Fresh-wood bending: linking the mechanical and growth properties of a Norway spruce stem
TREE PHYSIOLOGY VOLUME
Fresh-wood bending: linking the mechanical and growth properties of ? a Norway spruce stem
TOR LUNDSTR?M 0
URS HEIZ 0
MARKUS STOFFEL 1
VERONIKA ST?CKLI 0
Corresponding author ()
0 WSL, Swiss Federal Institute for Snow and Avalanche Research SLF , 7260 Davos Dorf , Switzerland
1 Laboratory of Dendrogeomorphology, University of Fribourg , 1700 Fribourg , Switzerland
annual ring width; bending mechanics; bulk density; heartwood; knottiness; latewood; sapwood; stem growth; water content
Summary To provide data and methods for analyzing stem
mechanics, we investigated bending, density and growth
characteristics of 207 specimens of fresh wood from different
heights and radial positions of the stem of one mature Norway
spruce (Picea abies L. Karst.) tree. From the shape of each
stress?strain curve, which was calculated from bending tests
that accounted for shear deformation, we determined the
modulus of elasticity (MOE), the modulus of rupture (MOR), the
completeness of the material, an idealized stress?strain curve
and the work involved in bending. In general, all mechanical
properties increased with distance from the pith, with values in
the ranges of 5.7 ?18 GPa for MOE, 23 ? 90 MPa for MOR and
370 ? 630 and 430 ?1100 kg m ? 3 for dry and fresh wood
densities, respectively. The first three properties generally decreased
with stem height, whereas fresh wood density increased.
Multiple regression equations were calculated, relating MOR,
MOE and dry wood density to growth properties. We applied
these equations to the growth of the entire stem and considered
the annual rings as superimposed cylindrical shells, resulting in
stem-section values of MOE, MOR and dry and fresh densities
as a function of stem height and cambial age. The standing tree
exhibits an inner stem structure that is well designed for
bending, especially at a mature stage.
An important factor determining the stability of trees is the
way in which stems bend under the pressure of the wind or
when impacted by rock falls or avalanches. To understand and
predict how stems bend, information about the material
properties of living stems is required. In particular, information is
needed concerning the relationships between properties of
growth, which are affected by forest management practices,
and properties of bending mechanics.
For dry and manufactured wood, the effects of growth on
mechanics are relatively well understood, as they are relevant
to the wood and paper industry. Research on the effects of
thinning regimes, fertilization and genetic plant manipulation on
mechanical wood properties has greatly contributed to this
( Kollmann 1968, Schmidt-Vogt 1991, Niemz
1993, Lindstrom 1996, 1997, Saren et al. 2001)
. In brief,
because the properties of tracheids correlate with age and
properties of annual rings
( Dinwoodie 1961, Fioravanti 2001)
most important variables to consider when estimating the
mechanical properties of clear dry wood of conifers are the
annual ring width, or the percentage of latewood, and tree age
( Deresse and Shepard 1999, Sirvi? 2001)
An increase in the water content (u) of fresh wood
(abbreviations and their definitions are listed in the Appendix) lowers
strength and stiffness, and generally reduces the modulus of
elasticity (MOE) and the modulus of rupture (MOR) by a
magnitude that depends on dry wood density
( Dinwoodie 2000)
although low-quality timber seems to be less affected
. An increase in u also lowers the longitudinal
shear modulus (G ) and resistance (?), which affect MOE and
MOR. The negative effect of an increase in u from the usual
value of 12% to the point of fiber saturation (about 50%) is, in
order of importance: MOR, ?, G and MOE
( Kollmann 1968,
Green et al. 1999)
, whereas an increase in u above about 50%
has no significant effect on any of these properties. However,
high u values affect the bulk density of fresh wood and thus the
mass of the living stem. Stem mass has a significant influence
on the root?soil moment and the stem stresses in a leaning tree
( Neild and Wood 1999)
, as well as on tree structural dynamics
when the stem is subjected to acceleration
( Milne 1991,
Dorren and Berger 2006)
. For healthy conifer wood, the
change in u that occurs in the transition from sapwood to
heartwood as a result of torus closure of tracheids is the only factor
that influences the mechanical properties of wood once the
differentiation of cambial xylem initials is complete
The presence of knots, whether dead or alive, greatly
reduces resistance to tension, but does not significantly affect
compression or shear strength
( Kollmann 1968)
. In timber
from Picea abies L. Karst. and Pinus sylvestris L., knots can
lower MOR by up to 75%
( Kucera 1973)
, but their influence
on deformation and resistance is lower in intact stems than in
(Green et al. 1999)
showed that dead and live knots reduce bending resistance to
about the same extent, although the Nordic timber-grading
( FSS 1997)
penalize, to some extent, dead and unsound
knots compared with live knots. In general, knots also appear
to slightly reduce the MOE of dry wood
( Kollmann 1968,
The inner-stem structure of Norway spruce displays
gradients in annual ring width and percentage of latewood
(www.ncdc.noaa.gov / paleo / treering.html). Although a multitude of
growth patterns can be found depending on growth conditions,
ring width generally decreases and the percentage of latewood
increases from the pith toward the bark, with the trends
reversing with tree height
( Trendelenburg and Mayer-Wegelin
. The radial and longitudinal trends in growth pattern are
more evident in mature trees, with the exception of suppressed
( Deresse and Shepard 1999)
. Water content also exhibits
gradients within the stem. In Norway spruce, heartwood water
content ranges between 30 and 50%, whereas sapwood
displays a season-independent water content that is commonly
above 150% in early spring and below 100% in late summer
( Trendelenburg and Mayer-Wegelin 1955, Schmidt-Vogt
. The presence of knots within the stem results from the
self-pruning of dead branches at the crown base. The relative
height at which self-pruning occurs increases as trees compete
for light, resulting in radial and longitudinal gradients in knot
(Ikonen et al. 2003)
Analysis of wood bending includes deformation caused by
longitudinal tension, compression and shearing
( Newlin and
, and related failure mechanisms. Tests on clear,
dry and fresh wood show that elasticity along the grain in
tension equals that in compression, but that tension failure
stresses are higher than those in compression
( Natterer et al.
. Because of differences in failure stresses and gradients
in the growth pattern and local material heterogeneities, a
gradual redistribution of compression, tension and shear
stresses occurs as the bending load increases
. This redistribution is marked for wood with continuous
wood fibers, as was shown in studies of dry logs
et al. 1998, Natterer et al. 2000)
. Compared with sawn logs,
intact logs have a 10% higher MOE, on average, and a 15 ? 25%
higher MOR ( Natterer and Sandoz 1997).
Turning living tree stems into dry timber, and especially into
clear wood, has several effects on material properties related to
bending. The overall goal of this study was to analyze stem
mechanics. Specific objectives were to: (1) relate dry and fresh
wood density and bending properties of a stem with radial and
vertical position in the stem; (2) develop statistical regression
models linking the mechanical properties of stem wood with
radial growth properties; and (3) apply the regression models
based on the data in (1) to the entire stem to determine
stemsection values of dry and fresh density, MOE and MOR as a
function of height and cambial age.
Materials and methods
Tree and test samples
We investigated bending mechanics, density and growth
properties of the fresh stem wood of one mature, co-dominant
Norway spruce tree growing at an elevation of 460 m near Z?rich
in Switzerland, on a dystric Cambisol
( FAO 1998)
, in a mixed
forest stand (
Lundstr?m et al. 2007
). The selected tree was
representative of other Norway spruce trees in the stand.
Characteristics of the test tree were: diameter at breast height
( DBH ), 400 mm; cambial age (AGE ) at breast height,
100 years; total height (H ), 35.0 m; length of the green crown
relative to H, 0.42; and bark thickness at breast height relative
to DBH, 6%. The tree was straight and apparently healthy.
The test Norway spruce tree was felled in mid-November
and the stem transported to the laboratory where it was placed
outside on level ground in the shade. The next day, stem
diameter and bark thickness were measured at 1-m intervals. The
stem was then subjected to five successive cutting steps (cf.
Figure 1): (1) three 3-m-long log sections (each comprising
log sections T3 + T2 + T1), with their base:top at stem heights
of 0.7:3.7, 10.1:13.1 and 19.2:22.2 m, respectively; (2) three
shorter log sections ( T1, T2, and T3) 0.5, 0.9 and 1.6 m in
length, respectively, with the section center at relative stem
heights ( zrel ) = z/H of 0.04 ( T3), 0.08 ( T2) and 0.10 ( T1) of the
first 3-m log, 0.31 ( T3), 0.35 ( T2) and 0.37 ( T1) of the second,
and 0.57 ( T3), 0.61 ( T2) and 0.63 ( T1) of the third 3-m log; (3)
quarter sections (A, B, C and D); (4) planks (I, II, ?.VI); and
(5) bending specimens, ranging from bark to pith (1, 2, ?8).
The bending specimens were 20 ? 20 ? 360 mm (127 pieces
cut from the three log sections denoted T1), 40 ? 40 ? 720 mm
(61, T2) and 80 ? 80 ? 1440 mm (19, T3), with a total of 113
bending specimens from the heartwood and 94 from the
sapwood. The transition between the water-conducting sapwood
and the heartwood was set to where there was a steep radial
crease in water content. Altogether 207 bending specimens
were obtained. For each specimen, annual ring width ( RW )
and percentage of latewood ( LW ) were measured with a
caliper and averaged over the cross section. The specimens were
grouped into four categories according to the approximate
orientation of the annual rings, with Cut = 1 corresponding to
almost straight rings, Cut = 4 to rings resembling a quarter of a
circle, and Cut = 2 and 3 to intermediate patterns. This
grouping depended on the cutting scheme position of the specimen.
For example, the specimens from the inner heartwood were
grouped in Category 4, and the ones from the outer sapwood in
Category 1 (cf. Figure 1). In addition to the bending
specimens, stem discs were cut from zrel = 0.0085 (i.e., at z =
0.30 m), 0.75, 0.90 and 0.99 to analyze radial growth.
Knot frequency was described by a three-number
classification. The first number refers to knottiness Q
cross section, which is the ratio between the sum of diameters
of knots on the compression- (upper) and tension- (lower) side
of the test specimen, along the middle 150 mm of the span, and
twice the specimen width. The second and third numbers were
calculated accordingly, but refer specifically to the knots on
the tension- (t) and the compression- (c) side of the specimen,
Qt and Qc, respectively. Although most knots were living, no
distinction was made between dead and live knots. Because
the annual ring curvature was generally small compared with
specimen size, the frequently used Knot Area Ratio
was less than or equal to 2Q. Wood defects other
than knots were not observed among the test specimens.
Tests and investigations
The specimens were tested in 3- and 4-point bending (187 and
20 specimens), according to the German industry code, DIN
includes all the mentioned DIN codes),
with an exception made for their water content u. For this
purpose, we used a hydraulic vertical press D-200-VS 91
connected to a data logger PK-SRG-5000 124 (Walter+Bai AG,
L?hningen, Switzerland). The load was applied with a
constant deflection rate so that the maximum force was obtained
within 90 ? 30 s. With load and deflection sampled at 1 Hz, the
tests were continued until complete failure of the specimen
occurred. After testing, the wood samples were cut close to the
failure zone to analyze u and the dry and fresh wood density
(?0, ?U ) according to DIN 52182 and DIN 52183. To
calculate ?0, we used the dry volume and the wet mass, and
corrected for tree volumetric shrinkage with the factor 1.10
( Kollmann 1968)
. The bending tests and the determinations of
u were completed within one day of specimen cutting and one
week of tree felling.
All data related to bending mechanics, such as MOE and
MOR, were based on the complete curves for bending stress
(?) as a function of bending strain (?), from no load up to
maximum load and then until the load returned to near zero. To
obtain these curves, we followed five steps ( Table 1) and used
E = Fs ??? 4C81I + C 2 ??? ; I =
C 2 = ? ?? E ?? L
? G ? bh
where F is the applied force ( N ), L, b and h are the length,
width and height of the bending specimens (mm), s is the
mid-span deflection ( Mm), LF is the distance between support
and force application (mm), C1 is a length factor that equals L3
in 3-point bending, C2 is a correction factor for shear
deformation, ? is a load coefficient that equals 8 / 27 and 16 / 81 in
3and 4-point bending, respectively
( Newlin and Trayer 1956)
and (E/G )? is an estimated relationship between the elasticity
in bending (E ) and shear (G ) along the grain. For sawn wood
of Norway spruce with a density = 400 ? 600 kg m ? 3, and u =
12%, E/G has a mean of 18 and is about 21 for green wood
50%: Kollmann 1968)
, which is the value that we adopted for
(E/G )?. The mass of the bending specimens contributed less
than 0.2% to MOR and was therefore omitted from the
analysis. Except for the analysis of MOE ( Table 1), ?(?) is used
throughout this study.
We also established characteristic ?(?) curves, where each
was the mean stress?strain curve for a specific group of
bending specimens. A distinction was made between three groups
of mean relative heights 0.10 (1), 0.37 (2) and 0.63 (3), and
between two radial ranges, the inner heartwood ( IH ) and the
sapwood (S). The outer heartwood (OH ), with properties between
the S- and IH-groups, was analyzed only individually. Each of
the six groups included between 18 and 32 ?(?) curves from
the small- and medium-sized bending specimens. The
stress? strain curves obtained were further described with
simplified ideal elastic?ideal plastic curves. The break point (?,?)
= (?el-pl,MOR??el-pl) between ideal elastic and ideal plastic
deformation was conditioned by an equal amount of total work
absorbed by the bent member up to ? = MOR, compared with
the original ?(?) curve.
The absorbed bending work (W ) ( Equation 2) and the
completeness of the material (?) ( Equation 3) were both
calculated from the load?deflection curve F(s):
W = ? F(s)ds
where Fmax is the maximum applied force ( N ), sMOR is the
value of s at MOR (mm) and WMOR is the work developed up to
MOR ( N m). The values of total work (Wtot) absorbed by the
bending specimen during the entire bending test were
compared for specimens of the same size. To compare with work
absorbed by specimens of other dimensions
(e.g., M?rki et al.
, Wtot was divided by the volume (V ) of the specimen
within the span. For calculations of W / V, only the specimens
tested in 3-point bending were included because a larger part
of the bending specimen is activated in bending stress in
4-point bending than in 3-point bending, which mobilizes
more W / V in 4-point bending. Moreover, work was also
expressed in terms of ? integrated with respect to ?, as denoted
by ??(?) d ?. This number characterizes the material and is
independent of the set-up and cross section of the test, just like ?.
We used Matlab 7.0 ( MathWorks) for all mechanics analyses.
The statistical relationships within and between the growth
characteristics and mechanical properties were explored with
S-Plus 2000 (Insightful Corporation). We first applied the
pair-wise test between variables to detect their possible
transformation and then performed forward and backward
stepwise linear regression of the model response variables
described by the explanatory variables.
To qualitatively rank the regression models, we used the
Akaike Information Criterion AIC
(Sakamoto et al. 1986)
the Mallows?s Cp statistic
( Daniel et al. 1980)
, where low
absolute values of AIC and Cp reflect high quality. The quality of
model i was expressed as AICrel,i = (1 / AICi )/(1 / AIC1 ) and
Cprel,i = (1 / Cpi )/(1 / Cp1), where i = 1 is the highest ranked
model. We also calculated the R 2 of each model, but as this
overestimates the quality of multivariate regression models of
correlated variables, it was given third priority in the model
ranking. We know of no mathematically correct and
straightforward method to calculate the amount of the total variance
that each Xj ( j = 1, 2,? n ? 1, n) explains in a multivariate
model of correlated variables (our case). Therefore, to rank the
importance of each Xj for its contribution to Y, we used the
difference in Cp value between the full, highest ranked model and
the model lacking the Xj (??Xj?, Equation 4):
n ?1 Cp(Model ? X j) ? Cp(Model)
+ Cp j = ?
The effect of each Xj on Y was expressed by the standardized
regression coefficient of Xj, aj = aj?SE( Xj ) / SE(Y ). Because the
investigated mechanical properties were obtained per tested
specimen, the growth characteristics needed to be averaged for
each bending specimen. For this reason, the large- and
medium-sized specimens were less useful for describing the
statistical relationships between the properties of growth and
mechanics than the small specimens. Consequently, we used only
the small specimens.
Application of statistical relationships to the mechanical properties of the entire tree stem
The growth data between pith and bark at seven values of zrel
were interpolated radially every millimeter and longitudinally
every percent of tree height (piecewise cubic interpolation).
The growth data at zrel = 0.10, 0.37 and 0.63 originated from
the 188 small- and medium-sized specimens, and those at zrel =
0.0085, 0.75, 0.90 and 0.99 from stem discs sampled at these
heights. Data on u were not obtained at the latter heights and
originate from neighboring Norway spruce trees with a growth
pattern similar to that of our test tree at zrel = 0.10, 0.37 and
0.63 (unpublished SLF ). Application of RW, u and Q to the
regression equations resulted in ?0, ?U, MOE and MOR within
the entire stem. The four mechanical properties were then
calculated for the stem section as a function of stem height and
AGE at breast height, with the relative distribution of u
approximated so as not to vary with AGE
( Trendelenburg and
Mayer-Wegelin 1955, Schmidt-Vogt 1991)
To calculate MOEsection ( Equation 5 ), we used cylindrical
shells ranging radially (x, which due to the circular shape,
equals r) from pith ( j = 1) to bark ( j = n) every mm and
longitudinally (z) from stem base (i = 1) to tree top (i = 101) every
percent of tree height.
MOE section, i =
4 ? MOE(x)i, j(xi4, j+1 ? xi4, j)
The calculation of MORsection assumes, in principle, that ?(x)
as MORsection is reached, is known. In default of this
knowledge, and ongoing from the strain?stress distribution of dry
timber as its cross section reaches MOR
( Kollmann 1968)
used two equations to calculate MORsection. The first (
Equation 6) applies a linear increase in longitudinal strain from pith
to bark, with a gradient governed by the strain at MOR for the
cylindrical shell at the radial distance of ?1xn, where ?1, is a
factor between 0 and 1 (method of linear strain).
Unlike this first approach, which requires knowledge of
?(?,x), the second approach ( Equation 7) is simpler. It applies
a degree of exploitation B(?2) of MORj, where B(?2) is an
array that increases from 0 to 1 between x = 0 and x = ?2xn and
then remains constant, at a value of 1, between x = ?2xn and xn
(method of scaled MOR). Referring, once again, to dry timber,
we set both ?1 and ?2 equal to 0.85.
Finally, the calculations of ?0section and ?Usection were simply
quadratic summations of dry and fresh wood densities from
the pith to the bark ( Equation 8).
1 n 2
2 ??(x )i, j(xi, j+ 1 ? xi2, j)
Failure mechanisms and behavior of bending stress?strain
The failure mechanism of the fresh bending specimen started
with the wood fibers buckling on the compression side,
followed by the fibers tearing apart on the tension side. The latter
often occurred close to knots if they were present in the
mid-span of the specimen. No signs of shear failure were
The complete ?(?) curves showed five general features
depending on the radial and height position in the stem (
Figure 2A and Table 2). First, all bending specimens, independent
of height or radial position, displayed a non-negligible
remaining resistance beyond the strain at MOR, ?MOR. Second,
generally, Wtot increased from the IH-wood to the S-wood. Third,
Wtot, MOE and MOR of the S-wood were almost identical at
stem heights 1 and 2, but lower at height 3, and the Wtot of the
IH-wood decreased with increasing stem height, whereas its
MOE and MOR changed little. Fourth, although ?(?) differed
in magnitude depending on height or radial position,
?(?) / MOR curves were similar in shape until ?MOR was
reached ( Figure 2B). Fifth, beyond ?MOR, ?(?) displayed a
more abrupt drop in the IH-wood compared with the S-wood.
Table 2 shows that MOE and MOR were both about 60%
higher and that Wtot/V is twice as large for the S-wood than for
the IH-wood. It was also apparent that the characteristics of the
simplified stress?strain curves were independent of the
position within the stem. At the individual specimen level, the
IH-wood displayed a step-wise decrease of ? after MOR (
Figure 2C). Generally, this stair-case pattern smoothed gradually
toward the bark. The SEs of the six mean ?(?) / MOR curves
( Figure 2B) were larger after MOR and for the IH-wood than
before MOR and the S-wood ( Table 2). There was a weak
general increase in ?MOR with MOR ( Figure 2C and Table 3).
Not accounting for shear deformation in the 3- and 4-point
bending tests led to underestimates of MOE of about 11 and
6%, respectively. Apart from this and Wtot/V, there were no
significant differences in bending properties between the 3- and
4-point bending tests.
Growth and mechanics measured across the stem
The measured properties of growth displayed characteristic
patterns from the pith to the bark: RW decreased, whereas LW
and u generally increased, with values in the range of
0.5 ? 6.5 mm for RW, 0.10 ? 0.71 for LW and 26 ?145% for u.
Knottiness followed a Gaussian curve, with its maxima further
from the bark at the stem base than higher up the stem. Values
of Q ranged from 0 to 0.9 (apart from one specimen with Q =
1.8) and averaged 0.12. The most obvious dependency on tree
height was the increase in u. The mechanical properties
followed similar patterns to the growth characteristics across and
along the stem. All mechanical properties generally increased
with distance from the pith, with MOR in the range
23 ? 90 MPa, MOE 5.7 ?18.1 GPa, Wtot/ V 35?219 kPa, WMOR/ V
20 ?128 kPa, and ?0 and ?U in the ranges 370 ? 630 and
430 ?1100 kg m ? 3, respectively. Except for W, all general
tendencies or means of measured values are shown in Figures 3
Relationships between the properties of mechanics and growth
As expected, several of the 13 variables of growth and
mechanics were strongly correlated ( Table 4). This resulted in
regression models for the response variables ?0, MOR and
MOE, with six or less significant variables. Altogether 40
useful models describing ?0 (9), MOR (20) and MOE (11) were
obtained. Nine of these models are described in Tables 5 ? 7.
Among the three response variables, ?0 was modeled with
dardized variable coefficient aj. The models (?MOR1,2) and the model variables Xj are listed top down according to the ranking and contribution to
the model, respectively. The degrees of freedom are 122. Significance: *, P < 0.05; **, P < 0.01; and ***, P < 0.001.
1 AICrel,i = (1/AICi)/(1/AIC1) and Cprel,i = (1/Cpi)/(1/Cp1), where AIC1 = ?1254 and Cp1 = 3.77E?04 are the criteria of the highest ranked model
(?MOR1) and i = 1, 2 refers to each of the two models.
2 +Cpj is the relative contribution of each variable to Cp of the model (Equation 4).
the highest quality, followed by MOR and then MOE. The
variable ?0 was strongly related to RW and LW ( Table 5), and
MOR was best described by ?0 ( Table 6). Although Qt
reduced MOR about twice as much as did Qc, the overall
frequency of knots was low, so knottiness contributed little to the
quality of the models. The exclusion of ?0 in the starting set of
variables involved log( RW) in the models describing MOR
( MOR2?4) because ?0 and log( RW) were strongly correlated.
Excluding ?0 also suggested models including AGE.
However, these models contained additional variables that
improved model quality little. Although Cut ( = 1, 2, 3 or 4)
tended to reduce MOR = ?18.9log(Cut) + 67.5 (R 2 = 0.35, both
coefficients P < 0.001), its influence was weak because of the
strong correlation with other variables in the multivariate
models ( Table 4) and the presence of mainly straight annual
rings (Cut = 1 or 2). The models describing MOE included
l o g ( RW), u and ?0 ( Table 7). Although MOE was also
reduced by Qc (P = 0.10), Cut and AGE, their impact on MOE,
like that of MOR, was weak (for the same reasons).
Growth and mechanics within the stem
The growth properties interpolated within the entire stem
( Figure 3) were applied to the regression models describing
mechanical properties within the stem ( Figure 4). These two
property groups display some general features. For example,
the zone between the IH-wood and the S-wood constitutes a
transition for all properties of growth and mechanics except
for Q. Between zrel = 0.1 and 0.5, Q appears to be primarily
influenced by the location of the lower part of the crown,
changing from zrel = 0.1 at AGE = 10 years to zrel = 0.5 at 35 years,
after which it increased little with height with increasing AGE.
Mechanics of the stem section
All mechanical properties of the stem section ( Figure 5)
decreased with stem height, except for ?Usection, which increased.
Within the stem section covered by the crown, MORsection, and
especially MOEsection, decreased with increasing height. At the
buttressed stem base, the large RW in the outermost stem
section reduced MORsection and MOEsection. At AGE = 100 years at
breast height, the section values of MOR ( Figure 5B) obtained
by the linear strain method ( Equation 6) were between 8 and
12% lower than those obtained by the scaled MOR method
( Equation 7). These differences, which were largest at the
stem base and in the crown, resulted from the different radial
1 AIC1 = 1222 and Cp1 = 1.104E+05.
2 +Cpj is the relative contribution of each variable to Cp of the model (Equation 4).
1 AIC1 = 2251 and Cp1 = 3.642E+08.
2 +Cpj is the relative contribution of each variable to Cp of the model (Equation 4).
1 AIC1 = 732.6 and Cp1 = 2342.
2 +Cpj is the relative contribution of each variable to Cp of the model (Equation 4).
distributions of bending stress ( Figure 6). Figures 5A?D show
that all section values decreased with AGE, independently of
Discussion and conclusions
The bending stress? strain curves, ?(?), obtained for fresh
Norway spruce wood differ markedly from those for dry wood
of the same species
( Kollmann 1968, M?rki et al. 2005)
fresh wood being more easily deformed and having a lower
MOR. In terms of energy absorption of fresh wood (i.e., area
under the ?(?) curve), these two behaviors have opposite
effects. A comparison of the values of energy absorption found
in our study with published values for dry wood of Norway
spruce undergoing bending
( Kollmann 1968, Brauner et al.
2005, M?rki et al. 2005)
suggests that the two effects are of
similar magnitude (a factor of 2). This implies that fresh and
dry wood of Norway spruce absorb about the same amount of
energy while bending to complete failure. Nevertheless, when
analyzing the bending behavior of living tree stems, it is
essential to consider the ?(?) of fresh wood, and not of dry wood.
Most mechanical properties related to the stress? strain
curve display a strong gradient in the radial direction because
of the gradient in growth properties. The S-wood and the
IH-wood (i.e., juvenile wood) form two distinct categories,
and the OH-wood provides the transition zone. A similar
radial tendency for MOE and MOR, but not for dry wood
density, was also recorded by
Huang et al. (2005)
in logs of red
cedar. Compared with S-wood, the IH-wood displays brittle
failure, with about 40% lower MOR and MOE and about half of
the total energy absorption per volume in bending. This means
that S-wood maintains its resistance and energy absorption at
strain values beyond MOR, unlike IH-wood. This difference
between S-wood and IH-wood may be a result of a ?global
effect? found in living stems as well as in the specimens: the
cross section of the bending specimen from the S-wood
included 10 ? 40 annual rings, and that of the IH-wood only 3 ?10
rings. The annual rings probably fail one after the other,
resulting in a gradual stress redistribution, which may be more
abrupt with five annual rings than with 30 rings. The
phenomenon may also be associated, in part, with the lower water
content of IH-wood compared with S-wood. Nevertheless, the
shapes of the ?(?)/MOR curves for IH-wood and S-wood were
similar up to MOR ( Figure 2B), resulting in similar idealized
stress?strain curves and values of completeness, which
simplifies the computation of ?(?) up to MOR for fresh wood at any
radial position on the stem.
Dry wood density was about 40% lower in IH-wood than in
S-wood. For fresh wood density, these proportions were
roughly inversed because of the water content. Sapwood
exhibits variations in water content during the year that can reach
several tens of percent
( Trendelenburg and Mayer-Wegelin
, and these variations affect the fresh wood density of the
living tree stem. Our Norway spruce tree was felled in the
middle of November when its water content probably
approximated the mean annual water content
Consequently, the stem section values of fresh density should
reflect the annual means.
Comparisons of the mechanical properties of the stem with
the literature are relevant only if the growth conditions and
properties of radial and apical growth are similar to those of
the tested Norway spruce. This applied both to the values
within the stem and those obtained for the stem section at
cambial ages of 25, 50, 75 and 100 years at breast height. The
100-year-old co-dominant Norway spruce we studied grew in
a relatively well drained soil and in competition in a fairly
dense stand, and is representative of the whole study stand and
of several other sites with similar climatic conditions in and
around the Alps (
, Schweingruber 1996) and even
(Core et al. 1979)
. It is evident that Norway spruce
stems with other growth patterns, e.g., trees growing alone, on
windy spots, on slopes at the tree line or that are overaged or
damaged, will exhibit radial as well as cross-sectional
mechanical properties that differ from the properties we
observed. For example, Norway spruce trees in the Black Forest
grow faster than the Norway spruce tree we investigated and
exhibit between 30 ? 40%
( Trendelenburg and Mayer-Wegelin
and 5 ?15% ( Br?chert et al. 2000) lower dry wood
density at equal cambial ages. For MOE,
Br?chert et al. (2000)
report 10 ? 30% lower values at the same cambial age, but the
MOE distribution within the stem was similar to that in our
study tree. We found no published studies on MOR of fresh
wood of Norway spruce combined with growth data.
The regression models help to identify growth properties
having large effects on MOR, MOE and dry wood density. Our
aim was to provide models for stems of a typical Norway
spruce tree. Trees growing under extreme conditions develop
special stem wood properties in addition to those considered in
this study such as compression wood, spiral or curly grain,
callus tissue, traumatic resin ducts and gelatinous tracheids,
which all influence wood mechanics
Mattheck 1998, Dinwoodie 2000)
. Previous studies have
shown that age positively influences mechanical properties
). Nevertheless, old trees with large radii and
widely spaced annual rings close to the bark do exist, but
multivariate studies of mechanics of fresh wood of old trees
are rare. We therefore decided not to present regression
models including age or the radial distance from pith, although
these variables slightly improved regression quality. The
correlations between variables ( Table 4) generally confirm
( Kucera 1973, Green et al. 1999, Dinwoodie
. The MOE and MOR values predicted by ?0, RW and Q,
with a correction made for u, are comparable with results from
( Kollmann 1968, Niemz 1993, Natterer et al.
. If we exclude trees with special growth properties, the
regression models ( Tables 5 ? 7) are likely to be generally
applicable to stems of Picea abies L. Karst. trees.
The calculated section values of MOR, MOE and wood
density ( Equations 5 ? 8) indicate that the radial and apical growth
of the Norway spruce stem we studied is well designed for
bending. The relatively high MOR and MOE values at the base
and low values toward the top should provide several
advantages for the resistance and stability of the tree, e.g., when it
interacts with strong winds. A flexible upper part of the stem
prevents the crown from catching wind flow, and the
stem-bending resistance is highest where it is most needed,
namely at the stem base, where the largest bending stresses
occur. The adaptive growth observed in the radial direction of the
stem economizes material in that the dense and strong wood is
located close to the bark where it most effectively provides
resistance and stiffness to the stem section. Although the
composite equations require validation (bending of entire logs),
they provide a further mechanism of how trees optimize their
growth, in addition to what has been found previously
and Ray 1996, Mattheck 1998)
The heterogeneity of wood within the stem of Norway
spruce and its mechanically optimized inner structure can be
accounted for using the relationships and methods presented
in this contribution. The methods can be used to analyze the
stability and energy dissipation of entire trees and the
influence of different thinning regimes on the mechanical
properties of the living stem.
The authors are indebted to the Board of the Swiss Federal Institutes
of Technology for financial support, as part of the ?Tree Stability and
Natural Hazards? (?Naturereignisse und Baumstabilit?t?) project. We
thank Werner Gerber for his precise and successful experimental
work and Silvia Dingwall for revising the text.
Levels of significance for P < 0.05, P < 0.01 and P < 0.001
Variable and intercept coefficients; standardized variable coefficient (cf. Statistical analysis)
Cambial age counted from pith
Statistical indices of quality of regression models and variables (cf. Statistical analysis)
Parameter describing the annual ring curvature within the cross section of specimens (cf. Figure 1)
Stem diameter on bark at breast height, i.e. 1.3 m
Secant modulus of elasticity due to pure bending = ?/?E
Apparent bending strain (simply ?bending strain?); ? without the contribution from shear deformation
? as ? reaches 0.4MOR (defined limit of elasticity); reaches MOR; and returns to 0.5 Mpa
The ideal elastic-ideal plastic ?(?)-curve has a break point at ? = ?el-pl and ? = ?el-plMOR
Applied force when testing bending specimens; maximum applied F
Shear modulus in the z-direction, i.e. along fibers
Total tree height
Completeness of the material in bending (cf. Equation 3)
Length, width and height of the bending specimen; distance between support and force application
Late wood portion = width of late wood/RW
Modulus of elasticity = the secant modulus of ?(?) as 0.4MOR is reached; MOE of stem section
Modulus of rupture = maximum value of ?(?); MOR of stem section
Integration of ?(?) up to MOR, i.e. area under ?(?)-curve
Knottiness of the specimen cross-section, on the tension (t) and on the compression (c) side
The radial coordinate of the stem section, ranging from pith to bark
Density of oven-dry and of fresh wood
Annual ring width
Mid-span deflection when testing bending specimens; s at MOR
Sapwood; inner heartwood
Standard error; R-squared value of a regression; P-value of significance
kg m ? 3
Shear stress and resistance in the z-direction
Wood water content = mass of water/oven-dry mass
Volume of bending specimen within the bending span
Work absorbed in bending, at MOR and at ?max
Cartesian coordinates of the standing tree stem: z is the height above stem base and x = 0 in the pith
Explanatory variable; response variable
Relative height above stem base = z/H
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