The application of the modified method of sections for calculations of curved flat rods

Abstract

In the paper there is presented the application of the modified method of sections (MMS) for calculation of flat curved rods approximated by the sequence of straight rods under the plain deforming in frames of the linear Bernoulli-Euler’s theory. It allows to simplify calculations of curved rods without using of the apparatus of the differential geometry. The orthogonal cross section is symmetrical relatively the plane of deformation to avoid warping. In the work there are presented the relations for transformation of defined internal force and kinematical parameters when there’s transition of the boundary of joining of two angle joint straight rods on plane. These relations are used for calculations by MMS of aggregate sequence of straight rods which approximate a curved rod. For checkups there was made the calculation of the quadrant circular arc using relations of the differential geometry and the calculation for the broken-line rod consisting of the aggregate sequence of five identical straight rods which approximate the quadrant circular arc for the same loadings. The kinematical parameters obtained in the particular point for the both calculations distinguish between themselves not more than for 1.5% which allows accurately to reveal the static redundancy using MMS. The areas of curvilinear diagrams of all internal forces equal to appropriate areas of diagrams obtained by MMS for the broken rod which proofs the conservation of the energy balance using MMS. The appropriate values of internal bending moments obtained on curvilinear diagrams and obtained by MMS for the broken rod are identical which enables to pass accurately the strength estimation or design using bending moments. The diagrams of axial and shear forces obtained by MMS for the broken rod are stepped where values of internal forces at each step approximately equal the mean values obtained from appropriate segments of the curvilinear diagram.