Responses of Hollow, Septate Stems to Vibrations: Biomechanical Evidence that Nodes Can Act Mechanically as Spring-like Joints

The hypothesis is proposed that nodes of hollow plant stemsact as spring-like joints by storing strain energy when stemsare bent and releasing this energy to elastically restore theoriginal postures of stems when bending forces are removed.This hypothesis was tested by subjecting stem segments consistingof four nodes and three intervening hollow internodes to axialcompressive loads and by determining the natural frequenciesof vibration of their nodes. Compression tests were used todetermine the critical load required to produce elasticallyrecoverable deformations for each of a total of 115 stem segmentsof the grassArundinaria técta(Walt.) Muhl. Each segmentwas observed to flex at or very near its nodes while internodesappeared to act as rigid bars. The natural (fundamental) frequenciesof vibrations of the nodes of these stem segments were subsequentlydetermined and equalled those predicted by engineering theoryassuming that nodes behave as spring-like joints. The data fromresonance frequency tests were then used to calculate the springconstants of stem segments (i.e. the force required to producea unit deflection in stems). These constants were found to agreewith those predicted by theory provided that nodes acted mechanicallyas spring-like joints. The transverse septa of the nodes of20 randomly selected stem segments were perforated with a needleand the spring constants of the impaired nodes were remeasuredand compared with those of the same stems before surgical manipulation.On average, nodal spring constants were reduced by 35%. Thisreduction agreed with the prediction that the perforation ofsepta would significantly reduce the ability of nodes to storestrain energy. Collectively, these results are interpreted tosupport the hypothesis that septate nodes can store and releasestrain energy. The hypothesis is discussed further in lightof the behaviour of a physical model which shows that nodal‘diaphragms’ can substantially stiffen a hollowcylindrical structure, although they are neither essential forthe storage of strain energy nor the subsequent elastic restorationof the model's shape once bending loads are removed. Plant stems; nodes; internodes; strain energy; elastic buckling; Brazier buckling; biomechanics     

The Growth Response of the Stems of Genetically Modified Tobacco Plants (Nicotiana tabacum'Samsun') to Flexural Stimulation

Genetically modified tobacco plants (Nicotiana tabacum‘Samsun’)with antisense cinnamyl alcohol dehydrogenase DNA, produce secondaryxylem of a reduced tensile stiffness. These plants were grownalongside control plants. The stems of the plants were flexedor protected from flexing over a period of several weeks. Thetensile moduli and second moments of areas of the differenttissues inside the stems were measured and used to calculatethe bending stiffness of the plants. In tobacco, the cylinderof xylem was found to be the most important tissue in determiningthe bending stiffness of the plants. The thickness of the xylemtissue cylinder increased when plants were subjected to flexuralstimulation. This increased the bending stiffness of the stems.The response to mechanical stimulation was found to be correlatedwith tissue strain and the genetically modified plants wereable to exactly compensate for the reduced modulus of theirxylem tissue by increasing the thickness of the xylem tissuecylinder more than in control plants.Copyright 1999 Annals ofBotany Company. Tobacco plants, stem bending, xylem tissue, second moment of area, thigmomorphogenesis, mechanical strain.     

Dynamic behaviour of inflorescence-bearing Triticale and Triticum stems

The mechanical response of cereal plant shoots to loads caused by wind and gravity in the field is swaying in flexure around the vertical or near vertical transient equilibrium position determined by the stationary component of the wind pressure. The aim of this work was to characterise the kinematic and dynamic attributes and their interrelations in freely swaying inflorescence-bearing stems of wheat (Triticum aestivum L.) and Triticale. The fundamental natural frequency of the stems appeared to be considerably lower than predicted from the theory of vibration using the model of a cantilever beam oscillator and assuming the spring constant to be equal to the force-deflection ratio. Because of the rate of deformation and visco-elastic behaviour of the plant material, a discrepancy of about 10% was found between the dynamic and static stem bending resistance. The presence of the tip inflorescence caused vibrating vertical stems to behave as compressed columns in which the effective spring constant was strongly biased by the apical load due to the weight of the inflorescence. At the late milk stage, in the freely swaying stems of wheat and Triticale, the resistance to dynamic lateral loads was reduced by about 30% as a result of compression exerted by the inflorescence. So the prominent effect of the tip inflorescence on the dynamic behaviour (the effective spring constant and the natural frequency) of the stem is attributed to the non-negligible magnitude of the inflorescence weight relative to the critical load producing elastic buckling in slender vertical structures. Stem softening as a consequence of increasing inflorescence weight is assumed to be one of the essential factors reducing the lodging resistance in cereal crops at the late milk stage. The feasibility of the compressed-column approach for predicting the dynamic bending performance of slender vertical plant organs is discussed. Received: 4 March 1998 / Accepted: 20 July 1998     

Stem biomechanics of three columnar cacti from the Sonoran Desert

The allometric relationship of stem length L with respect to mean stem diameter D was determined for 80 shoots of each of three columnar cactus species (Stenocereus thurberi, Lophocereus schottii, and S. gummosus) to determine whether this relationship accords with that predicted by each of three contending models purporting to describe the mechanical architecture of vertical shoots (i.e., geometric, stress, and elastic similitude, which predict L proportional to D(alpha), with alpha = 1/1, 1/2, and 2/3, respectively). In addition, anatomical, physical, and biomechanical stem properties were measured to determine how the stems of these three species maintain their elastic stability as they increase in size. Reduced major axis regression of L with respect to D showed that alpha = 2.82 ± 0.14 for S. thurberi, 2.32 ± 0.19 for L. schottii, and 4.21 ± 0.31 for S. gummosus. Thus, the scaling exponents for the allometry of L differed significantly from that predicted by each of the three biomechanical models. In contrast, these exponents were similar to that for the allometry previously reported for saguaro. Analyses of biomechanical data derived from bending tests performed on 30 stems selected from each of the three species indicated that the bulk stem tissue stiffness was roughly proportional to L2, while stem flexural rigidity (i.e., the ability to resist a bending force) scaled roughly as L3. Stem length was significantly and positively correlated with the volume fraction of wood, while regression analysis of the pooled data from the three species (i.e., 90 stems) indicated that bulk tissue stiffness scaled roughly as the 5/3-power of the volume fraction of wood in stems. These data were interpreted to indicate that wood served as the major stiffening agent in stems and that this tissue accumulates at a sufficient rate to afford unusually high scaling exponents tot stem length with respect to stem diameter (i.e., disproportionately large increments of stem length with respect to increments in stem diameter). Nevertheless, the safety factor against the elastic failure of stems (computed on the basis of the critical buckling height divided by actual stem length) decreased with increasing stem size tot each species, even though each species maintained an average safety factor equal to two. We speculate that the apparent upper limit to plant height calculated for each species may serve as a biomechanical mechanism for vegetative propagation and the establishment of dense plant colonies by means of extreme stem flexure and ultimate breakage, especially for S. gummosus.     

The mechanical role of bark

The ability of stem bark to resist bending forces was examined by testing in bending segments of Acer saccharum, Fraxinus americana, and Quercus robur branches with and without their bark. For each species, the bark contributed significantly to the ability of stem segments differing in age to resist bending forces, but its contribution was age-dependent and differed among the three species. The importance of the mechanical role of the bark decreased basipetally with increasing age of F. americana and Q. robur stem segments and was superceded by that of the wood for segments ≥ 6 yr old. A. saccharum bark was as mechanically important as the wood for stem segments 7 yr old but was not a significant stiffening agent for younger or older portions of stems. On average, the stiffness of the bark from all three species was 50% that of the wood. However, the geometric contribution to the flexural rigidity of stems made by the bark (i.e., the bark's second moment of area) was sufficiently large to offset its lower stiffness (Young's modulus) relative to that of the wood. A simple model is presented that shows that the bark must be as mechanically important as the wood when its radial thickness equals 32% that of the wood and its stiffness is 50% that of the wood. Based on this model, which is shown to comply with the data from three species purported to have stiff woods, it is evident that the role of the bark cannot be neglected when considering the mechanical behavior of juvenile woody stems subjected to externally applied bending forces.     

Computing factors of safety against wind-induced tree stem damage

The drag forces, bending moments and stresses acting on stems differing in size and location within the mechanical infrastructure of a large wild cherry (Prunus serotina Ehrh.) tree are estimated and used to calculate the factor of safety against wind-induced mechanical failure based on the mean breaking stress of intact stems and samples of wood drawn from this tree. The drag forces acting on stems are calculated based on stem projected areas and field measurements of wind speed taken within the canopy and along the length of the trunk. The bending moments and stresses resulting from these forces are shown to increase basipetally in a nearly log-log linear fashion toward the base of the tree. The factor of safety, however, varies in a sinusoidal manner such that the most distal stems have the highest factors of safety, whereas stems of intermediate location and portions of the trunk near ground level have equivalent and much lower factors of safety. This pattern of variation is interpreted to indicate that, as a course of normal growth and development, trees similar to the one examined in this study maintain a cadre of stems prone to wind-induced mechanical damage that can reduce the probability of catastrophic tree failure by reducing the drag forces acting on older portions of the tree. Comparisons among real and hypothetical stems with different taper experiencing different vertical wind speed profiles show that geometrically self-similar stems have larger factors of safety than stems tapering according to elastic or stress self-similarity, and that safety factors are less significantly influenced by the 'geometry' of the wind-profile.     

Oscillation frequencies of plant stems with apical loads

The frequency of free oscillations of plant stems with apical loads, as found in some cereals, is different depending on whether the stems are oriented vertically or horizontally. Neglecting the stem's own weight the differential equations describing the oscillation can be solved for both cases, although in the vertical orientation only for a limited set of conditions including constant bending stiffness along the stem. Comparison with experimental data shows that the difference between the oscillation frequencies in vertical and horizontal orientations can be attributed to the fact that in the vertical orientation the top load due to gravity induces a bending moment varying with the oscillation, while in the horizontal case this bending moment is nearly constant.     

On the mechanical properties of the rare endemic cactus Stenocereus eruca and the related species S. gummosus

We examined the hypothesis that the procumbent growth habit of the rare, columnar cactus Stenocereus eruca is in part the result of a diminution of the mechanical properties of stem tissues by comparing the properties of S. eruca plants with those of the putatively closely related semi-erect shrub S. gummosus. Intact stems and surgically removed anatomically comparable regions of the stems of both species were tested in bending and tension to determine their Young's modulus and breaking stress. A computer program was used to evaluate the contribution of each region to the capacity of entire stems to resist bending forces. Our analyses indicate that the principal stiffening agent in the stems of both species is a peripheral tissue complex (= epidermis and collenchyma in the primary plant body) that has a significantly higher tensile breaking stress and greater extensibility for S. gummosus than that of S. eruca. Computer simulations indicate that the wood of either species contributes little to bending stiffness, except in very old portions of S. gummosus stems, because of its small volume and central location in the stem. These and other observations are interpreted to support the hypothesis that S. eruca evolved a procumbent growth habit as the result of manifold developmental alterations some of which reduced the capacity of tissues to support the weight of stems.     

The modular hip revision stem made of titanium alloy--influence and optimisation of bending stiffness]

The MRP-Titan Revision stem has proved to be a highly successful implant system for revision arthroplasty of the hip. Good and excellent clinical and radiological results with spontaneous filling of bony defects have been reported, The observation of atrophy of the proximal femur associated with stem diameters > 17mm prompted us to examine the bending stiffness of stems of various diameters. To determine their static bending characteristics, the stems were tested under axial pressure loads in accordance with Euler's buckling case. Dynamic tests were performed with the mono-axial servohydraulic test equipment MTS 810. From a stem diameter of 18 mm upwards, deflection of the stem under loading decreased disproportionately, in direct correlation with the stem stiffness. By optimising the geometry and varying the alloy it is possible to obtain a constant ISD factor for the modular MRP-Titan revision stem CONCLUSION: The MRP-Titan revision stem is a reliable implant system for revision arthroplasty of the hip. Clinical findings of atrophy of the proximal femur associated with stem diameters > 17 mm was found to be correlated with a disproportionate increase in bending stiffness. The aim of further developments will be to reduce the stiffness of larger-diameter stems by making changes to the design and/or to the alloy (Ti15Mo, Ti13Nb13Zr, Ti12Mo6Zr2Fe2).     

The Mechanical Roles of Clasping Leaf Sheaths: Evidence fromArundinaria tecta(Poaceae) Shoots Subjected to Bending and Twisting Forces

Representative shoot segments of the grass speciesArundinariatéctaconsisting of one intact internode and its subtendingnode and clasping leaf sheath were tested to determine the mechanicalinfluence of the leaf sheath on the ability of stems to resistbending and twisting forces. These segments were also used tomeasure shoot morphometry and composite tissue Young's and shearmoduli (EandG,respectively) to simulate the global deformationpatterns attending bending and twisting by means of finite elementanalyses. On average, leaf sheaths contributed 33% of the overallbending stiffness and 43% of the overall torsional stiffnessof stem segments. Comparisons betweenEandGof isolated internodesand leaf sheaths indicated that sheaths were composed of stiffertissues measured either in bending or twisting. Thus, leaf sheathscould act as an external cylindrical brace composed of stiffermaterials than those of the internodes they enveloped. The magnitudesof internodalEandGwere correlated with internodal shape suchthat the ability of internodes to resist twisting relative tothe ability to resist bending forces decreased as internodesbecame more slender or developed thinner walls (both of whichoccur in an acropetal direction from the base to the tip ofshoots). Finite element simulations predicted that, in bending,the leaf sheath laterally braces internodal walls as they tendto ovalize in cross section and push against its inner surfacewhich ovalizes to a lesser extent in the plane normal to thecurvature of shoot flexure. In twisting, the successive ovalizedtransections of internodal walls assumed a helical pattern alongthe length of shoot segments. This helical deformation patternwas attended by an inner lateral contraction of internodal wallsthat was less developed in the leaf sheath that thus provideddecreasing mechanical support to the internode as the lateralcontraction of internodal walls amplified. The twisting of internodesand sheaths was also predicted to concentrate tensile and shearstrains in the nodal diaphragm. Here stress intensities sufficientto produce tissue shear failure were concentrated at two opposingpoints on the surface of the diaphragm. Finite element analysesthus identified a potential weak point in the mechanical constructionof hollow, septate shoots that are, nevertheless, more thanadequately stiff to support their own weight, yet sufficientlyflexible to twist without irreparable damage in normal winds.Copyright1998 Annals of Botany Company Plant stems; nodes; internodes; leaf sheaths; elastic moduli; wind lodging; biomechanics.     

Biomechanics of the columnar cactus Pachycereus pringlei

We report the longitudinal variations in stiffness and bulk density of tissue samples drawn from along the length of two Pachycereus pringlei plants measuring 3.69 and 5.9 m in height to determine how different tissues contribute to the mechanical stability of these massive vertical organs. Each of the two stems was cut into segments of uniform length and subsequently dissected to obtain and mechanically test portions of xylem strands, stem ribs, and a limited number of pith and cortex samples. In each case, morphometric measurements were taken to determine the geometric contribution each tissue likely made to the ability of whole stems to resist bending forces. The stiffness of each xylem strand increased basipetally toward the base of each plant where stiffness sharply decreased, reaching a magnitude comparable to that of strands 1 m beneath the stem apex. The xylem was anisotropic in behavior, i.e., its stiffness measured in the radial and in the tangential directions differed significantly. Despite the abrupt decrease in xylem strand stiffness at the stem base, the contribution made by this tissue to resist bending forces increased exponentially from the tip to the base of each plant due to the accumulation of wood. A basipetal increase in the stiffness of the pith (and, to limited extent, that of the cortex) was also observed. In contrast, the stiffness of stem rib tissues varied little as a function of stem length. These tissues were stiffer than the xylem in the corresponding portions of the stem along the upper two-fifths of the length of either plant. Tissue stiffness and bulk density were not significantly correlated within or across tissue types. However, a weak inverse relationship was observed for these properties in the case of the xylem and stem rib tissues. We present a simple formula that predicts when stem ribs rather than the xylem strands serve as the principal stiffening agents in stems. This formula successfully predicted the observed aspect ratio of the stem ribs (the average quotient of the radial and tangential dimensions of rib transections), and thus provided circumstantial evidence that the ribs are important for mechanical stability for the distal and younger regions of the stems examined.     

Structural development of the shoot and root systems of two winter wheat cultivars, Triticum aestivum L.

The structural development of the stems and basal anchorageroots of Galahad and Hereward winter wheat cultivars (Triticumaestivum L.) were investigated and related to their mechanicalfunction. Stem and root morphology, anatomy and mechanical propertieswere examined from tillering (March) up to maturity (August),together with plant weight distribution. This allowed us tocalculate a ‘factor of safety’ against root andstem failure throughout development. As the plants grew taller the stem and the anchorage ‘coronalroots’ increased in bending strength countering the increasingmechanical demands. The bending strength, in turn, was correlatedwith the amount of lignified material around the stem and rootperimeter. Structural development ceased by ear emergence, whenthe plant was at its tallest, but because the ear weight continuedto rise the ‘self-weight’ moment pushing the plantover continued to increase. This meant that the ‘safetyfactors’ of both cultivars against both root and stemmechanical failure decreased throughout development. In bothcultivars the safety factors against root failure were lowerthan for stem failure, and Galahad had lower factors of safetythan Hereward. All these findings were consistent with resultsof field trials; failure tends to occur late in development,during grain filling, and is localized to the root system, whilstGalahad is more prone to lodging than Hereward. The pattern of mechanical development of winter wheat seemsto be one which would maximize its reproductive success, maintainingits structural integrity especially early in development whileinvesting in a minimum of structural material. Key words: Safety factor, anchorage, lodging, biomechan-ics, structural development     

Cell wall proteome analysis of Arabidopsis thaliana mature stems

Plant stems carry flowers necessary for species propagation and need to be adapted to mechanical disturbance and environmental factors. The stem cell walls are different from other organs and can modify their rigidity or viscoelastic properties for the integrity and the robustness required to withstand mechanical impacts and environmental stresses. Plant cell wall is composed of complex polysaccharide networks also containing cell wall proteins (CWPs) crucial to perceive and limit the environmental effects. The CWPs are fundamental players in cell wall remodeling processes, and today, only 86 have been identified from the mature stems of the model plant Arabidopsis thaliana. With a destructive method, this study has enlarged its coverage to 302 CWPs. This new proteome is mainly composed of 27.5% proteins acting on polysaccharides, 16% proteases, 11.6% oxido‐reductases, 11% possibly related to lipid metabolism and 11% of proteins with interacting domains with proteins or polysaccharides. Compared to stem cell wall proteomes already available (Brachypodium distachyon, Sacharum officinarum, Linum usitatissimum, Medicago sativa), that of A. thaliana stems has a higher proportion of proteins acting on polysaccharides and of proteases, but a lower proportion of oxido‐reductases.     

Effects of Vibration on Mechanical Properties and Biomass Allocation Pattern ofCapsella bursa-pastoris(Cruciferae)

The herbaceous dicot speciesCapsella bursa-pastoris(Cruciferae)was used to determine the influence of chronic mechanical perturbationon the biomass allocation pattern (i.e. dry weight distributionamong roots, stems and reproductive structures) and the mechanicalproperties of roots and stems (i.e. tensile breaking stressand Young's modulus). It was hypothesized that mechanicallystimulated plants would allocate more of their total biomassto root systems and less to shoots compared to control plantsand that the breaking stress (a measure of strength) and Young'smodulus (a measure of material stiffness) would increase forroots and decrease for stems because these responses would adaptivelyreduce the bending moment at the base of shoots and increasethe anchorage strength of root systems. It was also hypothesizedthat mechanical perturbation would maladaptively reduce therelative fitness of individuals by reducing biomass allocationto their reproductive organs and the ability to broadcast seedsby means of elastic stem flexure. These hypotheses were testedby vibrating cultivated plants for 60 s every day during thecourse of growth to maturity and comparing their dry weightdistributions and the mechanical properties of their body parts(measured in tension) to those of undisturbed control plants.Based on a total of 51 experimentally manipulated and 44 controlplants for which mechanical properties were successfully tested,chronic organ flexure resulted in more massive root systemsand less massive vegetative shoots, increased the magnitudesof root breaking stress and Young's modulus and had the reverseeffect on stems, reduced the dry weight of reproductive structuresat maturity, delayed the formation of the first mature flowerand fruit, and accelerated the on-set of plant senescence comparedto control plants. These responses to chronic organ flexureare interpreted to be vegetatively adaptive, since they reducethe probability of stem and root failure as a consequence ofwind-pressure or foraging, and to be reproductively maladaptive,since they reduce reproductive effort and the ability to mechanicallydischarge seeds.Copyright 1998 Annals of Botany Company Adaptation, biomass allocation, biomechanics, elastic properties, roots, stems, thigmomorphogenesis.     

Initiation and progression of mechanical damage in the intervertebral disc under cyclic loading using continuum damage mechanics methodology: A finite element study

It is difficult to study the breakdown of disc tissue over several years of exposure to bending and lifting by experimental methods. There is also no finite element model that elucidates the failure mechanism due to repetitive loading of the lumbar motion segment. The aim of this study was to refine an already validated poro-elastic finite element model of lumbar motion segment to investigate the initiation and progression of mechanical damage in the disc under simple and complex cyclic loading conditions. Continuum damage mechanics methodology was incorporated into the finite element model to track the damage accumulation in the annulus in response to the repetitive loading. The analyses showed that the damage initiated at the posterior inner annulus adjacent to the endplates and propagated outwards towards its periphery under all loading conditions simulated. The damage accumulated preferentially in the posterior region of the annulus. The analyses also showed that the disc failure is unlikely to happen with repetitive bending in the absence of compressive load. Compressive cyclic loading with low peak load magnitude also did not create the failure of the disc. The finite element model results were consistent with the experimental and clinical observations in terms of the region of failure, magnitude of applied loads and the number of load cycles survived.     

Modelling primary and secondary growth processes in plants: a summary of the methodology and new data from an early lignophyte

A mathematical method, based on polar coordinates that allow modelling of primary and secondary growth processes in stems of extant and fossil plants, is summarized and its potential is discussed in comparison with numerical methods using digitizing tablets or electronic image analysing systems. As an example, the modelling of tissue distribution in the internode of an extant sphenopsid (Equisetum hyemale) is presented. In the second half of the paper we present new data of a functional analysis of stem structure and biomechanics of the early lignophyte Tetraxylopteris schmidtii (Middle Devonian) using the polar coordinate method for modelling the tissue distribution in stems of different ontogenetic age. Calculations of the mechanical properties of the stems, based on the modelling of the tissue arrangement, indicate that there is no increase in structural bending modulus throughout the entire development of the plant. The oldest ontogenetic stage has a significantly smaller bending elastic modulus than the intermediate ontogenetic stage, a 'mechanical signal', which is not consistent with a self-supporting growth form. These results, and the ontogenetic variations of the contributions of different stem tissues to the flexural stiffness of the entire stem, are discussed in the evolutionary context of cambial secondary growth.     

Safety and streamlining of woody shoots in wind: an empirical study across 39 species in tropical Australia

? Wind is a key mechanical stress for woody plants, so how do shoot traits affect performance in wind? ? We used a vehicle mounted apparatus to measure drag, streamlining and mechanical safety in 127 vertical lead-shoots, 1.2 m long, across 39 species in tropical Australia. ? Shoot dimensions and stem tissue properties were closely coupled so that shoots with low stem specific gravity or larger projected area had thicker stems. Thicker stems provide larger second moment of area (I), which increased shoot safety and bending stiffness but impeded shoot reconfiguration in strong winds, including frontal area reduction. Nonetheless, increasing I also improved streamlining. Streamlining was unrelated to traits except I. Stem tissue material properties only had small effects. Higher modulus of rupture increased shoot safety and higher Young's modulus impeded shoot reconfiguration. ? We found no conflict between bending stiffness and streamlining for woody shoots. Stiffness might help streamlining by increasing damping and stability, thereby reducing flagging in wind. Tissue-level traits did influence shoot-level mechanical safety and behaviour, but shoot geometry was much more important. Variable shoot and stem traits, which all influenced shoot biomechanics, were integrated in shoots to yield a relatively narrow range of outcomes in wind.     

Identification of biomechanical factors involved in stem shape variability between apricot tree varieties

BACKGROUND AND AIMS: Stem shape in angiosperms depends on several growth traits such as elongation direction, amount and position of axillary loads, stem dimensions, wood elasticity, radial growth dynamics and active re-orientation due to tension wood. This paper analyses the relationship between these biomechanical factors and stem shape variability. METHODS: Three apricot tree varieties with contrasting stem shape were studied. Growth and bending dynamics, mechanical properties and amount of tension wood were measured on 40 1-year-old stems of each variety during one growth season. Formulae derived from simple biomechanical models are proposed to quantify the relationship between biomechanical factors and re-orientation of the stems. The effect of biomechanical factors is quantified combining their mechanical sensitivity and their actual variability. RESULTS: Re-orientations happened in three main periods, involving distinct biomechanical phenomena: (a) passive bending due to the increase of shoot and fruit load at the start of the season; (b) passive uprighting at the fall of fruits; (c) active uprighting due tension wood production at the end of the season. Differences between varieties mainly happened during periods (a) and (b). CONCLUSIONS: The main factors causing differences between varieties are the length/diameter and the load/cross-sectional area ratios during period (a). Wood elasticity does not play an important role because of its low inter-variety variability. Differences during period (b) are related to the dynamics of radial growth: varieties with early radial growth bend weakly upward because the new wood layers tend to set them in a bent position. The action of tension wood during period (c) is low when compared with passive phenomena involved in periods (a) and (b).     

Stresses on joints of Opuntia laevis: forces necessary for mechanical failure and role of lignified xylem cells as resistance components

Species of Opuntia exhibit a wide range of morphologies. Understanding these morphologies may require knowledge of the mechanical stresses on joints of stem segments and as well as the internal components in joints that withstand joint failure (separation of the terminal cladode from the sub-terminal cladode after weights were applied perpendicularly to the long axis). Results of stress testing terminal cladodes of Opuntia laevis provided the following conclusions: (1) amounts of applied stress for joint failure were not related to the amounts of stress on joints before stress testing; (2) breaking strength (failure stress) was accurately determined for joints from linear plots of M (bending moment) versus I/c (section modulus) [breaking stress for O. laevis was 2.77 kPa]; (3) bending moments at failure were twice as high for tensile portions than for compressive portions of joints; and (4) bending moments at failure were positively correlated with amounts of lignified xylem cells in joints [for each mm² of lignified xylem cells in joints there was an increase of 0.06 N m of bending moment]. These data support the overall hypothesis that bending stresses are the main stresses at joints of Opuntia laevis and that lignified xylem cells are the main components that resist joint failure. Moreover, since tensile portions have more lignified xylem cells than other stem portions, tensile portions can resist more applied stress.