Stand variation in Pinus radiata and its relationship with allometric scaling and critical buckling height

Abstract

Background and Aims

Allometric relationships and the determination of critical buckling heights have been examined for Pinus radiata in the past. However, how they relate to more mature Pinus radiata exhibiting a wide range of stem diameters, slenderness and modulus of elasticity (E) at operationally used stand densities is largely unknown. The aim of this study was to examine the relationship between Pinus radiata stand structure variables and allometric scaling and critical buckling height.

Methods

Utilizing a Pinus radiata Nelder trial with stand density and genetic breed as variables, critical buckling height was calculated whilst reduced major axis regression was used to determine scaling exponents between critical height (H_crit), actual height (H), ground line diameter (D), slenderness (S), density-specific stiffness (E/ρ) and modulus of elasticity (E).

Key Results

Critical buckling height was highly responsive to decreasing diameter and increasing slenderness. Safety factors in this study were typically considerably lower than previously reported margins in other species. As density-specific stiffness scaled negatively with diameter, the exponent of 0·55 between critical height and diameter did not meet the assumed value of 0·67 under constant density-specific stiffness. E scaled positively with stem slenderness to the power of 0·78.

Conclusions

The findings suggest that within species density-specific stiffness variation may influence critical height and the scaling exponent between critical height and diameter, which is considered so important in assumptions regarding allometric relationships.

INTRODUCTION

The vertical stems of terrestrial plants must mechanically sustain their own weight against the influence of gravity. They also must be sufficiently stiff to resist bending and avoid breaking when subjected to large externally applied mechanical forces (Niklas, 1993). As such, modulus of elasticity (E), a property also known as stiffness, and density (ρ) are important mechanical properties. These properties are of interest because, in theory, the quotient of E and ρ (i.e. the density-specific stiffness, E/ρ) determines the extent to which vertical stems can grow before they reach their critical buckling height (i.e. the height at which elastic buckling is predicted to occur) (Niklas and Buchman, 1994).

Previous research (Watt et al., 2006a, b; Waghorn et al., 2007c) suggests that E may be regulated by stem slenderness. When light-demanding species such as Pinus radiata are subject to competition from neighbouring stems, rapid height growth is important to ensure that they are not overtopped. Under high levels of competition, trees become etiolated as priority is given to height growth at the expense of diameter increment. The critical height (H_crit) that a vertical tree stem can reach before it undergoes elastic buckling is given by Euler's buckling formula:

(1)

where C is the constant of proportionality, E is modulus of elasticity, ρ is the density of green wood and D is stem diameter (Greenhill, 1881).

The value for C can vary between 0·79 and 1·97 depending on assumptions about loading conditions and tree taper (Niklas, 1997). Regardless of the numerical value of C, eqn (1) predicts that the scaling of H obtains the proportionality H ∝ D^2/3, provided E/ρ and the safety factor remain constant (Niklas, 1993). The safety factor defines the margin by which critical height exceeds actual height. The scaling exponent α is thus the ratio of the relative growth rate in H with respect to the relative growth rate in D (Niklas, 2004). This assumption forms the basis of the elastic similarity model proposed by McMahon (1973) and McMahon and Kronauer (1976) which predicts the scaling of tree height based on diameter (i.e. H ∝ D^{α = 2/3}). However, if E/ρ is not a constant, then the scaling exponent α for the proportional relation H ∝ D^α depends upon the scaling of E/ρ with D (Niklas, 1993). The testing of this assumption has typically been carried out at a very broad level, investigating variation between genera (Niklas, 1993) and between species (Niklas, 1994a). More recently, this assumption has been tested on 4-year-old Pinus radiata across a wide environmental range (Watt et al., 2006b); however, how it relates to more mature Pinus radiata exhibiting a wide range of stem diameters, stem slenderness and E at a single site has not been studied.

For a given E, actual height will approach the critical height as stem slenderness (S) increases. Low slenderness ratios (H/D) indicate that very large self-loads are required to induce elastic buckling, whilst high slenderness ratios indicate that smaller self-loads are required to produce elastic buckling. These generalities exist because, for any columnar support member, the slenderness ratio is proportional to (E/P)^1/2, where E is the modulus of elasticity and P is the maximum self-load that a column can support, i.e. H³/D² ∝ (E/P)^1/2. Thus, the mechanical stability of very slender columnar stems requires either tissue with high E or stems with low P (Niklas et al., 2006). The Greenhill (1881) equation shows that trees can increase their critical height to avoid buckling as slenderness increases by increasing their density-specific stiffness (E/ρ). As green density is relatively constant at the tree level in younger trees composed mainly of sapwood on single sites (Lasserre et al., 2005; Waghorn et al., 2007a), this is accomplished primarily through increases in E.

Even though the height of a tree, H, may never exceed the critical buckling height, H_crit, the degree to which H approaches H_crit may influence wood properties. The safety factor is defined as the quotient of H_crit and H. If growth in size attains a safety factor less than unity, then the stem can be predicted to deform elastically under its own weight. Safety factors greater than unity, therefore, indicate that an individual stem can sustain greater mechanical loadings than those imposed by its own biomass (Niklas, 1994a). Rearrangement of eqn (1) in terms of the safety factor (H_crit/H) as:

(2)

shows that, for a given diameter, increases in slenderness need to be accompanied by increases in E to maintain a constant safety factor (Watt et al., 2006b).

Using measurements taken from trees grown across a stand density gradient, the primary objective of this study was to determine the validity of the elastic similarity model for these data by determining how H_crit and the safety factor scale with diameter.

MATERIALS AND METHODS

Site location

Measurements were taken from 17-year-old Pinus radiata trees that had been grown in a Nelder experiment (Nelder, 1962) located at Burnham, approx. 18 km south-west of Christchurch (latitude 43°36·5′S, longitude 172°17·75′E, altitude 70 m a.s.l.), New Zealand. The site was situated on Lismore stony silt loam soil (New Zealand Soil Bureau, 1968) and experienced a mean annual precipitation of 650 mm, in which seasonal water deficits occurred during January to March, when evapotranspiration exceeded rainfall (G. Furniss, University of Canterbury, New Zealand, pers. comm.).

Experimental plot

The experiment comprised seedlings of three breeding series (850, 870, 268), and within the 268 series, cuttings of two physiological ages (1-year-old cuttings and 3-year-old cuttings) which were taken from parents that had been grown to ages of 1 and 3 years. None of the breeding series were bred for improved wood stiffness. The Nelder contained 45 spokes separated by 8° intervals in ten circular rings with high initial stocking rates (2551 stems ha⁻¹) present at the centre of the Nelder and low initial stocking rates (209 stems ha⁻¹) present on the outer ring of the Nelder. Each breeding series/cutting treatment occupied nine of the 45 spokes. The trees had not received any silvicultural treatment at any stage prior to examination (Waghorn et al., 2007b).

Seventy-two Pinus radiata trees representing the five different breeding series/cutting treatments and five initial stand densities of 275, 364, 635, 1457 and 2551 stems ha⁻¹ were selected. The initial stand densities chosen cover the range of operational stand densities at which Pinus radiata is grown in New Zealand, which typically range from 250 to 1200 stems ha⁻¹. Three repetitions of each of the breeding series/cutting treatments by stand density interactions were sought. The selected trees were all completely surrounded by neighbouring trees within the Nelder trial plot.

Measurements

Measurements of tree height, diameter at breast height and ground line diameter were taken prior to felling. Stem slenderness was determined as the ratio of tree height to ground line diameter. All 72 stems were felled and de-limbed and then cut into 2-m-long logs up the length of the stem. Assessment of acoustic velocity for the bottom 2-m log from the 72 felled trees was carried out using HITMAN, an acoustic resonance instrument, which provided a volume-weighted average of velocity (Harris and Andrews, 1999; Kumar, 2004). Observations by Lindstrom et al. (2002) found correspondence between resonance-generated E and E from traditional static bending to be very strong (R² = 0·98) and relatively unbiased (y = 1·04x). The resonance measurement of velocity is a near perfect spatial average of the log, both in log length and cross-sectional area (Harris et al., 2002).

Green dynamic modulus of elasticity (Pa) was determined using the following equation:

(3)

where V is velocity obtained using HITMAN (km s⁻¹) and ρ is green density (kg m⁻³).

Green density was calculated from a sample of 38 discs taken at 1·4 m above ground level from selected trees representing a combination of breeding series/cutting treatments and initial stand densities. After acoustic velocity was assessed, 30-mm discs were taken from the 38 trees. The samples were placed in sealed plastic bags and frozen until measurements were taken. In the laboratory, the samples were defrosted and green density was determined as green weight/green volume using the immersion technique. From analyses of these measurements, green density was not found to vary significantly with either breed or stand density, and the mean green density was 935 kg m⁻³ (± 8·2 kg m⁻³). This figure along with the HITMAN velocity readings was used to determine E and critical height for all stems.

Data analysis

All analyses were undertaken using SAS (SAS Institute, 2000). Using a MIXED model with a cross-over design, an analysis of variance was used to examine the main and interactive effects of stand density and genetics on E at the whole-tree level. Allometric analysis was used to examine relationships between D, S, H, H_crit, E and E/ρ at age 17. Critical height was determined using eqn (1), with a value for C of 0·792 (Niklas, 1994b). For dimensional consistency, eqn (1) requires that density-specific stiffness be expressed in units of metres, by converting the value of ρ which was 935 kg m⁻³ into Newton m⁻³ (i.e. 1 kg weight force = 9·8067 N).

Regression models of the form

(4)

were used to determine the parameters α and β for the allometric relationships defined as logY and logX (Watt et al., 2006b). Reduced major axis regression (RMA) analyses were used to determine the scaling exponents and allometric constants (i.e. α_RMA and logβ_RMA, respectively) for the logY vs. logX allometric trends observed. RMA was used as the objective of the regression analysis was to determine functional rather than predictive relationships between two biological variables (Niklas et al., 2006). The regression parameters were computed using the formulae α_RMA = α_OLS/r and logβ_RMA = – α_RMA , where α_OLS is the ordinary least squares (OLS) regression exponent, r is the OLS correlation coefficient and and denote the mean values of variables logY and logX (Niklas, 1994b; Watt et al., 2006b). The 95 % confidence intervals (CI) for α_RMA and logβ_RMA were, respectively, determined using the formulae 95 % CI = α_RMA ± t_{N −2} (MSE/SS_X)^1/2 and 95 % CI = logβ_RMA ± t_N–2 {MSE[(1/n) + (logX²/SS_X)]}^1/2, where t_N–2 is the t-value, MSE is the OLS regression model mean square error, SS_X is the OLS sums of squares and n is the sample size (Niklas, 1994b; Watt et al., 2006b).

RESULTS

Variation in measured variables

Variation in diameter at breast height and slenderness was significantly (P < 0·0001) affected by initial stand density. Both diameter and stem slenderness exhibited an approx. 2-fold variation between 275 and 2551 stems ha⁻¹ (Table 1). The influence of stand density on tree height was not significant (P > 0·05).

Table 1.

Variation in diameter, height and stem slenderness across the five stand densities

Initial density (stems ha−1)	Mean diameter (cm)	Mean height (m)	Stem slenderness (m m−1)
275	35·3 (1·1)	18·1 (0·7)	51 (2)
364	35·3 (0·9)	19·2 (0·3)	55 (1)
635	30·8 (1·1)	19·2 (0·4)	63 (2)
1457	24·4 (0·8)	18·7 (0·3)	77 (2)
2551	18·5 (1·1)	18·3 (0·5)	103 (5)

The influence of genetics on stem dimensions was minimal compared with stand density. Genetics did not significantly (P > 0·05) affect diameter, tree height or stem slenderness. No significant interactions between genetics and stand density were observed for any of the stem dimensions examined.

E was significantly influenced by initial stand density (P < 0·0001) and, to a lesser extent, genetics (P = 0·003). There was a positive relationship between stand density and E, which exhibited a 37 % increase in E from 275 stems ha⁻¹ to 2551 stems ha⁻¹. The 870 breeding series exhibited the highest E of 6·6 GPa, which exceeded E for the 3-year-old cuttings, 268 breeding series, 1-year-old cuttings and 850 breeding series by 5, 10, 13 and 18 %, respectively.

Critical height and allometric scaling relationships

Critical buckling height was well above that of actual tree height and these differences diverged with increasing ground-line diameter (Fig. 1). Only one tree at the highest stand stocking (2551 stems ha⁻¹) was deemed to be especially close to unity (H = H_crit). This was due to the very low stem ground-line diameter that the tree displayed (12·5 cm). This low diameter combined with the tall height of the tree meant that stem slenderness was high (141 m m⁻¹), considerably more than any other tree. Otherwise, with increasing diameter and thus reduced slenderness, critical buckling height diverged from actual height.

Fig. 1.

Log-log (base 10) plot of ground-line diameter against actual height and estimated critical buckling height, as indicated in the key.

Regression of critical buckling height against diameter yielded a scaling exponent of 0·55, which was lower than the scaling exponent of 0·67 predicted with constant E/ρ by eqn (1), as (E/ρ)^1/3 scaled with D to the power of –0·25. The scaling exponent between actual height, H, and D was 0·30. E scaled negatively with stem diameter to the power of –0·75, but positively with stem slenderness to the power of 0·78 (Table 2).

Table 2.

Summary statistics of reduced major axis (RMA) regression of logY vs. logX

Variable		αRMA	logβRMA	r2
logY	logX
Hcrit	D	0·55 (0·45–0·64)	0·66 (0·21–1·10)	0·87
H	D	0·30 (0·07–0·53)	0·81 (0·30–1·33)	0·08
(E/ρ)1/3	D	–0·25 (–0·44 to –0·06)	2·28 (1·10–3·46)	0·39
E	S	0·78 (0·60–0·96)	–0·69 (–1·07 to –0·32)	0·42
E	D	–0·75 (–0·93 to –0·56)	1·81 (1·45–2·16)	0·39

The scaling exponent (α_RMA) and allometric constant (logβ_RMA) are presented with 95 % confidence intervals in parentheses.

Measurements showed that the average safety factor (H_crit/H) of the stems in this study was 1·60, in which only five of the 72 stems had a safety factor greater than two (i.e. H_crit was double that of H). The safety factor ranged from 1·07 to 2·50 and increased with increasing diameter and decreasing stem slenderness. The regression of log(H_crit/H) against logD indicated that the safety factor was moderately correlated with stem diameter (r² = 0·39). Stem slenderness and the safety factor were highly correlated (r² = 0·79; Fig. 2).

Fig. 2.

Relationship between the safety factor (H_crit/H) and stem slenderness (H/D).

DISCUSSION

Results from this study showed that the scaling exponent between H_crit and D of 17-year-old Pinus radiata was lower than predicted (0·55 vs. 0·67) and thus contradicted the assertion that trees obtain elastic similarity (H ∝ D^2/3) (McMahon, 1973; McMahon and Kronauer, 1976). Furthermore, the safety factor (H_crit/H) increased with increasing D violating the assumption of constant safety factors which form the basis of the elastic similarity model. The low scaling exponent of 0·30 observed between H and D occurred as the low 1·6-fold variation in H was accompanied by a higher 4·1-fold variation in D across all stand densities and breeds on the trial site.

The average safety factor of the stems in this study was 1·60, with a range of 1·07–2·50. While these safety factors were comparatively low, they did exceed unity, which was consistent with the field observations as buckling was not found to occur throughout the trial. Safety factors typically reported in the literature range from 4·0 to 10·0 (McMahon, 1973; McMahon and Kronauer, 1976; Niklas, 1993, 1994a, b; Niklas and Buchman, 1994). This is because incremental gains in H_crit decline with increasing D as H scales to the power of 2/3. As a result, safety factors will also decline unless accompanied by substantial increases in E/ρ. The relatively low safety factors found in this study were likely attributable to the higher stocking rates indicative of plantation-grown trees, which induce a more unstable etiolated form than occurs in open-grown trees.

The results in this study showed that stems of greater slenderness had lower safety factors due to the fact that D is the dominant factor in eqn (1). However, E was found to increase with slenderness. This possibly suggests reductions in the safety factor, associated with increases in slenderness, induced increases in E to reduce the risk of stem buckling. The relationship found between E and stem slenderness in this trial has a sound theoretical basis. Using a variant of the Euler buckling formula, Greenhill (1881) showed that E scaled positively with the maximum slenderness that can be attained before buckling occurs. This suggests that trees with high slenderness increased E to further increase the threshold at which buckling occurs. The strong relationship between slenderness and safety factor was not surprising as slenderness is the dominant term in eqn (2).

The mechanisms by which slenderness affects E were not examined in this study. However, the fact that tall slender trees do not typically fall over due to their own weight strongly suggests that tall stems are composed either initially of stiff tissues, which is not the case for Pinus radiata or that the mechanical and physical properties of stem tissues ontogenetically change with age (Niklas and Buchman, 1994) such that stems become stiffer as they increase in height. Juvenile wood in Pinus radiata is generally characterized by low density, thin cell walls, short tracheids with large lumens, high grain angle and high microfibril angle. These properties typically show marked improvement as the transition between juvenile and mature wood occurs such that values of E rapidly increase outwards from the pith (Watt et al., 2010). It can be shown that, for a given mass, a more slender tree will have a greater level of compressive stress in its stem (Niklas, 1994b, p. 164). It is hypothesized that trees are able to sense and respond to this higher level of compressive stress and produce new wood with higher E, possibly by manipulating the angle of cellulose microfibrils in the secondary cell wall (Watt et al., 2006b). This increase in E not only acts to reduce the compressive strains experienced by the stem, but in turn also acts to increase the critical compressive stress that the stem can withstand before buckling occurs (Watt et al., 2006b). This type of thigmomorphogenic response to mechanical stresses caused by self-imposed or dynamic loads such as wind commonly occurs in a large number of tree species (Jacobs, 1954; Jaffe, 1973).

Wind-induced alterations to stem biomechanical properties have been well documented (Nicholls, 1982; Telewski and Jaffe, 1986; Pruyn et al., 2000). These responses are thought to be developmental adaptations that reduce bending stresses acting on the stem by increasing the second moment of cross-sectional area and reducing speed-specific drag of the crown, resulting in reductions in stem slenderness through decreased height and increased diameter (Watt and Kirschbaum, 2011). However, the influence of stand density on slenderness is also likely to be partially mediated through the effects of stand density on stem movement. Research has shown that high stand densities lower stem deflection through both reducing wind speed within the canopy and damping stem oscillations through increasing the number of collisions with neighbours (Cremer et al., 1982). A number of studies have found that increased stem movement induced by mechanical perturbation or wind has reduced E in a variety of species (Telewski and Jaffe, 1986; Pruyn et al., 2000). As a result, it is likely that both morphology and E of trees at the higher stand densities in this study responded more to self-load, whereas the trees at the lower stand densities probably responded more to external forces such as wind.

The wide variation in scaling between H and D found here suggested that assumptions underpinning the elastic similarity model are violated when applied to measurements from Pinus radiata and possibly other plantation-grown species. Perhaps the most restrictive aspect of the elastic similarity model is that it is based on buckling under self-load as the limiting process and does not consider the influence of lateral external loads, predominantly caused by wind (Watt and Kirschbaum, 2011). Lichtenegger et al. (1999) showed mathematically that increases in tree height for a given diameter (H/D) are possible under lower external loads and in trees with higher strength, a property that is strongly and positively related to E (Niklas, 1992). This is supported by empirical observations (e.g. Putz et al., 1983) showing that trees that fail by stem fracture in wind storms have lower wood density, strength and E than trees that are uprooted.

Many of the key results presented herein and the inferences drawn from them are based on the Euler column formula. Given we were primarily interested in the idea of biomechanical ‘risk-taking’ by the trees in the context of stand density and genetics, the Euler formula provided an ideal medium to do this. The Euler formula, however, must be viewed as a pedagogical tool that offers insights into the relationships among variables that are much more complex in most real biological contexts as the formula requires that plant shape be idealized into a simplified and hence manageable form. While a simple ‘critical height’ concept has heuristic value, it obscures the fact that variation across stand densities and genetics may result in biologically meaningful variation in slenderness ratios and the probability of falling. Interestingly, previous studies have shown that, although the Greenhill model of the Euler formula is based upon unrealistic idealisations of tree form, it performs strongly compared with more realistic and comprehensive models (Holbrook and Putz, 1989) and is a good estimator of the self-supporting habit of stems (Jaouen et al., 2007).

The Euler formula assumes that columns are perfectly straight and uniform in cross-section and that the column must be constructed from an isotropic material, i.e. it must have a uniform E throughout. It also assumes that the weight of the column must be significantly less than the weight it supports, i.e. the column is essentially considered to be weightless, and that the force inducing elastic buckling is distributed over the full extent of the vertical stem (Niklas, 1992; Jaouen et al., 2007). Clearly, stems are rarely if ever ideal columns. They typically taper and lack a uniform E, and they are anything but weightless. Furthermore, safety factors that are often reported for stems are calculated on the basis of the stresses resulting from static loads (McMahon, 1973; Niklas, 1994b), although these are misleading because most healthy stems mechanically fail as a result of wind loading rather than supporting their own weight. As such the formula is more appropriate for essentially untapered stems such as those found at the highest stand densities in the trial. Even so, the Euler column formula provides a useful tool for examining relationships between biological variables.

In summary, results indicated that, for mature Pinus radiata grown at operationally used stand densities, a constant E/ρ could not be assumed, as the scaling exponent in this study between critical H and D was 0·55. This is lower than the value of 0·67 assumed under constant E/ρ. It was demonstrated that for a given E, actual H approached the critical H as stem slenderness increased. The results potentially suggested that increases in slenderness induced increases in E to reduce the risk of stem failure.