Shape optimization in exoskeletons and endoskeletons: a biomechanics analysis

Abstract

This paper addresses the question of strength and mechanical failure in exoskeletons and endoskeletons. We developed a new, more sophisticated model to predict failure in bones and other limb segments, modelled as hollow tubes of radius r and thickness t. Five failure modes were considered: transverse fracture; buckling (of three different kinds) and longitudinal splitting. We also considered interactions between failure modes. We tested the hypothesis that evolutionary adaptation tends towards an optimum value of r/t, this being the value which gives the highest strength (i.e. load-carrying capacity) for a given weight. We analysed two examples of arthropod exoskeletons: the crab merus and the locust tibia, using data from the literature and estimating the stresses during typical activities. In both cases, the optimum r/t value for bending was found to be different from that for axial compression. We found that the crab merus experiences similar levels of bending and compression in vivo and that its r/t value represents an ideal compromise to resist these two types of loading. The locust tibia, however, is loaded almost exclusively in bending and was found to be optimized for this loading mode. Vertebrate long bones were found to be far from optimal, having much lower r/t values than predicted, and in this respect our conclusions differ from those of previous workers. We conclude that our theoretical model, though it has some limitations, is useful for investigating evolutionary development of skeletal form in exoskeletons and endoskeletons.

1. Introduction

In many organisms, the principal load-bearing structures take the form of long, hollow tubes. In what follows, we will use the word ‘bone’ in a general sense to describe these structures, including the limb segments of exoskeletons as well as endoskeletons. Among these are the exoskeletal bones of arthropods, including crustaceans and insects, and the limb bones of humans and other vertebrates. Found in more than 80 per cent of all known species, the arthropod's exoskeleton ‘concept’ has been very successful in evolutionary terms and is also much older that the endoskeleton concept found in vertebrates. Both types of bones must be strong enough to withstand applied forces, and their morphogenesis and maintenance represents a significant metabolic cost to the animal; so one might assume that evolutionary selection has operated to optimize the use of bone material. Looking at the different evolutionary ‘approach’ of load-bearing structures in arthropods and vertebrates prompts the question: ‘is it better to have compact, thick-walled internal bones or broad, thin-walled external ones?’

Given a bone of a certain length, L (which is largely determined by other factors [1]), one can postulate that there will be an optimal shape for the cross section, that allows the bone to withstand applied forces without failure while minimizing the bone's total volume and weight. Thus, we may ask whether there has been an evolutionary adaptation towards this optimal shape.

There has been relatively little work done in the past to address this question. Currey and Alexander investigated the matter in some detail for the case of endoskeletons: their work is well summarized in Currey's book [2]. Currey also made a comparison of failure conditions for endoskeletons and exoskeletons [3]. More recently, Wegst & Ashby [4] considered a similar problem: shape optimization of plant stems. These previous studies had some limitations: Wegst & Ashby considered only bending, not axial loading. Currey did not consider the anisotropic mechanical properties of bone. Neither study considered that certain failure modes may combine synergistically in such a way that failure occurs at a lower applied load than predicted for either mechanism individually. In addition to mammalian bone, Currey considered arthropod limbs and sea urchin spines, but was limited by the available experimental data for those cases, while Wegst & Ashby considered only data from bamboo. In recent years, experimental data have become available, which are sufficiently comprehensive to allow a detailed study of two particular arthropods: the blue crab and the locust. These data will be described in detail below.

The aims of the present study were:

• — to develop a more sophisticated theoretical model considering both bending and axial loading, a wider range of failure modes and the possibility of interactions between failure modes;

• — to carry out stress analysis to determine the nature of applied forces on two particular arthropod limbs: the locust tibia and blue crab merus; and

• — given the findings of the two studies above, to comment on whether these limb bones, and also those of vertebrates, have evolved to the optimum possible shapes.

1.1. Theoretical model

A typical bone will be modelled as a straight, hollow tube of length L, having a constant, circular cross section of outer radius r and wall thickness t. Forces—causing either axial compression or bending—are assumed to act on the ends of the tube. This model obviously contains many simplifications, which will be discussed in detail below. The material will in general be anisotropic: we assume transverse isotropy with a Young's modulus of E_L in the longitudinal direction and E_T in the transverse direction, and corresponding strength (i.e. stress to failure) values σ_L and σ_T.

1.2. Failure modes

We can identify five possible failure modes (figure 1), as follows:

Figure 1.

Different failure modes of a long, thin-walled tube. Numbers correspond to the descriptions in the text: (1) fracture; (2a) Euler buckling; (2b) local buckling; (2c) buckling by ovalization; (3) splitting. ‘In bending, local buckling is predicted to occur at a slightly lower bending moment than ovalization buckling’.

1.2.1. Fracture

Failure by fracture will occur if the stress at any point in the bone exceeds its strength. This may occur either by yielding or by brittle fracture. For end loading, the stress will be highest in the longitudinal direction; so it is the longitudinal strength (σ_L) that is relevant here. In general, this may be different depending on whether the local stress is tensile or compressive. For axial loading with a force F on a tube having a cross section of area A, failure is predicted to occur when

1.1

In bending, failure by fracture will occur at a bending moment M_f given by

1.2

Here I is the second moment of area, which is a function of the radius r and thickness t of the tube, given by (π/4) × (r⁴−(r − t)⁴). Provided r is much greater than t, we can write this in a simplified form:

1.3

1.2.2. Buckling

Buckling failures occur when elastic deflections become unstable; the only material parameters that are relevant are the elastic moduli (E_L and E_T) and Poisson's ratio υ. Buckling occurs in relatively thin, flexible structures loaded in compression or bending. It is a complex phenomenon that is notoriously difficult to predict accurately. For our purposes, we can identify three types of buckling (figure 1):

1.2.2.1. Euler buckling

This occurs in slender columns loaded in axial compression. The entire column becomes unstable and deflects laterally at a critical stress given by

1.4

This equation applies when the two ends of the column are free to rotate. Following Currey [3], we have assumed that this is the most appropriate condition for bones; if the ends are more restrained, this will tend to increase the failure stress.

1.2.2.2. Local buckling

A thin tube loaded in compression may develop wrinkles owing to local instabilities. A good example is an aluminium drinks can that is pressed on its ends. This phenomenon is also elastic buckling, but differs from Euler buckling in that it does not depend on the length of the tube, but only on the ratio r/t. Accurate prediction of the stress needed to cause local buckling is difficult: the most commonly used equation [5] is:

1.5

However, it is generally accepted that the stress predicted by this equation is too large (when compared with experimental results) by about 40–60% [6]; so a reduction of 50 per cent was incorporated into our predictions.

1.2.2.3. Buckling by ovalization

Brazier noted that buckling can occur in a thin-walled tube subjected to bending because the cross-section changes shape [7]. An initially circular tube becomes elliptical, reducing its I value and thus potentially precipitating unstable elastic deformation (figure 1). The critical value of the bending moment at which this occurs is:

1.6

However, during bending, one side of the tube will be in compression and therefore also at risk of local buckling, which is predicted to occur at a slightly lower bending moment, corresponding to equation (1.6) with the factor 0.987 reduced to 0.939 [8]. Following Wegst & Ashby [4], we modified the above equations in the case of bending to take account of anisotropy, replacing E with .

1.2.3. Splitting

The deformation of the tube during loading gives rise to circumferential stresses and strains, which, though they are smaller than the longitudinal ones, can cause failure if the material is sufficiently anisotropic, with low transverse strength σ_T. For pure bending in a tube made from anisotropic material, the value of M at which splitting occurs was found [4] to be

1.7

1.3. Interactions between failure mechanisms

Interaction may occur between these different failure mechanisms, under conditions where the critical loads are similar for more than one mechanism. This problem is well known to engineers, especially for the case of Euler bucking and yielding in metallic materials. The load-bearing capacity of the structure close to the buckling/yielding transition has been found in practice to be considerably lower than that predicted by either of two equations separately [6]. As a result, empirical equations have been developed to make more realistic predictions. The simplest form suitable for our purposes calculates the total failure force F_t from the individual failure forces for buckling (F_b) and fracture (F_f) as follows:

1.8

This equation is generally regarded as being conservative, i.e. it provides a realistic lower bound to the predicted failure force [6]. It alters the predicted strength (i.e. failure force) but not the prediction for the optimum r/t ratio. It is well established in engineering situations dealing with metallic materials but has not previously been applied to bone. It seems likely that similar interactions will occur in bone: the general principle here is that when the stress is close to the fracture stress, nonlinear deformation processes such as plastic deformation and microdamage will occur, making the material more flexible, effectively lowering E and thus precipitating the buckling instability. This nonlinear material behaviour is known to occur in bone, both the mineralized collagen of mammalian bone and the chitin-based bone of arthropods. Therefore, we propose that equation (1.8) can be applied to predict lower-bound behaviour for interactions between fracture and buckling, including all three different mechanisms of buckling. However, interactions of this kind would not be likely to occur between the failure modes of fracture and splitting (because they involve deformation and damage on orthogonal planes) or between the different forms of buckling.

2. Experimental data

To test the predictions of our theoretical model, we used data from the literature and our own recent experiments. Limbs from two representative and relatively well-studied arthropod species were selected: the merus of the blue crab Callinectes sapidus and the tibia of the locust Schistocerca gregaria. The merus is the fourth most distal segment in the crab's multi-segmented limb: we analysed the walking legs, in which the merus is the largest segment. The locust's tibia has essentially the same anatomical position as the tibia in vertebrates, connecting the foot to the femur: we analysed the hind legs, which are the largest legs and are involved in jumping. The exoskeletons of both species are made of cuticle, a biological composite material consisting of chitin microfibrils embedded in protein layers. Unlike insect cuticle however, crustacean cuticle can contain a ceramic phase in the form of calcium salts [9]. Representing vertebrate bones, we selected the human femur, whose structure and properties have been previously studied in detail [10]. We required data on material properties and geometry for input into our predictions. In order to test our results, we needed measurements of actual r/t values for comparison to our predicted optimum values, and data on the measured load-carrying capacity of these bones when tested in bending and axial compression.

Table 1 summarizes the data used; in some cases, the appropriate parameters were not available and had to be estimated (as discussed below), in which case the value is given in brackets. The merus of the blue crab was tested by Hahn & LaBarbera [11], who loaded these limbs to failure in cantilever bending. Young's modulus and strength of material samples from the merus of this same species were measured in a different study [12], which found values of 318 and 9.9 MPa, respectively. These properties were measured in the transverse direction only; however, there is evidence to suggest that the material is isotropic. Hahn & LaBarbera also studied another limb segment, the dactylus. Observing material sections under polarized light, they found that the dactylus contained highly oriented fibres running longitudinally, while material from the merus (which they found to have six times lower strength) exhibited no preferential fibre alignment. Another study [13] found that material from the carapace of the blue crab (which has a similar strength to material from the merus) showed no anisotropy. Consequently, we have assumed the material to be isotropic in our predictions.

Table 1.

Experimental results taken from the literature: material properties, bone geometry and failure test results from whole bones. Numbers in brackets are estimates.

	crab merus	locust tibia	human femur
material properties
EL (GPa)	(0.318)	3.05	17.9
ET (GPa)	0.318	(1.5)	10.1
σL (MPa)	(9.9)	95	135a
σT (MPa)	9.9	(47)	53
whole bone failure stresses
axial (MPa)		10.7
bending (MPa)	6.0	72.1
geometry
L (mm)	25	20	450
A (mm2)	15.25	0.193	603
r/t	8.3	11.04	2

^aA higher value of 150 MPa was used for the longitudinal strength in compression.

Data on the locust tibia come mostly from our own experiments, recently published elsewhere [14]. We conducted axial compression and cantilever bending tests of whole tibiae cut off close to the joints. We calculated the longitudinal Young's modulus from the results of the bending tests. We did not measure longitudinal strength: two values of longitudinal tensile strength can be found in the literature [15,16] from which we used a value of 95 MPa. No results exist for transverse material properties, owing to the practical difficulties involved. We expected the material to show significant anisotropy; so we chose values for the transverse properties that gave the largest degree of anisotropy that would not result in a prediction of splitting failure, because we did not observe this failure mode experimentally. This resulted in a value of 2 for the longitudinal/transverse ratios of both stiffness and strength, which is in fact similar to the measured anisotropy ratios for vertebrate bone.

Much data are available for bones from humans and other mammals. We chose to use the data of Reilly & Burstein [17], whose careful and systematic study has been frequently cited, and provides results for longitudinal and transverse properties from the same human bone samples, giving an accurate estimate of anisotropy. Skeletal materials may have different strengths in tension and compression. For the arthropod materials, no data were available; so we assumed no difference. It is well known that bone is somewhat stronger in compression; so, guided by the wealth of data in Currey [2], we increased Reilly & Burstein's value of σ_L from 135 to 150 MPa for our analysis of axial loading.

3. Results

Different values of r/t will give rise to different failure modes, described by one or the other of the above equations. It is convenient to visualize this by plotting graphs of the load-carrying capacity of the bone as a function of r/t. For the case of axial tension, an appropriate parameter to describe load-bearing capacity is the stress at failure, because this gives the failure force for a constant area A, and therefore for a constant bone weight, assuming a given length. The length L needs to be specified in the particular case because it affects the Euler buckling stress but not the other failure stresses. For the case of bending, a convenient parameter is M/A^3/2 [4], as this gives the bending moment at failure for constant weight.

3.1. Example 1: the crab merus

Figure 2 shows the results of our predictions for the failure of the blue crab merus. When loaded in bending (the upper graph), the line corresponding to fracture increases with increasing r/t, reflecting the fact that increasing the bone's radius will tend to increase its second moment of area, I, even though t must be reduced to maintain constant volume. In contrast, the prediction line for buckling shows a decrease with increasing r/t, reflecting the instability of thin-walled structures. The optimum value of r/t occurs at the point where these two lines cross, and has a value of 9.96. The prediction line for splitting is not shown, as this failure mode was not predicted to occur for any value of r/t. This is to be expected, given the isotropic material properties.

Figure 2.

(a) Predictions for the blue crab merus loaded in bending and axial (b) compression. The vertical lines illustrate the optimum r/t ratio to resist axial compression (4.3) and bending (9.96). The experimental data point [11] shows that the actual r/t value lies between these two predictions. The experimental bending strength is close to the predicted value: no data are available for axial compressive failure. (Online version in colour.)

Also shown in the figure is a data point corresponding to the experimentally measured r/t value and failure load for the blue crab merus [11] loaded in cantilever bending. The average value of r/t measured by these workers, 8.3, is quite close to our predicted optimum value of 9.96. Hahn & LaBarbera reported that most of their samples failed by buckling, while a few failed by fracture, implying conditions close to the buckling/fracture transition. The experimental failure load lies just above our combined fracture/buckling prediction line, which we expected would be a lower bound value. When converted to a stress at failure, our predicted value is 5.6 MPa, very close to the experimental result of 6 MPa. The curve is very flat in this region; so the failure stress at the optimum r/t of 9.96 is only predicted to be 1 per cent higher than it is at r/t = 8.3.

Figure 2 also shows predictions for axial compressive loading for the crab merus (on the lower graph). Three failure modes are represented: Euler buckling, which tends to increase with r/t reflecting the increase in I; local buckling, which decreases with r/t, and fracture, which here takes a constant value independent of r/t. Combined failure lines are drawn for the fracture/Euler bucking and fracture/local buckling interactions. The optimum r/t is found at the intersection point between these two lines, which occurs at a value of 4.3, at a stress of 7 MPa; at this point, failure will be dominated by the fracture mode because both buckling modes occur at much higher stresses.

The optimum values for r/t for the two loading cases are very different: 9.96 and 4.3. However, the predicted stress to failure for the two cases is not so very different. Increasing r/t from 4.3 to 8.3 reduces the axial failure stress slightly, from 7 to 5.6 MPa, which is the same as the failure stress in bending.

From this analysis, it emerges that the experimentally measured value of 8.3 represents a good compromise for a structure that needs to resist both bending and axial loading. The resistance to bending loads is as high as possible; the resistance to axial loads is somewhat less than the best possible value attainable, but is equal to the bending resistance. Lower values of r/t would cause failure to occur preferentially in bending, while higher values would encourage axial failure.

3.2. Example 2: the locust tibia

Figure 3 presents results for the locust tibia using the same approach as outlined above for the crab merus. The main differences in this case are the relatively high longitudinal strength of the material and the long, slender shape of the limb, which encourages buckling. For axial loading, the optimum r/t value is very high at 29, while for bending it is 7.2, a reverse of the situation for the crab merus where the optimum r/t for axial loading was less than that for bending. Our experimental value for the locust tibia, r/t = 11.04 [14] lies much closer to the prediction for bending than that for axial loading. At this r/t value, the failure stress is predicted to be 12.3 MPa for axial loading and more than three times higher, at 38 MPa, for bending. These predictions imply that the locust tibia is optimized for bending, not for axial loading.

Figure 3.

(a) Predictions and experimental results for the locust tibia loaded in bending and (b) axial compression. The vertical lines illustrate the optimum r/t ratio to resist axial compression (29) and bending (7.2). The experimental data points show that the actual r/t ratio lies closer to the bending prediction than that for compression. The axial failure stress is well predicted; in bending, the tibia is somewhat stronger than predicted. (Online version in colour.)

Our experimental data points (also shown in figure 3) merit some further comment. In axial compression, the experimental failure stress was 10.7 MPa, which is close to the predicted value of 12.3 MPa. In bending, however the experimental failure stress was 72 MPa, which is considerably higher than the predicted value of 38 MPa. One reason for this is that the shape of the cross section is not circular, as we assume in our model. As figure 4 shows, it is distinctly elliptical in shape, with the long axis of the ellipse coinciding with the direction of bending in our experiments, which is also the primary bending direction in vivo. This did not greatly alter the I value: its measured value was 0.038 mm⁴, while if the cross section had been circular, it would have been 0.032 mm⁴. However, this change in shape may be effective in delaying ovalization: when ovalization occurs during loading, it distorts the tube in the opposite direction; so in this tibia, bending will initially act to make the section more circular in shape. If we were to allow for this in the model, the prediction line for buckling on figure 3 would move up, giving a value for the optimal r/t much closer to the experimental value.

Figure 4.

(a) Simplified cross section of the locust tibia taken from a MicroCT scan. The vertical direction in this picture corresponds to the dorsoventral direction in which bending normally occurs in vivo. (b) Illustration of transverse cut along the lateral-medial plane, showing slight curvature. (c) A side view of the tibia shows no noticable curvature, illustrating that the leg can be modelled as a simple, straight thin-walled tube. (Online version in colour.)

In conclusion, the locust tibia has an r/t value that is close to optimal for resisting bending forces, and it appears to have adjusted its detailed shape to improve resistance to ovalization during bending in the dorsal/ventral plane.

3.3. Example 3: vertebrate bone

Figure 5 shows predictions using typical material properties for human cortical bone and the dimensions of the human femur (from table 1). In bending, the predictions for splitting and buckling were very similar in the region where the lines fall below the line for fracture; so it is not clear which failure mode would occur. In any case, an optimum r/t value of about 30 can be predicted. For axial compression, the optimum value was 11.8. There was also some difference between the predicted failure stresses: 114 MPa for axial loading but only 67.5 MPa for bending. So, while the absolute numbers are different, the overall picture here is similar to that for the crab merus. We might therefore expect a value of r/t lying between these two optima, but closer to the bending solution, in the region 20–25. As with the other examples, the value for axial loading is strongly influenced by the bone length, or more accurately by the ratio between the length and the I value of the cross section. If we had chosen a more slender bone, then the optimum r/t value would have been higher: the value given here is probably one of the lowest likely to occur for mammalian bones.

Figure 5.

(a) Predictions for the human femur in loaded in bending and (b) axial compression. The vertical lines illustrate the optimum r/t ratio to resist axial compression (11.8) and bending (30). Experimental values for vertebrate long bones [2] show a typical value for r/t of 2, much less than these predicted values. (Online version in colour.)

In any case, actual measured r/t values are much lower. Currey reports data on long bones from many different species [2]; the most common value for land-based mammals is r/t = 2, with most bones lying between 1.5 and 3. For birds, a wide range of values from 2 to 8 were found. The only higher values recorded were those for the extinct pterodactyl Pteranodon, which had values from 10 to 20. According to our predictions, an optimized femur, i.e. one having the optimum r/t value, would be 2.8 times stronger in bending and 1.3 times stronger in axial compression than a femur having the typical r/t value of 2.

3.4. Stress analysis

The work described earlier has shown that the optimal shape for a bone will be different, depending on whether the bone is subjected to axial loading, bending or a combination of the two. To complete the picture, therefore, we must ask the question: what types of loading do these bones experience during normal use?

A lot of work has been done to estimate the loadings on vertebrate bones, especially human long bones. Despite this, there is still considerable controversy about the relative contributions of bending and axial compression to the overall stress. Early studies tended to conclude that bending was the dominant type of loading [18]. A previous study from our own research group found approximately equal contributions from bending and axial loading in the human tibia [19]. The main difficulty in making these predictions is to allow for the role of the muscles. Forces in the muscles act to stabilize the joints at the ends of the bones, preventing unwanted rotation. Muscles can act to reduce bending stresses, but in so doing, they will induce compression along the bone axis. A recent theoretical study constructed a detailed analysis using a computer model in which muscle loads were adjusted systematically to try to minimize the bending component in a human bone [20]. Even in this idealized study, it was not possible to achieve perfect axial compression: about 22 per cent of the stress was still caused by bending.

To our knowledge, no such studies have been carried out for arthropod exoskeletons. So, as part of the present work, we conducted some preliminary analyses using the same principles as applied to vertebrate bones in earlier studies. For convenience, the details of these analyses are given in the appendix below. The results are as follows.

The locust tibia experiences its most severe loading during jumping. Bennet-Clark [21] conducted a very detailed study of the relevant anatomy and measured the muscle forces involved; Sutton & Burrows [22] investigated the mechanics of jumping in detail. From these studies, we estimated the ground reaction force and calculated that this would cause a bending stress in the tibia of 42.2 MPa, which is approximately half the bending strength of this material (appendix A). Although the muscle forces are considerable, they act to cause compression on the femur, not in the tibia. Some axial compressive stresses arise in the tibia from the ground reaction force: we estimated a very small value of 0.25 MPa.

For the blue crab merus, there has been no detailed study equivalent to that of Bennet-Clark's for the locust tibia. We carried out an analysis of the stresses during lateral walking in water (appendix A), when muscles in the legs are used to overcome the fluid drag force on the animal's body. We estimated a bending stress of 0.91 MPa and an axial compressive stress of 0.46 MPa. So, in contrast to the locust tibia, the bending and axial stresses are of the same order of magnitude. The total stress, 1.37 MPa, is a significant proportion of the strength (table 1), giving a safety factor (i.e. the ratio of strength to stress) that is similar to that found in vertebrate bones.

Another type of loading that also occurs in bones is torsion: we did not consider this specifically in the present study, though our theoretical model could certainly be extended to do so. Predictions for torsion will probably be very similar to those for bending, because both are affected by the second moment of area, but torsional failure may be more sensitive to anisotropy. In any case, there are no experimental data available at present for torsional failure in arthropod bones.

4. Discussion

This study has examined two very different arthropod bones: the locust tibia and the crab merus. Our analysis was more comprehensive than any previous study: we considered both axial and bending loads, and we included five different modes of failure and also modelled the reduction of failure load resulting from interactions between these different modes. We had the opportunity, not available to previous researchers, of comparing our predictions with experimental measurements for the same bones in the same species.

We showed that there is no single, optimum value of r/t because the best value for axial loading is different from that for bending; so the best choice of r/t will depend on the relative amounts of bending and axial loading that occur during the most strenuous activities to which the bone is subjected. A compromise solution is therefore to be expected, and in both arthropods studied, we did find that the experimentally measured r/t value lay between the values predicted for bending and for axial loading. However, for the locust tibia, we found that the experimental value was much closer to the bending prediction, making this limb much stronger in bending than in compression. For the crab merus, the measured r/t value lay between the predictions for the two different loading modes, and we predicted that the limb would have identical strength for bending and axial loading. These findings are in complete agreement with the results of our stress analyses, which showed that, during the most strenuous activities for each animal, the locust tibia is loaded almost exclusively in bending (see also [23]), while the crab merus experiences similar levels of bending and axial compression.

Our study had a number of limitations. We did not consider long-term failure modes such as fatigue and creep. As such, our study is most relevant to failure under single, short-lived loading events, such as jumping, fighting and rapid motion to escape prey. Bone is known to fail by fatigue owing to repeated low-stress cycles in vivo and can be induced to fail by creep, though it seems that creep only dominates at loads higher than physiological levels [24]. Our theoretical model could easily be extended to consider fatigue and creep; however, there are no experimental data available for these failure modes in arthropod cuticle: a serious gap in our current knowledge of these animals. Vertebrate bone has a sophisticated system of remodelling [25] allowing it to repair damage and thus perform safely at higher stress levels, without failing. Bones also have other functions aside from structural ones; for example, vertebrate bone is an important organ of the endocrine system. We assumed all cross sections to be circular: this was a necessary simplification to allow us to study the problem widely and compare different species, but it ignores the fact that real bones often have other cross sectional shapes, notably ellipses and triangles, in response to preferential loading in certain planes and directions. The elliptical shape of the locust tibia certainly has a positive effect as noted earlier.

A further limitation is that we considered arthropod bones to be made from a single layer of cuticle, when in fact they consist of several layers—in particular, a relatively hard, stiff exocuticle and a much softer endocuticle [9]. This layering can affect other aspects of the mechanics of cuticle structures [26], but it is probably not a serious limitation in our case because the analysis is self-consistent: we used material property values that also assumed a single layer. It would be very interesting to predict the optimum values for the thicknesses of these different layers, though at present there is not enough experimental data to make the attempt worthwhile. We assumed that bones are hollow, empty tubes when in fact they contain muscles (in arthropods) and fat (in vertebrates), and in some cases, contain gas under positive or negative pressure, all of which will affect their load-carrying capacity to some extent. Other researchers have explored the positive effects of filling tubes with contents of lower elastic modulus [8,27], which tend to have a strengthening effect, dependent on various factors, including the adhesion between tube and contents. Our predictions will clearly be affected by the material property values used, which can be expected to vary somewhat and, in some cases, had to be estimated as no experimental data were available. It is perhaps worth noting that while our predictions of a bone's overall strength will be strongly affected by the absolute material properties, our prediction of the optimum r/t ratio will be controlled mainly by two ratios: the stiffness to strength ratio (in the cases of fracture and buckling) and the anisotropy ratio (which controls splitting).

There have been very few previous studies covering the same ground as the present one. As mentioned already, Wegst & Ashby applied similar reasoning to the case of plant stems in bending, though without considering interactions between failure modes. Currey's paper is mostly an analysis of axial loading, with some consideration of bending. He did not consider anisotropic material properties; he was aware of interactions between failure modes but did not include them in his calculations. He analysed 18 different arthropod limbs, concluding that some would fail by Euler buckling, some by local buckling and some by fracture. His analysis, though very insightful, was limited by a lack of material property data for most of the species considered: the literature shows that material properties for cuticle can be very different in different species, and even in different limb segments from the same species [11]. He also concluded that vertebrate bones never fail by local buckling but that some of the longer, more slender bones could fail by Euler buckling. He pointed out the potential advantages for vertebrates of increasing their r/t ratios. A recent paper [28] examined the human femoral neck, which consists of a thin cortical shell supported by an internal network of trabecular bone. In a theoretical analysis that considered fracture, local buckling and Euler buckling and that took account of the supporting effect of the trabecular bone, they concluded that buckling would not normally happen but could occur in older people as a result of osteoporosis. No experimental data were given to support these findings.

For a typical vertebrate bone, our analysis has shown that the optimal r/t value is much higher than generally occurs in nature. With an increased r/t ratio, bones could be approximately twice as strong; alternatively they could have the same strength while being considerably lighter, requiring less energy to be expended during their morphogenesis and maintenance and during locomotion. There are many examples of apparently ‘sub-optimal’ structures in nature, which often can be a result of non-obvious evolutionary constraints. It is clear that the vertebrate endoskeleton could not function with very high r/t values without drastic changes in the form of joints, muscles, etc., essentially turning the endoskeleton into an exoskeleton. Indeed, our analysis illustrates some of the advantages of the exoskeletal form. It may also be worth noting that vertebrates, though they constitute most of the larger animals, appear to be less ‘successful’ than invertebrates and other organisms with exoskeletons, in terms of the number of species, the number of individuals and the range of ecosystems inhabited. One thus might speculate that the possession of an endoskeleton made of less-efficient bones may be a limiting factor for evolutionary success.

However, other studies have come to different conclusions. A number of studies, of which the most comprehensive is that due to Currey & Alexander [29], have concluded that the optimum r/t value for mammals is 2 (based on strength) or 4 (based on stiffness). The essential difference between this analysis and the present one is that the medullary cavity was assumed to be filled with marrow, which makes a significant contribution to the total weight, especially for larger r/t values. But there is no reason to suppose that the bone would be full of marrow, because only a certain amount is needed.

5. Conclusions

• — The optimum ratio of radius to thickness, r/t, is different for different bones and also differs depending on whether the bone is being loaded in bending or in axial compression.

• — For the locust tibia, the experimentally measured r/t ratio is close to the predicted optimal value when the limb is loaded in bending, but far from optimal when axial loading is applied. This finding concurs with our stress analysis that predicts that the limb is loaded almost exclusively in bending during jumping.

• — For the merus of the blue crab, the experimentally measured r/t ratio lies between our predicted optimal values for bending and axial loading, giving exactly equal resistance to these two loading modes. This concurs with our stress analysis showing that similar levels of bending and compressive stresses occur during locomotion.

• — Vertebrate bones such as the human femur are concluded to be far from optimal in respect of their geometry: increasing r/t would give these bones significantly more resistance to failure for a given weight.

Appendix A. Stress analysis of the locust tibia and crab merus

Figure 6a illustrates the general principles that can be used to estimate the stress in a bone or any skeletal segment. Consider two body parts, here labelled ‘segment 1’ and ‘segment 2’, connected by a joint whose centre of rotation is at the point J. These segments are subjected to a set of forces and restraints representing the most strenuous activities normally carried out. In the example here, we imagine that the right-hand end of segment 2 is fixed and that segment 1 experiences a force F applied at a point X located at a distance L₁ from the joint. F₁ is that part of F that acts perpendicular to the line XJ and thus tends to cause rotation about J. To prevent this, a muscle located inside segment 2 pulls with a force f_m. The line of action of the muscle is located a distance d from the joint centre. The value of f_m can be found by equating moments about J, thus:

A1

Figure 6.

General principles used in estimating stresses in bones and other limb segments. (Online version in colour.)

Since d is relatively small compared with L₁, the muscle force will be large compared with the applied force F. These forces between them create both bending stress (σ_B) and compressive stress (σ_C) in segment 2. Because in this case, the muscle runs parallel to the limb axis, it causes a compressive force in the limb but no bending. As shown in figure 6b, the force F can be resolved into a component causing axial compression in segment 2 (F_C) and a component causing bending (F_B), with a bending moment of F_BL₂.

The total compressive stress is given by

A2

where A is the cross-sectional area of segment 2. The bending stress is given by

A3

where r is the distance from the neutral axis to the outside of segment 2 (equal to the radius if this segment is a circular tube) and I is the second moment of area of this segment evaluated for bending in the relevant plane.

We now apply these general principles to the two particular cases being studied here. Figure 7 shows the assumed geometry and loading for the crab merus. The greatest forces arise during fast locomotion under water. The body of the animal experiences a gravitational force that is small (typically 10% of body mass [30]) as a result of buoyancy. For a typical mass W of 148 g [11], this gives a force of 0.145 N. Martinez studied crabs moving under water and found that the peak drag force at a maximum speed was close to the animal's body mass; so we assume a value of 1.45 N [31]. Muscles act inside the merus creating moments about the joint centres at either end; because they take up most of the available space, we assumed that the average distance from the joint centre d was equal to half the limb radius. The several limb segments distal to the merus can be simplified into one single, rigid segment, assumed fixed where it meets the ground. Using body measurements reported in the literature [11,12], we assumed that the merus had a length of 20 mm, a radius of 4.5 mm, a cross section of 15.25 mm² and a second moment of inertia (I value) of 135 mm⁴. The distance from the centre of mass of the body to the proximal joint was estimated as 40 mm, the merus was taken to be inclined at an angle of 45° to the horizontal and the joint angle at the distal end of the merus was taken to be 90° [30]. Martinez et al. observed that the average number of legs in contact with the ground during locomotion under water was 2.5, sometimes reducing to as little as 2; so, in estimating peak stresses, we divided the total body force by two to find the force on each merus.

Figure 7.

Assumed geometry and loading for stress analysis of the crab merus. (Online version in colour.)

The calculated drag and gravity forces combined give an axial compressive force on the merus of 0.31 W, while muscle action causes a much larger force of 4.52 N. Together, they create a compressive stress of 0.46 MPa. Bending is caused solely by the applied forces, which give rise to a bending stress of 0.91 MPa.

For the locust tibia (figure 8), the greatest stress arises during jumping. Bennet-Clark found that the large extensor muscle located in the femur applies a peak force of 15 N at an angle of 30° to the axis of the tibia [21]. From our own measurements reported previously [14], we found average values of r, A and I for the tibia to be 0.594 mm, 0.193 mm² and 0.038 mm⁴, respectively. Assuming a mechanical advantage (the distance from the joint centre to the muscle attachment divided by the distance from the joint centre to the ground) of 50 (after Bennet-Clark), we estimated a ground force of 0.15 N per leg, which agrees well with Alexander's figure of 18 times body weight [32]. Examining video film of locusts jumping, we estimated the angle between the tibia and the ground at take-off to be 65°. This is somewhat larger than previous estimates [22] but the difference will not greatly affect the results. There are no muscles inside the tibia; so stresses are entirely due to this ground force. The estimated bending and compressive forces were 0.05 and 0.15 N, respectively. Because the tibia is relatively long and thin, the resulting compressive stress of 0.25 MPa is much less than the bending stress of 42.2 MPa.

Figure 8.

Assumed geometry and loading for stress analysis of the locust tibia. (Online version in colour.)