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The ‘Goldilocks’ effect: preservation
bias in vertebrate track assemblages
P. L. Falkingham1,*, K. T. Bates2, L. Margetts1,3
and P. L. Manning1,4
1
School of Earth, Atmospheric and Environmental Science, University of Manchester,
Williamson Building, Oxford Road, Manchester M13 9PL, UK
2
Department of Musculoskeletal Biology, Institute of Aging and Chronic Disease, University
of Liverpool, Sherrington Buildings, Ashton Street, Liverpool L69 3GE, UK
3
Research Computing, University of Manchester, Devonshire House, Oxford Road,
Manchester M13 9PL, UK
4
Department of Earth and Environmental Science, University of Pennsylvania,
Philadelphia, PA 19104, USA
Finite-element analysis was used to investigate the extent of bias in the ichnological fossil
record attributable to body mass. Virtual tracks were simulated for four dinosaur taxa of
different sizes (Struthiomimus, Tyrannosaurus, Brachiosaurus and Edmontosaurus), in a
range of substrate conditions. Outlines of autopodia were generated based upon osteology
and published soft-tissue reconstructions. Loads were applied vertically to the feet equivalent
to the weight of the animal, and distributed accordingly to fore- and hindlimbs where rel-
evant. Ideal, semi-infinite elastic–plastic substrates displayed a ‘Goldilocks’ quality where
only a narrow range of loads could produce tracks, given that small animals failed to
indent the substrate, and larger animals would be unable to traverse the area without becom-
ing mired. If a firm subsurface layer is assumed, a more complete assemblage is possible,
though there is a strong bias towards larger, heavier animals. The depths of fossil tracks
within an assemblage may indicate thicknesses of mechanically distinct substrate layers at
the time of track formation, even when the lithified strata appear compositionally homogeneous.
This work increases the effectiveness of using vertebrate tracks as palaeoenvironmental indi-
cators in terms of inferring substrate conditions at the time of track formation. Additionally,
simulated undertracks are examined, and it is shown that complex deformation beneath the
foot may not be indicative of limb kinematics as has been previously interpreted, but instead
ridges and undulations at the base of a track may be a function of sediment displacement vectors
and pedal morphology.
Keywords: footprint; finite-element analysis; trackway; computer
modelling; dinosaur
1. INTRODUCTION
The sample we have of the body fossil record is notor-
iously incomplete [1–4] and may be fundamentally
biased by environmental and taxon-specific factors
that potentially hamper our interpretation of ecological
and evolutionary dynamics through deep time [5–8].
Interdependent environmental and taxon-specific
biases are equally likely to affect the ichnological, or
trace fossil, record. The potential for systematic bias
towards ichnofossils produced by larger animals has
previously been recognized in the field of vertebrate
palaeoichnology, and particularly in the dinosaur
track record [9]. Sites preserving only the tracks of
very large saurischian dinosaurs (i.e. track lengths
greater than 0.5 m) are generally recognized as size-
biased assemblages and presumed to represent
sedimentary conditions in which only animals above a
certain threshold of body mass were capable of
producing recognizable tracks [9]. However, beyond
this supposition, the influence of body size on the
recorded diversity of vertebrate ichnofossil assemblages
is poorly understood both qualitatively and quantitat-
ively. It is therefore imperative that the process of
track formation and the variables associated with
environment and animal biology are investigated.
The relationship between the size of an animal and
the load applied to the substrate is not straightforward.
Given that pressure is a measure of force over area, the
resultant pressure exerted on the sediment surface is a
function not only of the animal’s mass (as weight),
but also the geometry of the autopodia. Quadrupedal
animals benefit from more feet in contact with the
ground, further reducing the load on the substrate
*Author for correspondence (peter.falkingham@manchester.ac.uk).
Electronic supplementary material is available at http://dx.doi.org/
10.1098/rsif.2010.0634 or via http://rsif.royalsocietypublishing.org.
J. R. Soc. Interface (2011) 8, 1142–1154
doi:10.1098/rsif.2010.0634
Published online 13 January 2011
Received 15 November 2010
Accepted 21 December 2010 1142 This journal is q2011 The Royal Society
beneath any single foot, as compared with a similar-
sized biped. In addition to size, foot morphology also
plays an important role in determining the magnitude
of deformation expressed as the depth of a track. Differ-
ing shapes present different paths for sediment
movement, resulting in variable distributions of force
that affect the extent to which any given foot may
indent a substrate [10,11]. These considerations are
not trivial if data on fossil track occurrences and abun-
dances are to be used in ‘higher level’ [12]
palaeobiological and palaeoecological inferences. Allen
[13] noted over a decade ago that a widespread under-
standing of track formation lagged behind knowledge
of anatomical aspects and distributions of fossil
tracks, and despite a number of rigorous experimental
studies in the intervening years [14–20], this still
remains the case.
Among vertebrates, the Dinosauria represent a
useful model system for studying track formation;
their high taxonomic diversity and long evolutionary
history yield an array of disparate foot morphologies
and a huge range in body mass with which to test for
possible biological factors underpinning preservational
bias. The group contains small and large obligate
bipeds, quadrupeds and supposed intermediate loco-
motor strategists (e.g. facultative bipedalism [21–23])
that may have exerted different underfoot pressures
according to foot geometry and body shape (i.e. mass
distribution). Coupled with a vast quantity of research
describing dinosaur tracks spanning more than a cen-
tury and a half [24 –26], dinosaurs provide the ideal
basis on which to further our understanding of fossil
track formation, and the size-related biases associated
therewith.
In this paper, information on foot anatomy and mass
distribution from osteological evidence and soft tissue
reconstructions are integrated with geotechnical
theory and computer simulation to explore the poten-
tial for size bias in the vertebrate track record. In
addition, features related to underfoot pressure and
foot morphology were examined in surface and subsur-
face planes (true tracks and undertracks). Previous
work on track formation using computer simulation
has explored independently the effects of substrate con-
sistency [27], foot anatomy [11] and force [28]. This
paper aims to present a combined study in which the
quantifiable variables of track formation are considered
as a whole system, in the hope of elucidating aspects of
preservational bias inherent in the fossil track record.
2. METHODS
The following experiments used parallel finite-element
analysis (FEA) software developed by authors
Margetts and Falkingham, using the freely available
ParaFEM libraries (www.parafem.org.uk) to model
track formation [11,27–29]. A number of dinosaur
tracks were simulated over a range of substrates in
order to explore bias in their formation resulting from
substrate- or taxon-specific factors.
2.1. The virtual foot
Four dinosaurs (Struthiomimus, Edmontosaurus,
Tyrannosaurus and Brachiosaurus) were chosen to
create a varied virtual track assemblage on a cohesive
substrate, representing a range of body masses, and
including obligate bipeds, an obligate quadruped and
a facultative biped (table 1). These particular taxa
were also chosen because they all have published data
on body mass and centre of mass (CM) position
[30,31], and represent a wide range in size, mass
and autopodial morphology. The taxa were not
selected in order to create some geotemporally correct
track assemblage.
A track is formed through the interaction of three
factors; force, foot anatomy and substrate [32,33].
Force applied and foot anatomy are both dependent
upon the track maker. To apply a reasonable force,
the body mass for each dinosaur was taken from the lit-
erature ([30,31]; see table 1). The reader is directed to
Bates et al.[30] for a comprehensive discussion on the
confidence of CM and body mass reconstructions. Ani-
mals spend a very small proportion of their time moving
at anything more than a walking speed, and it would
therefore be expected that most tracks are made by
walking animals. Indeed, this is corroborated by the
numbers of trackways showing walking, rather than
running gaits [34,35]. Stride length is positively cor-
related with speed [36,37], meaning that at low
speeds, the hip joint and CM will move a shorter dis-
tance horizontally from the contact between the foot
Table 1. Mass, weight, foot metrics and pressures used to represent various dinosaur taxa used in this study. Data for
Struthiomimus, Edmontosaurus and Tyrannosaurus from Bates et al.[30], and data for Brachiosaurus from Henderson [31].
trackmaker mass (kg) force (kN) foot length (m) foot surface area (m
2
) pressure (kN m
22
)
Struthiomimus 423 4.15 0.336 0.026 161.21
Edmontosaurus (biped) 813 7.98 0.29 0.052 151.92
Edmontosaurus Quadruped manus 813 2.55 0.12 0.011 241.39
Edmontosaurus Quadruped pes 813 5.42 0.29 0.052 103.31
Tyrannosaurus 7654 75.09 0.72 0.234 320.26
Brachiosaurus manus 25 922 95.11 0.6 0.144 662.29
Brachiosaurus pes 25 922 159.19 0.87 0.401 396.58
Edmontosaurus
a
813 7.98 n.a. 0.063 126.46
Brachiosaurus
a
25 922 254.29 n.a. 0.545 466.59
a
Edmontosaurus and Brachiosaurus are also shown with pressure values from manus and pes combined.
The ‘Goldilocks’ effect P. L. Falkingham et al. 1143
J. R. Soc. Interface (2011)
and the ground, resulting in a smaller angle of ground
reaction force (GRF) [37]. As such, for the purposes of
this paper, a purely vertical component to the applied
force was assumed. Force distributed through feet in
contact with the ground was taken as the weight of
the animal, calculated as mass gravity (9.81 m s
22
).
An animal of 100 kg would therefore exert a vertical
force upon the ground of 981 N.
For a biped, maximum force is transmitted through a
single foot when the opposite foot is raised, so the
pressure applied in this case was equal to the weight of
the animal divided by the surface area of a single foot.
This is to approximate the peak force at any one time
during limb support. In the case of quadrupedalism,
CM plays a role in determining how much of the animal’s
weight is distributed to the fore- and hindlimbs, after
which the two sets of limbs can be treated separately
as bipeds [37]. This is a simplification of the loads experi-
enced by the autopodia of a quadruped during
locomotion, but provides reasonable input values for
the purposes of this paper. CM estimates for Edmonto-
saurus and Brachiosaurus were taken from the
literature (see [30]forEdmontosaurus CM and [31]for
Brachiosaurus) and used to apportion force between
fore- and hindlimbs. The amount of the animal’s
weight given to each pair of limbs was equal to the rela-
tive position of the CM between the pelvic and pectoral
girdles, i.e. a CM 60 per cent of the way from the pectoral
girdle to the pelvic girdle would imply a weight distri-
bution of 60 per cent to the hindlimbs, and 40 per cent
to the forelimbs [31](figure 1). Treating the animal as
two linked bipeds with appropriate weights was sufficient
for the purposes of these experiments (see [37]forwalk-
ing models of quadrupeds represented as two bipeds in
tandem). While this may be a very simplified solution
that ignores the effects of complex gaits, walking vel-
ocities and limb kinematics of dinosaurs are unknown
and employing the loading regime outlined above
avoids incorporating additional unfounded assumptions
into the simulations. Consideration is given to the effects
of duty factor and locomotion later in the discussion.
Hadrosauridae have been interpreted as primarily
bipedal with facultative quadrupedalism at either low
[21,23] or high [38] speeds, based on anatomical features
in the forelimbs suggestive of either mode of loco-
motion, and trackway evidence also supporting both
gait reconstructions [35,39,40]. Edmontosaurus tracks
were therefore simulated as being made by both a
bipedal animal and a quadrupedal animal.
The indenters, or ‘virtual feet’, were created by produ-
cing outlines around ventral views of reconstructed
skeletal autopodia (figure 2). Skeletal geometry was
scaled to the same size as the specimens used by Bates
et al.[30] and Henderson [31] so as to remain consistent
with mass estimates. The outlines were then increased
in size to account for soft tissue. The outline of the
Edmontosaurus manus does not follow the osteology as
closely as the other indenters, instead being based on
the exceptionally preserved hadrosaur body fossil MRF
03(thoughscaledtothespecimenusedby[30]), as fig-
ured in Sellers et al.[38, fig. 5], where the manus soft
tissue takes a ‘mitten’-like form over the skeleton. This
is supported by hadrosaur manus tracks illustrated by
Lockley & Wright [39], and those described as ‘crescent
shaped’ by Currie [41]. The indenters representing the
manus and pes of Brachiosaurus were generated as in
Falkingham et al.[28] from reconstructions by Wright
[42]. These outlines defined the nodes and elements
that would be loaded on the FE substrate volume
(figure 2). For each animal, a volume of substrate was cre-
ated for each foot to be indented into. Only one pes
needed to be indented for each bipedal condition, and
only one manus and one pes for the Brachiosaurus and
quadrupedal Edmontosaurus.
acetabulum(a)
(b)
(c)
100 N
100 N
50 N50 N
100 N
100 N
30 N70 N
CM = 0%
CM = 50%
CM = 70%
0 N
glenoid
Figure 1. Loads beneath fore- and hindlimbs as determined by
CM position. A CM position of 50% gleno-acetabular position
applies equal load to both fore- and hindlimbs. As the CM is
positioned more anterior or more posterior, more load is
applied to the fore- or hindlimbs, respectively.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2. Foot outlines used to create indenters. (a)Edmonto-
saurus manus, (b) pes, (c)Brachiosaurus manus, (d) pes
(adapted from [42]), (e)Struthiomimus and ( f)Tyranno-
saurus. Scale bars, 0.1 m. Edmontosaurus pes shows how
foot outline was derived based on osteology.
1144 The ‘Goldilocks’ effect P. L. Falkingham et al.
J. R. Soc. Interface (2011)
2.2. The virtual substrate
An elastic-perfectly plastic von Mises model was applied
in order to model a cohesive clay-like substrate. The
mechanical properties of the substrate were defined by
the undrained shear strength (C
u
), Young’s modulus
(E) and Poisson’s ratio (v). These parameters relate,
respectively, to
— The strength of the substrate, that is, how much
stress is needed before failure of the sediment ( per-
manent deformation). Essentially a measure of
cohesion between grains, shear strength is most
strongly affected by water content [10,43]. Typical
values of C
u
in sediments located on the tidal
banks of the Bahia Blanca Estuary, Argentina,
were shown to range between 50 and 150 kN m
22
in the surface 1 m [44]. Values of C
u
according to
the British Standards for Geotechnical Engineering
are summarized along with field testing methods
in table 2.
— The stiffness of the substrate—how much defor-
mation is recoverable through elastic behaviour
before (and after) plastic deformation takes place.
The value of Young’s modulus is typically 1000
the value of C
u
in cohesive substrates [45].
— The compressibility of the substrate. In an entirely
incompressible substrate, v¼0.5. Such a substrate
will not change in volume when deformed, resulting
in expansion equal to compression along an axis per-
pendicular to that of the primary stress [10]. An
incompressible substrate could be considered to be a
fully saturated sediment, in which void space air has
been completely replaced by water (note: though
water is technically compressible to some extent, at
the magnitude of forces dealt with here, it can
safely be considered incompressible). Typical values
for saturated clay or mud would be 0.4–0.5 [46].
Many palaeontological FEA studies concerning
stress within bone use elastic models, in which there is
a linear relationship between stress and strain
(figure 3a), determined by E. The introduction of a fail-
ure criterion (C
u
), however, produces an elastic-perfectly
plastic model, whereby initial loading deforms the
material in a recoverable elastic manner (line O–Y0in
figure 3b) until the load is sufficient to plastically
deform the substrate. Further loading equals or exceeds
Table 2. Undrained strength classification of clays according
to BS 8004:1986, along with simple field tests (from [10]).
stiffness
state
undrained strength
(kN m
22
) test
hard .300 can be scratched by
thumb nail
very stiff 150–300 can be indented by
thumb nail
stiff 75 – 150 can be indented slightly
by thumb
firm 40 – 75 thumb makes
impression easily
soft 20–40 finger pushed in up to
10 mm
very soft ,20 finger easily pushed in
up to 25 mm
strain
Y¢Y¢¢ P
U
bearing capacity
first yield
v = 0.49
v = 0.1
O
stress
(a)
(b)
(c)
(d)
stressstressstress
Figure 3. (a) Elastic stress– strain relationship, (b) elastic-per-
fectly plastic stress– strain relationship. Initial elastic
deformation occurs along line O–Y0until stress exceeds the
strength of the substrate, at which point failure occurs and
deformation takes place along line Y0–P. If loading is halted
at Y00, and then removed, elastic recovery occurs along line
Y00 –U, parallel to initial deformation O– Y0.(c) The effects
of an elastic-perfectly plastic model distributed over a sub-
strate volume, in which parts are in plastic failure, and
others are in the elastic region. (d) The effects of Poisson’s
ratio on the overall form of the stress– strain relationship
within a substrate; dotted line, v¼0.49; solid line, v¼0.3;
dashed line, v¼0.1.
The ‘Goldilocks’ effect P. L. Falkingham et al. 1145
J. R. Soc. Interface (2011)
the bearing capacity and results in failure, where the sub-
strate can no longer support the load (line Y0–P in
figure 3b). When the load is removed, recovery occurs
along a line parallel to the original elastic deformation
(line Y00 –U in figure 3b). On the scale of individual
elements, this relationship is clear and well defined, but
over an entire mesh, where some elements may be in a
plastic state and others in an elastic one, the relationship
becomes less defined, with a curved portion where plastic
deformation occurs (figure 3c).
A soft clay-like substrate was generated in the FEA
simulations using 20-node hexahedral elements. The
20-node element is required in this case because of
the nature of the deformation; indenting into soft
substrate causes a large gradient of deformation from
negative vertical displacement beneath the edge of the
indenter, to positive vertical displacement adjacent to
the indenter. Eight-node elements lack the numerical
flexibility to deal with such a gradient, and so by
increasing the number of nodes defining the element,
a more accurate solution can be found. The volume of
substrate modelled was equal to four times the foot
length in all dimensions in order to avoid boundary
effects [13].
2.3. The process of indenting
While the foot of an animal can move at joints and the
soft tissue is deformable to an extent, as a whole, the
foot can be considered rigid compared with the non-
rigid substrate. In order to create a rigid loaded area,
rigid-body interface elements were generated above
the area on the mesh that would be loaded [47], essen-
tially creating a solid meshed foot on the surface of
the virtual substrate. The weight of the animal was
then applied to the interface elements to generate a uni-
form load over the foot. The ‘foot’ was loaded vertically
as a static analysis (i.e. independent of loading rate),
and then removed vertically in order to allow the sub-
strate to recover the elastic part of the deformation,
as would be the case in the formation of a real track.
The effects of loading rate are considered later in §4.
For each indenter, substrates were generated with a
high C
u
, and this was incrementally lowered until the
substrate could no longer support the load (i.e. bearing
capacity was exceeded). In all cases, Ewas equal to
1000C
u
, and v¼0.4. Maximum depth of indentation
beneath the virtual foot was recorded in each
experiment, as this value is a fair indication of the
degree to which a track is observable. Additionally,
surface tracks and undertracks were visualized and
qualitatively observed.
3. RESULTS
If the body mass and total foot surface area of the
models are logged, it can be shown that foot surface
area is proportional to mass
0.7
(see electronic
supplementary material, S1). This is close to the
relationship predicted by isometric scaling, where
surface area is proportional to mass
2/3
[48,49], and
suggests that as animals increase in size, the pressure
exerted on the substrate (discounting the effects of
locomotion) increases at a proportionally greater rate.
The small sample size used in this study means that
any difference in this relationship between quadrupeds
and bipeds cannot be observed, nor can the effects of
allometric scaling be explored.
In homogeneous substrates, there is a very narrow
range of C
u
values for any given pressure that allow
the formation of observable surface tracks (figure 4).
If C
u
is higher than this value, indenters fail to
deform the substrate to an appreciable degree, attaining
maximum track depths of less than a millimetre. Given
that many of the autopodia used were tens of centi-
metres in length, such deformation cannot realistically
be considered to be an observable track. Lower values
of C
u
than this narrow range cannot support the
applied load, and the substrate fails. The maximum
load a substrate can support beneath an indenter can
be approximated by calculating the bearing capacity
beneath a circular indenter under the specified load
using the following equation [50]:
bearing capacity ¼Cuð2þ
p
ÞS;ð3:1Þ
where Sis a shape factor equal to 1 þ0.2 (breadth/
length).
Using this equation, it can be seen that a substrate for
which C
u
¼100 kN m
22
will fail when the load on a circu-
lar indenter (S¼1.2) reaches 616.8 kN m
22
.The
800
(a)
(b)
700
600
500
400
300
200
pressure (kN m–2)
pressure (kN m–2)
100
0 20406080
no tracks
(substrate too soft;
firm subsurface
layer required)
no tracks
(substrate too firm)
shallow tracks
Cu (kN m–2)
Cu (kN m–2)
100 120 140 160 180
Figure 4. (a) Bubble plot of maximum track depth for a given
load on varying substrates. Size of bubbles qualitatively rep-
resent maximum depth of track. Line shown denotes the
predicted minimum C
u
required to support any given load
applied to a perfectly circular indenter. (b) Diagrammatic of
results. Tracks are only formed to any significant depth at
values approximately equal to those defined by the line, or
to the left of the line only if a firmer underlying layer is pre-
sent. Below and to the right of the line, loads are
insufficient to produce tracks of significant depth.
1146 The ‘Goldilocks’ effect P. L. Falkingham et al.
J. R. Soc. Interface (2011)
approximated failure point for circular indenters at any
given load is plotted as a line in figure 4.Thisprediction
is not the true value of bearing capacity for any specific
track, however, owing to variations in foot morphology.
However, as can be seen from figure 4,thisapproximation
is sufficiently close as to highlight the relationship.
4. DISCUSSION
The range of C
u
in which tracks of significant depth can be
generated is very small regardless of foot morphology or
load (figure 5). This limits tracks made in truly homo-
geneous substrates to an extremely narrow range of
pressures and subsequently producer sizes and foot mor-
phology. Below the minimum value of C
u
,atrackwillbe
formed providing there is a firmer substrate layer beneath.
If there is no firmer subsurface layer, the substrate cannot
support the load, and the animal in question will be
unable to traverse the area without becoming mired.
A key observation is that in simulating track
formation in a homogeneous semi-infinite elastic-
perfectly plastic substrate, generation of tracks to any
significant depth is difficult to achieve—many tracks
were so shallow that for real tracks of a similar depth,
it would be unreasonable to expect discovery in the
field. This is because the load required to plastically
deform the substrate and the maximum load that the
substrate can support are very close, implying that a
very specific pressure is required to generate a track in a
homogeneous substrate. There is therefore a ‘Goldilocks’
quality to homogeneous substrates regarding possible
track formation. A faunal assemblage represented by
tracks at a given tracksite will be strongly biased towards
(g)
0.002 0.004 0.006
maximum depth (m)
80
60
40
20
0
Cu (kN m–2)
80(a)(b)
(c)(d)
250
200
150
100
50
0 0.002 0.004 0.006 0.008
150
100
50
0 0.002 0.004 0.006
60
40
20
0 0.0005
0.005 0.01 0.015
0.001
Cu (kN m–2)Cu (kN m–2)
150
100
50
0
(e)(f)
0.0002 0.0004 0.0006 0.0008
0.005 0.01 0.015
maximum depth (m) maximum depth (m)
80
60
40
20
0
Cu (kN m–2)
80
60
40
20
0
Figure 5. Graphs showing maximum track depth and substrate C
u
for each indenter. The horizontal dashed line depicts theor-
etical bearing capacity for a circular indenter applying the same pressure. (a)Struthiomimus,(b)Tyrannosaurus,(c)
Brachiosaurus manus, (d)Brachiosaurus pes, (e)Edmontosaurus manus (quadrupedal loading), ( f)Edmontosaurus pes (quad-
rupedal loading), and (g)Edmontosaurus pes (bipedal loading). Each graph shows there is a very narrow range in which tracks
are generated, but the substrate is still able to support the load.
The ‘Goldilocks’ effect P. L. Falkingham et al. 1147
J. R. Soc. Interface (2011)
the largest animals the substrate can support, resulting in a
very low diversity of recorded body sizes. Taxa exerting
more pressure beneath their feet than the substrate can
support will avoid the area or become mired, while animals
producing less pressure than is required to create a track
will not leave observable impressions.
More commonly, substrates are polyphasic, with het-
erogeneous mechanical properties varying vertically
and laterally. If a substrate is underlain by a firmer
layer (e.g. compacted sediment or rock), then tracks
will be formed if the surface layer fails. If we consider
the scenario of a series of substrate layers becoming pro-
gressively firmer with depth (figure 6), it is observed
that any animal creating sufficient load as to deform
the uppermost substrate will generate a track. Refining
this stratification such that C
u
increases gradually with
depth results in the intuitive case that heavier animals
generate deeper tracks. It can be seen from figure 4b
that this being the case, there is a much larger range
of possible track-bearing substrates for animals exerting
a greater pressure (i.e. by being larger or moving faster).
Given the shallow nature of the tracks modelled in
homogeneous substrates, it becomes apparent that
most real tracks must therefore be formed in mechani-
cally heterogeneous substrates, or in relatively shallow
homogeneous substrates underlain by rock.
If a substrate is stratified with mechanically distinct
layers, then the depths and surface areas of present
tracks can be used to infer the depth and mechanical
properties of these layers at the time of track formation.
A track-bearing surface on which small and medium
tracks are impressed to a similar depth, yet on which
tracks made by larger animals appear deeper, will indi-
cate a mechanically homogeneous surface layer as deep
as the small and medium tracks (after accounting for
subsequent weathering/erosion). Such consideration of
tracks as palaeo-penetrometers may prove useful in
interpretations of palaeoenvironment, at least in so far
as determining substrate conditions at the time of
track formation.
Examining the biases inherent in track formation as
a consequence of animal size permits discussion of gen-
uine and artificial signals regarding diversity as
interpreted from tracksites. A track site limited to
large producers, e.g. a purely sauropod track assem-
blage, is likely to be a preservational artefact, or
indistinguishable from such. Smaller animals may
have been abundant at such a site, but unable to pro-
duce tracks in the substrate. Allen [51] noted in his
discussion of the Flandrian deposits of the inner Bristol
Channel and Severn Estuary that the fauna represented
by tracks lacked records of smaller mammals such as
foxes and dogs, arguing this was a preservational
rather than an ecological issue. The results presented
here support this hypothesis, and may be applicable
to other fossil tracksites dominated by large fauna
such as Fumanya, Spain [52,53], or the tracksites at
Paluxy River, Glen Rose, TX, USA [54]. In the case
of the Glen Rose tracks, the site is dominated by large
sauropod and medium to large theropod tracks. One
trackway has been interpreted as showing the inter-
action of a theropod and sauropod [55], implying that
the trackways were contemporary. Given the consider-
able depth of both the theropod and sauropod tracks,
and the lack of tracks from smaller animals, the track-
ways appear to be consistent with the Goldilocks
effect. Both the theropod and sauropod exceeded the
bearing capacity of the surface mud, and indented
deep tracks until supported by firmer substrate layers
beneath the surface. However, the depth of the soft sur-
face mud may have been too great for smaller animals to
safely traverse while leaving tracks, resulting in the for-
mation and subsequent preservation only of the largest
animals present. A similar case can be made for the
sauropod trackways at Fumanya, where deep sauropod
tracks dominate. Shallow theropod tracks were pre-
viously reported from the site, but have since been
subjected to weathering and are no longer present
[52]. In this case, only larger animals were able to pro-
duce tracks, resulting in an impoverished track
assemblage, whose low diversity has been exacerbated
further by weathering and the complete removal of
the shallower tracks left by smaller animals.
Given that there is a strong bias towards greater
underfoot pressures, the preservation potential of
track assemblages representing mixed age groups
(herd behaviour) is greatly reduced; there will be a
strong bias towards preserving only the largest mem-
bers of the group. If the adults within a group are
c
b
a
load
xyz
Cu
0.05 m
0.05 m
Cu = x
Cu = y
Cu = z
Figure 6. Hypothetical scenario in which three substrate layers
are considered, where C
u
increases with depth. Animals pro-
ducing loads that cause the surface layer to fail, but not the
subsequent layer (a,load ,b) will create tracks of 0.05 m
maximum depth. Animals producing loads sufficient to
deform layer two, but insufficient to deform layer three (a,
b,load ,c) will generate tracks to 0.1 m depth. Animals
producing loads above the bearing capacity of layer three
(load .c) will be unable to traverse the substrate, while
animals producing less pressure than is required to deform
the surface layer (load ,a) will not generate tracks.
1148 The ‘Goldilocks’ effect P. L. Falkingham et al.
J. R. Soc. Interface (2011)
particularly large, as in the case of sauropods for
instance, the range of substrates traversable by the
group will be constrained by the minimum substrate
strength that can support the largest animals. As
such, tracks from smaller individuals become far less
likely to form and subsequently preserve, because the
substrates over which the animals move may not be
soft enough to record the passage of smaller, juvenile
forms. This reduction in preservation potential of
mixed-age herds is supported by the fossil record;
Myers & Fiorillo [56] noted that of 13 sauropod track-
way associations indicating gregarious behaviour, only
three sites contained tracks from multiple age groups.
The presence on a single track-bearing surface of
both small and large true tracks (rather than trans-
mitted or undertracks), indented to approximately the
same depth (evidently halted by a firmer subsurface
layer), is likely to be more indicative of true diversity
in the area at the time of track formation (providing
effects of time averaging can be removed). Presence of
small, shallow tracks and large, deep tracks may not
be indicative of true diversity, however, if the large
deep tracks are particularly deep. In such a scenario,
it is possible that medium-sized animals produce too
great a pressure underfoot to be supported by the soft
surface layer, but sink too far before reaching a suppor-
tive layer as to be able to traverse the area. This
highlights the importance of considering not just the
tracks present at a tracksite, but their total three-
dimensional morphology, including foot anatomy and
track depth, in order to make interpretations about
faunal diversity.
The Goldilocks effect described in this study is miti-
gated when a substrate is exposed for a period of time
during which the mechanical properties alter, such as
when a substrate is drying out. Changes in mechanical
properties will undoubtedly be the rule, rather than the
exception, but the rate at which these changes occur
will determine the overall applicability of the Goldilocks
effect to the fossilized tracksite. In cases where the sub-
strate dries out over a relatively prolonged period,
the Goldilocks effect will be applicable over short time
spans, but will not be evident over the recording life
of the substrate, or in the preserved track surface. The
presence of sedimentary features such as drying cracks
or displacement rims that are unique to some trackways
and not others may shed some light on the preserva-
tional context, and as to whether the Goldilocks effect
noted here applies to a given track assemblage.
The experiments carried out in this study have used
body mass to apply a force through the autopodia in a
number of taxa. Loading in this way assumes a direct
relationship between body mass and force, and was car-
ried out as a rate-independent (i.e. static) analysis as
the focus of the work was to explore bias relating to
size, not necessarily locomotor mode. This was done
to avoid incorporating unfounded assumptions into
the simulations, given that the habitual gaits of dino-
saurs are unknown. Nevertheless, consideration must
be given to the effects of locomotion, duty factor and
limb kinetics and kinematics. As an animal begins to
move, the GRF gains a horizontal (forward – backward)
component in order to move the animal forwards [57].
This force vector may also incorporate a lateral com-
ponent depending on the animal’s gait. As speed
increases, the magnitude of the GRF also increases. In
terms of pressure applied, the pressure beneath an ani-
mal’s foot will increase as speed increases. As an animal
increases in speed, the minimum C
u
required to support
the load also increases, such that a substrate that pre-
viously would be incapable of recording an animal
standing or moving slowly may fail beneath the foot
of a running animal.
As an animal traverses a substrate, the rate of load-
ing is intrinsically linked to the speed and duty factor of
the animal, with loading rate increasing as duty factor
decreases. An increased loading rate results in more
resistance from the substrate, and the result is that
deformation occurs to a lesser extent (see electronic
supplementary material, S2). However, at higher
speeds, an animal exerts a greater force upon the sub-
strate, as noted above. As such, although the loading
rate increases with speed, the effects on substrate dis-
placement will be mitigated by the increased load
applied by the foot. It is important to note that the ana-
lyses carried out in this study were static, and thus did
not account for the effects of rate-dependent loading
and thixotropy. Exploring this complex interplay
between loading magnitude and loading rate is
beyond the scope of this study, and is impossible with-
out comprehensive locomotor reconstructions of the
animals in question. Instead, the Goldilocks effect can
be considered to be the base mechanic around which
other confounding factors such as limb dynamics, loco-
motion and substrate thixotropy have an effect. To
provide some insight into these complex issues, and
their relationship to the Goldilocks effect, the results
from a series of simple dynamic simulations are pro-
vided and discussed in electronic supplementary
material, S2.
Given the static vertical loading conditions employed
in this study, the relationship between size and track-
forming potential could be predicted to a reasonable
degree with simple mechanics and geotechnical
theory, as evidenced by the close correlation between
equation (3.1) and the results (figure 4a). However,
FEA provides benefits over simple mechanics. Simulat-
ing track formation allows for differing foot
morphologies to be tested, which was shown to be a
potentially important factor by Falkingham et al.[11].
FEA also allows us to explore the full three-dimensional
volume of simulated tracks. The importance of under-
standing the relationship between foot morphology,
three-dimensional deformation and undertrack pro-
duction will be demonstrated by the discussion of
features observed in the models.
5. DISCUSSION OF INDIVIDUAL TRACK
FEATURES
The simulations undertaken for this study present an
opportunity to investigate track features at the original
track surface, and in subsurface undertracks. Specific fea-
tures related to autopodia morphology and undertrack
depth are discussed here. In order to generate deeper
The ‘Goldilocks’ effect P. L. Falkingham et al. 1149
J. R. Soc. Interface (2011)
tracks and associated undertracks, failure was allowed to
occur, but was halted when maximum depth reached
0.05 m, approximating a firm subsurface layer.
The values of mass, foot morphology and CM pos-
ition used for the Edmontosaurus produce differing
pressures between manus and pes (151.93 kN m
22
and
103.31 kN m
22
, respectively). These differing pressures
imply that substrates of different C
u
are required to sup-
port the loads, which in turn creates a range of
substrates (C
u
¼20–40 kN m
22
), where the manus
causes the substrate to fail, but the pes does not. In
such substrates, if underlain by a firmer layer, only
the manus will generate tracks. This is the same mech-
anism as described in detail by Falkingham et al.[28]
for sauropod manus-only trackways. It is interesting
to note that the resulting pressure beneath the pes
when bipedal locomotion is assumed is less than for
the manus in a quadrupedal mode of locomotion.
Depending on the validity of the mass, CM and foot
outline input parameters used here, this may support
the hypothesis that differing modes of locomotion may
have been used for traversing different substrates. The
values employed in this study would suggest quadru-
pedalism to be potentially more advantageous on
firmer substrates, where the manus will not sink,
while a bipedal mode of locomotion would allow traver-
sal of softer substrates. However, this suggestion is
based only on underfoot pressures, and further factors
such as stability will ultimately determine gait. It
may therefore be unwise to infer otherwise unknown
locomotor styles from trackways in which the substrate
conditions at the time of track formation cannot be con-
strained. The reader is directed to Wilson et al.[58] for
discussion of a trackway in which an ornithopod track-
maker transitions between bipedal and quadrupedal
gaits as substrate changes.
As described above for the Edmontosaurus track
simulations, and as described by Falkingham et al.
[28], there are a range of substrates in which only the
manus, and not the pes, of Brachiosaurus produce
enough pressure to deform the substrate. When the
subsurface undertracks generated by manus and pes
are visualized, important features can be observed.
The bowl-like form of the Brachiosaurus pes
(figure 7) is reminiscent of a number of reported
sauropod tracks (e.g. [59,60]). The simulations here
indicate that such a bowl-like form is potentially
characteristic of undertracks. This is consistent with
the assumption that the plantar surface of the foot
was approximately flat, given that the shape of foot
required to form bowl-like tracks would prove unstable
on firm ground.
When the Brachiosaurus manus track is observed as
a subsurface undertrack at approximately 0.2 m depth,
a ridge running transversely across the track can be seen
(figure 8a,b). This ridge appears superficially similar to
the undulating track surface hypothesized to result
from three-phase movement of the foot [16,61]. How-
ever, with full control of all input variables, it is
known that in this case the loading was carried out in
an entirely vertical manner, evenly distributed through
a flat foot, and so the ridge cannot be a function of limb
kinematics or foot anatomy. Instead, this ridge is
produced through the displacement of sediment accord-
ing to Prandtl theory [10]. As substrate is deformed by a
load, it is pushed down and out from beneath the inden-
ter (figure 8c). The base of the actively deforming zone
of substrate undulates against the rigid, non-moving
zone [13]. A cross section through this area results in
a subsurface track containing a ridge of non-deformed
substrate. This effect is seen in the Brachiosaurus
manus track because of the round shape of the indenter.
This ridge is very subtle, and its absence from the other
simulated tracks implies that its formation is closely
linked with indenter morphology. Dynamic limb
motion consisting of more complex, non-vertical loading
will deform this ridge accordingly (e.g. see theropod
track in [16]); however, fossil track evidence indicates
that, as is the case here, sauropods placed the manus
vertically, at least when traversing soft substrates [62].
Further study is required to fully understand
this phenomenon and to avoid erroneous interpretation
of tracks with undulating bases. Note that some
previously described ridges in sauropod tracks
(e.g. [63]) appear more defined and morphologically
different to those outlined here, and we do not suggest
this mode of ridge formation for those cases.
surface
0.06 m
0.12 m
0.2 m
0.4 m
Figure 7. Series of undertracks as generated by the Brachio-
saurus pes, seen in isometric view. Note the bowl-like form
of successive undertracks, as compared with the flat interior
and distinct outlines of the uppermost tracks. Darker shading
represents deeper parts of the track.
1150 The ‘Goldilocks’ effect P. L. Falkingham et al.
J. R. Soc. Interface (2011)
Both theropod tracks (Struthiomimus and Tyranno-
saurus) indented to a considerably greater degree at the
posterior of the virtual foot (figure 9). The pes of the
Edmontosaurus also exhibited this feature, albeit to a
lesser extent. This effect is a function of the shapes of
the indenters as seen in Falkingham et al. [11]. The
appearance of a deeper posterior track portion under uni-
form loading of a flat indenter has important
consequences for interpretations of limb kinematics from
fossil tracks. Commonly, the morphology of real tridactyl
tracks is deeper beneath the distal areas of the digits as a
result of the increased pressure as the animal kicks off, but
in the simulated case there is no such loading regime. The
development of a deeper track beneath a larger, more
compact part of the foot would mean that the ‘two-
phase’ interaction of the foot (weight bearing and toe
off) described by Thulborn & Wade [61] could poten-
tially produce a track with the appearance of a ‘three-
phase’ foot–substrate interaction (which precedes the
above phases with touch-down), where the heel and
toes are deeper than the centre portion of the track
[16,64]. It has been proposed that the two-phase and
three-phase modes of locomotion represent knee-based
and hip-based retraction of the limb, respectively ([64]
and references therein), and that the associated pressures
across the foot differ accordingly. However, if vertical
loading, as has been used here, produces a deeper ‘heel’ in
tridactyl tracks without a ‘heel-down’ kinematic phase,
then attempting to differentiate between the locomotor
modes of theropod dinosaurs and birds from fossil tracks
may be considerably more difficult than has previously
been assumed without ongoing experimental studies.
The resultant tracks from these simulations are rel-
evant for studies of tracks where interpretations of
locomotion have been made based on the ‘pitch’ of the
track [57,65]. Such interpretations must consider as an
alternative, or at least confounding factor, varying shear
strength (as a function of water content) throughout the
total substrate layer at the time of track formation, result-
ing in track pitch altering systematically along a trackway.
Alternatively, variations in track ‘pitch’ may be influ-
enced by grain size or compositional differences, given
that sand responds in the opposite manner to mud [11],
allowing greater deformation beneath digits.
6. CONCLUSIONS
The simulation of tracks from a series of dinosaur taxa
ranging in size from 400 to 25 000 kg shows a linear
relationship between body mass and substrate shear
strength required to produce observable tracks. The
point of failure for a given track and subsequently the
shear strength of the substrate at the time of track for-
mation can be approximated by calculating the bearing
capacity required for a circular indenter of equal size
and load. Variations around this approximation are
due to the effects of foot shape.
Tracks of significant depth are not possible in homo-
geneous, cohesive substrates without the presence of a
firmer subsurface layer, because failure of the substrate
will result in the animal being unable to traverse the
area. A homogeneous cohesive substrate will only record
tracks from the largest animals that substrate can support
without failing. There is, however, a strong bias towards
tracks made by larger animals if there is a firmer substrate
beneath a softer layer. This Goldilocks effect means that
for a homogeneous substrate, loading conditions (that is,
the animal size, locomotion and foot morphology) must
be ‘just right’ in order for the animal to be able to traverse
the area but still form tracks. This has wide-ranging
implications for interpretations of palaeodiversity and
palaeoecology based on vertebrate track assemblages
preserved in lithified muds and silts.
anterior
(a)
(b)
(c)
anterior
Figure 8. (a) Isometric and (b) cross-section views of sauropod
manus undertrack at a depth of 0.2 m (track generated in sub-
strate of C
u
¼110 kN m
22
and halted when track depth
reached 0.05 m). Note the transverse ridge running medio-later-
ally through the track, appearing similar to the three-phase
track described by Manning ([16,fig.6a]; [64, fig. 12.7]).
Darker shading represents deeper parts of the track. (c)Theor-
etical displacement beneath a strip load in a cohesive substrate.
If a track is exposed in a layer corresponding to that marked,
tracks may appear to contain an internal ridge running across
thewidestpartofthetrack((c) modified from [13]).
A(a)(b)
A
A
P
PA
P
P
Figure 9. (a)Tyrannosaurus and (b)Struthiomimus tracks
viewed in plan and cross-section though digit III. Vertical dis-
palcement is greater at the posterior of the track owing to the
compact nature of the indenter in this area. Darker shading
represents deeper parts of the track.
The ‘Goldilocks’ effect P. L. Falkingham et al. 1151
J. R. Soc. Interface (2011)
Presence of small and large tracks indented to the
same depth on a single track-bearing surface (assuming
time-averaging/transmitted tracks can be accounted
for) offer the highest possibility of presenting a true rep-
resentation of faunal diversity in the area at the time of
track formation. Caution is strongly advised in making
any interpretations of faunal diversity or population
dynamics from track assemblages where all tracks
have been produced by similar-sized producers. Such
assemblages most likely represent a strongly biased
preservation, or an ‘instantaneous’ event. Substrates
which have dried out over relatively long time periods
will provide a fuller record of faunal diversity in the
area, but will be subject to biases and other erroneous
data associated with time-averaging.
Specific features regarding track and undertrack for-
mation have been noted for this range of indenters
based on dinosaur taxa. Bowl-like sauropod pedal
impressions may be indicative of being undertracks,
potentially of significant depth. Internal ridges may
form in tracks due to the vectors of displacement
beneath a uniform load (as in the Brachiosaurus
manus), or as a result of autopodia morphology causing
non-uniform displacement under uniform loading (as in
the tridactyl tracks). That these features can be formed
independent of limb kinematics is of great importance,
and highlights the need for further experimental work
to clarify the specifics of their formation.
The approach used here, of computer simulation using
FEA, has allowed the generation of tracks and associated
undertracks for a range of animal sizes that would be diffi-
cult to replicate using physical modelling. Employing
computational methods has also catered for constancy in
input variables between experiments, and has provided
the ability to easily and systematically manipulate those
variables. We recognize that this study makes a number
of assumptions and simplifications in terms of loading,
and expect subsequent research to build on the methods
used here to produce more complex models. Many of
the conclusions and observations recorded here are related
to the mechanics of substrates under load, and we hope
that this will encourage further research into the effects
of complex limb kinematics and kinetics on track
formation, in light of the confounding geotechnical effects
described here.
P.L.F. and K.T.B. were funded by the Natural Environment
Research Council (NER/S/A/2006/14033 and NER/S/A/
2006/14101, respectively). FEA simulations were run on the
HPCx supercomputing service, using Engineering and
Physical Sciences Research Council grant EP/F055595/1,
awarded to L.M. We would also like to thank James Jepson
for commenting on an early draft of the manuscript, and
also Jeff Wilson and one anonymous reviewer for their
constructive, helpful comments.
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