Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017
J. R. Soc. Interface
doi:10.1098/rsif.2012.0567
Published online
Shape optimization in
exoskeletons and endoskeletons:
a biomechanics analysis
David Taylor* and Jan-Henning Dirks
Mechanical Engineering Department, Trinity Centre for Bioengineering,
Trinity College Dublin, Dublin 2, Ireland
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.
Keywords: exoskeleton; arthropod; bone; strength; fracture; buckling
1. INTRODUCTION
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
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
*Author for correspondence ([email protected]).
Received 17 July 2012
Accepted 22 August 2012
1
This journal is q 2012 The Royal Society
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2 Shape optimization in exo/endoskeletons
D. Taylor and J.-H. Dirks
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 EL in the longitudinal direction
and ET in the transverse direction, and corresponding
strength (i.e. stress to failure) values sL and sT.
2a
We can identify five possible failure modes (figure 1),
as follows:
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 (sL) 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
Ff ¼ sL A:
ð1:1Þ
In bending, failure by fracture will occur at a bending
moment Mf given by
Mf ¼
sL I
:
r
ð1:2Þ
Here I is the second moment of area, which is a function
of the radius r and thickness t of the tube, given by
( p/4) (r 4 2(r 2 t)4). Provided r is much greater
than t, we can write this in a simplified form:
3
I ¼ pr t:
ð1:3Þ
1.2.2. Buckling
Buckling failures occur when elastic deflections become
unstable; the only material parameters that are relevant
are the elastic moduli (EL and ET) and Poisson’s ratio
y . Buckling occurs in relatively thin, flexible structures
loaded in compression or bending. It is a complex
2b
2c
3
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’.
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
sb ¼
1.2. Failure modes
J. R. Soc. Interface
1
EI p2
:
AL2
ð1:4Þ
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:
E
t
slb ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
2
r
3ð1 y Þ
ð1:5Þ
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:
0:987rt 2 E
Mob ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi :
1 n2
ð1:6Þ
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Shape optimization in exo/endoskeletons
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
pffiffiffiffiffiffiffiffiffiffiffiffito take account of anisotropy,
replacing E with EL ET .
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 sT. For pure bending in a tube made
from anisotropic material, the value of M at which splitting occurs was found [4] to be
sT r
:
ð1:7Þ
Ms ¼ 0:18A3=2 ðsT EL Þ1=2 1 1:24
ET t
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 Ft
from the individual failure forces for buckling (Fb)
and fracture (Ff ) as follows:
Ft ¼
1
:
1=Fb þ 1=Ff
ð1:8Þ
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
J. R. Soc. Interface
D. Taylor and J.-H. Dirks
3
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
material properties
(0.318)
EL (GPa)
0.318
ET (GPa)
sL (MPa)
(9.9)
sT (MPa)
9.9
whole bone failure stresses
axial (MPa)
bending (MPa) 6.0
geometry
L (mm)
25
15.25
A (mm2)
r/t
8.3
locust tibia
human femur
3.05
(1.5)
95
(47)
17.9
10.1
135a
53
10.7
72.1
20
0.193
11.04
450
603
2
a
A higher value of 150 MPa was used for the longitudinal
strength in compression.
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 multisegmented 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
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D. Taylor and J.-H. Dirks
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.
J. R. Soc. Interface
(a)
normalized bending moment at failure
M/A3/2 (MPa)
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.
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 sL
from 135 to 150 MPa for our analysis of axial loading.
20
18
16
14
12
10
8
6
4
2
0
(b)
30
stress at failure (MPa)
4 Shape optimization in exo/endoskeletons
buckling
fracture
combined fracture/buckling
experimental
local buckling
Euler buckling
fracture
combined (fracture/Euler)
combined (fracture/local)
25
20
15
10
5
0
5
10
15
r/t
optimum for
compression
optimum for
bending
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.)
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.
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
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Shape optimization in exo/endoskeletons
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 ¼ 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.
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
J. R. Soc. Interface
5
normalized bending moment at failure
M/A3/2 (MPa)
(a)
100
90
80
70
60
50
40
30
20
10
0
(b)
140
stress at failure (MPa)
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 ¼ 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.
D. Taylor and J.-H. Dirks
buckling
fracture
combined
(fracture/buckling)
splitting
experimental
local buckling
Euler buckling
fracture
combined (fracture/Euler)
combined (fracture/local)
experimental
120
100
80
60
40
20
0
5
10
15
optimum for
bending
20
25
30
35
r/t
optimum for
compression
40
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.)
(a)
dorsal
(b)
medial
lateral
lateral medial
(c)
ventral
proximal
distal
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.)
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
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6 Shape optimization in exo/endoskeletons
D. Taylor and J.-H. Dirks
normalized bending moment at failure
M/A3/2 (MPa)
300
(b)
300
stress at failure (MPa)
(a)
250
buckling
fracture
combined
(fracture/buckling)
splitting
250
200
150
100
50
0
fracture
combined
(fracture/Euler)
combined
(fracture/local)
200
150
100
50
0
5
10 15 20 25 30 35 40 45 50
optimum for r/t optimum for
compression
bending
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.)
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.
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
J. R. Soc. Interface
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.
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 landbased mammals is r/t ¼ 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.
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Shape optimization in exo/endoskeletons
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
J. R. Soc. Interface
D. Taylor and J.-H. Dirks
7
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
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8 Shape optimization in exo/endoskeletons
D. Taylor and J.-H. Dirks
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
J. R. Soc. Interface
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.
We are grateful to IRCSET, the Irish Research Council for
Science Engineering and Technology, for provision of
financial support.
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 fm. The line of action of the muscle
is located a distance d from the joint centre. The
value of fm can be found by equating moments about
J, thus:
F1 L1 ¼ fm d:
ðA 1Þ
Since d is relatively small compared with L1, the muscle
force will be large compared with the applied force F.
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Shape optimization in exo/endoskeletons
9
muscles
(a)
segment 1
J
drag force
d
L1
X
F
D. Taylor and J.-H. Dirks
merus
gravity force
muscle
F1
Figure 7. Assumed geometry and loading for stress analysis of
the crab merus. (Online version in colour.)
F2
segment 2
(b)
extensor
muscle
segment 1
J
femur
L2
joint centre
X
F
FC
tibia
FB
segment 2
ground
force
Figure 6. General principles used in estimating stresses in
bones and other limb segments. (Online version in colour.)
These forces between them create both bending stress
(sB) and compressive stress (sC) 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
(FC) and a component causing bending (FB), with a
bending moment of FBL2.
The total compressive stress is given by
sC ¼
ðFC þ fm Þ
;
A
ðA 2Þ
where A is the cross-sectional area of segment 2. The
bending stress is given by
sC ¼
FB L2 r
;
I
ðA 3Þ
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
J. R. Soc. Interface
Figure 8. Assumed geometry and loading for stress analysis of
the locust tibia. (Online version in colour.)
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 458 to the horizontal and the joint angle at
the distal end of the merus was taken to be 908 [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.
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 308 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
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10
Shape optimization in exo/endoskeletons
D. Taylor and J.-H. Dirks
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 658. 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.
REFERENCES
1 Kennedy, C. H. 1927 The exoskeleton as a factor in limiting and directly the evolution of insects. J. Morphol. 44,
267– 312. (doi:10.1002/jmor.1050440204)
2 Currey, J. 2002 Bones: structure and mechanics.
Princeton, NJ: Princeton University Press.
3 Currey, J. D. 1967 Failure of exoskeletons and endoskeletons.
J. Morphol. 123, 1–16. (doi:10.1002/jmor.1051230102)
4 Wegst, U. G. K. & Ashby, M. F. 2007 The structural efficiency of orthotropic stalks, stems and tubes. J. Mater.
Sci. 42, 9005– 9014. (doi:10.1007/s10853-007-1936-8)
5 Timoshenko, S. & Gere, J. M. 1961 Theory of elastic
stability, 2nd edn. New York, NY: McGraw-Hill.
6 Rees, D. W. A. 1997 Basic solid mechanics. London, UK:
Macmillan.
7 Brazier, L. G. 1927 On the flexure of thin cylindrical
shells and other ‘thin’ sections. Proc. R. Soc. Lond. A
116, 104 –114. (doi:10.1098/rspa.1927.0125)
8 Karam, G. N. & Gibson, L. J. 1995 Elastic buckling of cylindrical shells with elastic cores—I. Analysis. Int. J. Solids
Struct. 32, 1259–1283. (doi:10.1016/0020-7683(94)00147-O)
9 Vincent, J. F. V. & Wegst, U. G. K. 2004 Design and
mechanical properties of insect cuticle. Arthropod Struct.
Dev. 33, 187 –199. (doi:10.1016/j.asd.2004.05.006)
10 Martin, R. B., Burr, D. B. & Sharkey, N. A. 1998 Skeletal
tissue mechanics. New York, NY: Springer.
11 Hahn, K. & LaBarbera, M. 1993 Failure of limb segments
in the blue crab Callinectes sapidus (Crustacea: Malacostraca). Comp. Biochem. Physiol. A 105, 735–739.
(doi:10.1016/0300-9629(93)90276-A)
12 Taylor, J. R. A., Hebrank, J. & Kier, W. M. 2007 Mechanical properties of the rigid and hydrostatic skeletons of
molting blue crabs, Callinectes sapidus Rathbun. J. Exp.
Biol. 210, 4272–4278. (doi:10.1242/jeb.007054)
13 Dendinger, J. E. & Alterman, A. 1983 Mechanical properties in relation to chemical constituents of postmolt cuticle
of the blue crab, Callinectes sapidus. Comp. Biochem.
Physiol. A Physiol. 75, 421 –424. (doi:10.1016/03009629(83)90104-4)
14 Dirks, J. H. & Taylor, D. 2012 Fracture toughness of
locust cuticle. J. Exp. Biol. 215, 1502–1508. (doi:10.
1242/jeb.068221)
15 Jensen, M. & Weis-Fogh, T. 1962 Biology and physics of
locust flight. V. Strength and elasticity of locust cuticle.
Phil. Trans. R. Soc. Lond. B 245, 137 –169. (doi:10.
1098/rstb.1962.0008)
J. R. Soc. Interface
16 Ker, R. F. 1977 Some structural and mechanical properties
of locust and beetle cuticle. D.Phil Thesis, University of
Oxford, UK.
17 Reilly, D. T. & Burstein, A. H. 1975 The elastic and ultimate properties of compact bone tissue. J. Biomech. 8,
393– 405. (doi:10.1016/0021-9290(75)90075-5)
18 Carter, D. R., Harris, W. H., Vasu, R. & Caler, W. E.
1987 The mechanical and biological response of
cortical bone to in vivo strain histories. In Mechanical
properties of bone (ed. S. Cowin), pp. 81 –92. New York,
NY: ASME.
19 Taylor, D. & Kuiper, J. H. 2001 The prediction of
stress fractures using a ‘stressed volume’ concept.
J. Orthop. Res. 19, 919–926. (doi:10.1016/S0736-0266
(01)00009-2)
20 Sverdlova, N. S. & Witzel, U. 2010 Principles of determination and verification of muscle forces in the human
musculoskeletal system: muscle forces to minimise bending
stress. J. Biomech. 43, 387–396. (doi:10.1016/j.jbiomech.
2009.09.049)
21 Martinez, M. M., Full, R. J. & Koehl, M. A. R. 1998
Underwater punting by an intertidal crab: a novel gait
revealed by the kinematics of pedestrian locomotion in
air versus water. J. Exp. Biol. 201, 2609–2623.
22 Martinez, M. M. 2001 Running in surf: hydrodynamics of
the shore crab Grapsus tenuicrustatus. J. Exp. Biol. 204,
3097–3112.
23 Bennet-Clark, H. C. 1975 The energetics of the jump of
the locust Schistocerca gregaria. J. Exp. Biol. 63, 53 –83.
24 Alexander, R. M. 1995 Leg design and jumping technique
for humans, other vertebrates and insects. Phil.
Trans. R. Soc. Lond. B 347, 235–248. (doi:10.1098/rstb.
1995.0024)
25 Sutton, G. P. & Burrows, M. 2008 The mechanics of
elevation control in locust jumping. J. Comp. Physiol. A
194, 557–563. (doi:10.1007/s00359-008-0329-z)
26 Bayley, T., Sutton, G. P. & Burrows, M. 2012 A buckling
region in locust hindlegs contains resilin and absorbs
energy when jumping or kicking goes wrong. J. Exp.
Biol. 215, 1151–1161. (doi:10.1242/jeb.068080)
27 Carter, D. R. & Caler, W. E. 1985 A cumulative damage
model for bone fracture. J. Orthop. Res. 3, 84–90.
(doi:10.1002/jor.1100030110)
28 Taylor, D., Hazenberg, J. G. & Lee, T. C. 2007 Living with
cracks: damage and repair in human bone. Nat. Mater. 2,
263– 268. (doi:10.1038/nmat1866)
29 Dirks, J.-H. & Durr, V. 2011 Biomechanics of the stick
insect antenna: damping properties and structural correlates of the cuticle. J. Mech. Behav. Biomed. Mater. 4,
2031–2042. (doi:10.1016/j.jmbbm.2011.07.002)
30 Karam, G. N. & Gibson, L. J. 1995 Elastic buckling of
cylindrical shells with elastic cores—II. Experiments.
Int. J. Solids Struct. 32, 1285 –1306. (doi:10.1016/00207683(94)00148-P)
31 Thomas, C. D. L., Mayhew, P. M., Power, J., Poole, K. E. S.,
Loveridge, N., Clement, J. G., Burgoyne, C. J. & Reeve, J.
2009 Femoral neck trabecular bone: loss with aging and
role in preventing fracture. J. Bone Mineral Res. 24,
1808–1818. (doi:10.1359/jbmr.090504)
32 Currey, J. D. & Alexander, R. M. 1985 The thickness of
the walls of tubular bones. J. Zool. Lond. A 206, 453–
468. (doi:10.1111/j.1469-7998.1985.tb03551.x)