ORIGINAL PAPER
Biomechanical modeling of gravitropic response of branches:
roles of asymmetric periphery growth strain versus self

weight
bending effect
Yan

San Huang
•
Li

Fen Hung
•
Ling

Long Kuo

Huang
Received: 3 December 2009 / Revised: 29 August 2
010 / Accepted: 7 September
2010
。
Springer

Verlag 2010
Abstract
Bending movement of a branch depends on the
mutual interaction of gravitational disturbance,
phototropic response, and gravitropic correction. Four
factors are involved in gravitropic correct
ion: asymmetric
growth strain, eccentric growth increment,
heterogeneous longitu
dinal elasticity (MOE), and initial
radius which are asso
ciated with reaction wood
production. In this context, we have developed a simpli
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Communicated by R. Matyssek.
Y.

S. Huang (
&
) Department of Forestry, National Chung
Hsing University, 250, KuoKuang Rd., Taichu
ng 402,
Taiwan e

mail: yansanhuang@ntu.edu.tw
L.

F. Hung
﹒
L.

L. Kuo

Huang Institute of Ecology
and Evolutional Biology, National Taiwan University,
1, Roosevelt Rd. Sec. 4, Taipei, Taiwan
growth strain after defoliation and spring

back strain dur

ing de
foliation. The curvature change can be calculated
by using measured growth strain and spring

back strain
after defoliation. These results show that full

leaf
branches of
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123
et al. 1999). The accumulation of growth stress is the
optimized result of physiological adjustments to environ

mental stress (Niklas 1992;
Mattheck and Kubler 1995).
Maturation stress appears because of dimensional
changes of the cells during the maturation of the
secondary cell wall. However, the newly formed cells are
glued onto the rigid trunk inside which resists their
deformation. The
constrained strains are then
transformed to stresses owing to the stiffness of the ligni
ﬁ
ed cell wall. Two main hypotheses attempt to explain
the origin of growth stresses:
(1) Boyd (1972) proposed the lignin

swelling hypothesis
based on the observed swel
ling of the thickness of cell
walls during ligni
ﬁ
cation, to explain the mechanism of
incidence of longitudinal, tangential and radial growth
stresses, while (2) Bamber (1978, 1987) postulated that
the cellulose micro
ﬁ
bril (CMF) of the growing cell wall
con
tracts during crystallization. Growth stress is
produced in the longitudinal direction of the CMF when
the cell wall is constrained by the matured rigid xylem
inside.
Increased growth stress is found at speci
ﬁ
c locations
in leaning trunks or branches whe
re eccentric growth
occurs (Onaka 1949). Growth stress tends to reorient the
trunk or to maintain the branch in an equilibrium position
(or curving up), thereby maximizing exposure to sunlight.
In coniferous trees, compression wood is formed on the
lower s
ide of a tilted trunk, whereas in dicotyledonous
trees, tension wood is formed on the upper side
(Okuyama et al. 1986; Timell 1986a, b). By producing
reaction wood, up

righting of leaning trunks or branches
is achieved.
The mechanism of the reorientation
of slanted trunk or
the upward bending of branch depends on the
asymmetric growth stress distribution around the
periphery of the stem as a result of the occurrence of
reaction wood on one side. This asymmetric stress
produces a force couple which induces
the inner stem to
produce a bending moment to balance it. This moment is
transferred to the curving phe
nomenon of stem which
can be observed from its outward appearance (Fournier
et al. 1994).
In conifers, both a pulling force produced from tensile
stres
s in the opposite wood on the upper side of leaning
stem and a pushing force produced from compression
stress in the compression wood on the lower side
produce the upward bending movement. In
dicotyledonous trees, in contrast, only a pulling force
produced
from tensile stress in the tension wood on the
upper side of stem can be uti
lized (Archer 1986).
Therefore, on this point, conifers seem to be superior to
dicotyledonous trees in the mechanism of their righting
process.
Trees are capable of reorienting
trunks and branches
when stems are mechanically displaced from their
original growth position (Archer and Wilson
1970; Wilson
and Archer 1977; Niklas 1992). This orientation involves
the production of reaction wood at speci
ﬁ
ed positions.
All the
available
evidence concerning reaction wood appears to
support the hypothesis that the stimulus that largely
induces its formation is gravity (Wilson and Archer
1977;
Niklas 1992), and high concentrations of plant growth
regulators have been found on the lower side
of leaning
trunks (Savidge et al. 1982, 1983). Yamamoto et al.
(2002) investigated the interaction between bending
moment, due to self

weight of branches, and recovery
moment, resulting from asymmetric growth stress
distribution around the cross

section.
They concluded
that the growth stress gen
erated in the reaction wood is
suf
ﬁ
cient to counteract the pull of gravity on branches.
Fournier et al. (1994) pre
sented a biomechanical model
to explain the mechanism in the upward bending process
of a leaning st
em. The deri
vation of this model was
based on the assumptions that the longitudinal growth
strain distribution on the periphery of stem takes the form
of a sinusoidal wave, the cross

section is circular with
uniform growth increment, and the stem has a
ho
mogeneous modulus of elasticity. Coutand et al.
(2007) applied Fournier
’
s model to inspect the
gravitropic response of poplar trunks. Their mechanical
analyses showed that the effects of reaction wood on
negative gravitropism involve three factors: (1) the
radial
growth rate, (2) a growth stress difference between
reaction wood and opposite wood, and (3) the radius of
the trunk at the onset of the righting movement. The
combination of these three parameters causes a
continuous change of the cur
vature of st
em over time
that can be observed as a change in the trunk form.
Nevertheless, the effect of the self

weight of the trunk on
gravitropic processes is not involved in the model.
Although maximum bending stresses due to self

weight
are theoretically not loca
ted in the peripheral youngest
wood because of continuous growth increment, the effect
of self

weight moment on reorientation cannot be
ignored. A
ﬁ
rst analytical model to compare
gravitational disturbance and gravitropic correc
tion was
suggested by Fourn
ier et al. (2006), and the application
to experimental data has been performed on poplar
(Alme
´
ras and Fournier 2009) and various tropical
species (Alme
´
ras et al. 2009).
Recently, Alme
´
ras and Fournier (2009) developed a
somewhat sophisticated theoretica
l analysis on the
effects of peripheral growth stresses and self

weight on
the gravitropic action of tilted stems. In this
biomechanical model, heterogeneous elasticity of the
cross

section and a varied growth increment around the
periphery were con
sidere
d. Furthermore, bending
moment due to self

weight was calculated using trunk
taper analysis. The gravitropic performance was used to
indicate the dynamics of stem bending during growth.
Most previous research empha
sized growth stress in the
trunk, with on
ly a few studies focusing on growth stress
in branches (Ohsako and Yamada 1968; Yoshida et al.
1992a, b, 1999). In this study
123
we establish a new model based on Fournier
’
s model
considering both growth stress and self

weight moment.
The concept of spr
ing

back was introduced to tree reori

entation model for the
ﬁ
rst time. We de
ﬁ
ne
spring

back as the elastic recovery of bending when
self

weight moment disappears and spring

back strain as
the released self

weight bending strain. For inspecting
the validit
y of model, growth strain and spring

back strain
on branches of Taiwan red cypress (
Chamaecyparis
formosensis
Matsum.) and
ﬂ
amegold (
Koelreuteria
henryi
Dummer.) were used.
Biomechanical modeling
A brief description of Fournier
’
s model
Using beam theory
, Fournier
’
s model (1994) assumed
that the surface growth stress at the periphery of a tilted
trunk varies sinusoidally with angular positions along the
cir
cumference, and the variation of curvature of a tilted
trunk between time interval, which is relate
d to the
parameters such as the half difference between upper
and lower side growth strain of stem (
a
), initial radius of
stem with cir
cular cross

section (
R
), and uniform radial
growth incre
d
C
g
d
M
g
4
a
¼¼
ð
4
Þ
d
R EI
d
RR
2
where
h
is the circumferential position to begin at upper
side of stem,
a
L
(
h
) is the growth strain on the periphery
at position
h
,
a
=
(
e
gu

e
gl
)/2 is the half difference between
the upper and the lower side growth strain in the growth
increment,
i
a
¼
e
gu
þ
e
gl
=
2 is the average growth strain
in the growth increment,
E
is the longitudinal Young
’
s
modulus (MOE), d
C
g
is the change in curvature due t
o
asymmetric growth strain,
I
=
p
R
4
/4 is the moment of
inertia relative to the centroid of a circle, d
M
c
is the
couple moment of asymmetric growth stress in the
growth incre
ment due to the formation of reaction wood,
and d
M
g
is the balancing bending momen
t against d
M
c
.
The ratio d
C
g
/d
R
is the rate of change in curvature
due to asymmetric growth strain associated with growth
increment. From the last equation, it can be seen that the
value of d
C
g
/d
R
is proportional to
a
and inversely pro

portional to
R
2
. It
is obvious that the effect of self

weight
was not considered in the above model.
Addition of self

weight moment to Fournier
’
s model
The branch form is affected by the interaction between
the bending moment due to self

weight and that due to
the asymmetr
ic distribution of growth stress (Fig.
1). In
this study, assuming that the self

weight bending
moment of a branch (
M
s
) is proportional to that of a long
cone, and that between time
t
1
and
t
2
a uniformly
increased layer of wood (thickness d
R
) has been adde
d
along the periphery of a branch, the self

weight bending
moment of a branch can be expressed as
M
s
¼
1
k
p
R
2
l
q
gc
sin
/
ð
5
Þ
3 where
l
is length of branch (measured from branch end
to position of growth strain measurement), volume of
con
e
=
p
R
2
l
/3,
c
is
the distance between the branch
center of mass and the position of growth strain
measurement (
c
=
l
/ 4 for a cone),
q
is green density of
wood (green mass of wood/green volume of wood),
g
is
gravitational accelera
tion,
k
is the ratio of total mass to
wood
mass, with
k
[
1.
/
is the tilt angle between branch
and vertical (Fig. 1). If the branch is assumed to grow
isometrically, then
d
M
s
4d
R
¼
ð
6
Þ
M
s
R
Using beam theory (Timoshenko and Young 1962)
M
s
can be also expressed as
EI
b
M
s
¼
ð
7
Þ
R
Fig. 1
Bending tendency of branch.
W
self

weight of branch,
M
g
upward bending moment due to growth stress (
M
g
[
0),
M
s
down

ward bending moment due to self

weight of branch (
M
s
\
0).
/
is the
tilt angle between branch and vertical
123
ment (d
R
), which can be brie
ﬂ
y expressed as
a
L
ð
h
Þ
¼
i
a
þ
a
cos
h
ð
1
Þ
d
M
g
¼
d
M
c
¼
d
R
Z
2
p
R
cos
ha
L
ð
h
Þ
ER
d
h
ð
2
Þ
0
d
M
g
¼
p
E
a
R
2
d
R
ð
3
Þ
Similarly,
e
0
0
g
l
¼
e
gl
d
R
l
d
R E
l
i
E
ð
21
Þ
t
h
e
adjust
ed
paramet
er
a
0
0
with
respe
ct
t
o
a
can
b
e
expressed as
�
�
a
00
¼
ð
e
00
gu
e
00
gl
Þ
=
2
¼
e
gu
d
R
u
d
R E
u
i
E
e
gl
d
R
l
d
R E
l
i
E
.
2
ð
22
Þ
Bran
ch
D
(cm)
b
(
l
e
)
a
(
l
e
)
a
0
(
l
e
)
a
00
(
l
e
)
a
00
(
i
E
=
E
)
b
–
a
b
–
a
0
b
–
a
00
(
i
E
=
E
)
d
C
s
/d
R
d
C
g
/d
R
d
C
/d
R
no.
(
l
e
)
(
l
e
)
(
l
e
)
(
l
e
)
(m

2
)
(m

2
)
(m

2
)
II

1
1.35

3,182

2,858

3,323

2,8
80

3,024

324
141

157

346
313

32.8
II

2
1.40

2,512

2,540

2,819

2,499

2,623
28
307
111

318
322
4.0
II

3
2.10

1,969

1,912

2,139

1,889

1,983

57
170
14

221
221

0.5
II

4
2.07

1,370

1,464

1,700

1,47
4

1,548
94
330
178

95
105
10.3
II

5
1.08

1,976

2,083

2,268

2,029

2,130
107
292
155

332
364
32.0
II

6
1.17

3,319

3,197

3,473

3,110

3,266

122
154

53

501
482

18.4
where
b
=
(
e
su

e
sl
)/2 is the half difference in spring

back
strain (Huang et al. 2005) between the upper and the
lower sides on the periphery of branch at speci
ﬁ
ed
strain measurement position.
Consider
ing the relation between Eqs. 6 and 7, the
self

weight bending moment due to growth increment
d
M
s
can be expressed as
4d
R
d
R
d
M
s
¼
M
s
¼
4
EI
b
ð
8
Þ
RR
2
The measured spring

back strain includes the effect of
tilting angle, so that Eq. 7 is applicable to ti
lting
branches.
According to beam theory, change of curvature is
determined by the sum of d
M
s
and d
M
g
(see Eq. 3). The
rate of change in curvature associated with growth incre

ment d
C/
d
R
can be expressed as
d
C
s
d
M
s
4
b
¼¼
ð
9
Þ
d
R EI
d
RR
2
dC dC
s
þ
dC
g
4
ð
ba
Þ
¼
¼
ð
10
Þ
dRdR R
2
where d
C
s
/
d
R
is the rate of change in curvature
due to spring

back (self

weight) associated with growth
increment. Equation 10 assuming isometric growth is a
particular case of Alme
´
ras and Fournier
’
model (2009);
therefore, after changes,
it is consistent with their
equation. Our approach requires destructive
measurements and neglects branch taper. Nevertheless,
it can estimate the spring

back strain in branches in a
straightforward manner, and the resulting model is
laconic. Gravitropic pe
rformance
P
G
(Alme
´
ras and
Fournier 2009) is de
ﬁ
ned as
d
M
g
a
P
G
¼¼
ð
11
Þ
d
M
s
b
Modi
ﬁ
cation with eccentric growth ring
In the case of eccentric growth increment in the radial
direction and homogeneous MOE distribution on the
cross

section, the parameter
a
can be corrected simply
by transforming the growth increment on the upper side
(d
R
u
) and on the lower side (d
R
l
) to a uniform d
R
=
(d
R
u
?
d
R
l
)
/
2 and keeping the force acting on the area of
growth increment with arc angle d
h
at upper and lower
sides unch
anged (Fig. 2); then the adjusted growth
00
strain on the upper side
e
gu
and on the lower side
e
gl
can
be expressed as
0
E
e
R
d
h
d
R
¼
E
e
gu
R
d
h
d
R
u
ð
12
Þ
gu
0
d
R
u
e
¼
e
gu
ð
13
Þ
gu
d
R
Fig. 2
Reference model for a branch of radius
R
submitted to
eccentric growth increment. The
upper side
radial increment d
R
u
and the
lower side
radial increment d
R
l
are transformed to average
and uniform growth increment
d
R
=
(d
R
u
?
d
R
l
)/2
Similarly,
0
d
R
l
e
¼
ð
14
Þ
gl
e
gl
d
R
the adjusted parameter
a
0
with respect to
a
can be
expressed as
00
d
R
u
d
R
l
a
0
¼
ð
e
gu
e
gl
Þ
=
2
¼
e
gu
e
gl
2
ð
15
Þ
d
R
d
R
the adjusted d
C
0
/
d
R
and gravitropic performance
P
0
G
can be expressed as
d
C
00
4
a
g
¼
ð
16
Þ
d
RR
2
d
C
0
d
C
s
þ
d
C
g
0
4
ð
ba
0
Þ
¼¼
ð
17
Þ
d
R
d
RR
2
0
a
P
0
G
¼
ð
18
Þ
b
Modi
ﬁ
cation with eccentric growth ring
and heterogeneous elasticity
In case of heterogeneous elasticity over the
cross

section and varied growth increments around the
periph
ery,
E
is the longitudinal MOE of the inner part of
branch,
E
¼
i
ð
E
u
þ
E
l
Þ
=
2 is the mean value, and
E
l
and
E
u
are the lower
and upper side values of that of the new growth ring,
respectively. Again, using the principle of force
equalizing,
E
u
and
E
l
are tr
ansformed to
E
and the
adjusted growth
i
00 00
strain on the upper side
e
gu
and on the lower side
e
can be
gl
expressed as
0
0
E
i
e
R
d
h
d
R
¼
E
u
e
gu
R
d
h
d
R
u
ð
19
Þ
gu
00
d
R
u
E
u
e
¼
ð
20
Þ
gu
e
gu
i
d
RE
123
0
0
d
C
00
/
d
R
and the adjusted gravitropic performance
P
G
can
be expressed as
d
C
00 00
E
i
4
a
E
g
¼
ð
23
Þ
d
RR
2
d
C
00
d
C
s
þ
d
C
g
00
4
ba
00
E
i
E
¼¼
ð
24
Þ
d
R
d
RR
2
0
0
E
i
a
P
0
0
E
¼
ð
25
Þ
G
b
The term
E
i
=
E
in Eqs.
23, 24 and 25 is related to
radial
i
gradient in MOE. If
E
¼
E
¼
E
u
¼
E
l
and d
R
=
d
R
u
=
000
¼
P
00
d
R
l,
then
a
=
a
=
a
and
P
G
¼
P
G
0
G
. When only eccent
ric
growth ring is considered, parameter
a
0
is precisely equal
to the half of the gravitropic ef
ﬁ
ciency
e
r
used by Alme
´
ras and Fournier (2009). However, in the case that
eccentric growth ring and heterogeneous elasticity are
considered, parameter
a
00
is n
o more equivalent, because
Alme
´
ras and Fournier (2009) assumed d
R
,
e
, and
E
varied sinusoidally along the periphery, while we assume
that the resulting force acts in this way. The expression
and
0
application of
a
or
a
00
is simpler than those of
e
r
in Alm
e
´
ras and Fournier
’
s model (2009), though we may loose
the fact of separating it into two distinct components: one
depending only on growth increment, and the other on
elastic heterogeneity of wood.
Modi
ﬁ
ed model for defoliating action of
deciduous trees
For a circular cross

section of stem with an initial radius
(
R
), and uniform radial growth increment (d
R
), if
b
and
a
are the spring

back and the growth strain parameters of
the branch in full

leaf condition as described above, then
again we have Eq. 10 f
or d
C/
d
R.
During the defoliating season,
b
1
is the half difference
of spring

back strain between upper and lower sides
mea
sured on the lea
ﬂ
ess branch, and
a
1
is that of
growth strain. With the spring

back strain during the
defoliating process expressed a
s
e
sudef
and
e
sldef
,
respectively, we obtain
��
����
a
1
¼
ð
e
gudef
e
gldef
Þ
=
2
¼
e
gu
e
sudef
e
gl
e
sldef
=
2
¼
a
b
def
ð
26
Þ
b
1
¼
½
ð
e
su
e
sudef
Þð
e
sl
e
sldef
Þ
l
=
2
¼
bb
def
ð
27
Þ
b
1
a
1
¼
ba
ð
28
Þ
where
b
def
=
(
e
sudef

e
sldef
)/2 is the
b
during defoliating
action,
and
e
gudef
and
e
gldef
are, respectively, the
measured growth strains on upper side and lower side of
the branch after defoliation. It is clear that growth strain
is the sum of measured growth strain after defoliation
and spring

back strain during defoliat
ion. This means
that growth strain is superimposed by the spring

back
strain during defoliation. The self

weight bending
moment of defoliated branch due to growth increment
d
M
1s
can be expressed as
d
M
1s
¼
p
E
b
1
R
2
d
R
ð
29
Þ
So that d
C/
d
R
can be expressed as
d
C
4
ð
ba
Þ
4
ð
b
1
a
1
Þ
¼¼
ð
30
Þ
d
RR
2
R
2
The rate of change in curvature of a defoliated branch
associated with growth increment d
C
1
/
d
R
and gravitropic
performance
P
1G
can be expressed as
d
C
1
d
M
1s
þ
d
M
g
4
ð
b
1
a
1
b
def
Þ
¼¼
ð
31
Þ
d
R EI
d
RR
2
d
M
g
a
a
1
þ
b
def
P
1G
¼¼¼
ð
3
2
Þ
d
M
1s
b
1
b
1
In the case of eccentric growth increment and MOE
variation on the cross

section, a similar relation can be
obtained as below:
d
C
0
4
ð
b
1
a
0
Þ
1
¼
ð
33
Þ
d
RR
2
0
a
P
0
¼
ð
34
Þ
1G
b
1
00
E
i
dC
00
4
ð
b
1
a
Þ
1
E
¼
ð
35
Þ
dR R
2
0
0
E
i
a
P
0
0
E
1G
¼
ð
36
Þ
b
1
where
b
1
,
a
1
,d
C
1
0
/d
R
,d
C
1
00
/d
R
,
P
0
1G, and
P
00
1G are the
defoliated condition of
b
,
a
,d
C
0
/
d
R
,d
C
00
/
d
R
,
P
0
G, and
P
00
G, respectively.
Materials and methods
ment (d
R
), which can be brie
ﬂ
y expressed as
a
L
ð
h
Þ
¼
i
a
þ
a
cos
h
ð
1
Þ
d
M
g
¼
d
M
c
¼
d
R
Z
2
p
R
cos
ha
L
ð
h
Þ
ER
d
h
ð
2
Þ
0
d
M
g
¼
p
E
a
R
2
d
R
ð
3
Þ
Similarly,
e
0
0
g
l
¼
e
gl
d
R
l
d
R E
l
i
E
ð
21
Þ
t
h
e
adjust
ed
paramet
er
a
0
0
with
respe
ct
t
o
a
can
b
e
expressed as
�
�
a
00
¼
ð
e
00
gu
e
00
gl
Þ
=
2
¼
e
gu
d
R
u
d
R E
u
i
E
e
gl
d
R
l
d
R E
l
i
E
.
2
ð
22
Þ
Bran
ch
D
(cm)
b
(
l
e
)
a
(
l
e
)
a
0
(
l
e
)
a
00
(
l
e
)
a
00
(
i
E
=
E
)
b
–
a
b
–
a
0
b
–
a
00
(
i
E
=
E
)
d
C
s
/d
R
d
C
g
/d
R
d
C
/d
R
no.
(
l
e
)
(
l
e
)
(
l
e
)
(
l
e
)
(m

2
)
(m

2
)
(m

2
)
II

1
1.35

3,182

2,858

3,323

2,8
80

3,024

324
141

157

346
313

32.8
II

2
1.40

2,512

2,540

2,819

2,499

2,623
28
307
111

318
322
4.0
II

3
2.10

1,969

1,912

2,139

1,889

1,983

57
170
14

221
221

0.5
II

4
2.07

1,370

1,464

1,700

1,47
4

1,548
94
330
178

95
105
10.3
II

5
1.08

1,976

2,083

2,268

2,029

2,130
107
292
155

332
364
32.0
II

6
1.17

3,319

3,197

3,473

3,110

3,266

122
154

53

501
482

18.4
Mean

2,212

2,185

2,457

2,164

2,
272

27
245
60

263
263
0.4
SD
1,117
987
1,102
971
1,020
238
235
223
167
160
31.1
To explain the above modeling, experimental data from
previous studies (Huang et al.
2005) were
used for quan

titative evaluation of the modi
ﬁ
ed model. Six horizontal
branches were selected from young trees of
Chamaecyparis formosensis
with DBHs ranging from 4
to 20 cm to mea
sure growth strains at 4 to 10 different
distances away from
123
the tru
nk. After removing the bark, strain gauges were
glued in the longitudinal direction on the upper side and
the lower side at a speci
ﬁ
c position on the branch of
standing trees. After the gauges had been zeroed, the
branch was cut down and kept vertical to m
easure
spring

back strains (
e
su
and
e
sl
) caused by loss of
self

weight moment. Thereafter, the released surface
growth strains (
e
gu
and
e
gl
) were measured by
cross

cutting the branch at a position 5 mm behind and
in front of the strain gauge.
In order to
realize the effect of defoliation,
ﬂ
amegold
tree (
Koelreuteria henryi
Dummer.) which is a deciduous
dicotyledonous tree endemic to Taiwan was studied.
Four slightly tilted
ﬂ
amegold branches on the campus
of National Taiwan University, Taipei (25
�
00
0
N,
121
�
27
0
E) were chosen. Three branches (ND1
–
ND3)
were naturally defoliated by change of season; the
spring

back strain (
e
1su
and
e
1sl
) and growth stain (
e
gudef
and
e
gldef
) of three or four sites each were measured on
February 2009. One arti
ﬁ
cially defoliated b
ranch (AD)
was measured on December 2008. The defoliation
spring

back strains (
e
sudef
and
e
sldef
) were measured after
all leaves on the branch were removed before the branch
was cut down.
Results and discussion
Evaluation of gravitropic response with exp
erimental
data
Table
1 shows computed average values of 4
–
10 sites
for each parameter related to gravitropic response in
Chamaecyparis formosensis
from our previous data
(Huang et al. 2005). The parameters
b
and
a
are
associated with gravitational disturb
ance and gravitropic
correction, respectively.
b
is always negative indicating a
downward bending tendency, while

a
is usually positive,
indicating
an upward bending tendency. As shown in Fig.
3, a
strong linear relationship between
a
and
b
existed. This
fact intensively indicated the in
ﬂ
uence of self

weight on
asymmetric strain distribution on the periphery. The neg

ative values of
a
and
b
both decreased dramatically as
the distance from trunk increased to about 50 cm and
then decreased gradually (Fig. 4
). Observation of branch
cross

section showed that the pith became more
eccentric as its distance from the trunk decreased,
indicating intensive development of compression wood
that can induce a large compression stress (Huang et al.
2005). It is suggested
that the generation of growth
stress on the lower side of bran
ches that developed a
large mass of compression wood is affected by
gravitational bending moment due to branch self

weight
which is responsible for the large spring

back strain (Fig.
5). In ot
her words, the gravitational stimulus is
responsible for high growth stress of compression wood.
In this study, the situation of heterogeneous MOE on the
cross

section and varied growth increment on the
periphery was also considered. In the case of
gymnos
perm trees, MOE of compression wood is lower
than that of normal wood (Alme
´
ras et al. 2005; Boyd
and Foster 1974; Panshin et al. 1964). Owing to
eccentric growth increments, the annual ring width of
compression wood is larger than normal wood. Referring
t
o experimental data of Alme
´
ras et al. (2005) and Huang
et al. (2005), mean values of
E
u
=
E
¼
1
:
19,
E
l
=
E
¼
0
:
81, d
R
u
/
d
R
=
0.65, d
R
l
/
d
R
=
1.35
ii
i
0 00
(
i
and
E
=
E
¼
1
:
05 for gymnosperms were used to
calculate the parameters
a
,
a
00
and
a
E
=
E
). Parameter
a
on
ly argues asymmetric growth strain on the periphery,
whereas effects of eccentric growth increment and both
this and MOE
00
(
i
variation are associated with
a
0
and
a
E
=
E
), respectively.
Eccentric growth increases the value of
a
0
and
a
00
due to
the increas
ed growth in compression wood, whereas
MOE variation decreases
a
00
due to the low MOE in
compression wood. Table 1 shows that the mean values
of
b
,
a
,
a
0
,
a
00
,
Table 1
Experimental data of
Chamaecyparis formosensis
Matsum. used for calculating the curva
ture change of branches (Huang et al. 2005)
Branch diameter at base (
D
), spring

back strain parameter (
b
), asymmetric growth strain parameter (
a
,
a
0
and
a
00
), mean modulus of
elasticity in the
new growth increment (
E
i
), homogenized elasticity of the cross

section (
E
), rates of gravitational disturbance
(d
C
s
/
d
R
) and gravitropic correction (d
C
g
/
d
R
), and net rate of curvature change (d
C/
d
R
). Besides
D
, all values of each branch are averages
compu
ted from 4 to 10 measuring sites. The values of mean and SD are computed from all measuring sites of six branches
123
ment (d
R
), which can be brie
ﬂ
y expressed as
a
L
ð
h
Þ
¼
i
a
þ
a
cos
h
ð
1
Þ
d
M
g
¼
d
M
c
¼
d
R
Z
2
p
R
cos
ha
L
ð
h
Þ
ER
d
h
ð
2
Þ
0
d
M
g
¼
p
E
a
R
2
d
R
ð
3
Þ
Similarly,
e
0
0
g
l
¼
e
gl
d
R
l
d
R E
l
i
E
ð
21
Þ
t
h
e
adjust
ed
paramet
er
a
0
0
with
respe
ct
t
o
a
can
b
e
expressed as
�
�
e
gu
d
R
u
d
R E
u
e
gl
d
R
l
.
2
Fig. 3
The relationship between growth strain parameter (
a
) and
spring

back parameter (
b
)
Fig. 4
The relationship between the growth strain parameter (
a
),
spring

back parameter (
b
) and the distance from the measuring
sites to the trunk
00
(
i
and
a
E
=
E
) were

2,212, SD 1,117,

2,185, SD 987,

2,457, SD 1,102,

2,164, SD 971, and

2,272, SD 1,020,
respectively. The effect of eccentric growth increment
has a positive ef
ﬁ
ciency
of 12.5% to increase
a
0
and
a
00
values based on an
a
in which only asymmetric growth
strain is considered, while variation of MOE at the
periphery has a negative ef
ﬁ
ciency of

13.4% and
decreases
a
00
and
00
(
ii
a
E
=
E
) values. On the other hand, the ratio
of
E
=
E
has
00
(
i
ef
ﬁ
ciency of 4.9% to increase the value of
a
E
=
E
). The
total effect of MOE is a decreased 8.5% ef
ﬁ
ciency.
Asymmetric growth strain distribution apparently was the
main source of gravitropic correction. It is important that
Fig. 5
The rela
tionship between the released growth strains at the
lower side of the branch (
e
gl
) and the spring

back strain parameter
(
b
)
Fig. 6
The relationship between d
C
s
/
d
R
,d
C
g
/d
R
,d
C/
d
R
and the
distance from the measuring sites to the trunk. d
C
s
/
d
R
: rates of
gravit
ational disturbance, d
C
g
/
d
R
: rates of gravitropic correction,
and d
C/
d
R
: net rate of curvature change
the negative effect of MOE gradient between normal wood
and compression wood de
creases the value of
a
00
and
00
(
i
a
E
=
E
) which in turn lowers the ef
ﬁ
ciency of gravitropic
correction. However, the existence of juvenile wood in
the
00
(
i
inner part of stem would increase the value of
a
E
=
E
),
because the juvenile wood has larger micro
ﬁ
br
il angle
(MFA), shorter
ﬁ
ber length, and therefore lower MOE
than the new growth ring.
As to why the evolution of gymnosperms adopted such a
disadvantageous biomechanical strategy to produce high
MOE in normal wood and low MOE in compression
wood, two com
ments can be made (Alme
´
ras et al.
2005). The
ﬁ
rst concerns the physical and mechanical
properties of wood. Several researchers (Boyd and
Foster 1974; Yamamoto et al. 1991; Huang et al. 2005)
showed that the MFA of compression wood is signi
ﬁ
cantly larger t
han that of nor
mal wood. The magnitude of
growth strain and MOE mainly depends on MFA in the S
2
layer of the cell wall. Compression strains are induced if
MFA is large, whereas
123
high longitudinal MOE is obtained if it is small. It seems
that it is no
t possible for compression wood
ﬁ
ber to
have both high MOE and large compression strain for ef
ﬁ
cient gravitropic correction. The second comment is
that upward movement is not the unique mechanical
function of the stem. In fact, its primary role is to suppo
rt
the tree without mechanical failure. In gymnosperms,
compression wood with high lignin and low cellulose has
larger compression strength parallel to grain than normal
wood (Panshin et al. 1964). In tilted stems or branches,
self

weight and external load
s such as snow and wind
pressure statically and dynamically induce compression
stress on the lower side. The compression wood with
high crushing strength pro
tects wood from compression
break. Self

optimization of gymnosperm
’
s adaptation to
ecological envi
ronment is thus achieved. In dicotyledons,
tension wood occurs on the upper side, and tensile
stress usually appears on both the upper and lower
sides. Because the tensile strength of wood is 2
–
3 times
higher than its compression strength, tensile stress c
an
therefore compensate for the induced compres
sion
stress to reduce the risk of failure.
00
(
i
The value of (
b

a
E
=
E
)) is closer to (
b

a
) than (
b

a
0
),
so that the unadjusted value (
b

a
) is essentially
satisfactory for evaluating the net gravitropic t
endency.
The tendency is toward upward movement for positive
values and downward bending for negative values. The
value of (
b

a
0
) showed a greater tendency for upward
bending because the effect of heterogeneous elasticity
was not considered in the calcula
tion. These results are
con
sistent with those of Alme
´
ras and Fournier (2009),
who found the mean form factor
f
for gymnosperms close
to unity. However, this is true only for gymnosperms, in
which the effect of MOE and eccentricity roughly com

pensate eac
h other. In angiosperm trunks, these two
factors both improve the ef
ﬁ
ciency, so that
a
is not such
a good approximation for the gravitropic ef
ﬁ
ciency, i.e.,
the factor
f
is signi
ﬁ
cantly larger than 1. In the case of
branch, however, the condition is differ
ent from that of
trunk. This will be discussed later.
Rates of gravitational disturbance and
gravitropic correction
Rates of gravitational disturbance expressed as d
C
s
/
d
R
had a mean value of

260, SD170 (Table
1). This rate of
distur
bance represents the
effect of the self

weight
moment bending a branch down. This value is always
negative, indicating downward bending tendency. Rates
of gravitropic correction expressed as d
C
g
/
d
R,
d
C
0
g
/
d
R,
and d
C
00
g
/
d
R
had mean values of 260, SD160; 290,
SD170; and 270, S
D160, respectively. The value of d
C
00
g
/
d
R
is close to d
C
g
/
d
R
, so that d
C
g
/
d
R
alone can in
practice be used to estimate the rate of correction. The
positive mean values indicated upward
bending tendency. The net rates of gravitropic response
expressed as
d
C/
d
R
, which is the sum of d
C
s
/
d
R
and
d
C
g
/
d
R
, had a mean value of 0.4, SD31. Again, the net
tendency is toward upward movements for positive
values and downward bending for negative values. The
in
ﬂ
uence of distance from trunk for each measured
positions on
the values of d
C
g
/
d
R,
d
C
s
/
d
R,
and d
C/
d
R
is
shown in Fig.
6. Because the computed rates of
gravitropic correction and gravitational disturbance had
the same order of magnitude, the studied branch
samples were close to a state of biome
chanical
equilibrium.
It was obvious that the values of d
C/
d
R
were
distributed close to the zero line. According to Alme
´
ras
and Fournier
’
s analysis (2009) for the rate of disturbance
and correction of nine poplar trees with mean DBH
=
0.24 m, they obtained mean
a
=
319, d
C
s
/
d
R
=

0.08,
d
C
g
/
d
R
=
0.088, and d
C/
d
R
=
0.008. The absolute values
of d
C
s
/
d
R
and d
C
g
/
d
R
were also close to each other. The
studied trees were also close to equilibrium condition.
Equilibrium is necessary for a branch to maintain a
suitable angle from trunk
to receive suf
ﬁ
cient sun light
for photo
synthesis. This is the most common condition
observed in the
ﬁ
eld. The values of the rate of
curvature change for poplar were very small compared
with Taiwan red cypress because of small values of
a
and large radius
of trunk (Alme
´
ras and Fournier 2009).
Basic wood properties are density, stiffness, growth
strain, and their distribution within the cross

section.
Radial growth increases the area and the moment of
inertia of the cross

section, and self

weight bending
moment increases with the increase of growth (radius
and length). The long

term tendency of a growing branch
depends on how the gravitropic correction balances the
gravitational disturbance. Reaction wood formation often
accompanied by eccentric radial gro
wth mainly controls
the stem

right
ing mechanism. The ef
ﬁ
ciency of this
righting process depends on a combination of at least
four factors: asym
metric growth strains, heterogeneous
elasticity, eccentric radial growth, and initial radius in the
righting pr
ocess. A biomechanical analysis based on
beam theory was done by Alme
´
ras et al. (2005) to
quantify the individual in
ﬂ
uence of each factor. They
emphasized that asymmetric growth stress distribution at
the periphery is the critical factor of gravitropic
co
rrection. For dicotyledons, asymmetric growth strain
alone explains on average 57% of the total reorientation.
Eccentric growth is commonly related to heterogeneity of
growth strains. This factor allows a 31% increase in ef
ﬁ
ciency of correction. Asymmetric
MOE allows 13%
increase of reorientation. In the case of gym
nosperms,
asymmetric growth strains alone explain on average
90% of the total reorientation. The effect of eccentric
growth increases the ef
ﬁ
ciency of correction by 26%.
MOE asymmetry has a nega
tive effect, with 24%
decreased ef
ﬁ
ciency.
123
Gravitropic performance
Changes in curvature due to self

weight and asymmetric
growth stress distribution depend on the change of
growth increment d
R
. The gravitropic performance
P
G
expressed as the ratio o
f
a
to
b
indicates the dynamics of
bending direction at the cross

section:
[
1 for an upward
bending growth,
\
1 for a downward bending growth, =1
for the stem section in biomechanical equilibrium without
curva
ture change during growth, and a negative value
for
a downward bending growth both by self

weight and
growth stress (epinasty). In this special case,
compression wood may occur on the upper side of
branch in conifers or ten
sion wood may occur on the
lower side of that in dicoty
ledons. For all 36 meas
ured
positions, 50% of those indicated upright movement. The
means of
P
G
,
P
0
G and
P
00
G
were 1.028, SD0.128; 1.149, SD0.124; and 1.066,
SD0.124, respectively. According to Eqs. 11, 18 and 25,
G
:
P
00 000
(
i
P
G
:
P
0
G
=
a
:
a
:
a
E
=
E
), respectively. The perc
ent
age of
upward bending for
P
0
G further increased to 83% as a
result of eccentric growth increment and then returned to
64% if both eccentric growth and heterogeneous MOE
were considered. As a whole, the measured samples of
branches were all around the
biomechanical equilibrium
condition. For simplicity, the gravitropic ef
ﬁ
ciency can
be estimated using
a
which is a critical parameter to
determine biomechanical response, and the rates of
correction and disturbance can be estimated from Eqs.
4
and 9,
resp
ectively. Because
b
is an independent parameter in
the estimation of gravitropic performance, the adjustment
of parameter
a
to
a
0
and
a
00
induces the adjustment of
P
G
to
P
0
G and
P
00
G. The value of
P
00
G is close to
P
G
, so that
P
G
is suf
ﬁ
cient for the est
imation. It should be mentioned
that the estimation of
P
G
is based on Eq.
11, where the
ratio
a
/
b
comes from Eqs. 3 and 8 which are
model

dependent. The estimation of d
M
g
is based on the
assumption of a con
tinuous sinusoidal variation of
a
. An
assumption
of dis
continuous variation of
a
described by
Alme
´
ras and Fournier (2009) obtained an estimate of
d
M
g
which was
1.27 times that obtained with a sinusoidal variation when
half of the ring has maximal
a
and the other has minimal
a
. On the other hand, the e
stimation of d
M
s
is based on
the simple assumption that the volume of a branch is a
long cone. Alme
´
ras and Fournier (2009) described a
more elaborate estimation of stem volume by analysis of
a tapered stem. The estimation of
P
0
G and
P
00
G further
con
sid
ers the in
ﬂ
uences of eccentric growth increment
and MOE variation.
Effects of defoliation on the gravitropic response
Measured strain data of
Koelreuteria henryi
Dummer.
were shown in Table 2. Our analysis has shown that
growth strain is the sum of measu
red growth strain after
defoliation and spring

back strain during defoliation. As
shown in Table 3, the mean
a
1
of the naturally defoliated
branches
Table 2
Measured strain data of
Koelreuteria henryi
Dummer.
Branch
D
(cm) Natural defoliation strain (
l
e
)
Arti
ﬁ
cial defoliation strain (
l
e
)
Spring

back strain after Growth strain after disc Spring

back strain after Spring

back strain Growth strain after branch cut
cut leaves removed after branch cut disc cut
Upper
e
1su
Lower
e
1su
Upper
e
gudef
Lower
e
gldef
U
pper
e
sudef
Lower
e
sldef
Upper
e
su
Lower
e
sl
Upper
e
gu
Lower
e
gl
AD 3.86

186 354

454 704

1,652

719 ND1 4.84

77 168 250

1,435 ND2 4.46

185 257 63

480 ND3 4.88

248 283

76

670
AD
arti
ﬁ
cial defoliated branch,
ND
natural defoliated branches The val
ues of
each branch are averages computed from 3 to 4 measuring sites
Table 3
Experimental data of
Koelreuteria henryi
Dummer. used for calculating the curvature change of branches
Branch
a
(
l
e
)
a
1
(
l
e
)
b
def
(
l
e
)
b
(
l
e
)
b
1
(
l
e
)d
C/
d
R
(m

2
) d
C
1
/
d
R
(m

2
)
P
G
P
1G
AD

467

197

270

579

309

1.22 1.88 0.84 1.58 ND1 843

123

10.66 ND2 272

221

5.38 ND3 297

265

5.44
Asymmetric growth strain parameter of defoliated branch (
a
1
), spring

back strain parameter during defoliating action (
b
def
) and after
defoliati
on (
b
1
), net rate of curvature change after defoliation (d
C
1
/
d
R
), and gravitropic performance of full

leaf branch (
P
G
) and defoliated
branch (
P
1G
). The values of each branch are averages computed from 3 to 4 measuring sites
123
are positive due to overla
y of spring

back strain. In
full

leaf condition, the tendency is for downward bending
with
P
G
=
0.84, whereas after defoliation the tendency
becomes upward bending (
P
1G
=
1.58) due to weight loss.
There
fore, the weight of leaves plays an important role in
g
ravitational disturbance. The curvature change can be
calculated by using measured strain and spring

back
strain after defoliation. The values of d
C
/d
R
are all
negative indicating downward bending tendency.
After defoliation d
C
1
/d
R
becomes positive indica
ting
upward bending tendency. The spring

back strain during
arti
ﬁ
cial defoliation of a branch (
b
def
=

270) and that after
defoliation (
b
1
=

309) have the same order of magnitude,
indicating that the bending moment of leaf and that of
branch wood are roughl
y equal. Based on experi
mental
data of Alme
´
ras et al. (2005) and Alme
´
ras and Fournier
(2009), mean values of
E
u
=
E
¼
1
:
25,
E
l
=
E
¼
ii
0
:
75 and
E
i
=
E
¼
1
:
15 for poplar were used to calculate
the
0 00
(
i
parameters
a
,
a
00
and
a
E
=
E
). Our measurement obtaine
d
d
R
u
/
d
R
=
0.88, 0.88, 0.97, d
R
l
/
d
R
=
1.12, 1.12, 1.03 for
site 1, 2, and 3, respectively. The mean
a
,
a
0
,
a
00
, and
00
(
i
a
E
=
E
) are

467,

350,

636, and

731, respectively.
Asymmetric growth strain is still the main factor of cor

rection. Eccentric growt
h increment decreases ef
ﬁ
ciency of correction by 25% with respect to
a
,
asymmetric MOE on the periphery increases ef
ﬁ
ciency
by 61%, and MOE dif
ference between growth increment
and cross

section (
E
i
=
E
) increases ef
ﬁ
ciency by 20%.
The positive ef
ﬁ
ciency of
MOE is different from that of
gymnosperms as described above because the
longitudinal MOE of tension wood is usually larger than
in normal wood. For most woody angiosperms, the
growth eccentricity occurs on the upper side of the
inclined trunk. However, b
ased on our
ﬁ
eld observation,
the branch of most angiosperms shows increased growth
increment on the lower side, especially on the part near
the trunk. This phenomenon was also observed in the
branches of
Viburnum odoratissimum
var.
awabuki
(Wang et al. 20
09). The intriguing issue still awaits future
exploration.
Conclusion
In this study, we have developed a simpli
ﬁ
ed
biomechan
ical model, based on basic beam theory, to
calculate the bending moment and rate of curvature
change due to asymmetric growth str
ess and self

weight
(associated with growth increment and reaction wood
formation) of bran
ches. Using our previous data, we
have shown that the curvature change of gravitropic
correction and gravita
tional disturbance are of the same
order of magnitude in
quantity for Taiwan red cypress.
The four factors
(asymmetric growth strain, heterogeneous elasticity,
eccentric radial growth, and initial radius in the righting
process) are now included in the derivation of formulas.
Among these factors, asymmetric gr
owth strain is clearly
shown to be the main factor in
ﬂ
uencing the
reorientation.
In gymnosperms, eccentric growth increases the ef
ﬁ

ciency of gravitropic correction, while MOE variation
decreases the moment induced in the growth ring and
hence lowers the
ef
ﬁ
ciency of correction. The
gravitropic performance of the measured branches
shows that their situations are close to equilibrium. In
deciduous dicotyle
donous trees, the effect of defoliation
on measured growth strain and curvature is emphasized.
Our ana
lysis has shown that growth strain is the sum of
measured growth strain after defoliation and spring

back
strain during defoliation. The rate of curvature change in
full

leaf condition can be calculated by using measured
growth strain and spring

back strai
n after defoliation. The
ef
ﬁ
ciency of reorientation of a defoliated branch is
greater than that of a full

leaf branch due to the loss of
some weight. Our experimental data are in agreement
with the model. Of course, this model is also suitable in
the case
of tilted trunks. It is important that self

weight
bending moment can be directly measured by
spring

back strain. Since the model is simple, it can be
easily followed by biological researchers and can be use

fully applied in forestry research and other
ﬁ
el
ds.
Acknowledgments
The authors would like to thank Mr. Ching

Chu
Tsai for technical assistance.
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123
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