ESTIMATION OF YOUNG'S MODULUS AND

FAILURE STRESSES IN THE HEN'S EGG SHELL

M. A. TungMember C.S.A.E.

Agricultural Engineering DepartmentUniversity of British Columbia,

Vancouver 8, B.C.

INTRODUCTION

The number of cracked eggs received by grading stations hasdoubled in the past fifteen years. Acurrent estimate of the annual lossto Canadian egg producers is fourmillion dollars (10). The growingproblem of cracked eggs is aggravated by the trend toward higheregg production per hen and moreextensive use of egg-handling equipment. The successful design of egg-handling equipment requires reliabledata on physical and mechanical properties of eggs.

Rehkugler (5) measured Young'smodulus and failure stresses of tenegg shells using ring or semicircularportions of shell. The results showedconsiderable variation and the authoracknowledged the need for furtherstudy of egg shell mechanical properties. Sluka et al. (8) and Hammerleand Mohsenin (3) subjected shellsto internal hydrostatic pressure andcalculated stresses in the shell atfailure. Voisey and Hunt (11) usedRehkugler's values for Young's modulus and ultimate strength along withshell theory to predict force at failureof shells loaded between flat platesbut found their estimates to be considerably lower than the measuredforces at failure. An alternate approach to this problem is the estimation of egg shell elastic propertiesfrom deformation behavior. Thispaper describes the use of shelltheory, shell geometry and deformation characteristics to estimateYoung's modulus and failure stressesof egg shells.

THEORY

Analysis of stresses in the egg shellwas simplified for this work byassuming the egg to be spherical sothat existing solutions could be used.Reissner (6) provided solutions fora spherical shell loaded at a point

by

L. M. StaleyMember C.S.A.E.

Agricultural Engineering DepartmentUniversity of British Columbia,

Vancouver 8, B.C.

and for a spherical shell with loaduniformly distributed over a smallcircular area. For a point load, totaldeformation of the shell is

dt - PR V3(l-V*) 12 ET2

where d t — total deformation in

the direction of applied force

P = applied forceR = radius of curvature of

shell

V — Poisson's ratio ofshell material

E = Young's modulus ofshell material

T = shell thickness

If deformation, load, radius of curvature and shell thickness are measured,and a value for Poisson's ratio assumed, Young's modulus may be determined using

E = PR V3(l-V2) 22 dt T

Stresses in meridial and latitudinaldirections (membrane stresses) atthe axis of load may be calculatedfrom

-PV3(1-V2) 38 T2

Bending stresses may also be computed for positions in the shell adjacent to the loading axis; however,they become infinite at the apex.

For a load uniformly distributedover a small circular area, deformation of the shell is

ds = 4 PR V3(l-V2)n ET2

ker' u +[ ker' u +J_1L u u2 J

CANADIAN AGRICULTURAL ENGINEERING, VOL. 11, No. 1, MAY 1969

J. F. RichardsDepartment ofPoultry Science

University of British Columbia,Vancouver 8, B.C.

where u = r ^12(1-V*) 5VRT

ds = shell deformation inthe direction of applied force

r = radius of the circularloading area

ker' = is a Kelvin-type Bes-sel function (1, 2)

Young's modulus is then

E 4 PR V3(l-V2)n ds t2

ker' u -f*" ker' u +JlL u u2 J

Membrane stresses at the centre ofthe loading area are

-P V3(l-V2)n t2

[" ker' u +L u ih]

and bending stresses areSb= ±3P(1 + V)

n t2

[^]where kei' is a Kelvin-type Besselfunction.

Timoshenko and Goodier (9) provide a method for calculating contactarea between a sphere and flat plate.The radius of the contact area is

r — [0.75 P R (k +ks)]l/3 9

where kp = 1 —VP 10~e7~

ks = 1-V' 11Es

Vp= Poisson's ratio of platematerial

Vs = Poisson's ratio ofsphere material

Ep = Young's modulus ofplate material

Es = Young's modulus ofsphere material

Deformation in the direction of applied force, due to local flatteningat the two loading surfaces is

df =[~9 F (kp + ks)2 R

fll/3... 12

The total deformation is due to localflattening at the loading surfaces andbending of the shell, that is,

dt = df + ds .13

Estimates for Young's modulus ofegg shell may be used in equation 11and compared with calculated valuesfrom equation 6 in a trial-and-errorsolution. Equation 9 must be usedwith reservation for approximationof the loading area. Unlike a solidsphere, a shell experiences slightbending that increases its radius ofcurvature, and also there is a parabolic load distribution at the contactsurface rather than a uniformly distributed load as is assumed in shelltheory.

PROCEDURE

A total of 299 eggs were collectedfrom fifty Single-Comb White Leghorn pullets during August and September, 1967. An attempt was madeto obtain equal numbers of eggs fromall birds. A standard laying rationwas fed and individual cages wereused to allow identification of eggsproduced by each hen. Egg diameterat the equator was measured beforeeach egg was compressed at 44 ± 2microns per second parallel to itsmajor axis between flat plates in theapparatus (figure 1) described byRichards and Staley (7). The eggwas held in an upright position witha pliable foam rubber support whileforce and deformation were measured with an X-Y recorder. Egg contents were discarded and shell membranes removed in boiling aqueoussodium hydroxide. The shells wererinsed in water, dried and the average thickness at the egg equator wasmeasured using a dial micrometerfitted with a spherical anvil.

Figure 1. Compression testing machine.

Half the diameter of the egg atthe equator was assumed to be theradius of shell curvature. Young'smodulus and Poisson's ratio for thesteel loading plates were 21,100kg/mm2 and 0.25 respectively. Poisson's ratio for the egg shells wasestimated as 0.25. It should be notedthat varying the estimate for Poisson's ratio by ± 0.05 alters the calculated value of Young's modulus byabout 1.5 percent.

A computer programme was written to calculate Young's modulusand membrane stresses for a pointload (equations 2 and 3) andYoung's modulus, membrane stressesand bending stresses for uniformly

distributed load (equations 6 to 8).Polynomial approximations were usedin place of the Bessel functions(1). Means and standard deviationswere calculated for the pooled dataand analyses of variance were usedto test whether eggs from differentbirds possessed different mechanicalproperties.

RESULTS AND DISCUSSION

Estimates for Young's modulus ofindividual egg shells, using point loadand distributed load assumptions, differed by less than three percent. Because of the wide variation in theestimates of this modulus betweeneggs, the simpler point load calculations should be satisfactory for mostpurposes. On the basis of assumptions made and precision of measurement, error limits on these calculations are estimated at ± 25 percent.That is, if egg shells satisfy the conditions imposed by the theories leading to equations 2 and 6, Young'smodulus for an egg shell is within25 percent of the calculated value.

The values obtained from Young'smodulus (table I) are about threetimes greater than the average valueof 1520 kg/mm2 reported by Rehkug-ler (5). They are, however, onlyslightly lower than the average published values (4) for limestone (5900kg/mm2) and marble (5600 kg/mm2) which are similar in chemicalcomposition to egg shells. A lowerYoung's modulus for egg shells, thanfor limestone or marble, may be dueto the presence of numerous poresand an organic matrix in part of theshell. The range of Young's modulus

TABLE I. MEANS AND STANDARD DEVIATIONS

Mean

Force at failure, kg 3.186

Deformation at failure, mm 0.131

Radius of curvature, mm 21.76

Shell thickness, mm 0.314

2Young's modulus (point load), kg/mm 1+571.0

Membrane stresses " -6.758

2Young's modulus (distributed load),kg/mm 4694.0

-6.665Membrane stresses

Bending stresses " -'15.66

CANADIAN AGR'CULTURAL ENGINEERING, VOL. 11, No. 1, MAY 1969

S.D.

0.778

0.025

0.655

0.028

909.4

1.329

939.7

1.307

8.699

TABLE II. F - RATIOS FROM ANALYSES OF VARIANCE TO TEST DIFFERENCES AMONG BIRDS AND DWERENCES AMONG EGGS OFINDIVIDUAL BIRDS

Among Birds 1 Amc ng Egg2

s

Force at failure 8 .08.'. J.

5 .44 A A

Deformation at failure 2 .90 A A 5 .00 A A

Radius of curvature 36 14A .'.

6 .91 A A

Shell thickness 14 24 A A 15 46.'. ...

Young's modulus (point loa d) 4 62 7 28 A A

Membrane stresses 6 58.'. A

0 32 N.S

Young's modulus (distri buted load) 4 44 •:•. a 7 47 A A

Membrane stresses 6 59 A * 0 33 N.S

Bending stresses 5 84A J.

0 64 N.S

1 Degrees of freedom, 49/244

2 Degrees of freedom, 5/244

** P < 0.01

for 95 percent of the shells tested lieswithin 2800 to 6500 kg/mm2.

Membrane stresses in meridial andlatitudinal directions were considerably lower than bending stresses nearthe axis of load when the shells werecrushed between flat plates. Sincethe greatest stresses in the shell weredue to bending, failure may well haveoccurred when the material at theinner shell surface reached its ultimate tensile stress as was suggestedby Voisey and Hunt (11). The valuesof maximum stresses in the shell maynot be valid estimates of the ultimatetensile strength of shell material because of interacting bending andmembrane stresses. Nevertheless,failure stresses were found to be considerably higher than those reportedby Rehkugler (5), Sluka et al. (8)and Hammerle and Mohsenin (3).

Analysis of variance (table II)showed highly significant (P ^ 0.01)differences among birds and amongsuccessive eggs from individual birdsfor Young's modulus. Stresses atfailure were highly significantly different when the eggs were consideredon a bird-to-bird basis; however,there was no significant differenceamong successive eggs from the samehen.

SUMMARY

Shell theory was used with physical properties and crushing strengthof 299 eggs to estimate Young's modulus and failure stresses of egg shells.Average values for Young's moduluswere 4571 and 4694 kg/mm2 whenassuming a point load and a loaduniformly distributed over a smallarea respectively within estimatedlimits of ± 25 percent. Membranestresses at failure for point loadsaveraged —6.76 kg/mm2. For loadsdistributed over a small area averagemembrane stresses were —6.67 kg/mm2 and bending stresses were ±46.66 kg/mm2 at the shell apex.

ACKNOWLEDGEMENTS

The authors acknowledge financialsupport for this investigation by theNational Research Council of Canada. The technical assistance of MissL. Robinson is also appreciated.

REFERENCES

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CANADIAN AGRICULTURAL ENGINEERING, VOL. 11, No. 1, MAY 1969

4th Ed. The MacMillan Company, New York, 324-327. 1961.

3. Hammerle, J. R. and N. N. Mohsenin. Determination and Analysesof Failure Stresses in Egg Shells.J. Agric. Eng. Res. 12(1): 13-21.1967.

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8. Sluka, S. J., E. L. Besch and A.H. Smith. Stresses in ImpactedEgg Shells. Trans. Amer. Soc.Agric. Eng. 10(3): 364, 365, 369.1967.

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