Estimating stem and root-anchorage flexibility in trees

This paper describes a nondestructive method for distinguishing root flexibility from stem flexibility in living trees. It is used here for Sitka spruce (Picea sitchensis (Bong.) Carrière.), but is applicable to any species where the main stem is normally straight and near-vertical. Well-known engineering equations permit the calculation of deflected shape for a vertical cantilever with arbitrary distribution of mass and bending stiffness, when infjected to a lateral force. The equations are used to calculate stem deflections of four Sitka spruce trees for which the stem and branch mass distribution and stem taper have been measured. Free parameters in the mathematical model are a nominal value of Young's Modulus E (assumed uniform and isotropic over the cross section and height of the tree stem) and a root-anchorage stiffness k. The former allows the stem to curve, whereas the latter represents the flexibility of the roots and allows the stem to tilt elastically at ground level. For each of the four trees, the calculated deflection curve is compared with actual deflections measured when the living tree is pulled by a rope at a specified point. By adjusting both E and k, iteratively, a best fit solution is obtained. This provides a simple and effective way to determine both stem stiffness and root hinge stiffness from a single experiment on a living tree.     

The elasticity and vertical distribution of stress within stems of Picea sitchensis

Eight 22-year-old Picea sitchensis (Bong.) Carr. trees growing in a Scottish forest were bent using a winch and cable. The shapes of the bent stems were modeled using the structural theory of a cantilever beam, with non-uniform stem taper and large deflections taken into account. The Young's modulus of elasticity for the stems was estimated to be 2.0-6.4 GPa, which is lower than that reported for sawn green timber of this species, but similar to that for intact green stem sections. The longitudinal distribution of stress along the stems had a maximum at a height that was dependent on taper. Maximum stress occurred higher up in the more tapered stems.     

Young's modulus of sections of living branches and tree trunks

Young's modulus along the grain (elasticity, E) was measured on 10 sections of branches and three tree trunks, with bark, of Picea sitchensis (Bong.) Carr., Pinus contorta Dougl. ex Loud., Larix decidua Mill. and Betula pendula Roth. syn. verrucosa Ehrh. The sections were simply supported and corrections were made for taper and deflection due to shear. The E values for trunks were at the lower end of the range reported for green timber (2.4-7.5 GPa), and those for branches were still lower (0.7-4.6 GPa). Values of E for branches decreased with decrease in specific gravity, which corresponded with an increase in percentage water content. When E values were calculated using underbark diameters they fell more closely within the range reported for green timber.     

Creep of the beam in Japanese conventional structures composed of green and kiln-dried timber II: predictive model for relative deflections

We conducted creep tests to evaluate creep behaviors of conventional Japanese framing (jikugumi) structures as reported in a previous article. We measured beam deflections of two structures: one of them was composed of only green timbers (G) and the other with only kiln-dried timbers (D). Besides the two structures, we prepared green and kiln-dried beams to measure moisture content (MC), weight, and dynamic Young’s modulus (E _f) by the longitudinal vibration method. We attempted to predict deflections of beams in the structures by using experimental data for single beam specimens. The proposed simple predictive model was derived from two equations: a relation between MC and equilibrium moisture content calculated with temperature and relative humidity, and a relation between MC change and relative deflection change. Beam deflections were traced for 2.5 years, while the predictions were based on experimental data from loading to the 11th day of the test. It was assumed that sensitivity of deflection change to MC should differ during desorption or adsorption. Although annual cyclic changes were observed in E _f, there was no obvious relationship between E _f and beam deflection. Part of this article was presented at the Annual Meeting of the Architectural Institute of Japan, Kyushu, September 1998     

A three-dimensional paradigm of fiber orientation in timber

Knowledge of the three-dimensional orthogonal directions of wood material at any position within a tree is necessary for the understanding of strength reducing effects of knots and essential for the continuation of research in areas which relate small clear wood specimen behavior to the behavior of full size structural timber. A complete three-dimensional paradigm describing the geometry of knots and related fiber distortion, initially derived to predict the strength-reducing behavior of knots in structural timber of Norway Spruce with the finite element method, is presented in this article. Besides strength prediction analyses, it is believed that the paradigm may be useful in other areas of research on structural timber that are effected by fiber orientation, such as drying and form change of structural timber. The paradigm generates fiber orientation in any position within a log or lumber from assumed fiber patterns in planes parallel to the longitudinal direction of the original tree. Fiber patterns in the radial and tangential directions are derived from physical restraints related to fiber production within the annual increase surfaces of the tree and from theories of knot formation. The adaptability of the paradigm allows practically any softwood knot to be modeled with an accuracy that is limited only by input-data. The knot-axis may be non-linear, and the knot cross-section oval with its vertical and horizontal axis increasing from the pith of the stem at chosen rates. Spiral grain may also be included in the paradigm and vary with the annual growth layers. Investigations presented in this article showed that generated fiber orientations for Picea abies complied well with measured fiber distortions, and that the general trends of fiber orientation, explained by the applied knot formation theory, is reflected in the measured specimens. Received 12 May 1999     

Diameters and dry weights of tree shoots: effects of Young's modulus, taper, deflection and angle

The structural theory for cantilever beams was used to calculate the diameters and dry weights of wood that unbranched shoots must produce to support their own weights. The study was done on Picea sitchensis (Bong.) Carr., Pinus contorta Dougl., Larix decidua Mill. and Betula pendula Roth. syn. verrucosa Ehrh. The weights of wood increased in a non-linear fashion with increase in shoot length. A large investment in wood (as measured by diameter and dry weight) was required to maintain a small endpoint deflection (1-5% of the length). By contrast, the degree of linear taper had only a small effect on support costs, as did the Young's modulus of the wood (over the range 1-4 GPa) and the angle of the shoot from the horizontal (over the range 0-45 degrees ). Current year's shoots on young trees of P. sitchensis and P. contorta incur a high support cost in order to maintain small (1-5%) deflections throughout the year: similar shoots of L. decidua and B. pendula have smaller support costs because they deflect by about 20% after leaf expansion in spring.     

A method for 3D reconstruction of tree crown volume from photographs: assessment with 3D-digitized plants

We developed a method for reconstructing tree crown volume from a set of eight photographs taken from the N, S, E, W, NE, NW, SE and SW. This photographic method of reconstruction includes three steps. First, canopy height and diameter are estimated from each image from the location of the topmost, rightmost and leftmost vegetated pixel; second, a rectangular bounding box around the tree is constructed from canopy dimensions derived in Step 1, and the bounding box is divided into an array of voxels; and third, each tree image is divided into a set of picture zones. The gap fraction of each picture zone is calculated from image processing. A vegetated picture zone corresponds to a gap fraction of less than 1. Each picture zone corresponds to a beam direction from the camera to the target tree, the equation of which is computed from the zone location on the picture and the camera parameters. For each vegetated picture zone, the ray-box intersection algorithm (Glassner 1989) is used to compute the sequence of voxels intersected by the beam. After processing all vegetated zones, voxels that have not been intersected by any beam are presumed to be empty and are removed from the bounding box. The estimation of crown volume can be refined by combining several photographs from different view angles. The method has been implemented in a software package called Tree Analyzer written in C++. The photographic method was tested with three-dimensional (3D) digitized plants of walnut, peach, mango and olive. The 3D-digitized plants were used to estimate crown volume directly and generate virtual perspective photographs with POV-Ray Version 3.5 (Persistence of Vision Development Team). The locations and view angles of the camera were manually controlled by input parameters. Good agreement between measured data and values inferred from the photographic method were found for canopy height, diameter and volume. The effects of voxel size, size of picture zoning, location of camera and number of pictures were also examined.     

Support costs of different branch designs: effects of position, number, angle and deflection of laterals

The structural theory for cantilever beams was used to calculate the dry weight of wood that a branch of Picea sitchensis (Bong.) Carr. must produce to support its own weight plus that of laterals in different positions and numbers, and with different angles and deflections. The endpoint of the branch was maintained at 2% of its length, and constant values or functions (measured on real branches) were assumed for Young's modulus, taper and self weights of wood and foliage. Support costs (branch wood dry weights) were minimized when laterals were positioned so that the center of gravity was close to the base of the branch, and when the branch was angled upward from the horizontal. Small savings in support costs also resulted when laterals were not forward pointing, were angled upward and had endpoint deflections of at least 20% of their length.     

林分径阶蓄积量与材种出材量测算的研究

研究了林分径阶蓄积量与材种出材量的估测模型,其基本原理是应用相对树高曲线法测定林分径阶蓄积量,应用一致性削度方程法计算径阶单木材种出材率;相对树高曲线法与一致性削度方程法是基于胸径和树高两个测树因子的二元立木材积测定系统,通过建立较完善的数学模型改进计算方法,提高估测效果.     

Branch breakage under snow and ice loads

Measurements were made on branches and trunks of Picea sitchensis (Bong.) Carr. to determine the relationship between (i) the bending moment at the bases of branches that cause breakage, and (ii) midpoint diameter cubed. The theory for cantilever beams was then used to calculate the basal bending moments and midpoint diameters of branches with different numbers of laterals and endpoint deflections, given previously measured values of Young's modulus, taper and weights of foliage and wood. Snow and ice loads (equal to 2 and 4 g cm(-1) of shoot, respectively) were then included in the calculation to determine whether the basal bending moments exceeded the breakage values. The likelihood of breakage increased with an increase in (i) number of laterals, and (ii) endpoint deflection under self weight (without snow or ice)-features that had previously been shown to lessen the amount of branch wood required to support a unit of foliage. However, branches which deflected moderately (> 10% of their length) under their own weight deflected greatly under snow or ice loads and might shed powdery snow before breakage occurs.     

The contribution of structural indices to the modelling of Sitka spruce (Picea sitchensis) and birch (Betula spp.) crowns

Crown dimensions are important for the quantification of tree interactions in some growth models. This study investigates the potential for structural indices and other spatial measures to improve the prediction of crown radius and crown length for birch (Betula spp.) and Sitka spruce (Picea sitchensis (Bong.) Carr.) in forests in Wales. Crown dimensions were measured for 125 birch and 154 spruce in six fully stem-mapped research plots. These data were used to test the performance of a crown radius model and a crown length model which estimated crown dimensions on the basis of allometric relationships with stem dimensions. Spatial data from the six plots were used to calculate the structural indices mean directional index, diameter correlation index, species mingling, dbh and height dominance, and dbh differentiation, as well as the Hegyi competition index, and basal area of neighbours and larger neighbours, for each crown measurement sample tree, using various numbers of nearest neighbours. Two non-spatial indices, BAL and BALMOD, were also calculated for all sample trees for comparison. These spatial and non-spatial variables were then incorporated into modified crown dimension models. Model performances, in terms of efficiency and relative bias, were compared to determine whether the inclusion of spatial or non-spatial variables resulted in any improvements over models using tree dimensions alone. Crown length and radius were found to be correlated with most of the spatial measures studied. Models incorporating spatial variables gave improvements in performance over allometric models for every data set, and performed more consistently than models containing non-spatial variables. The greatest improvements were achieved for suppressed birch in unthinned forests which had irregularly shaped and strongly displaced crowns. The spatial variable contributing to the most efficient model for each data set varied widely. This points to the complexity of tree spatial interactions and indicates that there is a great deal of scope for investigating other structural indices and crown dimension model forms.     

A flexible stem taper and volume prediction method based on mixed-effects B-spline regression

Modelling stem taper and volume is crucial in many forest management and planning systems. Taper models are used for diameter prediction at any location along the stem of a sample tree. Furthermore, taper models are flexible means to provide information on the stem volume and assortment structure of a forest stand or other management units. Usually, taper functions are mean functions of multiple linear or nonlinear regression models with diameter at breast height and tree height as predictor variables. In large-scale inventories, an upper diameter is often considered as an additional predictor variable to improve the reliability of taper and volume predictions. Most studies on stem taper focus on accurately modelling the mean function; the error structure of the regression model is neglected or treated as secondary. We present a semi-parametric linear mixed model where the population mean diameter at an arbitrary stem location is a smooth function of relative height. Observed tree-individual diameter deviations from the population mean are assumed to be realizations of a smooth Gaussian process with the covariance depending on the sampled diameter locations. In addition to the smooth random deviation from the population average, we consider independent zero mean residual errors in order to describe the deviations of the observed diameter measurements from the tree-individual smooth stem taper. The smooth model components are approximated by cubic spline functions with a B-spline basis and a small number of knots. The B-spline coefficients of the population mean function are treated as fixed effects, whereas coefficients of the smooth tree-individual deviation are modelled as random effects with zero mean and a symmetric positive definite covariance matrix. The taper of a tree is predicted using an arbitrary number of diameter and corresponding height measurements at arbitrary positions along the stem to calibrate the tree-individual random deviation from the population mean estimated by the fixed effects. This allows a flexible application of the method in practice. Volume predictions are calculated as the integral over cross-sectional areas estimated from the calibrated taper curve. Approximate estimators for the mean squared errors of volume estimates are provided. If the tree height is estimated or measured with error, we use the “law of total expectation and variance” to derive approximate diameter and volume predictions with associated confidence and prediction intervals. All methods presented in this study are implemented in the R-package TapeR.     

Variability with xylem depth in sap flow in trunks and branches of mature olive trees

Knowledge of sap flow variability in tree trunks is important for up-scaling transpiration from the measuring point to the whole-tree and stand levels. Natural variability in sap flow, both radial and circumferential, was studied in the trunks and branches of mature olive trees (Olea europea L., cv Coratina) by the heat field deformation method using multi-point sensors. Sapwood depth ranged from 22 to 55 mm with greater variability in trunks than in branches. Two asymmetric types of sap flow radial patterns were observed: Type 1, rising to a maximum near the mid-point of the sapwood; and Type 2, falling continuously from a maximum just below cambium to zero at the inner boundary of the sapwood. The Type 1 pattern was recorded more often in branches and smaller trees. Both types of sap flow radial patterns were observed in trunks of the sample trees. Sap flow radial patterns were rather stable during the day, but varied with soil water changes. A decrease in sap flow in the outermost xylem was related to water depletion in the topsoil. We hypothesized that the variations in sap flow radial pattern in a tree trunk reflects a vertical distribution of water uptake that varies with water availability in different soil layers.     

Simulating the dynamic behavior of Douglas-fir trees under applied loads by the finite element method

The finite element method of structural analysis was used to model the dynamic behavior of three 20-year-old Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) trees subjected to applied loading. Detailed measurements of stem and branch geometry were made for each tree, enabling the first-order branches of each tree to be represented as individual cantilever beams attached to the stem. Three values for branch modulus of elasticity (E) were assumed: 4, 5 and 6 GPa. For two trees with relatively large crown masses (175 and 250 kg), significantly improved estimates of natural frequency were obtained when the branches were modeled as separate cantilever beams rather than as a series of discrete masses attached to the stem. Closest agreement with the results from field sway tests was found when branch E was 4 GPa. Oscillations of individual branches contributed to the damping of tree oscillations--a phenomenon known as structural damping--with the contribution increasing as branch E decreased. When branch E was 4 GPa, the phase difference between the oscillation of the stem and that of some branches was almost 180 degrees. We applied a series of forces separately to the stem and branches of each tree and determined the mechanical transfer function for each loading case. These transfer functions were similar to the theoretical transfer function for a damped harmonic oscillator, but showed a smaller tree response at higher loading frequencies, particularly when branch E was 4 GPa. Branch structural properties, particularly modulus of elasticity, appear to be important in defining overall tree behavior under applied loading.     

树木风振机理研究综述

概述了近几十年来树木风振模型的发展,提出建立树冠、树干和树根三者结合的整体有限元模型,同时考虑其非线性效应是今后的研究方向,并且还需开展对树冠结构减振效应的量化研究,建立树干风振恢复力模型以及树木风振的随机振动模型.     

Changes in stem taper for birch plants growing in tree shelters

Plantings with pendula (Betula pendula Roth) and pubescent (Betula pubescens Ehrh.) birches were studied on five localities (Lat. 57–60 ° N) in Sweden. Seedlings were planted in 1988 or 1989. The experiment contained 50 plants per parcel and five replications per treatment. Among treatments harrowed area and birches covered with tree shelter were included. In spring 1993 five plants growing on harrowed area, with and without tree shelter, respectively were randomly chosen on each parcel. A total of 25 birches per treatment, species and locality were used. The trial was replicated in 1994 and 1995. The plant height was estimated and diameter at 20 and 60% of tree height was measured each year for 4 yr. The stem taper was calculated by the quotient D₆₀/D₂₀. The stem taper for birches without tree shelter was 0.5 for both species on all localities. The quotients for birches, which have grown in tree shelter, were 0.6–0.75 the first years of the trial. Then the quotients decreased and 2–3 years later they were on the same level as for birches planted without tree shelters. The results of the studies indicate that tree shelters should be removed two to three years after the plant has reached the top of the shelter. The height of the tree shelter might be shortened to 0.8–1.0 m to stabilise the plant earlier than with the conventional shelter height of 1.2 m.     

The anatomical traits of trunk wood and their relevance to oak (Quercus robur L.) vitality

Oaks’ decline in vitality is attributed to a complex process that involves interactions of several factors leading to increased trees’ mortality. This study investigates the structure of trunk wood of oaks with reference to its physiological role in hydraulic conductivity. On the basis of the crown condition, the oaks were classified into three health groups: healthy trees, declining trees and dead trees. Anatomical traits of wood, such as annual ring width, vessel density, vessel diameter of earlywood and theoretical hydraulic conductivity, were measured and calculated. The narrowest annual rings formed by the cambium were observed in dead oaks. These trees were also characterized by the smallest diameter of earlywood vessels, not only in the period of occurrence of dieback symptoms, but also during their whole life. It is suggested that the formation of narrow annual rings and earlywood vessels of small diameter increases susceptibility of a tree to decay. A reduced vessel diameter implies changes in hydraulic conductivity of oak trunks and thus impairs the water transport, which affects the health of trees. The process of oak decline is considered to have characteristics of natural selection and leads to the elimination of the weakest trees.     

Estimation of taper curve using stand variables and sample tree measurements

The aim of the study was to develop methods for estimating the taper curves for trees tallied in a forest inventory. The average stem form in a stand was described by the principal components of the stand effects in the stem dimensions measured in the polar coordinate system. Measurements of diameter at breast height, diameter at a height of 6 meters, and height taken from trees on the sample tree plots were used for determining the first four principal components. Regression models were derived to predict the principal components from the site and growing stock variables. These models were used to estimate the taper curves of the tallied trees. Use of the principal components estimated by the regression models gave less reliable results than use of the principal component estimates based on measurement of the height of one randomly chosen tree on the sample plot. The best result was found with combined use of the principal component estimates and one height measurement per sample plot.     

Derivation of stem taper from the pipe theory in a carbon balance framework

A dynamic tree growth model is described. The model derives the development of stem taper and vertical distribution of branch basal area from the pipe model, assuming that reuse of active pipes is regulated by foliage dynamics in a vertically explicit crown with a foliage distribution of constant shape. Based on empirical findings, the pipe model was modified slightly to allow the foliage/sapwood ratio to vary as a function of distance from the treetop. Growth was derived from carbon balance in a stand of different size trees that may shade each other. The model was applied to old and middle-aged trees growing in dense and sparse stands of Scots pine for which stand-level measurements are available as a chronosequence, but individual trees have been measured only once. Measured trees were compared with corresponding simulated trees for stem taper and vertical distribution of branch basal area. The results indicated that the pipe model assumptions, combined with a model of tree growth, are capable of producing realistic predictions of the vertical distribution of stem and branch diameter in trees of different sizes in the stand. A comparison of the results with a simple form of the uniform stress theory showed good agreement between the two models. However, a significant difference was found between the measured relative contribution of heartwood to total stem diameter and the predicted share of disused pipes in the stem. A possible explanation for this discrepancy is that the transition from sapwood to heartwood is gradual rather than abrupt as assumed in the model. A modification of the pipe model to incorporate a gradual transition is outlined.