Download The mass distribution along the neck was

“Some sauropods raised their necks –
Evidence for high browsing in Euhelopus zdanskyi”
Andreas Christian –
Electronic supplementary Material
In this supplement the basic data, equations and methods are presented that are used for reconstructing
the habitual neck posture of Euhelopus zdanskyi at rest and for calculating energy expenditures for
Euhelopus zdanskyi and Brachiosaurus brancai.
The method (Preuschoft-method) used for the reconstruction of the neck posture is outlined below. For
a detailed description of the method including tests on extinct and extant vertebrates see Christian &
Preuschoft (1996) and Christian (2002). Basic data on the dimensions, the body mass and the mass
distribution along the neck of Brachiosaurus are taken from Henderson (2004), Christian & Dzemski
(2007) and Gunga at al. (2008).
The neck, as well as the trunk and tail of a terrestrial vertebrate experiences forces and
torques, that are a function of the posture and the distribution of body mass. Bending moments along
the longitudinal body axis act primarily in a sagittal plane unless rapid lateral accelerations take place.
The pattern of bending moments along the vertebral column depends on mass distribution, posture,
and the forces exchanged between the animal and the substrate (ground reaction forces). In the neck,
bending moments are usually highest at the base and decrease towards the head. In a vertical position,
however, the neck experiences weak bending moments and weight forces are predominant.
As long as the neck is not orientated backwards, bending moments along the neck must be
counteracted at the intervertebral junctions by tension in epaxial muscles, tendons, or ligaments that
are located dorsal to the vertebral centra (Preuschoft 1976, Alexander 1985, Christian & Preuschoft
1996). A muscle, tendon, or ligament force Fm acting in a sagittal plane above the transverse axis of
an intervertebral joint produces a torque Fmh about that joint, where h is the lever arm of the force Fm.
The lever arm is the distance between the line of action of the force and the axis of the joint. The
transverse axis of an intervertebral joint can be assumed to pass through the centre of the intervertebral
disc (Preuschoft 1976, Alexander 1985). In most neck regions, the lever arms of the epaxial forces can
be estimated to be equal to the vertical distances between the centres of the intervertebral discs and
tips of the neural spines (Preuschoft 1976, Christian & Preuschoft 1996).
The pulling force, Fm, of the epaxial muscles produces a compressive force of the same
magnitude between the vertebral centra (Preuschoft 1976, Alexander 1985, Christian & Preuschoft
1996). This compressive force acts on the cartilage in the intervertebral joint. The muscle force, F m, at
a given position in the vertebral column can be calculated by Fm = constant ∙M/h (Preuschoft 1976,
Alexander 1985, Christian & Preuschoft 1996). M is the bending moment in the sagittal plane at the
position investigated. The total compressive force, F, acting on the intervertebral cartilage is the sum
of two components: first, the muscle force, Fm, due to the bending moment, M, as described above,
and second, the weight force, Fg, of the fraction of the neck cranial to the position investigated
multiplied by the cosine of the angle  between the plane of the intervertebral joint and the horizontal
plane (see Preuschoft 1976, Christian & Preuschoft 1996, Christian & Heinrich 1998). Thus:
F = Fm + Fgcos 
Forces in the neck
φ
Fg·sin φ
weight force Fg
lever arm of
muscle force
intervertebral joint
muscle force Fm
total compressive force
F = Fm + Fg·sin φ
Forces different from static or quasistatic forces are neglected, assuming, that forces due to
accelerations or other activities are not predominant. This assumption seems reasonable for sauropods
except the foremost neck region that might have been involved in quick movements for positioning the
head. This assumption has been corroborated in studies on long necked mammals (giraffes and camels,
Christian (2002)), despite the occasional use of the head for combat in these animals. Under the
assumption of equal safety factors, the highest regularly occurring compressive forces, F, acting on the
intervertebral cartilage along the neck should be proportional to the transverse cross-sections, A, of the
intervertebral joints (Preuschoft 1976, Christian & Preuschoft 1996, Christian & Heinrich 1998).
Consequently, the stress on the intervertebral cartilage, equivalent to the force F divided by the crosssectional area A of the intervertebral discs, should be constant along the neck. This assumption was
also shown to be reasonable (Christian 2002).
The cross-sectional area of the intervertebral cartilage is calculated by assuming an elliptical
shape of the joints, with the transverse and dorsoventral diameters of the cranial surface of the
adjacent vertebral centrum used as the major axes. In sauropod necks, caudal and cranial surfaces of
contiguous vertebral centra are approximately proportional, so that two sources of measurements are
available. The distance of each intervertebral joint from the occipital condyle is be measured along the
vertebral centra.
For various hypothetical neck postures, the compressive forces, F, is calculated along the neck
and divided by the cross-sectional areas, A, of the intervertebral joints. A hypothetical posture will be
rejected if the stress (F/A) is not approximately constant along the neck. The stress patterns are
conclusive without further statistical processing of the data. For a numerical comparison, the variation
in stress along the neck can be described by the ratio between the standard deviation (SD) of the stress
values and the mean stress.
The Preuschoft method was shown to be a robust and reliable instrument for the reconstruction
of the habitual neck posture of long-necked terrestrial vertebrates (Christian 2002). It can be applied
even if the distribution of mass along the head and neck and the lever arms of the neck muscles and
ligaments are only roughly estimated, because it is not affected by systematic errors in estimates of
segment masses, lever arms, muscle forces or cross-sectional areas of intervertebral discs. Therefore,
by using the Preuschoft method, it is of no importance whether reconstructions of the head and neck
are too heavy, or too light, or whether estimates of lever arms or cross-sectional areas are too high or
too low, as long as the errors are similar for different segments along the neck. However, with lower
estimates of neck mass, the head mass becomes more important. Lower estimates of the neck mass
compared to the mass of the head generally yield lower habitual postures of the neck. In the caudal
region of the neck, increased lever arms of epaxial forces have to be taken into account because of
muscles and ligaments that spun well above the neural spines.
For Euhelopus the mass distribution along the neck was reconstructed under the assumptions of an
elliptical cross-section, with the main axis measuring 1.2 times the height and 1.25 the width of the
skeleton, and a low neck density of only 0.6 gcm-3 as usually suggested for sauropods because of
apparently large air volumes in the necks (Wedel 2005, 2009; Henderson 2004, 2006; Wilson &
Upchurch 2009). Additional mass was added to the foremost neck section because of muscles that
connect the neck with the head. The volume of the head was approximated by an ellipsoid with a
density of 1.0 gcm-3. Body mass was assumed to equal 3.8 metric tons (Mazzetta et al. 2004).
Segment
Length [m]
Mass [kg]
head
0.39
13.0
Neck 1
0.13
2.75
Neck 2
0.15
2.75
Neck 3
0.225
4.0
Neck 4
0.26
5.75
Neck 5
0.27
6.5
Neck 6
0.28
7.5
Neck 7
0.285
8.5
Neck 8
0.3
10.0
Neck 9
0.3
12.5
Neck 10
0.33
15.5
Neck 11
0.32
16.5
Neck 12
0.31
17.0
Neck 13
0.3
18.5
Neck 14
0.3
22.0
Neck 15
0.24
22.5
Neck 16
0.214
25.0
Neck total
4.605
210.25
Estimated mass distribution along the neck of Euhelopus zdanskyi
For Brachiosaurus, the neck was also assumed to be very light with an overall mass of neck and head
of roughly 2.5 tons. The body mass was assumed to equal about 26 tons (Henderson, 2004). This is
less than the estimates by Mazetta et al. (2004) (39.5 tons) and Gunga et al. (2008) (38 tons) but
appears more reasonably in the light of recent reconstructions of body density (e.g., Wedel 2005,
2009; Henderson 2004, 2006).
Energy costs of transport Eloc are calculated by Eloc = 10,8 · m-0,32 J kg-1 m-1 (m: body mass, SchmidtNielsen 1984). The basal metabolic rate BMR is assumed to follow the equation BMR = 3,6 · m0,71
Watts for homoiotherms (Seymour 2009 derived from the analysis by White et al. 2009). Energy
expenditures due to an increased blood pressure are calculated according to Seymour (2009), assuming
an increase in the metabolic rate by 1% for each 10 mm Hg arterial blood pressure above a basic value
of 100 mm Hg. The mechanical work Wli for lifting the neck mass is assumed to equal the energy
costs for raising the body mass (Schmidt-Nielsen 1984): Wli = mN · h · 27,3 J kg-1 m-1 (mN: head and
neck mass, h: upward shift of the centre of mass of the neck-head system). Energy expenditures for
lowering the neck and for accelerating the neck were low and are therefore neglected. The possibility
of elastic storage in ligaments or tendons is not taken into account.
References
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