J. Zool., Lond. ( A ) (1985) 205, 191-203

The effect of behaviour and body weight on the mechanical design of horns

A N D R E W K I T C H E N E R Biomechanics Group, University of Reading, Whiteknights, Reading, Berks RG6 2AJ

(Accepted I0 April 1984)

(With 4 figures in the text)

A review of the behaviour of bovids and cervids when fighting suggests that it is only the dimensions of the base of horns and antlers which are important mechanically in their design. A fourth power function of the diameter of the base of a horn (the second moment of area (I)) increases linearly with body weight so that there is the same maximum stress in the horn during fighting. This is consistent with the efficient use of materials and is found to be the case for different types of fighting. Sheep and goats fight most forcefully and have a higher ratio of I to body weight than antelopes and deer, which tend to use less forceful wrestling.

Contents Page

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I93 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 I

Introduction

Horns and antlers were first thought to have evolved as a means of defence against predators, but most recent studies show that this is not so. Their evolution as weapons in intraspecific combat is now generally accepted (Geist, 1966; Walther, 1966; Clutton-Brock, 1982; Packer, 1983). However, analysis of the function of antlers has been confused by allometric transformations of arbitrary measures of antler size plotted against body size (Huxley, 1932; Could, 1974; Clutton- Brock et al., 1980).

There has been no analysis of the mechanical design of horns or antlers except in a recent paper by Packer (1983) based on the maximum lateral force that the horn core can withstand in fighting. It distinguishes between two different designs of horn. Males use their horns in intraspecific combat and females use their horns against predators. However, this analysis takes no account of how horns are used in fighting or of interspecific differences in fighting.

Although other parts of horns may be used in less forceful types of fighting and sparring, in most cases horns are engaged most forcefully at their bases (Geist, 1966; Walther, 1966; Leuthold, 1977; Walther, 1978; Spinage, 1982). Therefore, a horn can be approximated to a short, straight, cylindrical beam fixed at one end and free at the other. It is possible to apply engineering beam theory to this model horn after taking into account the different speeds of loading in different types of fighting and the viscoelasticity of the materials used (Wainwright et al., 1976).

If this beam is not to break, the force per unit area (or bending stress) in the beam when loaded must not exceed the breaking stress (or bending strength) of that structure. In long beams, the stress measured in shear is unimportant compared to the stress measured in bending and so it is not

191 0022-5460/85/020191+ 13 $03.00/0 0 1985 The Zoological Society of London

192 ANDREW KITCHENER

usually considered. The problem with short beams is that the shear stress may be more important than the bending stress and may exceed the shear strength before bending strength is reached. If this were the case, it would be pointless considering stresses only in bending.

The full justification for the model used in this paper is given in the appendix. The maximum bending stress is in the order of 20 MPa, whereas the maximum shear stress in a horn during fighting is about 1 MPa. In bending, the ratio of maximum sustainable to maximum calculated stress (safety factor) in the horns of a waterbuck (Kobus ellipsiprymnus) is only about 1.5 compared to a much higher safety factor of greater than three for the horns of a gazelle (Gazella thornsoni). Unfortunately, there appears to be no value for the shear strength of bone in the literature, but dentine is of a similar composition, and has similar mechanical properties, to bone. The maximum shear stress in the horn during fighting is, at the most, about 1 MPa in the case of the waterbuck, which is much lower than the shear strength of dentine (70 + MPa), thereby giving a much higher safety factor in shear (Waters, 1980). On this basis, waterbuck would be more likely to break their horns than gazelles. The incidence of breakage of the horns of waterbuck is much higher (3.5%) than that for gazelles (0.8%) (Packer, 1983). A rough calculation for the largest antelope, the eland (Taurotragus oryx), gives an even lower safety factor consistent with the much higher incidence of horn breakage (6%) recorded for this antelope.

Failure in the horn is also related to critical crack length. The critical crack length of a structure describes how susceptible it is to surface defects, such as cracks which cause local concentrations of stress which may result in the propagation of a crack through the structure. If the crack reaches the length of the critical crack length, it is likely to propagate itself through the material (Gordon, 1978). Critical crack length is easily calculated using the maximum bending stress, stiffness (1.6 GPa) and the work needed to break the material (6.25 kJm-2) (Kitchener, unpubl.). The critical crack length of a waterbuck horn is 3.3 cm which is equal to the radius at the base of the horn, whereas it is more than 15 cm for the gazelle horn. This fits very well with the low safety factor of waterbuck horns. It would not make sense to overdesign a horn, so that it has a small critical crack length, if it has a low safety factor and is just as likely to break. This might result in an expensive, heavy and cumbersome structure. With a low safety factor it is best to have a critical crack length of about the same dimension as the horn.

The forces in fighting are predictable because the behaviour is ritualized, particularly in large species and those that fight most forcefully (Walther, 1966). They fight most frequently with animals of the same age and body size (Geist, 1971; Schaller, 1977; Nievergelt, 1981). Therefore, the force acting on horns during fighting is a function of the body weight, where the function is dependent on the type of fighting. The simplest design solution for a growing animal, or in the evolution of a larger or smaller body size, would be to change the dimensions of the horns to balance the changes in predictable force, but using the minimum amount of material necessary to keep the expected maximum stress during fighting below a certain value.

Beam theory (see appendix) gives the maximum bending stress in a horn during fighting as: F .d . r ’

I’ (1)

where: F = force acting on the horn d = distance between where force is applied and base r’ = radius of the base of the horn I’ = second moment of area of the base of the horn (Gordon, 1978).

T H E M E C H A N I C A L D E S I G N O F H O R N S 193

The length of horn used, d, is small because horns are engaged most forcefully at their bases. It will not change very much for changes in body size and can be regarded as a constant. The circumference at the base of horns does increase with body weight in bovids (e.g. nyala (Trugeluphus angusi) (Tello & Van Gelder, 1975); Wild goat (Cupru aegugrus cretica) (Papageorgiou, 1979); eland (Taurotragus oryx) (Jeffrey & Hanks, 1981)), and it is quite likely that Red deer (Cervus efaphus) antlers show firstly an increase and then a decrease in this dimension, as do antler weight and body weight with age (Huxley, 1932).

Therefore, to balance any changes in force due to growth or the evolution of a different body size, the radius of the base of the horn can be increased to keep the maximum stress of fighting the same. However, changes in radius need only be very small because second moment of area (a fourth power function of the radius) changes more rapidly than radius in equation (l), to result in this balancing effect. For example, if waterbuck and eland horns were to be mechanically equivalent, it would require a doubling of the second moment of area of the waterbuck horns. This needs an increase of less than one fifth of the radius of the waterbuck horns. Therefore, changes in radius are very small compared with the second moment of area, and the radius can be regarded as being virtually a constant. Second moment of area is the principal factor in the design of horns. In order to ensure that the maximum expected stress during fighting is the same, it would be expected that there should be a linear relationship between second moment of area and body weight for individuals within a species. This would also apply to different species, assuming that the type of fighting or forces generated, structure of the horn and mechanical properties of the materials were all similar. If horns were geometrically similar or the radius were still an important independent factor in the design of horns, a linear relationship between second moment of area and body weight would not be expected.

Materials and methods Second moments of area were calculated from a mean radius, which was derived from measurements of the

circumference at the base of the horn given by Ward (1903; 1935) and Best & Best (1977). This method results in an underestimate of I' for horns which depart from a circular cross-section (Table I). A satisfactory method

T A B L E I A comparison between a methodfor actualmeasurement of secondmoment of area (Purslow & Vincent, 1977) and the method used in this paper. Horns which depart

most from circularity have a lower estimate of second moment of area

Actual I (Purslow & Vincent, Estimated I

1977) (1') P Species (m') (m") I

Mouflon 1.09x 10-6 9.05 x 10 0.83

Urial 2 . 8 2 ~ 10 " 2.17 x lo-" 0.77

Feral goat 7.90 x 10 ' 6 . 8 7 ~ 10 0.87

Waterbuck 1-40 x 10 ' 1 . 2 0 ~ 10 0.86

Waterbuck 1.62 x 10 1.46 x 0.90

(Ovis musimon)

(0. orientalis)

(Capra hircus)

(Kohus ellipsiprymnus)

194 A N D R E W K I T C H E N E R

has not been found for determining the cross-sectional shape of a horn without removing it from the skull. This is undesirable in museums and private collections and therefore, actual measurements of second moment of area using the method of Purslow & Vincent (1977) are not possible. However, the method used here will always give a result which is a constant proportion of the actual measure, provided the cross- sectional shape of horns is the same in an adult male population. This method will be found to be unsatisfactory if animals with different cross-sectional shapes have the same linear relationship, and where the proportional measure of I’ is different.

These measurements are biased towards larger-than-average animals. However, this is countered by records of body weights in the general and older literature (e.g. Meinertzhagen, 1938; Dorst & Dandelot, 1972; Walker, 1975; Haltenorth & Diller, 1980) which are also biased towards larger-than-average animals. For example, Meinertzhagen (1938) gave a range of body weights for average-sized adult male eland as 728-942 kg, whereas Sachs (1967) measured more recently a range of 400-573 kg. It seems unlikely that eland have changed so dramatically in only 30 years. Earlier literature (Ward, 1903) records smaller individuals when populations were only just becoming known to game hunters and records were still being established. These were used in preference to later references (Ward, 1935; Best & Best, 1977) which were only used for species and subspecies poorly known to or unknown to science in the earlier reference.

Body weights are usually given in the literature as a range. The mean of the range values was calculated because it was considered that these represent the extremes of a normal distribution of body weights. Body weights were obtained from a number of sources (Table 11) because general references are unreliable (e.g. Dorst & Dandelot, 1972; Walker, 1975; Haltenorth & Diller, 1980). Ifpossible, body weights were taken from recent field studies, especially if they were associated with measurements of horns (e.g. Sachs, 1967; Tello & Van Gelder, 1975; Attwell, 1977; Papageorgiou, 1979). If several references gave mean weights, which were different, a mean of the mean weights was taken as a rough guide to body weight.

T A B L E I 1 The literature used as references for body weights

Tragelaphus angasi T. scriptus T. strepsiceros Bubalus depressicornis B. mindorensis Bos gaurus Bison bonasus Cephalophus maxwelli Damaliscus dorcas Connochaetes taurinus Raphicerus campestris Procapra spp. Oreamnos americanus Capra aegagrus cretica

Caprinae African bovids

Eurasian and Oriental bovids

Tello & Van Gelder, I975 Odendaal, 1977 Nowak & Paradiso, 1983 Nowak & Paradiso, 1983 Anon., 1978 Morris, 1947 Mohr, 1952 Ralls, 1973 Kettlitz, 1967; Keep, 1974 Attwell, 1977 Cohen, unpubl. Nowak & Paradiso, 1983 Rideout & Hoffmann, 1975 Papageorgiou, 1979

Geist, 1971; Schaller, 1977 Oboussier & Schliemann, 1966;

Sachs, 1967; Hofmann, 1973 Prater, 1965; Brink, 1968;

Lekagul& McNeely, 1977

Second moment of area was plotted against body weight on linear axes. Several graphs had to be produced because the measurements spanned several orders of magnitude. Allometric transformations of the data were not found to be useful because they obscured the obvious linear relationships. Regression analysis was carried out on any linear relationships which became evident from the graphs. The regression lines were

T H E M E C H A N I C A L D E S I G N O F H O R N S 195

compared where the slope and intercept for different lines were similar, in order to determine whether the different lines could be combined.

Unfortunately, there is no measure of the forces or energy generated in different types of fighting, which could act as a n independent index of fighting. Therefore, estimates of the energy and force produced in fighting were made from the following equations:

K.E. = J . m . v 2 (2) P . E . = m . g . h (3)

F = m . a (4) where:

K.E. = kinetic energy P.E. = potential energy

F = force m =body weight v = velocity g = acceleration due to gravity (1 0 m s 2 ) h =height a = acceleration (Weber et al., 1965).

Three obviously different types of fighting were compared. Sheep (Ovis) run a t each other a t full speed and collide with considerable force (Geist, 1971). The maximum velocity of a 100 kg bovid would be about 12 m.s-', based on the regression analysis of Alexander et al. (1977). The energy of fighting can be calculated usingequation (2). Goats (Capra) rear up on their hind legs and crash down on an opponent (Schaller, 1977). Assuming that this were a 100 kg body falling from its full body length to its shoulder height above ground with an acceleration due to gravity, the energy of fighting can be calculated from equation (3). Antelopes clash in a rather more gentle manner and wrestle with their horns, so it can be assumed that they arecolliding at no greater than their average velocity of 5 m.s-' (Pennycuick, 1975). In the latter 2 examples, there will be an overestimate of energy because only a proportion of the body weight is used in fighting. Force was calculated from equation (4) assuming deceleration in 0.1 sec.

Results and discussion There is a linear relationship between second moment of area (1') of the base of the horn and

body weight within a species, e.g. the nyala (Tragelaphus angasi) (Fig. 1). This supports the idea that in a growing animal the circumference of the horn grows just enough to balance the increase in body weight, so that the maximum stress in the horn during fighting is the same. If horns were to show only geometric similarity there would be an exponential relationship between I' and body weight, as indicated by the curve in Fig. 1 .

The same applies to different species, where the same design solution is used in conjunction with the evolution of a larger or smaller body size. Up to eight different linear relationships are evident (Figs 2-4; Table 111). The gradients are sufficiently different to make a comparison of linear regressions unnecessary, except in the case of the goats (Capra) (group no. 6), hartebeest (Alcelaphus and Darnaliscus) (group no. 4) and the cattle (Bos and Bubalis) (group no. 5). However, when these lines are compared, it is not found to be possible to combine the data from any two or all three lines (Table 111). Therefore, each group may be using its own unique horn design to resist different maximal stresses in its horns, or different groups may generate different forces during fighting, so that the maximum stress in each type of horn is similar.

Many authors have attempted to categorize and describe the fighting of bovids (e.g. Hediger, 1955; Geist, 1966; Walther, 1966; Ewer, 1968; Eibl-Eibesfeldt, 1975; Kurt, 1977; Leuthold, 1977).

196

gr A N D R E W K I T C H E N E R

I /

/ 0

81 I I 1 1 I J 80 100 120 Body weight (kg)

FIG. I . The relationship between second moment of area and body weight for the nyala (Tragelaphus angasi) data from Tello & Van Gelder, 1975). The dashed line shows the relationship between second moment of area and body weight if horns were geometrically similar.

However, it is not easy to generalize because bovids may display a variety of fighting techniques (Geist, 1971; Schaller, 1977; Walther, 1978). The commonest types offighting are as follows: sheep ram into each other from some distance apart (Geist, 1971); goats and ibex rear up on their hind legs and crash down on their opponents (Schaller, 1977); hartebeest and gnus (Connochaetes) clash repeatedly and wrestle on their metacarpals (Estes, 1969; Gosling, 1974; Leuthold, 1977). The majority of antelopes clash initially and then wrestle with their horns interlocked or crossed over at

TABLE I I I Regression analysis of the linear relationships between second moment of area and body weight. AN slopes are signiJicantly different from each other ( P i 0.05) and from zero

( P < 0.05). The sample size is n and the correlation coeficient is r

Range of body weights

Group n Slope Intercept (kg) r

1. Dwarf

2. Oryx, goat- antelopes,

antelopes 19 2.89 x 10 lo -5.95 x lo-" 2-25 0.9 I

deer 26 1.91 x Itg -3.48 x I t a 25-220 0.90 3. Antelopes 54 6 . 1 4 ~ le9 -9.13 x 10 a 5 - 700 0.99 4. Hartebeest 10 3.36 x I t a -3.10 x 100-250 0.84

6. Goats, gnus 17 3.98 x I t 8 - 1.30 x It6 30-300 0.97 7. African buffalo 3 1.16 x It' -2.79 x I t 5 250-800 0.99 8. Sheep 20 1.92 x 1 t 7 -7.30 x 10 30-200 0.96

5. Cattle 6 3.67~10-* - 9 . 1 O x l t 6 200-1100 0.98

d

1

m m 0 c 12

T H E M E C H A N I C A L DESIGN O F H O R N S

O0 0

0

c

E 8/

197

Body weight (kg)

FIG. 2. The relationship between second moment of area and body weight for dwarf antelopes (A) and duikers (0). The latter fit on to the ‘antelope’ line (Fig. 3).

0 l2 r H

O O H 0 0

c C O0 E 0

P A 1

O O H 8- E 0 0 c

c C O0

E E 4- 0 8.” 00

s o om ..“e - 0 0.

0 I I I I 1 I I I 30 90 150 210 270

0

0

0

0

0 0 .

0 O.

0

Body weight (kg)

FIG. 3. The relationship between second moment of area and body weight for antelopes (0) and deer, goat-antelopes and oryx (0). The beginning of the ‘goat’ (m) and ‘sheep’ (0) lines are also apparent.

their bases (Walther, 1958; 1966; 1978; Leuthold, 1977). African buffaloes clash violently after charging at each other, whereas other cattle tend to wrestle more (Sinclair, 1977). Bovids with short horns tend to use lateral displays of the body in agonistic encounters and rarely use their horns, except in attacks against the body of an opponent (Geist, 1966; Leuthold, 1977; Dittrich &

198 A N D R E W KITCHENER

8

H

Body weight (kg)

FIG. 4. The relationship between second moment of area and body weight for goats (m) and sheep (0). The sheep-like Caucasian ibexes (0) have an intermediate design of horn.

Boer, 1980). These categories correspond largely with the groups which have horns of different design.

However, apart from this rather subjective categorization, there is no numerical score of different types of fighting. Estimates of energy and force generated in fighting are given in Table IV. They are only a rough guide to the maximum energy and force produced by a single animal in different types of fighting, but they do correlate with the magnitude of the slopes in the regression analysis.

T A B L E I V Maximum theoretical energies and forces produced by three

different boviak in their respective types ojjghting

Speed Height Energy Force ( m d ) (m) @J) (kN)

~

Sheep 12 - 7.20 12

Antelope 5 - 1.25 5 ~ Goat 1.8 1.80 6

Therefore, each line can be seen to represent animals which produce different maximal forces by fighting differently. It is possible that all the designs of horns have similar maximal stresses in them during fighting. Some of the lines show a strong phylogenetic influence and it is not surprising that the design of horn and type of fighting are influenced in this way. However, this is not exclusively so.

The ‘sheep’ line (8) includes all the sheep which fight similarly. It includes, however, the Bharal (Pseudois) which is considered to be a goat, but which fights like a sheep as well (Schaller, 1977).

T H E M E C H A N I C A L D E S I G N O F H O R N S 199

The ‘goat’ line (6) includes the goats, a few hartebeest and the gnus. The fighting of goats contrasts markedly with that of hartebeest and gnus and yet they have a similar design of horns, thereby suggesting that the relative forces are similar. The Caucasian ibexes (Capra caucasica and C. cylindricornis) are said to have a sheep-like morphology (Geist, 1971) and they appear to have a horn design and perhaps a fighting technique which is intermediate between sheep and goats.

The ‘hartebeest’ line (4) has few data points, all of which are considered to be unreliable. Where reliable data have been available (Kettlitz, 1967; Sachs, 1967), the points have fitted on to the ‘antelope’ line (3 ) . The ‘cattle’ line (5 ) , although distinct, also suffers from having too few data. Bison (Bison) fall well below this line because, although they fight similarly to cattle, they use the thick hair on their heads to absorb the shock of impact instead of their horns (Geist, 1966).

The ‘antelope’ line (3) includes bovids with similar horn design and type of fighting, but the morphology of horns and phylogeny are quite diverse. This group includes the forest duikers (Cephalophus), the waterbucks (Kobus and Redunca), the impala (Aepyceros), most of the bushbucks (Tragelaphus and Taurotragus), some hartebeest (Damaliscus), the gazelles (Gazella and Procapra) and other gazelle-like forms (Ammodorcas, Antilope, Antidorcas, Saiga and Pantholops). Most of these species have complex horns, are territorial or maintain harems and exhibit sexual dimorphism (Leuthold, 1977; Spinage, 1982). Very few antelopes do not fit this line. These include the Greater kudu (Tragelaphus strepsiceros), the Bongo (T . eurycerus) and the Sable antelope (Hippotragus niger) which have a horn design comparable with that of hartebeest. The Sable antelope fights fiercely on its metacarpals by clashing its horns. The horns do not always clash at the base, so that a greater force acts with a greater bending moment resulting in a stress a t the base of the horn greater than might be expected (Kurt, 1977; Leuthold, 1977). The Nilgai (Boselaphus tragocamelus) has a horn design poorer than expected. It has very short horns which could be damaging in combat. Therefore, it uses an elaborate lateral display of the body and neck- fighting in agonistic encounters (Walther, 1958; Sankhala, 1977).

The Oryx (Oryx), Addax (Addax nasomaculatus), Dama gazelle (Gazella duma), goat-antelopes (Rupicaprinae) and deer (Cervidae) seem to fall on a common line (2) (Fig. 3 ) which represents a horn design less resistant to bending than that of the antelopes. The bovids on this line usually have simple horns, show little territoriality and are not greatly sexually dimorphic. They seem to rely on lateral displays of the body and rarely use their horns. If the horns are used, they are used directly against the body of an opponent where the shock of impact will be absorbed by the soft tissues of the opponent (Geist, 1966; Leuthold, 1977). The oryx (Oryxgazella callotis) does fight similarly to other antelopes and it seems paradoxical that they should have an inferior horn design. However, they are not territorial and form fairly loose, mixed herds where the frequency of very forceful fights may be low (Walther, 1978).

It is probably coincidental that deer should fall on this line. Their antlers are made of a different material and are different structures. Unfortunately, unlike horns, antlers cannot grow from their bases (Goss, 1983) and so, to maintain a mechanically equivalent structure with growth of body weight, antlers must be cast and regrown.

The bovids with horns which are least resistant to bending include the dwarf antelopes (Neotraginae), bush duikers (Sylvicapra), four-horned antelope (Tetracerus), and Rhebok (Peleu) which fall on a common line (1). Very little is known about fighting in this group, but horn-to-horn contact is unlikely. Agonistic encounters are probably settled by displays of the body (Leuthold, 1977; Dittrich & Boer, 1980).

The process of domestication in mammals has involved the artificial selection of placid, juvenile-like individuals (Clutton-Brock, 1981). There has been a reduction in size or even loss of

200 A N D R E W K I T C H E N E R

horns in some sheep (Ovis aries) and cattle (Bos taurus) breeds, except where there has been specific selection for large or decorative horns. Domestic cattle fall well below the wild cattle line in the analysis above, suggesting that behaviour and horn size are in some way linked. Placid domestic animals probably do not fight as forcefully or as frequently as their wild ancestors and, as a result, horn design has suffered. The Gayal (Bosfrontalis) is a semi-domesticated form of the Gaur (Bos gaurus), a wild ox from southern Asia, and its horns are much closer to those of a truly wild species, indicating natural selection for a more efficient fighting organ.

Therefore, a link has been established between an aspect of morphology and behaviour by means of a mechanical analysis. The design of horns is directly related to how big the owner is and how they are used. The most forceful fighters have horns which are most resistant to bending, but for each type of fighting the maximum stress is the same, regardless of how big an animal is.

I would like to thank Prof. Anne Alexander of the University of Natal, Prof. R. McNeill Alexander of the University of Leeds and Dr Ortie Bourquin, Principal Scientific Officer of the Natal Parks Board, for supplying references for body weights. I would also like to thank Dr Juliet Clutton-Brock and Mr Iain Bishop of the British Museum (Natural History) and Mr Richard Cindery of Whipsnade Zoo for supplying me with horns. Finally, I would especially like to thank my supervisor, Dr Julian Vincent of the Biomechanics Group, University of Reading, for discussion and constructive criticism of the ideas in this paper. Mechanical tests were carried out on an Instron tensile testing machine, the funds for which were provided by the Royal Society.

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Appendix When beams experience bending, two types of stresses are set up within the material. Bending stresses

(consisting of tensile and compressive stresses) reach a maximum on the tension and compression faces of the beam, whereas shear stresses reach a maximum along the central axis of the beam. Therefore, a beam can fail,

202 A N D R E W K I T C H E N E R

either because the bending strength is exceeded by the bending stress, or because the shear strength is exceeded by the shear stress. Usually the shear stress in beams is unimportant compared with the bending stress, but in short beams the shear stresses may become relatively more important, so that they must be considered as well.

Using estimates of the maximum forces acting on horns during fighting, calculations were made of the maximum bending stress, maximum shear stress, safety factors in bending and shear (the strength or rupture modulus divided by the shear or bending stresses) and maximum deflection of the horns at the point where they are engaged (Gordon, 1978). This was done for the horns of a waterbuck (Kobus ellipsiprymnus) (body weight of 200 kg) and the horns of a Thomson’s gazelle (Gazellu thomsoni) (18 kg). Their horns are roughly circular in cross-section, thereby simplifying calculations. Although horns are composite structures, being made up of bone and keratin, the stiffness in bending (1.6 GPa) and rupture modulus in bending (20 MPa) are about the same for each material at the base of the horn where the maximum stresses occur during fighting (Kitchener, unpubl.). Stiffness is a measure of how much a material deflects due to a load and rupture modulus is a measure of the highest stress that the material can withstand before failing, i.e. a measure of strength (Gordon, 1978). This means that the calculations of stresses and deflections can be made on an apparently homogeneous beam.

The maximum bending stress in the horn of a waterbuck is about 14 MPa which is only just exceeded by the rupture modulus of the horn. The maximum shear stress is only about 1 MPa, which is very small compared with theshearstrengthofdentine(70 + MPa)(Waters, 1980). Dentinehassimilarmechanicalproperties to, and is chemically similar in composition to, bone for which there is apparently no measure of shear strength in the literature. Therefore, the safety factor for the bending stress (1.45) is much lower than that for the shear stress (70+), so that only bending stresses need to be considered in the design of horns. The safety factors of bending stress (3) and shear stress (300 +) are much higher for the horns ofa gazelle, but bendingstress still has the lower safety factor and remains important. Maximum deflection of the horns at the point where they are engaged in fighting is less than 2 mm for both waterbuck and gazelle horns. This deflection would not be too great for the horns to be effective in fighting and this does not affect the assumptions made in using the model.

Having shown that bending stresses are more important than shear stresses in horns during fighting, and that the bending strength is the most important consideration in their design, it is now allowable to apply beam theory to the short, straight cylindrical beam which represents a model horn. Beam theory gives the stress in a cylindrical beam as follows:

M . r

1 u=-

where: u = bending stress

r = radius of the beam I = second moment of area.

M = bending moment

The bending moment is the product of the force acting on the beam and the distance between the point where the force is applied, and the point where the stress is measured. The second moment of area describes how the material is distributed in a cross-section of the beam and, hence, how well it resists bending. It is given by:

The maximum equal to:

stress (omax) that can occur in a horn is at base when it is loaded at the tip. The stress is

F.d.r‘

I’

T H E M E C H A N I C A L DESIGN OF H O R N S

where: F = force acting on the horn d = distance between where force is applied and base r’ = radius of the base of the horn I’ = second moment of area of the base of the horn (Gordon, 1978).

203