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     

Biomechanics and functional anatomy of hollow-stemmed sphenopsids. I. Equisetum giganteum (Equisetaceae)

The hollow stem of Equisetum giganteum owes its mechanical stability to an outer ring of strengthening tissue, which provides stiffness and strength in the longitudinal direction, but also to an inner lining of turgid parenchyma, which lends resistance to local buckling. With a height >2.5 m isolated stems are mechanically unstable. However, in dense stands individual stems support each other by interlacing with their side branches, the typical growth habit of semi-self-supporters.     

The influence of gravity and wind on land plant evolution

Aspects of the engineering theory treating the elastic stability of vertical stems and cantilevered leaves supporting their own weight and additional wind-induced forces (drag) are reviewed in light of biomechanical studies of living and fossil terrestrial plant species. The maximum height to which arborescent species can grow before their stems elastically buckle under their own weight is estimated by means of the Euler-Greenhill formula which states that the critical buckling height scales as the 1/3 power of plant tissue-stiffness normalized with respect to tissue bulk density and as the 2/3 power of stem diameter. Data drawn from living plants indicate that progressively taller plant species employ stiffer and lighter-weight plant tissues as the principal stiffening agent in their vertical stems. The elastic stability of plants subjected to high lateral wind-loadings is governed by the drag torque (the product of the drag force and the height above ground at which this force is applied), which cannot exceed the gravitational bending moment (the product of the weight of aerial organs and the lever arm measured at the base of the plant). Data from living plants indicate that the largest arborescent plant species rely on massive trunks and broad, horizontally expansive root crowns to resist drag torques. The drag on the canopies of these plants is also reduced by highly flexible stems and leaves composed of tissues that twist and bend more easily than tissues used to stiffen older, more proximal stems. A brief review of the fossil record suggests that modifications in stem, leaf, and root morphology and anatomy capable of simultaneously coping with self-weight and wind-induced drag forces evolved by Devonian times, suggesting that natural selection acting on the elastic stability of sporophytes occurred early in the history of terrestrial plants.     

Buckling of inner cell wall layers after manipulations to reduce tensile stress: observations and interpretations for stress transmission

The inner layer of the cell wall in tissues that are under tensile stress in situ, e.g. epidermis and collenchyma of etiolated sunflower hypocotyls, shows a pattern of transverse folds when the tissues are detached and plasmolysed. This can be observed by Nomarski imaging of inner surfaces of the outer cell walls and electron microscopy of longitudinal sections after peeling the epidermis and bathing it in plasmolysing solutions. The folds are apparently caused by buckling of the inner layer due to the longitudinal compressive force exerted on this layer by the outer wall layer, when it shrinks after the removal of the longitudinal tensile stresses. In these stresses, two components can be distinguished: the tissue stress, disappearing on peeling, and that caused directly by turgor pressure, disappearing in hyperosmotic solution. Investigation of the buckling indicates that the outer layer of the cell wall transmits in situ most of the longitudinal tensile stress in the wall. The common concept that the inner layer of the wall is the region bearing most stress and therefore regulating growth can still be valid with respect to the transverse stress component.     

Contributions to the Biomechanics of Plants. I. Stabilities of Plant Stems with Strengthening Elements of Different Cross-Sections Against Weight and Wind Forces*

Biophysical considerations allow estimates of the mechanical stresses on self-bearing vertical stems of plants. Even at moderate wind velocities the stresses induced by aerodynamic forces dominate over those induced by the own weight. Using polar coordinates, analytical expressions of cross-sectional area and axial second moment of area for centrisymmetric structures with symmetries threefold or higher are derived. Calculating the relative section modulus for various (centrisymmetric) arrangements of stabilizing structures leads to an estimate of the “mechanical effectivity” of these structures. If for plant stems, seen as composite materials, the second moments of area and the elastic moduli are known, the contribution of the different tissues to mechanical stability can be determined quantitatively. The mechanical design of early “vascular” land plants and of stems of (fossil) trees and lianas in different ontogenetic stages can be assessed.     

A mechanical perspective on foliage leaf form and function

The mechanical behaviour of large foliage leaves in response to static and dynamic mechanical forces is reviewed in the context of a few basic engineering principles and illustrated in terms of species drawn from a variety of vascular plant lineages. When loaded under their own weight or subjected to externally applied forces, petioles simultaneously bend and twist, and thus mechanically operate as cantilevered beams. The stresses that develop in petioles reach their maximum intensities either at their surface or very near their centroid axes, where they are accommodated either by living and hydrostatic tissues (parenchyma and collenchyma) or dead and stiff tissues (sclerenchyma and vascular fibres) depending on the size of the leaf and the species from which it is drawn. Allometric analyses of diverse species indicate size-dependent variations in petiole length, transverse shape, geometry and stiffness that accord well with those required to maintain a uniform tip-deflection for leaves with laminae differing in mass. When dynamically loaded, the laminae of many broad-leaved species fold and curl into streamlined objects, thereby reducing the drag forces that they experience and transmit to their subtending petioles and stems. From a mechanical perspective, the laminae of these species operate as stress-skin panels that distribute point loads more or less equally over their entire surface. Although comparatively little is known about the mechanical structure and behaviour of foliage leaves, new advances in engineering theory and computer analyses reveal these organs to be far more complex than previously thought. For example, finite-element analyses of the base of palm leaves reveal that stresses are decreased when these structures are composed of anisotropic as opposed to isotropic materials (tissues).     

SAFETY FACTORS IN VERTICAL STEMS: EVIDENCE FROM EQUISETUM HYEMALE

Predictions from a mechanical model for hollow vertical stems are tested against morphometric and mechanical studies of the vertical stems of Equisetum hyemale. The model predicts 1) that the wall thickness of hollow internodes must be at least 15% of the external radius of shoots, 2) that the elastic modulus of stems is quantitatively related to the ratio of apoplast (cell walls) to symplast (cytoplasm) areas in transverse sections through stems, and that (3) hollow stems are designed to sustain an additional and significant proportion of their own weight. The “safety factors” predicted for a hollow vertical stem are used to examine two adaptationist explanations for hollow stems: 1) “economy in design,” which argues that natural selection will favor a reduction in the metabolic cost in constructing an organ, and 2) “mechanical design,” which argues that stems are designed to maximize their mechanical stability during vertical growth. Evidence from E. hyemale indicates that 1) there is a developmental limit to the maximum allotment of biomass invested in the construction of stems, 2) as stem height increases, morphometric adjustments in internodal wall thickness occur which converge on predicted safety limits, and 3) the elastic modulus of stems changes as a function of the ratio of apoplast to symplast areas seen in transverse sections through shoots. Biomechanical and developmental evidence and the allometry of E. hyemale stems are consistent with the view that stems are designed for safety and are inconsistent with some predictions based on the economy in design.     

SOME OBSERVATIONS ON THE MECHANISM OF ORIENTATION MOVEMENT OF WOODY STEMS

The force that induces orientation movement of inclined or bent woody stems is generated in reaction wood even in young terminal stems with a large proportion of soft tissues to secondary xylem. Compression wood formed in pine as a response to inclination expands longitudinally after the stresses are released by sawing it from the stem. The increment of length of compression wood when sawed is equal to the decrement of its length which occurs during drying. This suggests that stresses developed by compression wood in the stem are related to imbibition of water by its cell walls. Not all compression wood develops tensile forces in the stem. Neutral compression wood was observed in the lower portion of inclined stems of pine. Tension wood in poplar develops contractile forces in the stem during its aestival maturation. However, when harvested before developing contractile forces in situ, it develops such forces during drying. This suggests that in poplar the mechanism which produces forces responsible for orientation bending also involves changes in cell wall hydration.     

Changes in the factor of safety within the superstructure of a dicot tree

The objective of this study was to determine whether the factor of safety for mechanical stability varied among stems differing in size and age within the superstructure of a large dicot tree. Two factors of safety were selected for study: the quotient of the critical buckling height and the actual length of stems, Hcrit/L, and the quotient of the modulus of rupture (the force per unit area required to break a stem) and the working stress (the force per unit area resulting from the biomass measured distal to a stem), M_R/σ_w. These two dimensionless safety factors were determined for a total of 420 shoot segments comprising much of the aboveground biomass of a Robinia pseudoacacia (Fabaceae) tree measuring 18.7 m in height and 1347 kg in mass, and 0.46 m in diameter (40 yr old) at 1.2 m from the ground. An S-shaped trend was observed when each of the two factors of safety was plotted as a function of stem age. Each factor decreased from a local maximum for the most distal (peripheral) stems in the canopy to a local minimum value for stems ∼10 yr old; each factor increased again to another local maximum for stems 11–18 yr old, and then decreased steadily toward the base of the trunk. This trend was the result of the allometric relationships among stem diameter, length, biomass, and material properties (stiffness and strength) with respect to stem age. Although they were disproportionately more slender than their older counterparts, peripheral stems were sufficiently stiff and strong to sustain the stresses resulting from their weight and that of foliage without deflecting under these loads, yet they were sufficiently flexible to easily bend and thereby presumably provide a mechanism to reduce the drag forces acting on the entire tree. In contrast, the internally imposed mechanical forces acting on progressively older stems increased at a greater rate than the observed rate of increase in stem stiffness, strength, or diameter. The probability of mechanical failure, which must be considered from a demographic perspective (i.e., an age-dependent phenomenon), thus increased from older branches to the base of the trunk. Reports of similar allometric trends based on interspecific comparisons among diverse dicot species comply with the allometry observed for the R. pseudoacacia tree and suggest that the S-shaped trend for the factor of safety holds for stems differing in age drawn from individual trees and for the trunks of conspecifics differing in age drawn from a dense population.     

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.     

The buckling of plant roots

The mechanical stresses required to buckle root tips were measured directly for seven plant species. For two of these, both seminal and primary lateral roots were measured. For four of the plant species investigated, the easier and more rapid method used to measure the buckling stresses of excised root tips gave results which agree closely with those obtained for the growing roots of intact plants. Values of buckling stress were also calculated from previously determined elastic properties of roots. The calculated and measured values of buckling stress are of the same magnitude only, and comparison for ten root types fails to show any consistent relationship between these two methods. From the results from three plant species it has been possible to define empirical functions to account for the observed changes in root buckling stress with air gap length and water stress. Buckling stresses are not significantly affected by the concentration of nitrate in the growth medium.     

A finite element analysis of hollow stemmed hip prostheses as a means of reducing stress shielding of the femur.

Stress shielding of the femur is known to be a principal factor in aseptic loosening of hip replacements. This paper considers the use of a hollow stemmed hip implant for reducing the effects of stress shielding, while maintaining acceptably low levels of stress in the cement. Using finite element modelling, the stresses in the proximal femur using different shapes of hollow stem were compared with those produced using comparable sizes of solid stem with different values of elastic modulus. A reduction in stress shielding could be achieved with a hollow stem. A cylindrical hollow stem design was then optimised in order to control the maximum allowable stress in the cement, the minimum allowable stresses in the bone, and a combination of the two. The resulting stems achieved an increase in proximal bone stress of about 15% for the first case and 32% for a model using high strength cement, compared with solid stems of the same nominal outside diameter. The gains of these theoretically optimised designs dropped off rapidly further down the stem. Linearly tapered hollow stems reached a 22% gain, which could be a good compromise between acceptable cement stresses and ease of manufacture.     

The allometry of saguaro height

The allometry of plant height H with respect to mean stem diameter D was determined based on 118 saguaro plants. The slope obtained for the reduced major axis regression analysis of the data was 2.36 ± 0.085, indicating that taller plants are disproportionately more slender than their shorter, presumably younger counterparts. The consequences of this positive, extremely anisometric relation on the elastic stability of stems were estimated by computing the critical buckling height H_crit for each of the 118 stems on the basis of the mean density-specific stiffness (i.e., the quotient of Young's elastic modulus E and bulk tissue density ρ) determined for a single section from a mature saguaro stem. E/ρ was nearly equivalent to that of tissue samples of sclerenchyma isolated from other plant species. Since the slope of H_crit vs. D equals ≈ 0.67 when E/ρ ≈ a constant, the safety-factor for saguaro stems (i.e., H_crit/H) appeared to be size-dependent such that it decreased with increasing plant height (i.e., H_crit/H ≈ D^-1.65). However, the mean safety-factor computed for the 118 saguaro specimens was 9.64, indicating that, on the average, plant height was well below H_crit. Additionally, circumstantial evidence suggests that saguaro stems become more stiff as they increase in size (and age) and that the rate of stem growth decelerates over time. The former would obtain a near size-independent safety-factor against elastic buckling while the latter protracts the time required to reach the critical buckling height. Comparisons among the allometries of H and H_crit for saguaro, other cacti, nonwoody, and highly branched tree species indicated that saguaro size overlaps with the lower size-range of the largest known dicot and gymnosperm tree specimens likely as a consequence of the high E/ρ of mature saguaro stems.     

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     

Structural Basis of Stress Concentration in the Cytoskeleton

Professor Y.C. Fung has shown that living tissues remodel extensively in response to mechanical forces such as blood pressure variations. At the cellular level, those mechanical perturbations must be perceived by individual cells. However, mechanisms of mechanochemical transduction in living cells remain a central challenge to cell biologists. Contrary to predictions by existing models of living cells, we reported previously that a local stress, applied via integrin receptors, is propagated to remote sites in the cytoplasm and is concentrated at discrete foci. Here we report that these foci of strains and stresses in the cytoplasm correspond to local peak deformation or local buckling of microtubules and are near the actin bundles of the cell. Multiple images at different z heights demonstrated more foci of concentrated displacements in the middle of the cell than at the apex or at the cell base. Together with previously published work, these findings underscore the importance of tensed bundled filamentous actin in intracellular mechanical stress distribution and signaling.     

Hollow occurrence and abundance varies with tree characteristics and among species in temperate woodland Eucalyptus

Tree hollows are a critical but diminishing resource for a wide range of fauna around the world. Conservation of these fauna depends on sustainable management of tree species that produce the hollows on which they depend. This study addressed the need for empirical data about intraspecific and interspecific variation in hollow occurrence and abundance in woodland trees in Australia. We measured and performed hollow surveys on 1817 trees of seven species of woodland Eucalyptus in central‐western New South Wales, Australia. Trees were surveyed at 51 one‐hectare sites and about 30% of trees surveyed had multiple stems. Generalized linear mixed models that accounted for nestedness of stems within trees and trees within sites detected a significant amount of variation in hollow occurrence and abundance. Models for individual tree stems of live trees showed hollow probability and abundance increased with diameter at breast height (DBH) and with increasing senescence (form). Stems of Eucalyptus microcarpa Maiden had a higher probability of having hollows than similar DBH stems of Eucalyptus camaldulensis Dehnh., Eucalyptus melliodora A.Cunn. ex Schauer or Eucalyptus populnea ssp. bimbil L.A.S.Johnson & K.D.Hill. Dead stems in live trees were more likely to have hollows than live stems of similar DBH. Each stem in a multi‐stemmed tree had a lower probability of hollow occurrence and lower abundance of hollows than single‐stemmed trees of similar DBH. For stems of dead trees, hollow occurrence and abundance increased with DBH and differed depending on stage of senescence. A comparison of our data with other studies indicates regional variation of hollow abundances within tree species.     

Myrmecotrophy: Plants fed by ants

Two plant genera with tubers specialized for occupation by ants absorb nutrients from waste materials accumulated by the resident colonies. The mineral resources of these host plants are augmented by colony foraging which functions as a second root system. This mutualistic interaction has become known as myrmecotrophy. Many other kinds of plant structure are apparent adaptations to accommodate ant colonies; these include pouches on leaves or petioles and hollow twigs, stems or thorns. Sometimes the ant species residing in these structures are aggressive towards enemies of the host plant and are important for plant defence. Recent research provides some evidence that myrmecotrophy may have a wider role in plant nutrition, at least when subsidizing the costs of plant defence.     

Macromolecular biophysics of the plant cell wall: Concepts and methodology

Plant cell walls provide form and mechanical strength to the living plant, but the relationship between their complex architecture and their remarkable ability to withstand external stress is not well understood. Primary cell walls are adapted to withstand tensile stresses while secondary cell walls also need to withstand compressive stresses. Therefore, while primary cell walls can with advantage be flexible and elastic, secondary cell walls must be rigid to avoid buckling under compressive loads. In addition, primary cell walls must be capable of growth and are subjected to cell separation forces at the cell corners. To understand how these stresses are resisted by cell walls, it will be necessary to find out how the walls deform internally under load, and how rigid are specific constituents of each type of cell wall. The most promising spectroscopic techniques for this purpose are solid-state nuclear magnetic resonance (NMR), and Fourier-transform infrared (FTIR) and Raman microscopy. By NMR relaxation experiments, it is possible to probe thermal motion in each cell-wall component. Novel adaptations of FTIR and Raman spectroscopy promise to allow mechanical stress and strain upon specific polymers to be examined in situ within the cell wall.     

Abiotic stresses increase plant regeneration ability

In disturbed habitats, vegetative regeneration is partly ruled by plant reserves and intrinsic growth rates. Under nutrient-limiting conditions, perennial plants tend to exhibit an increased allocation to storage organs. Under mechanically stressful conditions, plants also tend to increase allocation to below-ground biomass and storage organs. We tested whether those stresses acting differently on plants (nutrient level versus mechanical forces) led to similar effect on storage organs and regeneration ability. We measured, for an aquatic plant species, (1) the size and allocation to storage organs (stems) and (2) the regeneration ability of the storage organs. Plant stems were collected in 4 habitats ranked along a nutrient stress gradient, and having encountered null versus significant mechanical stress (flowing water). All stems were placed in similar neutral conditions and left for a period of 6 weeks before measuring their survival and growth. Dry mass allocation to the storage organ (stem) was higher in stressful habitats. Moreover, stress encountered by plants before the experiment significantly affected regeneration: stems of previously stressed plants (i.e. plants that had grown in nutrient-poor or mechanically stressful habitats) survived better than unstressed ones. Stems of plants having encountered mechanical stress before the experiment had increased growth in nutrient-rich habitats but reduced growth in the poorest habitats. These results demonstrate that regeneration could rely on the level of stress previously encountered by plants. Stress could lead to greater regeneration ability following mechanical failure. The possible mechanisms involved in these results are discussed.