Stability of pinned–rotationally restrained arches
László Péter Kiss
Available online 02 September 2020

Abstract
The article aims to find the buckling loads for pinned–rotationally restrained shallow circular arches in terms of the rotational end stiffness, geometry and material distribution. The loading is a concentrated vertical force placed at the crown. A geometrically nonlinear model is presented which relates not only the axial force but also the bending moment to the membrane strain. The nonlinear load-strain relationship is established between the strain and load parameters. This equation is then solved and evaluated analytically. It turns out that the stiffness of the end-restraint has, in general, a significant effect on the lowest buckling load. At the same time, some geometries are not affected by this. As the stiffness becomes zero, the arch is pinned-pinned and as the stiffness tends to infinity, the arch behaves as if it were pinned-fixed and has the best load-bearing abilities.

Mathematics Subject Classification
74G60; 74B15

Keywords
arch, buckling, stiffness, snap-through

DOI
https://doi.org/10.2298/TAM200402010K

An analytical mechanics approach to the first law of thermodynamics and construction of a variational hierarchy
Hamid Said
Available online 03 November 2020

Abstract
A simple procedure is presented for the study of the conservation of energy equation with dissipation in continuum mechanics in 1D. This procedure is used to transform this nonlinear evolution-diffusion equation into a hyperbolic PDE; specifically, a second-order quasi-linear wave equation. An immediate implication of this procedure is the formation of a least action principle for the balance of energy with dissipation. The corresponding action functional enables us to establish a complete analytic mechanics for thermomechanical systems: a Lagrangian-Hamiltonian theory, integrals of motion, bracket formalism, and Noether's theorem. Furthermore, we apply our procedure iteratively and produce an infinite sequence of interlocked variational principles, a variational hierarchy, where at each level or iteration the full implication of the least action principle can be shown again.

Mathematics Subject Classification
37K05; 80M30

Keywords
continuum mechanics, first law of thermodynamics, least action principle, dissipation, variational hierarchy

DOI
https://doi.org/10.2298/TAM200315011S