Abstract—A  mathematical  shape representative of the shell of the egg of a domestic fowl is proposed  and a stress analysis is carried out for the shell when it is subjected to internal pressure。 Algebraic expressions for the membrane solution are seen to give a close approximation to the complete analysis。 Copyright C 1996 Elsevier Science Ltd。83136

xeJ» c'rds:   modified ellipse, axial symmetry, membrane analjsis, internal  pressure。

OT A T I O N

dimensions defining curve of revolution

in-plane principal forces per unit length at a point on shell of revolution internal pressure

parallel circle radius for a point on shell of revolution principal radii of curvature at a point on shell of revolution thickness

cartesian co-ordinates for curve of revolution angle defined in Fig。 1

parameter used to define point on curve of  revolution

angle defining meridional position at point on shell of revolution with respect to 。x direction    indicate principal (meridional, circumferential) directions at point on shell of revolution

principal stresses at a point on shell of  revolution

I NT R O D U C T I O N

The subject of the present paper was prompted by studies (1 2) of the composition and fracture strength of eggshells of the domestic fowl。 Test methods used in previous investigations have involved bending of the shell, but further tests were to be conducted by subjecting eggs to internal pressure。 For this purpose it was necessary to predict the stress distribution in an egg of typical shape, and as before, elastic deformations were  assumed。

G E O M E T R Y

As a result of measurements on a number of eggs used in the tests by Entwistle et at。 [lj, the shape of an inpidual specimen was found to be very close to that of a shell of revolution, the curve of revolution (the meridian) being of modified elliptical form。

In establishing a suitable shape for the meridian the basic shape of an elliptical profile was taken in the parametric form

where 。x and J' are the Cartesian coordinates, while u and 6 are the semi-major and semi-minor axes, respectively [see Fig。 1(a))。

It is proposed  that the modification is as  follows

where eJh is small。

The  resulting  shape,  where  typical   measured   dimensions   a   e ——  34 mm,  2a ——  60 inin, 2b —— 45 mm have been used, is given in Fig。 1(b) and was found to be a very good representation of the profile of a real egg, where the above measurements (i。e。 a + e。 2a, 2b) may be subject to a little variation [3j。 In Fig。 1(b), tan b = J/。x。 Measured thicknesses were observed to vary only a little throughout any specimen  (13。

Fig。 1。  Shape of meridian (a) ellipse (b) modified  ellipse。

ST R ESS A NA LY S IS

Assessment  of the fracture tests  to be carried  out  on eggs  under  internal  pressure    required a theoretical analysis of the stress distribution in the shell, which was assumed to be of the form given by Eqn (1)。

A finite element analysis using elastic shell theory assumptions was carried out。 Apart from the applied load, the details and method of analysis were identical to those given by Entwistle ct a/。 (1)。 Changes in geometry during loading could be taken into account in the analysis, but for the load considered, their effect was insignificant。 The material was assumed to be homogeneous and isotropic。

Since the thickness to principal radius ratio was always very small in any position and there was no rapid change of shape or discontinuity in the shell, it was considered that membrane analysis would be likely to provide a solution with little significant   error。