Abstract
It is generally accepted that an optimal arch has a funicular (moment-less) form and least weight. However, the feature of least weight restricts the design options and raises the question of durability of such structures. This study, building on the analytical form-finding approach presented in Lewis (2016. Proc. R. Soc. A 472, 20160019. (doi:10.1098/rspa.2016.0019)), proposes constant axial stress as a design criterion for smooth, two-pin arches that are moment-less under permanent (statistically prevalent) load. This approach ensures that no part of the structure becomes over-stressed under variable load (wind, snow and/or moving objects), relative to its other parts—a phenomenon observed in natural structures, such as trees, bones, shells. The theory considers a general case of an asymmetric arch, deriving the equation of its centre-line profile, horizontal reactions and varying cross-section area. The analysis of symmetric arches follows, and includes a solution for structures of least weight by supplying an equation for a volume-minimizing, span/rise ratio. The paper proposes a new concept, that of a design space controlled by two non-dimensional input parameters; their theoretical and practical limits define the existence of constant axial stress arches. It is shown that, for stand-alone arches, the design space reduces to a constraint relationship between constant stress and span/rise ratio.
Highlights
The analysis presented here is inspired by the observation that the principle of constant stress governs the formation of natural objects, such as trees, bones or shells, all of which exhibit a minimal stress response to loading
The principle of constant stress ensures that no part of the structure gets over-stressed under variable load relative to its other parts—a phenomenon observed in natural objects, such as trees, bones or shells
This paper presents a model of a smooth two-pin, moment-less arch, shaped by a constant value of axial stress and statistically prevalent load, such as the weight of the structure
Summary
The analysis presented here is inspired by the observation that the principle of constant stress governs the formation of natural objects, such as trees, bones or shells, all of which exhibit a minimal stress response to loading. The proposed form-finding approach makes a clear distinction between the independent input variables: span, rise, loading and constant value of stress, and the dependent output variables: horizontal reactions, centre-line profile, offset from the centre (in the case of an asymmetric arch) and material distribution in the structure. It should be noted that the relationship for A0 given by equation (2.15), as well as equations (2.12) and (2.13) describing H, do not hold for the case w = 0, i.e. a stand-alone arch carrying its own weight only In this case, the cross-section area at the apex, A0, becomes an independent input variable, and the centre-line profile is independent of both w and H. A stand-alone arch requires a separate treatment, as discussed
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