On the distribution of tree growth stresses

Summary A plane strain pure bending elastic model is used to predict stress states in a cylindrically orthotropic cylinder due to asymmetric strain distributions continously induced at the periphery. A general solution for the stress states related to each component of the strain distribution expanded in a Fourier series is given. Details of solutions for eleven representative tree species are given. Also the change of curvature of a stem caused by a measured peripheral strain distribution is computed for a given period of growth.     

On the distribution of tree growth stresses

Summary A model for the build up of residual growth stresses in cylindrical tree stems is presented. By using Bessel functions the general equations can be solved to allow for surface strain distributions that vary both in the circumferential as well as longitudinal directions. An arbitrary surface strain distribution can be decomposed into Fourier components and the residual internal stresses can be found by superimposing the component solutions due to each new increment of growth. A numerical simulation leading to the residual stress distribution for a typical hardwood is presented.     

Growth strain in the trunk and branches of Chamaecyparis formosensis and its influence on tree form

Distributions of growth strains in branches, straight trunks and basal sweeping trunks of Chamaecyparis formosensis Matsum. trees were measured with strain gauges. Microfibril angles (MFAs) of the S2 layer of the cell wall were measured by the iodine deposition method and their relationships with growth strain examined. The magnitude of the compressive stress on the lower side of trunks with a basal sweep was greater than that of the tensile stress at the surface of straight trunks. However, transverse compressive stress was similar around the trunk regardless of whether normal wood or compression wood was present. The released surface growth strains varied with MFA. At MFAs of 20-25 degrees , growth stress changed from tension to compression, and compressive stress increased dramatically in the compression wood region. Branches suffer bending stress due to self-loading. This stress is superimposed on the growth stress. Growth strains on the upper or lower sides of branches were larger than those in the trunks, suggesting that generation of growth stress on the lower sides of branches with extensive compression wood is affected by the gravitational bending stress due to self-loading. We conclude that branch form is affected by the interaction between the bending moment due to self-loading and that due to the asymmetric distribution of growth stress. Growth strain distribution in a branch differed depending on whether the branch was horizontal, upward bending or downward bending.     

Evaluation of non-destructive methods of measuring growth stress in Eucalyptus globulus: relationships between strain, wood properties and stress

High levels of growth stress are implicated in causing end splitting of logs, deflection during sawing and deformation of boards as stresses are released during sawing operations. Level of stress is a function of strain and the elastic modulus of the wood (MOE). Levels of peripheral strain can be measured on standing trees and, if the MOE is known, stresses can be estimated. The validity of using peripheral strain measurements relies on underlying theoretical models that relate strain to expected patterns of stress distribution and levels of board deflection. This study evaluates these theoretical relationships by determining relationships of stress and strain with board deflection, end splitting and a range of wood properties.

Peripheral strain levels were extremely variable within the bottom log and little evidence was found for consistent patterns of variation, although measurements generally increased with increasing height above ground. Sampling on two sides of the standing tree at breast height appeared to be a suitable strategy, with the mean for these strain readings having a correlation (r) of 0.86 with the average strain in the bottom log.

Growth strain was not a reliable predictor of board deflection and cannot be recommended as a non-destructive sampling method. Overall there was a poor relationship between growth strain and board deflection. No consistent relationships were found between a range of wood properties and growth strain or board deflection across both sites. Stress levels were calculated for each tree as the product of growth strain and modulus of elasticity and the relationship between calculated stress and mean board deflection determined. No relationship was found at either site with correlations being very close to zero.

The underlying theoretical relationships between stress and strain were examined and several questions raised about the validity of such models.     

On the distribution of tree growth stresses

Summary Growth stress distributions in trees are derived using the hypothesis that longitudinal and circumferential growth strains are continuously induced at the periphery of the growing stem. A plane strain combined with pure torsion model is used to compute the internal stresses and strains due to forces and moments caused by the new growth increment. The twisting angle of the pure torsion model is caused by the shear stresses set up in the growth increment as the growth strains are induced along the grain axis and the coupling of axial and torsional effects due to the elastic constants for the inclined grain material. Detailed stress distributions are derived for a range of constant grain angle cases. The extreme sensitivity of the torsional shear distribution to small grain angles is noteworthy.This work was supported in part by NSF Grant ENG 74-02428     

Creep and stress relaxation behavior for natural cellulose crystal of wood cell wall under uniaxial tensile stress in the fiber direction

We investigated the temporal changes in creep and stress relaxation behavior in both microscopic crystalline cellulose and macroscopic strain of wood specimen using Japanese cypress (Chamaecyparis obtusa Endl.) to understand the viscoelastic properties of wood cell walls. Specimens 600 µm in thickness were observed by the X-ray diffraction and submitted to tensile load. The crystal lattice strain of (004) plane and macroscopic strain of specimen were continuously detected during creep and stress relaxation tests. It was found that the creep compliance based on macroscopic strain showed a gradual increase after instantaneous deformation due to loading and then the parts of creep deformation remained as permanent strain after unloading. On the other hand, crystal lattice strain showed a different behavior for macroscopic strain; it kept a constant value after instantaneous deformation due to loading and then increased gradually after a certain period of time. These differences between macroscopic and microscopic levels were never found in the stress relaxation tests in this study. Relaxation modulus at the macroscopic level only showed a decreasing trend throughout the relaxation process. However crystal lattice strain kept a constant value during the macroscopic relaxation process. In addition, the microfibril angle (MFA) of wood cell wall has a role of mechanical behavior at microscopic level; crystal lattice strains were smaller with increasing MFA at both creep and relaxation processes. Creep compliance and stress relaxation modulus at the macroscopic level decreased and increased with increasing MFA, respectively. Our results on the viscoelastic behavior at microscopic level evidenced its dependency on MFA.     

Theory to estimate the stress due to moisture in wood using a transmission-type microwave moisture meter

 In recent years a microwave transmission-type moisture meter has been developed in Japan. Its purpose is to measure the average moisture content of thick woods. Since its development I have realized that there is a negative correlation between the moisture content of wood and the power voltage of the meter. This realization suggests that an invisible stress has an effect on the attenuation constant of the wood. The presence of such a stress in the wood could easily be proven by the slicing technique. In this article a theory is presented to explain further the effect of this stress on the attenuation constant. The theory was applied to softwood specimens in various states of moisture. It was concluded that the calculated strain distributions of the various specimens approximated those of the experimental results. Thus, the proposed theory presented herein has validity or adaptability with regard to qualitatively understanding the stress. Future research efforts would also be expected to detect the stress in wood due to moisture. Received: November 30, 2001 / Accepted: March 18, 2002 Acknowledgments The author thanks Mr. K. Hayashida and Mr. T. Taniguchi (former students of Fukui University of Technology) for their assistance in the experiment. The author also thanks Dr. Okada of the Kawasaki Kiko Co. and Prof. Dr. Sobue of Shizuoka University for their valuable discussions and suggestions. Part of this report was presented at the Annual Meeting of the Central District of the Japan Wood Research Society, Gifu, September 2000 Correspondence to:T. Takemura     

Mechanical behavior of the crystal lattice of natural cellulose in wood under repeated uniaxial tension stress in the fiber direction

This study investigated the relationship between the cellulose crystal lattice strain (crystalline region) and the macroscopic surface strain in specimens of Chamaecyparis obtusa wood under repeated uniaxial tension stress in the fiber direction. Changes in the strain of the crystal lattice were measured from the peak of (004) reflection using the transit X-ray method. The macroscopic surface strain of each specimen was measured with a strain gauge. In both loading and unloading, the surface strain changed linearly with changes in stress. However, crystal lattice strain was not linear but exhibited changes along a curve with changing stress. Under stressed conditions, the crystal lattice strain was always less than the surface strain, regardless of the frequency of repetition in the loading and unloading cycle. The ratio of the crystal lattice strain to the surface strain showed a negative correlation for stress in both loading and unloading. That is, the ratio decreased with increasing stress, and finally tend to converge to a specific value. The ratio (I/I ₀) between the diffracted intensity (I ₀) in the (004) plane in the unloaded condition and the diffracted intensity (I) in the (004) plane in the loaded condition tend to converge on a specific value with increasing frequency of repetition. When the substantial tension Young’s modulus of the wood in the longitudinal direction decreased, the ratio of the strain of the crystal lattice to the surface strain also decreased. Moreover, the ratio decreased with increasing microfibril angle of the specimen.     

Fatigue of structural plywood under cyclic shear through thickness I: fatigue process and failure criterion based on strain energy

The fatigue behavior of plywood specimens under shear through thickness was examined on the basis of strain energy to obtain common empirical equations for the fatigue process and failure criterion under various loading conditions. Specimens were cut from commercial plywood panels of 9-mm thickness. Loading conditions were set as follows: a square waveform at a loading frequency of 0.5 Hz, a triangular waveform at 0.5 Hz, and a triangular waveform at 5.0 Hz. Peak stress applied was determined to be 0.5, 0.7, and 0.9 of static strength, that is, stress levels of 0.5, 0.7, and 0.9. The stress-strain relationships were measured throughout the fatigue test, and the strain energy was obtained at each loading cycle. Loading conditions apparently affected the relationship between stress level and fatigue life. On the other hand, the relationship between mean strain energy per cycle and fatigue life was found to be independent of loading conditions. Mean strain energy per cycle obtained as the fatigue limit was 5.85 kJ/m³ per cycle. Assuming that the accumulation of strain energy is a fatigue indicator, the fatigue process and failure criterion for the plywood specimens under the three loading conditions were commonly expressed by the relationship between cumulative strain energy and loading cycles.     

On the origin of growth stresses in trees

Summary A mechanism for growth stress generation is studied which involves a contractive strain in the microfibril direction and swelling strain in the transverse direction in the developing wall of wood cells. A cylindrically anisotropic elastic model is used to calculate the accumulation of residual stresses in the S₂ wall as it is formed. An explicit relation between the shrinkage/swelling strains in the growth increment of the cell wall and the resulting axial and circumferential stresses induced in the cell is derived. For gymnosperm cells the transition from tensile stress in normal wood cells to compressive stress in compression wood cells is found with increasing microfibril angle.     

A hygro-mechanical analysis of poplar wood along the tangential direction by restrained swelling test

A better understanding of the hygro-mechanical behaviour of poplar wood (Populus alba L.) is proposed by means of restrained swelling tests. Small clear specimens were tested along their tangential direction in controlled climatic conditions; a dry climate (30 % relative humidity and 30 °C temperature) and a humid one (80 % relative humidity and 30 °C temperature) were cyclically set and kept constant each during 7 days. In these conditions, three matched specimens were tested at the same time along the tangential direction: specimen A free to shrink and swell, specimen B free to shrink and prevented from swelling beyond its initial dimension, specimen C prevented from shrinking and swelling. A specific device was set up to monitor strain and/or stress on each specimen. During the adsorption phase, compression forces (on specimens B and C) were measured, and during the desorption step, compression set shrinkage (compression strain at zero load) and tension stress (on specimens B and C, respectively) were measured. In this paper, the dedicated experimental device is presented and a deep analysis of experimental curves is then proposed by means of a rheological approach. By the analysis, some basic data are obtained (swelling coefficient, swelling pressure and compression set shrinkage), and a complex behaviour of wood made of reversible and irreversible phenomena is characterized.     

Experimental study of the mechanical behaviour of thin slices of maturating green poplar wood using cyclic tensile tests

In this study, longitudinal cyclic tensile tests were performed on green wood samples of Populus cv I4551. Complex mechanical behaviour, such as permanent set in terms of displacement and strain-dependent stiffness, was found. A linear relationship between stiffness and strain enabled each sample to be characterized. A large-scale experimental campaign led to the observation of the evolution of temporal and spatial intra-ring mechanical behaviour of intra-ring sliced specimens during wood maturation and tree gravitropic response. The results showed that tension wood lamellas exhibit strain hardening behaviour whereas normal wood lamellas exhibit strain softening behaviour. Temporal variations in the strain hardening behaviour of tension wood lamellas occurred once cell wall deposition was achieved. Moreover, no correlation was found between infradensity and mechanical behaviour of lamellas. These results show that cells undergo temporal variations in their properties due to a complex maturation process.     

Nonlinear shear properties of spruce softwood: experimental results

Shear properties of clear softwood from Norway spruce were investigated by means of the Arcan shear test. The test enabled fairly detailed measurements of pure shear until failure in each of the three orthotropic shear planes, where video extensometry was used for strain measurements. A total number of 85 specimens were tested in 6 different configurations. A varying degree of nonlinearity was observed between the different configurations and material planes, especially for rolling shear. The stress–strain curves were adapted with linear, bilinear and Voce models. Compared to the linearized variant, it was found that the bilinear model generally increases the model accuracy by a factor of approximately two, whereas the Voce model showed even higher accuracy, although its adaptive robustness was somewhat lower. The parameters demonstrated low correlations to density, whereas the correlations with the initial shear moduli in many cases were considerably higher and significant. The correspondence with similar values reported in literature was found to be fairly good.     

Finite element analysis of stress and strain distributions in mortise and loose tenon furniture joints

We studied the effect of loose tenon dimensions on stress and strain distributions in T-shaped mortise and loose tenon （M＆amp;LT） furni-ture joints under uniaxial bending loads, and determined the effects of loose tenon length （30, 45, 60, and 90 mm） and loose tenon thickness （6 and 8 mm） on bending moment capacity of M＆amp;LT joints constructed with polyvinyl acetate （PVAc） adhesive. Stress and strain distributions in joint elements were then estimated for each joint using ANSYS finite element （FE） software. The bending moment capacity of joints increased significantly with thickness and length of the tenon. Based on the FE analysis results, under uniaxial bending, the highest shear stress values were obtained in the middle parts of the tenon, while the highest shear elastic strain values were estimated in glue lines between the tenon sur-faces and walls of the mortise. Shear stress and shear elastic strain values in joint elements generally increased with tenon dimensions and corre-sponding bending moment capacities. There was consistency between predicted maximum shear stress values and failure modes of the joints.     

A numerical study of the shape stability of sawn timber subjected to moisture variation Part 1: Theory

A three-dimensional theory for the numerical simulation of deformations and stresses in wood during moisture variation is described. The constitutive model employed, assumes the total strain rate to be the sum of the elastic strain rate, the moisture-induced strain rate and the mechano-sorption strain rate. Wood is assumed to be an orthotropic material with large differences between the longitudinal, radial and tangential directions in the properties found. The influence of the growth rings, the spiral grain and the conical shape of the log on the orthotropic directions in the wood is taken account of in the model. A finite element formulation is used to describe the deformation process and the stress development during drying.The research presented in this paper is a part of the national research programme in Sweden concerning wood physics and drying. It was financially supported by the Research Foundation of Swedish Sawmills and the Swedish Council for Forestry and Agricultural Research.     

疲劳／蠕变交互作用下塑木地板的断裂机理

对塑木地板进行抗弯性能测试,分别选取25％和75％两种应力水平进行疲劳／蠕变试验。研究了塑木地板在疲劳／蠕变复合作用下的断裂机理。结果表明：随着应力水平的增加,蠕变应变也随之增加,75％应力水平时产生的应变为25％应力水平时的3倍;其疲劳／蠕变曲线与纯蠕变曲线十分相似;在25％的应力水平作用后,材料的剩余抗弯强度为原来的94％～97％。     

The effect of saw kerf width on the value of the axial growth stress measured by slitting a log along its axis

The surface growth stress in logs can be determined by slitting the log axially and measuring the outward deflection of the two halves. The saw kerf removes stressed material which generates a compressive strain in the remaining material and thereby reduces the surface growth stress. The purpose of the work is to establish the magnitude of this error. The analysis assumes a uniform compressive stress in the core of the log and a logarithmic distribution in the outer region. The total axial force in the kerf before removal is calculated and, from this, the stress change in the material remaining after kerf removal is derived. Data are presented for a range of kerf widths and for three different growth stress distributions. Direct measurement of the kerf removal error was made by direct measurement of the contraction of the log and by strain gauges to measure the strain. The measured values were close to the calculated values for three growth stress distributions. The analysis yields acceptable values for the kerf removal error. The investigation concludes that for kerf widths not greater than one-tenth of the log diameter, the error in the measured tensile surface growth stress is less than 10%.     

Viscoelastic properties of wood from Chinese-fir and poplar plantations

Elastic and strength properties(proportional-limit stress(σ prop ),Young's modulus(E),breaking stress(σ max )in static bending parallel to grain in a longitudinal direction),as well as stress relaxation in air-dried condition and water-saturated conditions at seven different constant temperatures and increasing and decreasing temperatures were investigated for wood from Chinese-fir and poplar plantations.The results show that hygrothermal conditions considerably affect these mechanical properties.The higher the moisture content(MC)or temperature,the lower the strength of wood.Further investigation of the effects of constant temperature on stress relaxation indicates that high temperature specimens have low relaxation moduli and high fluidity.In the case of increasing temperature the range of the modulus of relaxation is larger than in the case of a reduction in temperature,while the residual moduli do not show large differences.This is because the modulus at high temperatures decreases more than that at low temperatures.The fluidity of specimens in a state of water desorption increases slowly at the beginning,increases quickly until the MC reaches an equilibrium moisture content(EMC)and then becomes stable,which is quite different from that in a water-saturated state.Fluidity in a desorption state is much higher than in a water-saturated state.This is probably due to the fact that the former is in an unstable state which can be interpreted as a state with internal strain and has therefore a greater potential to release strain.     

Growth strain in coconut palm trees

Until recently, growth stress studies have been made only on coniferous and dicotyledonous trees. Growth stress of trees is thought to be initiated in newly formed secondary xylem cells. This stress can accumulate for years and is distributed inside the trunk. Major characteristics of the trunk of monocotyledonous trees include numerous vascular bundles scattered inside the ground tissue and the lack of secondary growth for enlarging the diameter of the trunk. We used the strain gauge method to measure the released growth strain of the monocotyledonous woody palm, coconut (Cocos nucifera L.), and to investigate the surface growth strain of the trunk and central cylinder at different trunk heights. The internal strains of both vertical and leaning trunks were measured and compared with those of coniferous and dicotyledonous trees. We found that tensile stress existed longitudinally on the surface of vertically growing trunks, whereas compression stress was found at the bending position of leaning trunks. Compression stress was found in the outer part of the central cylinder, whereas tensile stress is generally found in the outer part of the trunk in coniferous and dicotyledonous trees. The distribution of strain in the palm trunk is similar to that of compression wood of the leaning trunk of a conifer. Specific gravity was greater in the outer part of the trunk than in the inner part of the trunk. This difference may be related to the distribution of growth stress.     

应力波在原木中传播理论的研究

对原木中微小单元进行动力学分析,确定应力波沿原木轴向传播的波动方程,应用分离变量法,得出应力波沿原木轴向传播的位移、速度、应力和应变方程。根据波动学理论中的反射和透射原理,从传播方向与界面垂直和不垂直2个方面得出界面两侧位移、速度、应力、应变的关系式。在此基础上,建立传播方向与界面垂直情况下的应力波沿原木径向或弦向传播的位移、速度、应力和应变方程。从这些方程可以得出:微小单元的位移、速度、应力和应变不仅与原木试件的弹性模量、密度有关,而且与小锤的敲击力、小锤与原木试件作用时间、敲击方向有关。