Annals of Botany 80 : 275–287, 1997
Relative Resistance of Hollow, Septate Internodes to Twisting and Bending
K A R L J. N I K L A S *
Section of Plant Biology, Cornell UniŠersity, Ithaca, New York 14853, USA
Received : 9 December 1996
Accepted : 2 April 1997
The aim of this paper was to examine the mechanical behaviour of hollow internodes with transverse nodal septa
subjected to bending and twisting and to determine the extent to which this behaviour agreed with predictions made
by the theory of elastic stability treating thin walled tubes or ‘ shells ’. This theory determined the experimental
protocol used in this study because it required the empirical determination of two important material properties of
stem tissues (i.e. the Young’s elastic modulus, E, and the critical shear stress, τ) and required the use of a dimensionless
grouping of variables as a descriptor of internodal shape (i.e. l #}2tR, where l is internodal length, t is wall thickness,
and R is external radius). All of these variables were measured for a total of 92 internodes (removed from the stems
of field grown plants from a total of six species) followed by correlation analyses to determine whether the ability of
internodes to resist twisting relative to bending (summarized by the quotient τ}E ) correlated with the shape descriptor
l #}2tR. Analyses of the data indicated that : (1) the extent to which nodal septa influenced internodal bending stiffness
declined as internodal length increased relative to wall thickness or external radius ; and (2) the ability to resist torsion
relative to the ability to resist bending declined as internodal length increased relative to wall thickness or external
radius. Both of these trends agreed well with the theory of elastic stability. Also, as theory predicts, the mechanical
behaviour of internodes was correlated better with the shape descriptor l #}2tR than with any measure of absolute
internodal size (e.g. l or t). Thus, internodal shape (in part defined by the spacing of nodal septa which influences l )
largely dictates the mechanical behaviour of stems subjected to twisting or bending.
# 1997 Annals of Botany Company
Key words : Hollow internodes, bending, torsion, elastic stability, biomechanics, plants.
INTRODUCTION
Comparatively few papers discuss the effects of nodal
transverse diaphragms or septa on the ability of otherwise
hollow stems to resist bending or torsion (Niklas, 1989 a,
1992 ; Spatz, Speck and Vogellehner, 1990 ; Schulgasser and
Witztum, 1992 ; Spatz, Boomgaarden and Speck, 1993 ;
Spatz and Speck, 1994 ; Spatz et al., 1995). Nevertheless,
many phylogenetically disparate taxa have converged on
hollow, septate stems (e.g. Asteraceae, Poaceae and
Equisetaceae), suggesting that this morphology confers
selective advantages under some environmental conditions.
What these advantages or selection pressures may be remains
problematic. Depending on the species, stems with hollow
internodes may provide a physiological advantage in terms
of carbon reallocation to meristematic regions during stem
growth (Carr and Jaffe, 1995), reduce the potential of frost
damage (Niklas, 1989 b), or foster beneficial symbiotic
relationships with insects (e.g. Fiala and Maschwitz, 1992 ;
see, however, Morrill et al., 1994).
Although these and other alternative explanations exist,
most authors favour the view that hollow stems provide a
mechanical advantage and that nodal septa act as mechanical braces to prevent local (Brazier) wall buckling (Spatz
et al., 1990 ; Niklas, 1992 ; Schulgasser and Witztum, 1992 ;
Spatz and Speck, 1994 ; Spatz et al., 1995). This view is
attractive not because hollow stems are necessarily cheaper
* Fax ­607 255 5407, e-mail kjn2!cornell.edu
0305-7364}97}090275­13 $25.00}0
(see Kull, Herbig and Otto, 1992), but because simple
calculations show that, for equivalent biomass, hollow
stems can grow 26 % taller than their solid counterparts and
because, for some species, nodal septa demonstrably stiffen
stems (Niklas, 1989 a, 1992).
Nevertheless, the mechanical behaviour of hollow, septate
stems is not well understood. In part this is attributable to
the complexity of the engineering theory treating thinwalled tubes or ‘ shells ’ (Timoshenko, 1923 ; Brazier, 1927 ;
Lundquist, 1932 ; Ades, 1957 ; Reissner, 1959 ; Seide and
Weingarten, 1961 ; Timoshenko and Gere, 1961 ; Gresnigt,
1986). This theory shows that the ability to resist bending or
torsion depends on the absolute magnitudes, as well as the
proportional relations, of many interdependent variables
that are often difficult to measure reliably. Indeed, small
measurement errors for any one of these variables can often
cascade into widely disparate estimates of the mechanical
performance of tubular structures—so much so, that
engineers often ignore the analytical solutions posited by
theory and rely instead on a strictly experimental approach.
Another concern about engineering theory is that many
of the assumptions made to derive analytical solutions may
be inappropriate for plants (see Nachtigall, 1994 for a
review). For example, the traditional theory of elastic
stability assumes that materials are isotropic in their
mechanical behaviour—i.e. materials are assumed to have
equivalent mechanical properties when they are loaded in
different directions (see Brazier, 1927 ; Timoshenko and
Gere, 1961 ; Cottrell, 1964 ; Gresnigt, 1986). In contrast,
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# 1997 Annals of Botany Company
276
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
many stem tissues differ in their mechanical behaviour when
they are pulled, compressed, or twisted in their radial,
tangential, or longitudinal planes of symmetry (Wainwright
et al., 1976 ; Hettiaratchi and O’Callaghan, 1978 ; Vincent,
1990 ; Niklas, 1992 ; Schulgasser and Witztum, 1992 ; Gibson
et al., 1995 ; Spatz et al., 1995). Mathematical tools exist to
cope with extremely anisotropic materials (see Hashin,
1965 ; Piggot, 1981 ; Guynn, Ochoa and Bradley, 1992 ;
Budiansky and Fleck, 1993 ; Karam and Gibson, 1995 ;
deBotton and Schulgasser, 1996 ; Schultheisz and Waas,
1996). However, they are rarely applied to botanical
structures. A final concern with engineering theory is that
size and shape are treated as essentially independent
variables, whereas stem size and shape may not be free to
vary independently because of plant growth and development. In summary, engineers and their theories often fail
to address the complexities typically facing biologists.
Clearly, one approach to evaluating the rigor of engineering theory as it relates to hollow, septate stems is to
compare its theoretical predictions to trends observed for
experimentally manipulated stems. This approach served as
the agenda of this paper whose objective was, first, to
measure empirically the variables predicted by theory to
influence the mechanical behaviour of thin walled tubes
with transverse diaphragms and, second, to compare the
observed trends among these variables with those predicted
by theory. A review of this theory (see Appendix) identified
the variables that had to be empirically measured, as well as
dictating the manner in which the data had to be analysed.
Specifically, the theory indicates that two material properties
of stem tissues are important : the Young’s elastic modulus,
E, and the critical shear stress, τ. Young’s elastic modulus
is a measure of a material’s stiffness in bending ; the critical
shear stress is a measure of the ability of a material to resist
twisting (Wainwright et al., 1976 ; Vincent, 1990 ; Niklas,
1992). From a biological perspective, these two material
properties are of special interest because virtually every
plant organ bends and twists simultaneously when
mechanically perturbed by an external force (Niklas, 1991 ;
Vogel, 1992, 1995 ; Ennos, 1993). Thus, it is reasonable to
suppose that the proportional relationship between the
critical shear stress and the Young’s elastic modulus of
internodal tissues may be more important than the absolute
values of these two material properties. For this reason, the
Young’s elastic modulus and the critical shear stress of stem
tissues were determined for internodes and were then used
to construct the dimensionless quotient τ}E.
In regard to bending stiffness, the influence of nodal septa
was evaluated by comparing the stiffness of stem segments
with intact and surgically impaired nodal diaphragms.
Nodal septa can act mechanically as transverse braces or
struts that restrict the ends of internodes from deforming,
and ultimately buckling, and thus increase the measurable
stiffness of internodes and intact stems. Surgical destruction
of nodal diaphragms by perforation with a needle should
thus reduce the effective stiffness of stem segments compared
to the same segments with intact nodal diaphragms.
A review of the engineering theory also indicated that
dimensionless groupings of ‘ structural ’ variables should
correlate well with τ}E. From a biological perspective, each
of these groupings of variables serves to describe the shape
of stem internodes (and indirectly the spacings of nodal
septa along the lengths of stems). But, according to theory,
the most important of these shape descriptors is l #}2tR,
where l is tube (internodal) length, t is wall thickness, and R
is external radius (see Appendix). Thus, special emphasis
was placed on the extent to which τ}E and l #}2tR were
correlated among the species examined in an attempt to
understand when, and if, nodal septa play an important
mechanical role in hollow stems.
MATERIALS AND METHODS
Biological materials
Stems from six herbaceous species with hollow internodes
were harvested during the summer of 1996 : Equisetum
arŠense, E. hyemale, Phleum pratense, Polygonum orientale,
Sambucus canadensis, and Triticum aestiŠum. The plants
used in this study grew under natural (field) conditions. The
only criteria used to select stems were that they had a
healthy and mature appearance and that they had straight,
unbent internodes.
Bending experiments
These experiments were used to determine the Young’s
elastic modulus, E, of stem internode tissues and to
determine the mechanical consequences of removing nodal
septa on the effective stiffness of internodal tissues.
This agenda required careful attention to the radius of
curvature used to bend a stem. The theory of elastic stability
(Timoshenko and Gere, 1961) states that the bending
moment, M, required to bend a straight thin-walled tube to
curvature, C, equals EIC, where I is the axial second
moment of area which is a measure of the contribution
made by cross sectional geometry to bending stiffness.
Because a tube’s wall will ovalize during bending, the axial
second moment of area of each transection through even a
slightly flexed tube equals 0±25π[a$b®(a®t)$(b®t)], where a
is the major semiaxis and b is the minor semiaxis of the
elliptical cross section, and t is the wall thickness of the tube
(Fig. 1 A). Thus, the Young’s elastic modulus of the materials
in the wall of a flexed tube can be determined from the
formula
M
.
(1)
E¯
0±25πC[a$b®(a®t)$(b®t)]
However, this elementary formula holds true only if the
radius of curvature, r (i.e. the reciprocal of C ), is large with
respect to the dimensions of the tube. In turn, this assumes
that the eccentricity, e, between the centroid and neutral
axes running the length of the flexed tube is negligible (see
Appendix). Consequently, it was necessary to calculate a
suitable radius of curvature based on the undeformed
transverse dimensions of each stem and to specify e such
that the assumption underpinning eqn (1) was not violated.
For the experiments reported here, the radius of curvature
was calculated by means of eqn (A 7) (see Appendix) for
which the difference between the centroid and neutral axes
was arbitrarily set as 5 % (i.e. e ¯ 0±05R).
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
A
l
2a
2b
t
r = 1/C
M = Pl
l
B
θ
t
l'
T = Pl'
R
F. 1. Diagrams illustrating experimental protocols used to test stem
internodes in bending and torsion. A, Protocol for bending internode
(with wall thickness, t) to curvature C (against a block of wood with
radius, r) by means of a mass-force P applied at the tip of the internode
with effective bending length l. The bending moment M, which is the
product of P and l, and deformations (determined by measuring the
major and minor axes, 2a and 2b, respectively) in representative
transverse cross sections were used to compute the Young’s elastic
modulus of stem tissues. B, Protocol for twisting an internode with
external radius R and wall thickness t to an angle θ by means of an
applied torque T, which equalled the product of a mass-force and its
lever arm l«. See text for further details.
Experimentally, stem segments were cut in the field and
their cut ends were coated with petroleum jelly to reduce
subsequent tissue dehydration. All stem segments were
examined in the laboratory within 24 h of being harvested.
Each segment consisted of a single internode flanked by its
two nodal septa and a portion of the adjoining internode at
either end (see Fig. 1 A). One end of the internode was firmly
clamped (by means of an adjustable rubber casket) to the
slightly concave rim of a vertically oriented circular block of
wood with uniform radius, r (see Fig. 1 A). The clamp was
applied to avoid overlap with the nodal septum at the end
of the segment and to avoid noticeable compression of the
internodal wall. Weights were then applied to a pan
suspended by a wire from the cantilevered free end of the
segment until the segment flexed snugly against the rim of
the circular block of wood. Thus, the curvature of flexure
equalled the reciprocal of the radius of the wooden block
(i.e. C ¯ 1}r). The bending moment (in units of N m)
required to achieve this curvature of flexure equalled the
product of the applied mass-force, P (in units of N), and the
free-length, l, of the segment (in units of m). Segments
277
whose walls creased or buckled locally during a bending
experiment were rejected.
To determine the axial second moment of area, I, of the
flexed stem segment, the major and minor axes (2a and 2b,
respectively ; see Fig. 1 A) were first measured at each end
and at the mid-length of the segment with a microscope
equipped with an ocular micrometer. The segment was then
unloaded, removed from the surface of the block of wood,
and free-hand sectioned at the mid-length and at each end
to measure the wall thickness, t, of the internode with a
microscope equipped with an ocular micrometer. These
three free-hand sections were made after the stem segment
was subsequently tested in torsion (see below). Averaged
values for a, b, and t (in units of m) were used to compute
an average I which in turn was used to compute E by means
of eqn (1). (The ovalization of internodal walls is not
constant along the length of an internode flanked by two
end-restrains. In general, the ovalization is expected to be
greater near the mid-length than at either end of the
internode, and thus the axial second moment of area of the
internode computed on the basis of averaged measurements
of a and b will be over-estimated. However, because the
curvatures of bent internodes were small and because the
radial and tangential deformations of the segment with
respect to the plane of bending were also very small (see
Appendix), this over-estimate of I was considered trivial.)
Data were rejected for any stem segment that failed to
elastically restore its original external transverse dimensions
after unloading.
The influence of nodal septa on the tissue stiffness
(Young’s elastic modulus) of segments was determined by
removing both nodal septa from the segment with a razor
blade. Each segment was then re-bent in the manner
previously described. Calculations for determining E compensated for the reduction in the free-length of each segment
due to the removal of the septa and the flanking portions of
adjoining internodes. Once again, data were rejected from
internodes that failed to elastically restore their original
external transverse dimensions after unloading.
Twisting experiments
The critical shear stress, τ, is the torque force, normalized
with respect to the surface area through which this force
acts, which produces a permanent (inelastic) deformation.
Elementary elastic theory indicates that the critical shear
stress is given by the formula τ ¯ RT}J, where R is the
external radius of the tube, T is the moment of torque (the
product of an applied load and its lever arm which produces
a given angle of twist θ), and J is the polar second moment
of area, also called the ‘ torsional constant ’ (see Niklas,
1992). For a hollow tube with wall thickness t, J ¯
π[R%®(R®t)%]}2. Thus, the critical shear stress measured in
torsion is given by the formula
τ¯
2RT
.
%
π[R ®(R®t)%]
(2)
This stress was determined for segments previously tested
in bending by removing both nodal septa with a razor blade
278
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
and inserting wooden pegs lightly coated with a fast-drying
glue into the lumens of both cut ends of a segment. Each peg
was specifically fabricated for each stem segment to provide
a snug fit by first measuring the lumen diameters with a
microcaliper. During each twisting experiment, the entire
span of the internode was supported by the slightly concave
surface of a horizontally oriented wooden brace. One of the
two wooden pegs was firmly held in the jaws of a small vice.
Weights were then gradually added to a container suspended
at the end of a cantilevered beam attached at a right angle
to the wooden peg at the free end of the specimen (see Fig.
1 B). The magnitude of the torque force was gradually
increased and the angle of twist θ was simultaneously
measured by means of a protractor sighted through a low
power magnifying glass. The moment of torque resulting in
the first noticeable ‘ crease ’ or ‘ crimp ’ in the twisted
segment was recorded. The applied force at the end of the
cantilevered beam was then slightly reduced, and the
instantaneous (external) transverse dimensions of the deformed segment were measured with a microcaliper. Data
were collected only from segments that failed to fully
elastically rebound to their original transverse dimensions.
As noted, the average wall-thickness (based on three
transverse free-hand sections), measured with a microcaliper, was used to determine the polar second moment of
area J.
The degree to which transverse tissue heterogeneity
resulted in tissue shearing at the surface of the pegs inserted
into stem segments was not evaluated in these experiments.
Typically, the stiffer tissues of internodal walls (e.g.
sclerenchyma) were located near the epidermis, while less
stiff tissues (e.g. parenchyma) were located near or at the
surface of the central pith cavity. If the inner tissues twist to
a greater degree than the outer tissues, then the angle of
twist measured at the surface of internodal walls will be
slightly underestimated. This bias was not evaluated,
although inspection of the ends of internodes at the
completion of twisting experiments revealed no obvious
tissue shearing failure.
Although the relationship between the torsional shear
modulus, G (i.e. a measure of tissue stiffness measured in
torsion), and the Young’s elastic modulus of internodal
tissues was not the focus of this study, the former was
determined for each internode tested in twisting based on
the formula
Tl
2Tl
.
(3)
G¯ ¯
%
θJ θ[R ®(R®t)%]
Because this formula assumes that the proportional (elastic)
limits of stem tissues were not exceeded, and because the
protocol used to determine the critical shear stress (see
above) violated this assumption, the values reported for G
are undoubtedly under-estimates of the actual torsional
shear moduli of stem segments.
Poisson’s ratios
The theory of elastic stability shows that the material
property called the Poisson’s ratio influences the mechanical
behaviour of thin walled tubes (see Appendix). Symbolized
by Š, this material property is expressed as the ratio of
negative lateral strain to the strain measured in the direction
of the applied force (see Niklas, 1992, p. 546). For a planar
section with two principal axes of symmetry, two Poisson’s
ratios can be calculated. For isotropic materials, these two
ratios have equivalent magnitudes. For anisotropic
materials, such as the majority of plant tissues, these ratios
can take on widely different values (see Schulgasser and
Witztum, 1992).
However, for the purposes of this study, the effect of
Poisson’s ratios on the interpretation of experimental
results was assumed to be minimal because the values of
(1®Š#) and (1®Š#)"/#, which figure prominently in equations
governing the behaviour of thin-walled tubes in torsion [see
Appendix, eqns (A 21) and (A 22)], do not differ significantly
from unit. For example, a reasonable estimate for many
types of plant tissues is that Š ¯ 0±3 (Niklas, 1992 ;
Schulgasser and Witztum, 1992). If so, then (1®Š#) ¯
0±91 and (1®0±3#)"/# ¯ 0±95. Thus, throughout this study,
the value of (1®Š#) or (1®Š#)"/# was set equal to unity.
Data analyses and presentation
A total of 645 stem segments were tested mechanically.
However, data for both the critical shear stress and the
Young’s elastic modulus of stem tissues were gathered only
for a total of 92 stem segments because the majority (i.e.
86 %) of the segments tested failed to elastically retrieve
their original dimensions in bending experiments or elastically returned to their original dimensions after being
twisted. Because the engineering theory evaluated in this
study places special emphasis on the relationship between
τ}E and l #}2tR, the data from only these 92 stem segments
were analysed statistically.
Although none of the variables measured in this study
could be assumed to be independent, the relationships
among variables were determined by ordinary least squares
regression (Model I) analyses whenever the objective was to
provide an estimate of a y-variant based on a specified value
for an x-variant. However, when the objective was to
determine the scaling (allometric) relationship between two
variables, reduced major axis regression and correlation
(Model II) analyses were used. This statistical protocol was
especially important in evaluating whether the autocorrelated relationship between experimentally determined
values for (l}t)#(τ}E ) and l #}2tR agreed with theoretical
predictions (see Fig. 7). Ordinary least squares regression
analysis was first used to determine the scaling exponent,
αOLS (the slope of the regression curve), and the correlation
coefficient, r, for the relationship between (l}t)#(τ}E ) and
l #}2tR, after which the scaling exponent for the equivalent
reduced major axis regression curve, α, was computed from
the formula α ¯ αOLS}r (see Niklas, 1992). The value of α
was then compared to that predicted by theory [see Appendix
eqns (A 23) and (A 24)] within the size ranges observed for
(l}t)#(τ}E ) and l #}2tR. All regression analyses and statistical
tests used the software package JMP# (version 3, SAS
Institute Inc., Cary, NC, USA).
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
30
279
1.0
25
Ewo = Ew
0.9
0.8
Ewo /Ew
Ewo (GN m–2)
20
15
0.7
10
0.6
5
0
5
10
15
20
25
0.5
102
30
103
RESULTS
105
l /2tR
Ew (GN m )
F. 2. Young’s elastic moduli of internodes whose nodal septa were
removed (Ewo) plotted against the Young’s elastic modulus of the same
internodes with intact nodal septa (Ew). Solid line indicates the
regression curve obtained from ordinary least squares regression
analysis.
104
2
–2
F. 3. The effect of removing nodal septa on the bending stiffness of
internodes as shown by plotting the dimensionless quotient of the
Young’s elastic moduli of internodes whose nodal septa were removed
(Ewo) and the Young’s elastic modulus of the same internodes with
intact nodal septa (Ew) (i.e. Ewo}Ew) against the internodal shape
descriptor l #}2tR, where l is internodal length, t is wall thickness, and
R is external radius (see Fig. 1). Data taken from Fig. 2. Solid line
indicates the regression curve obtained from ordinary least squares
regression analysis.
Bending experiments
5
4
G (GN m–2)
Ordinary least squares regression and correlation analyses
of log -transformed data from a total of 92 stem segments
"!
indicated that the Young’s elastic moduli of internodes with
and without their nodal septa (i.e. Ew and Ewo, respectively)
did not correlate with internodal external radius, R, or wall
thickness, t. For example, regression of Ew against R gave
r# ¯ 0±003, while Ew against t gave r# ¯ 0±007. In contrast, the
elastic moduli of internodes with and without nodal septa,
correlated with internodal length, l, at the 1 % level (r# ¯
0±32). The magnitudes of Ew and Ewo were far more
dependent on internodal shape than on individual measures
of internodal size. For example, regression of log "!
transformed data gave Ew ¯ 75 (l}t)−!±$% (r# ¯ 0±45) and
Ew ¯ 33 (l}R)−!±#) (r# ¯ 0±49), both of which were significant
at the 1 % level. Similar negative correlations were found for
the relationship between Ewo and l}t and between Ewo and
l}R, indicating that the elastic moduli of internodes with
and without nodal septa decreased as internodal length
increased with respect to wall thickness or external radius.
Values for Ew and Ewo were linearly and positively
correlated with one another. Specifically, ordinary least
squares regression analyses indicated Ewo ¯®0±12­0±80 Ew
(r# ¯ 0±96) (Fig. 2). Although this simple linear relationship
suggested that the removal of nodal septa resulted, on
average, in a 20 % reduction in stiffness, the magnitude of
this reduction was clearly dependent on internodal shape.
This was evident when Ew}Ewo was plotted against the shape
parameter l #}2tR (Fig. 3). Regression analyses of the data
indicated a strong and positive correlation between these
two variables (r# ¯ 0±66). Noting that the removal of nodal
septa has little effect on stiffness when Ew}Ewo E 1 and more
effect as Ew}Ewo gets progressively smaller, inspection of the
3
2
1
0
5
10
15
Ew (GN m–2)
20
25
F. 4. Shear modulus measured in torsion (G) plotted against the
Young’s modulus of the same internodes with intact nodal septa (i.e.
Ew). Solid line indicates the regression curve obtained from ordinary
least squares regression analysis.
bivariate plot indicated that the removal of nodal septa had
little or no effect on the stiffness of long and thin walled or
broad internodes and significantly more effect on the stiffness
of short and thick walled or narrow internodes (Fig. 3).
The effect of the removal of septa on internodal stiffness
was not significantly correlated with other internodal shape
parameters. For example regression of Ew}Ewo against t}R,
l}t, or l}R gave r# ¯ 0±000, 0±001 and 0±002, respectively.
280
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
A
10–1
10–2
100
τ /E
τ (GN m–2)
100
101
10–2
10–3
102
102
Ew (GN m–2)
10–1
B
F. 5. Critical shear stress (τ) plotted against the Young’s modulus of
the same internodes with intact nodal septa (i.e. Ew). Solid line indicates
the regression curve obtained from ordinary least squares regression
analysis.
–1
103
l /t
10
The Young’s elastic and shear moduli of internodes were
positively and linearly correlated with one another (Fig. 4).
Regression of G against Ew gave the linear regression
formula G ¯®0±01­0±20 Ew (r# ¯ 0±88, n ¯ 92), indicating
that, on average, the magnitude of the shear modulus was
20 % that of internodes with septa. (Because G was measured
for internodes with pegged ends, a legitimate comparison
between G and Ewo was not possible.) Ordinary least squares
regression analyses of log -transformed data also indicated
"!
that the shear modulus was significantly and negatively
correlated with three shape parameter : G ¯ 16 (l}t)−!±$&
(r# ¯ 0±31), G ¯ 7±1 (l}R)−!±$! (r# ¯ 0±31) and G ¯ 10
(l #}2tR)−!±") (r# ¯ 0±34). Based on its coefficient of correlation, the shape parameter l #}2tR explained more of the
variance in G than any other shape parameter. In contrast,
the shear modulus was not correlated significantly with any
measure of absolute internodal size. Owing to the correlation
between G and E, the shear modulus was not expected to
correlate significantly with internodal external radius, R, or
wall thickness, t. Indeed, regression of G against R gave
r# ¯ 0±010, while G against t gave r# ¯ 0±009. However, as
expected, G did correlate with internodal length, l, at the
1 % level (r# ¯ 0±33).
The critical shear stress, τ, was positively correlated with
the Young’s elastic moduli of internodes with nodal septa
(Fig. 5). Ordinary least squares regression of the log "!
transformed data showed that τ ¯ 0±004 E"w±%' (r# ¯ 0±62).
The critical shear stress was negatively and significantly
correlated with the shape parameters l}t and l #}2tR.
Specifically, ordinary least squares regression of log "!
transformed data gave τ ¯ 19 l}t−!±)( (r# ¯ 0±67) and τ ¯ 2±9
±
−
l #}2tR ! $& (r# ¯ 0±45). Although a comparatively small
amount of the variance in τ was explained by these
correlations, the statistical relationship between τ and each
of the two shape parameters was significant at the 1 % level.
τ /E
Twisting experiments
10–2
10–3
10–1
100
t /R
F. 6. Dimensionless quotient of the critical shear stress (τ) and the
Young’s modulus of the same internodes with intact nodal septa (i.e.
τ}E ) plotted against two dimensionless groupings of variables that
describe internodal shape : l}t (internodal length divided by wall
thickness ; A) and t}R (wall thickness divided by external radius ; B).
Solid lines indicate the regression curves obtained from ordinary least
squares regression analyses.
Relationship between τ}E and internodal shape
The dimensionless quotient of the critical shear stress and
the Young’s elastic modulus, τ}E, was significantly
correlated with internodal shape (Fig. 6). Specifically,
regression of log -transformed data showed that τ}E was
"!
negatively correlated with l}t (r# ¯ 0±45) and positively
correlated with t}R (r# ¯ 0±47). Thus, the critical shear
modulus decreased relative to the Young’s elastic modulus
as internodal length increased with respect to wall thickness,
while the shear modulus increased relative to the elastic
modulus as wall thickness increased with respect to external
radius. These two relationships were summarized when τ}E
was plotted against l #}2tR. The resulting negative correlation was statistically robust at the 5 % level (r# ¯ 0±57)
and indicated that the critical shear stress relative to the
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
104
(l}t)# (τ}E ) and l #}2tR (Fig. 7). Specifically, reduced major
axis regression, which was required because neither of the
two parameters could be assumed to independent variables,
gave the regression curve (l}t)# (τ}E ) ¯ 1±9 (l #}2tR)!±(# (r# ¯
0±93 ; n ¯ 92). Within the size ranges of 10# % (l}t)#
(τ}E ) ! 10% and 10$ % (l #}2tR) ! 10&, the standard error of
the scaling exponent 0±72 was ³0±02. Within the same size
ranges, the scaling exponent predicted from engineering
theory was 0±72³0±03 [see Appendix eqns (A 23) and
(A 24)]. Thus, the empirically observed trend between (l}t)#
(τ}E ) and l #}2tR was statistically indistinguishable from
that predicted by the engineering theory treating the
mechanical behaviour of thin-walled tubes.
A
103
102
(l/t)2 (τ /E)
101
Eqn (A23)
Eqn (A24)
100
104
101
102
103
104
105
B
α = 0.72
3
10
Eqn (A23)
Eqn (A24)
102
103
281
104
105
2
l /2tR
F. 7. Comparisons between observed trends in experimental data and
theoretical expectations of engineering theory treating the mechanical
behaviour of thin walled tubes [see Appendix, eqns (A 23) and (A 24)].
The expected relationships between the variables (l}t)# (τ}E ) and
l #}2tR are shown by thin curved lines in both graphs ; the observed
relationship between experimentally determined values for these
variables is indicated by the reduced major axis regression curve (the
dark line, B). Comparison of experimental data with theoretical
expectations over a larger size range than occupied by the data (A)
reveals a nonlinear log-log relationship between (l}t)# (τ}E ) and l #}2tR
within the lower size range. Comparison of experimental data with
theoretical expectations within the size range occupied by the data (B)
reveals a linear log-log relationship between (l}t)# (τ}E ) and l #}2tR.
Within this size range, the slope α of the reduced major axis regression
curve predicted by theory equals 0±72 ; the slope of the regression curve
for the data equals 0±72.
Young’s elastic modulus decreased as internodal length
increased with respect to either wall thickness or external
radius. No statistically significant correlation was observed
between τ}E and the absolute size of internodes. Specifically,
regression of τ}E against t, l, and R gave r# ¯ 0±000, 0±001,
and 0±003, respectively.
A strong and positive correlation was observed between
the autocorrelated dimensionless groupings of variables
DISCUSSION
The data gathered during this study do not permit a
legitimate discussion of the mechanical behaviour of entire
stems because only individual internodes were mechanically
tested and because it is reasonable to suppose that
neighbouring internodes along the lengths of intact stems
interact in mechanically complex ways (Niklas, 1989 a ;
Speck, 1994). Another concern with drawing generalizations
about stems is that only six species were mechanically
tested. Thus, the results of statistical analyses were probably
influenced by the choice of taxa (i.e. phyletic correlative
affects). Nevertheless, statistically robust correspondences
were observed between empirically observed and predicted
trends. This similitude indicates that the mechanical
behaviour of intact stems can be appreciated at least
qualitatively across a reasonably broad taxonomic spectrum
of species.
The most important conclusion that can be drawn from
this study is that the mechanical behaviour of hollow,
septate stems is correlated more with internodal shape than
with the absolute length, wall thickness, or external radius
of internodes. Here, ‘ shape ’ refers to any natural variable or
grouping of variables that derives meaning solely from the
physical system it describes, rather than from one or more
external (and thus arbitrary) standards for measuring size
(Ipson, 1960 ; Bookstein, 1978 ; Niklas, 1994). Although a
number of dimensionless groupings of variables can be used
to describe internodal shape (of which many are found to
correlate well with internodal resistance to torsional or
bending moments ; see Spatz et al. 1993 ; Spatz and Speck,
1994), the most consequential shape descriptor was found
here to be the quotient of the square of internodal length
and twice the product of internodal wall thickness and
external radius (i.e. l #}2tR). This descriptor of internodal
shape has the ability to summarize the simultaneous
influence of all three morphological features on the ability
of an internode to cope with the absolute magnitudes of
bending or twisting forces. It is easily intuited that, all other
things being equal, a short tube will resist twisting or
bending more than its longer counterpart, or that a thin
walled tube will twist or bend more easily than one with a
thicker wall (see Spatz and Speck, 1994 for experimental
confirmation). In contrast, it is difficult to predict the effects
of co-variation in length, wall thickness, and external radius
282
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
on the ability of internodes to resist twisting or bending.
Nevertheless, the data presented here indicate that the
largely unfamiliar dimensionless grouping of variables
l #}2tR provides a reasonable predictor of mechanical
behaviour.
Dimensionless groupings of variables were also more
successful in predicting the ability of hollow internodes to
cope with a twisting force relatiŠe to a bending force than
any of the three measures of internodal absolute size. This
is useful because most plant organs twist as they bend in a
manner that can shed externally applied loads such as
falling rain or snow or reduce wind drag (Schwendener,
1874 ; Vogel, 1989, 1992 ; Ennos, 1993 ; Niklas, 1996). Once
again the shape descriptor l #}2tR was found to be the most
useful dimensionless grouping of morphological variables in
predicting the relative ability of internodes to cope with
torsion relative to bending.
Another significant finding is that the mechanical influence
of nodal septa on the bending stiffness of hollow internodes
depends more on the shape than on the absolute size of
internodes. This resonates with previous reports that show
that the maximum bending moment and the mode of
mechanical failure of internodes depend on their length,
wall thickness, and external radius (Spatz et al. 1990, 1993 ;
Spatz and Speck, 1994 ; see also Mattheck, Bethge and
West, 1994). Once again, the shape parameter l #}2tR was
found to be an important predictor of the contribution
nodal septa make to bending stiffness. Specifically, the data
reported here indicate that nodal septa confer little or no
mechanical benefit to long internodes with either thin walls
or large external radii, whereas transverse diaphragms can
contribute up to 55 % of the effective stiffness of short
internodes with either thick walls or large external radii.
Curiously, although a correlation was expected between
internodal wall thickness and external radius, none was
found for any paired comparison between wall thickness,
external radius, or internodal length. Exploratory analyses
of co-variance did indicate that internodal length, wall
thickness, external radius, and tissue stiffness are highly
correlated (data not presented). This suggests that stem
growth and development can attain internodal shapes that
are mechanically efficacious providing that the material
properties of stem tissues are developmentally permitted to
co-vary with morphological variables.
The usefulness of l #}2tR in predicting mechanical performance, however, must be approached cautiously. Despite
their statistical robustness, the correlations presented here
had comparatively low coefficients of correlation, indicating
that only a small portion of the variance in the data was
explained. One possibility for this is that small error
measurements of internodal dimensions (especially length,
which is squared in the shape descriptor) or of the elastic
deformations resulting from mechanical loadings (which are
difficult to measure in the elastic range of material
behaviour) typically cascade into large over- or underestimates of mechanical performance. This explanation is
consistent with the engineering theory underpinning this
study. An alternative, but not mutually exclusive explanation, which is likewise consistent with the engineering
theory, is that very small ‘ structural imperfections ’ in
internodal walls result in comparatively large deformations
during mechanical testing. Both of these explanations are
likely and both highlight the difficulty in achieving accurate
predictions of the mechanical behaviour of thin walled
biological structures.
This and prior research into the mechanical behaviour of
hollow, septate internodes suggests that the considerable
morphological and anatomical heterogeneity normally seen
along the lengths of intact stems can evoke substantial
differences in the mechanical performance of different parts
of the same stem. For example, the internodes of the plants
examined during this study were typically shorter (and thus
nodal septa were more closely spaced) nearer the base than
the tip of stems. This pattern is predicted to result in a
basipetal increase in bending stiffness from the tip to the
base of stems and an opposing acropetal increase toward
the tip in the ability to accommodate large torsional
moments without incurring mechanical failure. These
opposing longitudinal trends could be advantageous by
permitting a stem to simultaneously cope with a large
bending moment exerted at its base and potentially large
torsional moments resulting from wind at its free, distal end.
Much more research is required to fully comprehend the
manifold tasks plant stems must perform and the ways in
which these tasks are performed before generalizations
about the mechanical significance of hollow, septate stems
are forthcoming. Future refinements of engineering theory
and experimental protocols are needed, especially those that
take into account normally occurring variations in wall
thickness and tissue material properties along the lengths of
individual internodes. An important aspect that has received
little or no attention in the literature is the influence of the
morphology and size of nodal septa on the torsional and
bending stiffness of stems. This feature was not examined
here, yet it is reasonable to suppose that thick or stiff nodal
diaphragms provide greater resistance to local (Brazier) wall
buckling than thin or flexible nodal septa. Also, current
engineering theory and practice indicate that tubes reinforced with circumferentially thickened ‘ flanges ’, which
mimic the appearance of plant stems with ‘ swollen
internodal joints ’, are far stiffer in bending and torsion than
their counterparts lacking flanges. This aspect of stem
morphology has received scant attention. What can be said
with some confidence at this time is that nodal septa play an
important mechanical role for comparatively short internodes with either thick walls or large external radii, but have
little mechanical consequence when they are sparsely spaced
along the lengths of hollow stems with thin or narrow walls.
A C K N O W L E D G E M E N TS
The author gratefully acknowledges Thomas Speck
(Botanischer
Garden,
Albert-Ludwigs-Universita$ t),
Dominick J. Paollilo, Jr. (Section of Plant Biology, Cornell
University), and an anonymous reviewer for their constructive and thoughtful comments on drafts on this paper
and especially for noting an egregious typographical error
in the equations pertaining the second moment of area for
an elliptical cross section.
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
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APPENDIX
Radius of curŠature in bending experiments
The flexural (bending) stresses that develop in a slightly
bent stem can be determined using the elementary flexure
formula M ¯ EIC, where M is the bending moment, E is
Young’s elastic modulus (i.e. the quotient of bending stress
and strain), I is the axial second moment of area, and C is
the curvature of the bent stem which is the inverse of the
radius, r, of curvature (See Fig. A1). However, this
elementary formula does not hold true when r is small with
respect to the dimensions of a bent stem because, under
these circumstances, the centroid axis (i.e. the axis running
the length of a stem defined by the centre of mass of each
transverse cross section) and the neutral axis (i.e. the axis
running the length of a stem defined by where bending stress
levels equal zero) do not coincide. Thus, to use the
elementary formula, r must be adjusted to the dimensions of
the stem to assure that the eccentricity, e, between the
centroid and neutral axes is small.
284
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
A
100
2R
t = 0.3 cm
0.2
z
z
B
a
c
2a
e
r
2b = (1 – ξ)2R
e
b d
rn
Radius of curvature, r (m)
y
x
0.1
0.005
x
rc
r
z
rn rc
10–1
100
101
R/t = γ
v
C
u
y
F. A2. Radius of curvature r plotted against the dimensionless shape
descriptor R}t for thin walled tubes with different wall thicknesses t.
The linear log-log lines are computed based of eqn (A 7).
x
w
z
a
where A is unit cross-sectional area. Because the integral
must vanish, it also follows that
t /2
c
b
Ny
Qy
Nx
Nxy
Nyx
1
rc
Qx
d
F. A1. Geometry and mechanical notation for a flexed thin walled
tube whose length, width, and depth are measured in the Cartesian
coordinate system x, y, z. A, Longitudinal and transverse sections
through an unflexed tube (internode) with length l, wall thickness t, and
external radius R. B, Representative longitudinal and transverse
section through a flexed tube whose uniform radius of curvature equals
r. In longitudinal section, the flexed tube has a centroid axis of radius
rc and a neutral axis (where bending stresses equal zero) with radius rn.
The difference (‘ eccentricity ’), e, between the central and neutral thus
equals rc®rn. In transverse section, the flexed tube has an elliptical
cross sectional geometry whose major axis 2a is in the plane of bending
and whose minor axis 2b is normal to the plane of bending. The minor
axis 2b of each elliptical transection, the elastic deformation ξ due to
flexure, and the external radius R of the unflexed tube are related such
that 2b ¯ (1®ξ)2R ; the major axis 2a of each elliptical transection, the
elastic deformation ξ due to flexure, and the external radius R of the
unflexed tube are related such that 2a ¯ (1­ξ)2R (not shown). C,
Representative portion of the wall of a flexed tube (see section abcd in
B) exhibiting longitudinal, tangential, and radial deformations (u, Š, w)
resulting from normal and radial shearing forces per unit distance (N
and Q, respectively) in the middle surface (t}2) of the shell-like wall.
For further details, see Appendix. Adapted from Timoshenko and
Gere (1961).
Referring to Fig. A 1, e equals the difference between the
radius of curvature of the centroid axis (rc) and the radius of
curvature of the neutral (rn). That is, e ¯ rc®rn. Adopting
the notation z ¯ rc®r, such that z is measured inward from
the centroid axis toward the neutral axis, it follows that
z®e
®1 dA ¯ & 0
dA,
& 0rr ®11 dA ¯ & 0rr ®e
r ®z1
®z 1
n
area
c
area
c
area
c
(A 1)
&
area
zdA
e
¯
rc
z
1®
rc
0 1
&
area
dA
.
z
1®
rc
0 1
(A 2)
Replacing the denominator in eqn (A 2) with the expansion
series
z −"
z
z #
z $
¯ 1­ ­
­
­ …,
(A 3)
1®
rc
rc rc
rc
0 1
01 01
and using only the first two terms in this series, eqn (A 2)
takes the form
& z 01­rz 1 dA Ee & 01­rz 1 dA.
area
c
area
(A 4)
c
Because the integral of z[dA is zero and because the integral
of z#[dA equals the second moment of area, I, the eccentricity
of the neutral axis is given by the approximate formula
eE
I
,
rc A
(A 5)
where A is now the total area of each transverse cross
section.
When a stem with hollow cylindrical internodes with
external radius R and wall thickness t is bent to a radius of
curvature r, the resulting elliptical cross sections through
the stem with a major semiaxis a and a minor semiaxis b
have I ¯ 0±25π[a$b®(a®t)$(b®t)], rc ¯ r­b, and A ¯
πt(a­b®t). Also, the relationship between the semiaxes of
the elliptical cross section and the original radius of the
unbent internode is given by a ¯ (1­ξ)R and b ¯ (1®ξ)R,
where ξ is the elastic deformation of the original teret cross
section resulting from flexure. As noted, for the elementary
flexure formula to hold true, e must be kept small. This
condition was met by setting a 5 % eccentricity of the
neutral axis and a 5 % elastic deformation of R as acceptable
285
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
experimental standards (i.e. e ¯ 0±05R and ξ ¯ 0±05, respectively). Solving eqn (A 5) with these boundary conditions
for r gives
r¯
E
when the products of the derivatives of N!xy with the
displacements u, Š, and w are neglected. Substitution of
9 0 1:
Et
¦u 1 ¦Š
®w1:
N ¯
Š ­
(1®Š#) 9 ¦x R 0¦θ
Et
¦u Š ¦Š
­
N! ¯
2(1®Š#) 0¦x R ¦θ1
Et$
¦#w Š ¦#w ¦Š
­
­
M ¯®
9
12(1®Š#) ¦x# R# 0 ¦θ# ¦θ1:
Et$
¦#w 1 ¦#w ¦Š
­
­
M ¯®
Š
12(1®Š#) 9 ¦x# R# 0 ¦θ# ¦θ1:
Et$
¦Š ¦#w
­
M! ¯
0
12R(1®Š) ¦x ¦x¦θ1
Nx ¯
5[a$b®(a®t)$(b®t)]
®b
t(a­b®t)
5[R%®(1±05R®t)$(0±95R®t)]
®0±95R.
tR(2R®t)
[γ%®(1±05γ®1)$(0±95γ®1)]
®0±95γt.
γ(2γ®1)
xy
y
xy
8
and using the notation
α¯
t#
12R#
φ¯
M(1®Š#) τ(1®Š#)
¯
2πR#Et
E
Buckling of a hollow stem subjected to a torque
Referring again to Fig. A 1, the general equations of
equilibrium for any small element through a twisted hollow
stem take the form
5
0
1
R
¦Ny
¦N
¦#Š ¦w
®
­R xy­Nyx
®Qy ¯ 0
¦x¦θ ¦x
¦θ
¦x
R
¦Qx
¦Q
¦Š ¦#w
®
­R y­(Nxy­Nyx)
­Ny ¯ 0
¦x ¦x¦θ
¦x
¦θ
0
(A 8)
6
7
1
0
1
¦Nx ¦N!yx
M
¦#u
¦#Š
®R
­
®
¯0
¦x#
¦x
¦θ 2πR# ¦x¦θ
R
¦Ny
¦N!
M ¦#Š ¦w
®
­R xy­
®Qy ¯ 0
¦θ
¦x πR# ¦x¦θ ¦x
R
¦Qx
¦Q
M ¦Š ¦#w
­
­R y­Nyx­
¯0
πR# ¦x ¦x¦θ
¦x
¦θ
0
¦#u (1®Š) ¦#u R(1­Š) ¦#u
¦w
­
­
®νR
¦x#
2 ¦θ#
2
¦x¦θ
¦x
1
5
¦#u
¦#Š
®R
¯0
0¦x¦θ
¦x#1
¦#u R#(1®Š) ¦#u R(1­Š) ¦#u ¦w
­
­
®
¦θ#
2
¦x#
2
¦x¦θ ¦x
­α
9¦θ¦#u#®R#(1®Š) ¦x¦#Š#
¦$w ¦$w
­R#
­
¦x#¦θ ¦θ$ :
¦#Š ¦w
®
­φR 0
¯0
¦x¦θ ¦x 1
(A 12)
9
7
6
¦Š
¦u
¦%w
¦%w
¦%w
­RŠ ®w®α R%
2R#
­
¦θ
¦x
¦x%
¦x#¦θ# ¦θ%
5
1
0
R#
­φR
R
(A 11)
where E is the Young’s elastic modulus of the stem’s tissues,
t is internodal wall thickness, and Š is the Poisson ratio of
stem tissues, gives the following expressions for eqn (A 9)
(see Timoshenko and Gere, 1961, p. 501).
8
where u, Š and w are small displacements from the original
cylindrical form assumed by a twisted cylindrical stem at
equilibrium, Nxy is the resultant shearing force, θ is the angle
of twist per unit length, R is the external radius of the
representative element of the stem, and Qx and Qy are the
shearing forces per unit distance (see Timoshenko and Gere,
1961).
Noting that Nxy ¯ Nyx ¯ M}2πR#­N!xy, where M is the
twisting moment per unit length of the cylindrical stem (i.e.
the applied torque), M}2πR# is the resultant shearing force,
and N!xy is the small change in the shearing force due to
buckling, eqn (A 8) has the form
(A 9)
8
7
6
(A 10)
7
6
x
(A 7)
Thus, a suitable radius of curvature depends both on the
absolute magnitude of internodal wall-thickness and on the
proportional relationship between internodal external radius
and wall-thickness.
¦N ¦N
¦#Š
¯0
R x­ x®RNxy
¦x#
¦x
¦x
5
y
(A 6)
This formula reveals that a suitable r depends on the
absolute size and shape of a representative cross section
through each stem. For example, recasting eqn (A 6) in
terms of the dimensionless quotient R}t ¯ γ and solving for
r based on different values for t (see Fig. A 2) :
r E 5t
Et
¦u Š ¦Š
­
®w
#
(1®Š ) ¦x R ¦θ
:
¦Š ¦$Š
­2φR 0 ­
¯0
¦x ¦x¦θ1
­(2®Š) R#
¦$Š
¦$Š
­
¦x#¦θ ¦θ$
8
286
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
Equation (A 12) indicates that the nodal lines on the
torqued surface of the twisted stem take the form of helices
rather than straight lines. Integration of eqn (A 12) shows
that the form of these helices is given by the three
displacements which take the form
0
λx
®nθ
R
1
Š ¯ B cos
0λxR®nθ1
w ¯ C sin
0λxR®nθ1 ,
9
A
5
8
10
20
5
10–2
(A 13)
6
7
8
where λ ¯ mπR}l, n is the number of torsion induced helical
‘ waves ’ that spiral around the length of the twisted
internode, m is the intensity of the torque measured along
length x, and A, B and C are numerical constants.
If the internode is very long, the constraints on circumferential twisting imposed by nodal septa will have little
effect on the magnitude of the critical shear stress, τ,
resulting from torsion. Under these conditions, substitution
of eqn (A 13) into eqn (A 12) gives the formulae
®A λ#­
l/D = ∞
l/D = 2
τ /E
u ¯ A cos
–1
10
: 9
:
5
:
(1­v)
(1®Š)
λn®B n#(1­α)­
λ#(1­2α)®2φλn
2
2
0
λ
¯0
n
9
λ
n
Aλn®Bn 1­αn#­(2®Š)λ#α®2φ
πR(1®Š#)
.
l
(A 17)
6
7
Expressing this equality in terms of the previous notation,
the critical moment of torque is given by the formula
:
­C [1­α(λ#­n#)#®2φλn] ¯ 0
100
F. A3. The quotient of the critical shear stress and Young’s elastic
modulus (τ}E ) plotted against the shape descriptor t}D for tubes with
different length to external diameter ratios (l}D). Solid lines are
computed on the basis of eqns (A 21) and (A 22) ; dashed line computed
for an infinitely long tube based on eqn (A 20) (see Brazier, 1927).
φ¯
1
­Cn 1­αn#­αλ#®2φ
–1
10
t /D
in the elastic limit of internodal tissues, φ must be very
small, and so λ must be very small. Thus, λ# can be
neglected in comparison to unity, and φ ¯ λ(1®Š#)}2.
Noting that λ ¯ mπR}l and that m ¯ 2,
(1®Š)
(1®Š)
n#®λnφ ­B
λn®λ#φ ®CŠλ ¯ 0
2
2
9
10–3
10–2
Mcr ¯
8
(A 14)
where φ is the angle of twist resulting from the applied
torque. Equation (A 14) consists of three linear formulae
that can be used to derive solutions for A, B and C different
from zero provided that their determinant equals zero.
Solving for the determinant and neglecting all terms
containing α#, α$, φ, φ#, and αφ, gives the following
expression for the angle of twist
2π#R$Et
.
l
(A 18)
This formula is the classical Greenhill solution for the
sideways buckling of an extremely long and thin rod
subjected to torque. Taking n ¯ 2 and λ ¯ 1 and solving for
λ when φmin, we find that
λ¯
0 1
48α !±#&
1®Š#
φ¯2
φ ¯ ²λ%(1®Š#)­α[2λ%(1®Š#)­(λ#­n#)%­(3­Š)λ#n#
Mcr ¯
®(2­Š) (3®Š)λ%n#®(7­Š)λ#n%­n%®2n'] (2λn&
5
9α3$ (1®Š#):! #&
±
(A 19)
6
7
πo2
EoRt&
3(1®Š#)!±(&
8
®2λn$­4λ$n$®2λ$n­2λ&n)−"´.
(A 15)
When n ¯ 1, eqn (A 15) becomes
λ%(1®Š#)­αλ%[λ%­4λ#­(2­Š) (1®Š)]
(A 16)
φ¯
2λ$(1­λ#)
and, neglecting all terms in the numerator containing α, it
reduces to φ ¯ λ(1®Š#)}2(1­λ#). Provided buckling occurs
and thus
τcr ¯
01
Mcr
E
t "±&
.
¯
±
2πR#t 3o2(1®Š#)! (& R
(A 20)
The relationship given by eqn (A 20) can be used for
estimating the critical shear stresses for any long cylindrical
internode subjected to torque provided internodal
dimensions and Young’s elastic modulus are known.
287
Niklas—RelatiŠe Resistance of Hollow, Septate Internodes to Twisting and Bending
For short internodes, the influence of nodal septa on
critical shear stresses cannot be neglected. Unfortunately,
the procedure required to determine the torsional displacements in internodal walls is complex (see Timoshenko
and Gere, 1961 pp. 504–505). However, provided a number
of mathematical simplifications are made, the following
relationships are predicted for short and moderately long
cylindrical tubes :
(1®Š#)
01 0 1
(
0
1*
l # τ
(1®Š#)"/#l# $/# "/#
¯ 4±6­ 7±8­1±67
2Rt
t E
(A 21)
for tubes with clamped ends, and
(1®Š#)
01 0 1
(
0
1*
l # τ
(1®Š#)"/#l# $/# "/#
¯ 2±8­ 2±6­1±40
2Rt
t E
(A 22)
for tubes with simply supported ends, where l is the length
of the untwisted tube. Both of these formulae indicate that
the magnitude of the critical shear stress normalized with
respect to Young’s elastic modulus, τ}E, is size-independent
and proportional to two dimensionless quotients l}D and
t}d, where D is the external diameter of the untwisted tube
(see Fig. A3).
Assuming that (1®Š#) E 1, eqns (A 21) and (A 22) become
0lt1# 0Eτ 1 ¯ 4±6­97±8­1±67 02Rtl # 1$ #:" #
0lt1# 0Eτ 1 ¯ 2±8­92±6­1±40 02Rtl # 1$ #:" # .
/
/
/
(A 23)
/
(A 24)
Plotting the expression (l}t)# (τ}E ) against (l #}2tR) indicates
that these two formulae predict very similar relationships
when (l #}2tR) & 10" which holds true for all the plant stems
tested in this report (see Fig. 7).