotheses of normal plane section refined version of the complete system of equations of one-dimensional theory of closed thin elastic bars of rectangular cross section is constructed. They are modeled by the locked curved rods. The bars are in a plane stress created by external force applied to lateral surfaces of the bar. Considering that the main axis of the cross section coincide with the direction of axes of natural trihedron of a fixed fiber of a bar and neglecting the deplanation of cross sections the formula to determine the internal longitudinal and lateral forces and bending moments, referred to the axis of the bar, are deduced. These formulas are equivalent to the differential equilibrium conditions of an element of bar. To determine the reactions of redundant constraint of static uncertainty closed bar in the conditions of a symmetric stress the force method is used. Physical relations between the longitudinal forces and bending moments and certain deformations of fixed fiber of a bar are built on the basis of consideration of deformation of its infinitesimal areas. Neglecting the influence of cross-axis forces and deformation of the cross section of the bar the conditions of compatibility between displacement points of random fiber and its deformation are recorded. These conditions allowed to express components of the displacement vector of the point of the bar through two one-valued functions, which depend on the external load. The special cases of problems for the bar, modeled by elastic line with tension (compressive) and bending stiffness or flexible elastic line are examined. Expressing symmetrical external load by finite trigonometric sums the method of calculation of mode of deformation of the bar with smooth contour without angular points is developed.
Authors and Affiliations
Andrii Syaski, Natalia Viktorivna Shevtsova
Keywords
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