Egg-inspired engineering in the design of thin-walled shelled vessels: a theoretical approach for shell strength

Abstract

A novel subdiscipline of bionics is emerging in the form of ‘egg-inspired engineering’ through the use of egg-shaped ovoids as thin-walled tanks and building structures. Hügelschäffer’s and Narushin’s models of egg geometry are highly applicable within this proposed subdiscipline. Here we conducted a comparative analysis between the two models with respect to some of the most important egg parameters. These included contents volume, shell volume, and the location of the neutral axis along the shell thickness. As a first step, theoretical studies using the Narushin’s model were carried out due to the lack (or limited amount) of data on the geometric relationships of parameters and available calculation formulae. Considering experimental data accumulated in the engineering and construction industries, we postulate a hypothesis that there is a correlation between location of the neutral axis and the strength of the walls in the egg-shaped structure. We suggest that the use of Narushin’s model is preferable to Hügelschäffer’s model for designing thin-walled shelled vessels and egg-shaped building structures. This is due to its relative simplicity (because of the requirement for only two initial parameters in the basic equation), optimal geometry in terms of material costs per unit of internal capacity, and effective prerequisites for shell strength characteristics.

A novel approach to egg and math: Improved geometrical standardization of any avian egg profile

“…In the course of our previous studies, 6,28,29,34,41,56 we accumulated a large number of photographic and digital images of various eggs obtained directly by us or available as open source. Thanks to this, we selected the egg images that can be classified as pyriform from 32 bird species.…”

“…Since, in order to carry out a comparative analysis of eggs of different species, shapes, and sizes, it is much more convenient and efficient to use not the initial sizes, but their certain ratios, 5,29,34 we presented numerical data characterizing each egg in the form of the following ratios: B/L, w/L, and D p /B. To make it clearer what is the difference between the Hügelschäffer model used to describe the blunt end from the Hügelschäffer model for the pointed end, we also gave the values of the ratio w p /L after the respective calculation of the value of w p according to Equation (4).…”

“…These have included (i) a comparative analysis of the universal formula with others developed before or after its publication [20][21][22][23][24][25][26][27][28] ; (ii) its use for deriving applied calculation formulae, for example, for computing egg volume or surface area that can be successfully used in biological, zootechnical, and/or other studies [29][30][31][32] ; (iii) technological developments 33,34 ; (iv) space research, for example, for building egg-F I G U R E 1 Combination of two Hügelschäffer's models conforming to (i) a classical ovoid with parameters B, L, and w (blue line) and (ii) a parabolic ovoid whose axial shift corresponds to half the egg length, that is, w p = L/2. shaped lunar habitat structures 35 and modeling broad-line regions when studying supermassive black holes 36 ; (v) computer games 37 ; and even (vi) interest in its inclusion in the school curriculum in mathematics as an adapted version of our research.…”

“…In addition, this theoretical finding served as an impetus for its implementation in diverse studies and disciplines. These have included (i) a comparative analysis of the universal formula with others developed before or after its publication 20–28 ; (ii) its use for deriving applied calculation formulae, for example, for computing egg volume or surface area that can be successfully used in biological, zootechnical, and/or other studies 29–32 ; (iii) technological developments 33,34 ; (iv) space research, for example, for building egg‐shaped lunar habitat structures 35 and modeling broad‐line regions when studying supermassive black holes 36 ; (v) computer games 37 ; and even (vi) interest in its inclusion in the school curriculum in mathematics as an adapted version of our research 38 …”

Hügelschäffer egg curve and surface

“…Analogous to the previous example, a Hügelschäffer egg-shaped curve model can be generated according to the profiles of egg-shaped pipe sewers obtained through composition of circular arcs with the shape index B/L = 2/3 or B/L = 3/4 as given in [51]. In recent years, the application of Hügelschäffer models, as well as other mathematical models for describing the shape of eggs -ovoid forms (curves and surfaces) in the poultry industry and food engineering is on the rise, as evident from papers [2], [14], [15], [17], [20], [22] - [32].…”