BASIC STRESS-STRAIN STATE OF A MULTILAYER SYMMETRIC CIRCULAR ARCH UNDER THE ACTION OF NORMAL FORCES IN THE MIDDLE SECTION. REPORT 1. ARCHES OF HIGH CURVATURE

Abstract

Circular arches are common elements of construction and machine building structures. They can be both
individual components and reinforcing elements of thin-walled casings. An important case of arches deformation
is the symmetrical folding in its own plane surface under the action of concentrated force. The resistance
of homogeneous isotropic arches is sufficiently investigated. However, the mechanics of multilayer
arches deformation is studied considerably less, which creates additional obstacles to introducing such elements
in designing practice. The purpose of this work is a theoretical study of the basic stress-strain state
(SSS) of a symmetrically fixed multilayer arch under the action of concentrated normal force in the middle
section by developing the analytical solution of the corresponding problem. The first part of the article deals
with the statement of the problem, the prerequisites, the main stages of developing its general solution and
the conditions that simulate different methods of fixing the ends of a solid arch and arch with a pinconnected
joint in the middle section. In the course of solving the problem, the studied arch symmetry enabled
to consider only its half separated over the middle section, taking into account the corresponding static
and kinematic conditions on the ends. The SSS of such element is similar to the SSS of a circular multilayer
bar with loads at the ends, which enabled to use the exact solution of elasticity theory of the corresponding
problem obtained by the authors. The solution developed in such a way corresponds to the exact solution of
the problem on the most part of the arch and, near the loaded section and the fixed ends, gives a simplified
description of the SSS without taking into account its local distortions. The obtained general correlations
depend on six unknown parameters in the middle section. For the purpose of their determining, static and
kinematic conditions which correspond to different methods of fixing the ends of the arch and the connection
of its halves were obtained. To demonstrate the possibilities and approbation of the obtained solution, the
authors gave the results of determining the SSS of a four-layer arch with rigid ends’ fixation with the ratio of
the mean radius to the section height that is equal to 1.75, as well as the results of additional calculations at
increasing the mentioned ratio to 15. The developed solution enables to identify the SSS of symmetric composite
arch elements and rings in order to study their static strength and stiffness and can also be used in
solving more complex deformation problems of multilayer curved elements of constructions.