Twist-to-bend ratios of woody structures

Flexural rigidity (El), or resistance to bending loads, andtorsional rigidity (GJ), or resistance to twisting loads, weremeasured on a variety of woody structures-bamboo culms, threekinds of hardwood trunks, two softwood trunks, two vines, andpine roots. The ratios of these rigidities, EI/GJ, was highestand relatively constant for bamboo and hardwoods, slightly lowerfor softwoods, and lower still for vines and roots. All valueswere substantially above those for circular cylinders of ordinaryisotropic materials; since all specimens were nearly circularin cross-section, the high values reflect the elastic moduliof wood rather than geometric factors. While all material showedsubstantial creep under torsional loading, only the vine, Wisteria,crept appreciably under flexural loading as well. Key words: Wood, trees, flexural rigidity, torsional rigidity, biomechanics     

Mechanics and Growth Form of the MossHylocomium splendens

The mossHylocomium splendenshas two different growth forms.Sympodial growth occurs where the apical meristem ceases activityannually and growth is continued by a lateral bud. Sympodialplants are vertical and self-supporting. Monopodial growth occurswhen the apex continues growth. Monopodial plants are prostrate.The aims of the study were to examine stem mechanics of thedifferent growth forms and to compare mechanical propertiesalong stems. Stems of annual segments were subjected to threepoint bending tests. In sympodial plants the stiffness of thestem material increased significantly with segment age. Flexuralrigidity increased significantly with age in segments from 1to 4 years old, and then declined. Segment diameter decreasedsignificantly with age in sympodial plants. Monopodial plantsshowed no significant effects of segment age on the diameter,material stiffness or flexural rigidity of stems. Sympodialsegments were significantly wider, stronger, more rigid andcomposed of stiffer material with a higher stress at yield thanmonopodial segments, but did not have a larger proportion ofstrengthening material. Sympodial stems had significantly morecellulose than monopodial stems. The mossHylocomium splendensshowsa range of mechanical adaptations, as a self-supporting or aprostrate plant, which suit it to life in very different environments.Copyright1998 Annals of Botany Company. Hylocomium splendens, growth form, mechanics, cellulose, stiffness, flexural rigidity, bending, monopodial, sympodial, adaptation, anatomy, stems, plant.     

Biomechanical Responses of Chamaedorea and Spathiphyllum Petioles to Tissue Dehydration

Data are presented for the mechanical responses to dehydrationof petioles from two monocotyledons (Chamaedorea erumpens andSpathiphyllum Clevelandii). These data were used to test thehypothesis that the mechanical properties (elastic modulus Eand flexural stiffness EI) of petioles from C. erumpens arealtered significantly less by dehydration than those of petiolesfrom Spathiphyllum. Dehydration, resulting from either dryingat room temperature or from submergence in various concentrationsof mannitol solutions, produced significant increases in E anddecreases in EI (due to geometric distortions) in both youngand mature Spathiphyllum petioles. Similar trends were observedfor young petioles of C. erumpens, but significantly less sofor mature petioles of this species. Regardless of petiolarage, E increased allometrically as a linear function of tissuedensity, which in turn correlated with the volume fraction oflignified tissues in petioles; however, the proportional increaseof E as a function of tissue density was significantly greaterfor C. erumpens petioles than for Spathiphyllum. Anatomicalanalyses of petiolar transections indicated that Chamaedoreapetioles had larger volume fractions of lignified tissues thanthose of Spathiphyllum and that these tissues were located tomaximize stiffness. These data (and previously reported allometricrelationships between EI and petiolar length) shed light onthe difficulties in evaluating the ‘costs’ of committingtissues to mechanical support. Petioles, biomechanics, leaf anatomy, monocotyledons     

The twist-to-bend compliance of the Rheum rhabarbarum petiole: integrated computations and experiments

Plant petioles can be considered as hierarchical cellular structures, displaying geometric features defined at multiple length scales. Their macroscopic mechanical properties are the cumulative outcome of structural properties attained at each level of the structural hierarchy. This work appraises the compliance of a rhubarb stalk by determining the stalk’s bending and torsional stiffness both computationally and experimentally. In our model, the irregular cross-sectional shape of the petiole and the layers of the constituent tissues are considered to evaluate the stiffness properties at the structural level. The arbitrary shape contour of the petiole is generated with reasonable accuracy by the Gielis superformula. The stiffness and architecture of the constituent layered tissues are modeled by using the concept of shape transformers so as to obtain the computational twist-to-bend ratio for the petiole. The rhubarb stalk exhibits a ratio of flexural to torsional stiffness 4.04 (computational) and 3.83 (experimental) in comparison with 1.5 for isotropic, incompressible, circular cylinders, values that demonstrate the relative structural compliance to flexure and torsion.     

The Mechanical Significance of Clasping Leaf Sheaths in Grasses: Evidence from Two Cultivars of Avena sativa

The elastic (Young's) modulus and flexural rigidity of internodeswith and without their clasping leaf sheaths were determinedfor culms from two cultivars (‘Astro’) and (‘Garry’)of Avena sativa L. Data indicate that early in the developmentof culms, leaf sheaths can have a higher elastic modulus thanthe internodes they envelope, and by virtue of their location,leaf sheaths contribute significantly to the flexural rigidity(hence, resistance to bending) of internodal segments. As culmsmature, the elastic modulus of leaf sheath and internodal tissuesreach parity. However, because of the acropetal pattern by whichnew internodes are produced by shoot apices, sheaths continueto provide mechanical support to distal internodes, particularlythe peduncle. Data for the two cultivars indicate that the elasticmodulus and flexural rigidity of culms can vary significantlywithin the species. Comparisons between the flexural rigidityof the two cultures and the resistance of stems to lodging indicatethat flexural rigidity is not significant to lodging. The engineeringprinciples relevant to the mechanical advantages conferred byclasping leaf sheaths are discussed within the context of grassshoot morphology. Biomechanics, leaf sheath, Avena, elastic modulus     

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.     

Shade induced changes in biomechanical petiole properties in the stoloniferous herb <Emphasis Type="Italic">Trifolium repens</Emphasis>

Increased cell number and cell length both contribute to shade induced elongation of petioles which enables stoloniferous plants to place their leaf lamina higher up in the canopy. Although petiole elongation is assumed to be beneficial, it may also imply costs in terms of decreased biomechanical stability. We test the hypothesis that shade induced elongation changes the biomechanical properties of petioles and that the underlying mechanisms, cell division and cell elongation, differentially affect biomechanical properties. This was done by subjecting 14 genotypes differing in the relative contribution of cell size and cell number to shade induced elongation responses to high light conditions and to simulated canopy shade. Developmental traits (cell size and cell number), morphological traits characterizing the petioles, as well as biomechanical characteristics were measured. Our results show that, comparable to stems of non-clonal plants, the rigidity of a petiole’s tissue (the Young’s modulus) increases, leading to increased flexural stiffness of petioles subjected to shading. Increased flexural stiffness proved to be associated with increased performance under shaded conditions. Our results also indicate that cell number affected the material properties and the flexural stiffness of petioles. However, the degree and pattern of the effects differed between light environments. Shade induced increase in cell number translated into shade induced increase of Young’s modulus and flexural stiffness. Genotypes producing relatively larger cells under shaded conditions experienced a decrease in tissue rigidity. In concert our results indicate that the pattern of selection on flexural stiffness, and thereby also on shade induced changes of cell number and cell size differs among light environments. An erratum to this article can be found at     

THE ELASTIC MODULI AND MECHANICS OF POPULUS TREMULOIDES (SALICACEAE) PETIOLES IN BENDING AND TORSION

Changes attending leaf development in the mechanical behavior and elastic moduli of the petioles of Populus tremuloides Michx. were examined in terms of total leaf weight (Wt), lamina and petiole weight (Wl and Wp), petiolar length (L), and lamina surface area (A). A primary concern was the extent to which two elastic moduli (Young's modulus E and the shear modulus G) of petioles changed over time and were correlated with one another. E and G were measured by means of multiple resonance frequency spectra, and together with the dimensions of cross sections through petioles, these two elastic moduli were used to estimate the stiffness of petioles in simple bending and torsion. Petiolar bending was predicted by means of a model incorporating expressions for both the bending stiffness (El) and torsional rigidity (GJ), where I is the second moment of area and J is the torsional constant. The predictions from these models were compared to observed petiolar bendings due to Wl. Additionally, the frequency of the oscillatory motion of leaves (placed in a wind tunnel with a 1 m sec^-1 ambient wind speed, directed normal to the blade of the leaf) was determined. Results indicate that L, A, Wl, Wp, and Wt were positively correlated with the age of leaves (crudely estimated as a function of leaf plastochron index, LPI); these morphometric parameters were also correlated with the magnitudes of E and G. Also, G was positively and linearly correlated with E, and was, on the average, an order of magnitude less than E. EI and GJ were positively correlated with LPI. The relationships among E, G, EI, C and Wl, Wp, Wt are discussed in terms of leaf allometries.     

The Mechanics of the Flower Stem of the Sedge Carex acutiformis

The mechanics of the triangular stems of Carex acutiformis wasinvestigated by subjecting sections to bending and torsionaltests. The stem was rigid in bending, being stiffened peripherallyby lignified material around the vascular bundles, but becauseof its triangular shape it was vulnerable to local buckling.Despite being and narrow the stem was able to support the seedhead, though it sagged appreciably towards the tip. In contrastthe stem had very low torsional rigidity, both because of itstriangular shape because the strands of lignified material wereisolated from each other. In its lowland habitat this allowsthe drooping stem to twist away from the light winds, so reducingdrag and the chances of self-fertilization. This method of reconfiguringis not possible in the shorter, stiffer mountain sedges whichmust withstand higher winds; many therefore have more circularstems which will be more efficient at resisting bending.Copyright1993, 1999 Academic Press Carex acutiformis, sedge, mechanics, bending, torsion     

Petiole mechanics, leaf inclination, morphology, and investment in support in relation to light availability in the canopy of Liriodendron tulipifera

To determine the role of leaf mechanical properties in altering foliar inclination angles, and the nutrient and carbon costs of specific foliar angle variation patterns along the canopy, leaf structural and biomechanical characteristics, biomass partitioning into support, and foliar nitrogen and carbon concentrations were studied in the temperate deciduous species Liriodendron tulipifera L., which possesses large leaves on long petioles. We used beam theory to model leaf lamina as a uniform load, and estimated both the lamina and petiole flexural stiffness, which characterizes the resistance to bending of foliar elements at a common load and length. Petiole and lamina vertical inclination angles with respect to horizontal increased with increasing average daily integrated photon flux density (Q_int). Yet, the light effects on lamina inclination angle were primary determined by the petiole inclination angle. Although the petioles and laminas became longer, and the lamina loads increased with increasing Q_int, the flexural stiffness of both lamina and petiole increased to compensate for this, such that the lamina vertical displacement was only weakly related to Q_int. In addition, increases and decreases in the petiole inclination angle with respect to the horizontal effectively reduced the distance of lamina load from the axis of rotation, thereby reducing the bending moments and lamina inclination due to gravity. We demonstrate that large investments, up to 30% of total leaf biomass, in petiole and large veins are necessary to maintain the lamina at a specific position, but also that light has no direct effect on the fractional biomass investment in support. However, we provide evidence that apart from light availability, structural and chemical characteristics of the foliage may also be affected by water stress, magnitude of which scales positively with Q_int.     

Biomechanical Attributes of the Leaves of Pine Species

The flexural stiffness EI, elastic modulus E, second momentof area I, length L, and weight Wt of foliage leaves from 26species of pine were measured to determine the manner in whichEI is scaled to leaf size (L or Wt, or both) and to determineif the biomechanical attributes of foliage leaves could be juxtaposedwith the systematic affiliations of species within Pinus Biomechamcaland morphometric data were explored based on a model from engineeringtheory that predicts the relationships among EI, Wt and L providedthat the maximum tip-deflection _max of an untapered cantileveredbeam sustaining its own weight is maintained as the beam eitherincreases in length or weight, or both, I e _max = Wt L²/8EIThe data show that EI disproportionately increased as eitherWt or L, or both increased The allometry of leaves did not conformto that predicted by the model Rather, EI was proportional tothe product of Wt and L raised to an exponent slightly greaterthan one Thus, Omax was predicted not to be maintained as eitherWt or L, or both, increased, as verified by observations Therelationship between El and Wt(L) differed for species typicallyproducing two leaves per fascicle and those bearing more thantwo leaves per fascicle Also, El is geometrically constrainedby the number of leaves produced per fascicle, principally interms of the effects of the number of leaves per fascicle onI Pinus, pine, leaves, biomechanics, Young's modulus     

Cofilin increases the bending flexibility of actin filaments: implications for severing and cell mechanics

We determined the flexural (bending) rigidities of actin and cofilactin filaments from a cosine correlation function analysis of their thermally driven, two-dimensional fluctuations in shape. The persistence length of actin filaments is 9.8 μm, corresponding to a flexural rigidity of 0.040 pN μm². Cofilin binding lowers the persistence length ∼5-fold to a value of 2.2 μm and the filament flexural rigidity to 0.0091 pN μm². That cofilin-decorated filaments are more flexible than native filaments despite an increased mass indicates that cofilin binding weakens and redistributes stabilizing subunit interactions of filaments. We favor a mechanism in which the increased flexibility of cofilin-decorated filaments results from the linked dissociation of filament-stabilizing ions and reorganization of actin subdomain 2 and as a consequence promotes severing due to a mechanical asymmetry. Knowledge of the effects of cofilin on actin filament bending mechanics, together with our previous analysis of torsional stiffness, provide a quantitative measure of the mechanical changes in actin filaments associated with cofilin binding, and suggest that the overall mechanical and force-producing properties of cells can be modulated by cofilin activity.     

Reorientation of daffodil(Narcissus: Amaryllidaceae) flowers inwind: drag reduction andtorsional flexibility

Daffodil flowers extend laterally from the long axes of their stems; as a result, wind on a flower exerts torsional as well as flexural stress on the stem. Stems respond by twisting, and thus flowers reorient to face downwind in moderate winds, in the process reducing their drag by ～30%. This repositioning is facilitated by the stems' relatively low torsional stiffness. Daffodil stems have a ratio of flexural to torsional stiffness of 13.27 ± 0.96 (SD), compared with 8.33 ± 3.20 (SD) for tulip stems, which bear flowers as symmetrical extensions of their long axes, and compared with 1.5 for isotropic, incompressible, circular cylinders.     

Twisting and bending of biological beams: distribution of biological beams in a stiffness mechanospace

Most biological beams bend and twist relatively easily compared to human-made structures. This paper investigates flexibility in 57 diverse biological beams in an effort to identify common patterns in the relationship between flexural stiffness and torsional stiffness. The patterns are investigated by mapping both ideal and biological beams into a mechanospace defined by flexural and torsional stiffness. The distribution of biological beams is not random, but is generally limited to particular regions of the mechanospace. Biological beams that are stiff in bending are stiff in torsion, while those that bend easily also twist easily. Unoccupied regions of the mechanospace represent rare combinations of mechanical properties, without proving that they are impossible. The mechanical properties of biological beams closely resemble theoretical expectations for ideal beams. Both distributions are potentially being driven by the interdependence of the material and structural properties determining stiffness. The mechanospace can be used as a broadly comparative tool to highlight systems that fall outside the general pattern observed in this study. These outlying beams may be of particular interest to both biologists and engineers due to either material or structural innovations.     

Flexural and torsional stiffness in multi-jointed biological beams

Flexibility, the ability to deform in response to loads, is a common property of biological beams. This paper investigates the mechanical behavior of multi-jointed beams, which are characterized by a linear series of morphologically similar joints. Flexural stiffness and torsional stiffness were measured in two structurally distinct beams, crinoid arms (Echinodermata, Comatulida) and crustacean antennae (Arthropoda, Decapoda). Morphological data from these beams were used to determine the relative contributions of beam diameter and joint density (number of joints per millimeter of beam length) to the flexural and torsional stiffness of these two structures. As predicted by beam theory, beam diameter influenced stiffness in both crinoid arms and crustacean antennae. In crinoid arms, increases in joint density were associated with decreases in stiffness, but joint density had no significant influence on stiffness in crustacean antennae. In both crinoid arms and crustacean antennae, the magnitudes of flexural and torsional stiffness, as well as the ratio of these two variables, were similar to previously reported values for non-jointed biological beams. These results suggest that the structural design of a biological beam is not a limiting factor determining its mechanical properties.     

Torsional motion of eosin-labeled F-actin as detected in the time-resolved anisotropy decay of the probe in the sub-millisecond time range

The internal motion of F-actin in the time range from 10(-6) to 10(-3) second has been explored by measuring the transient absorption anisotropy of eosin-labeled F-actin using laser flash photolysis. The transient absorption anisotropy of eosin-F-actin at 20 degrees C has a component that decays in the submicrosecond time scale to an anisotropy of about 0.3. This anisotropy then decays with a relaxation time of about 450 microseconds to a residual anisotropy of about 0.1 after 2 ms. When the concentration of eosin-F-actin was varied in the range from 7 to 28 microM, the transient absorption anisotropy curves obtained were almost indistinguishable from each other. These results show that the anisotropy decay arises from internal motion of eosin-F-actin. Analysis of the transient absorption anisotropy curves indicates that the internal motion detected by the decay in anisotropy is primarily a twisting of actin protomers in the F-actin helix; bending of the actin filament makes a minor contribution only to the measured decay. The torsional rigidity calculated from the transient absorption anisotropy is 0.2 X 10(-17) dyn cm2 at 20 degrees C, which is about an order of magnitude smaller than the flexural rigidity determined from previous studies. Thus, we conclude that F-actin is more flexible in twisting than in bending. The calculated root-mean-square fluctuation of the torsional angle between adjacent actin protomers in the actin helix is about 4 degrees at 20 degrees C. We also found that the torsional rigidity is approximately constant in the temperature range from 5 to approximately 35 degrees C, and that the binding of phalloidin does not appreciably affect the torsional motion of F-actin.     

<Emphasis Type="Italic">Arabidopsis</Emphasis> petiole torsions induced by lateral light or externally supplied auxin require microtubule-associated TORTIFOLIA1/SPIRAL2

Although rather inconspicuous, movements are an important adaptive trait of plants. Consequently, light- or gravity-induced movements leading to organ bending have been studied intensively. In the field, however, plant movements often result in organ twisting rather than bending. This study investigates the mechanism of light- or gravity-induced twisting movements, coined “helical tropisms.” Because certain Arabidopsis cell expansion mutants show organ twisting under standard growth conditions, we here investigated how the right-handed helical growth mutant tortifolia1/spiral2 (tor1) responds when stimulated to perform helical tropisms. When leaves were illuminated from the left, tor1 was capable of producing left-handed petiole torsions, but these occurred at a reduced rate. When light was applied from right, tor1 plants rotated their petioles much faster than the wild-type. Applying auxin to the lateral-distal side of wild-type petioles produced petiole torsions in which the auxinated flank was consistently turned upwards. This kind of movement was not observed in tor1 mutants when auxinated to produce left-handed movements. Investigating auxin transport in twisting petioles based on the DR5-marker suggested that auxin flow was apical-basal rather than helical. While cortical microtubules of excised wild-type petioles oriented transversely when stimulated with auxin, those of tor1 were largely incapable of reorientation. Together, our results show that tor1 is a tropism mutant and suggest a mechanism in which auxin and microtubules both contribute to helical tropisms.     

Leaf support biomechanics of neotropical understory herbs

Plants in light-limited tropical rainforest understories face an important carbon allocation trade-off: investment of available carbon into photosynthetic tissue should be advantageous, while risk of damage and mortality from falling debris favors investment into nonphotosynthetic structural tissue. We examined the modulus of rupture (σ(max)), Young's modulus of elasticity (E), and flexural stiffness (F) of stems and petioles in 14 monocot species from six families. These biomechanical properties were evaluated with respect to habitat, rates of leaf production, clonality, and growth form. Species with higher E and σ(max), indicating greater resistance per unit area to bending and breaking, respectively, tended to be shade-tolerant, slow growing, and nonclonal. This result is consistent with an increase in carbon allocation to structural tissue in shade-tolerant species at the expense of photosynthetic tissue and growth. Forest- edge species were weaker per unit area (had a lower E), but had higher flexural stiffness due to increases in stem and petiole diameter. While this is inefficient in requiring more carbon per unit of structural support, it may enable forest-edge species to support larger and heavier leaves. Our results emphasize the degree to which biomechanical traits vary with ecological niche and illustrate suites of characteristics associated with different carbon allocation strategies.     

Mechanical and anatomical adaptations in terrestrial and aquatic buttercups to their respective environments

The mechanical adaptations of the stems of four species of Ranunculusto their respective environments were studied by combining tensile,bending and flow-tunnel tests, with anatomical observation. Stems of the two terrestrial species, R. acris and R. repens,had high values for rigidity, EI, because they were stiffenedby large quantities of peripherally placed lignified material.This trend is less evident in R. repens, which had a lower rigidity,though it with stood a higher breaking strain than R. acris.This may adapt R. repens to its creeping habit and help it withstandtrampling. The aquatic R. peltatus and R. fluitans, which live in stilland fast-flowing water, respectively, are both more flexibleand have higher breaking strains, of 0.1–0.15, than terrestrialplants, which may allow them to withstand sudden tugs due toflow. R. peltatus maintains the central lumen, places structuralelements away from the centre, and has a higher rigidity thanR. fluitans, which may allow it to avoid self-shading, and supportitself when the water level falls. The stem of R. fluitans shows adaptations for withstanding dragfrom fast-flowing water. The stem has a low rigidity which allowsit to minimize inertial drag forces by aligning itself parallelto the direction of the local flow. However, the rigidity (andthe second moment of area, I) does not appear to be minimized.This may allow the plant to avoid drag due to flag-like fluttering.A weak region of the stem near the base may act as a ‘mechanicalfuse’ which protects the root system by allowing seasonalgrowth to be lost. Key words: Ranunculus, mechanics, flow, anatomy     

Effect of anisotropic bending rigidity and finite twisting rigidity on statistical properties of DNA model filaments

The persistence length and effective long-range bending rigidity are derived for a discrete model of an anisotropically bending filament and shown to be independent of the torsional rigidity. The twisting persistence length is found to be independent of the anisotropic bending rigidity. Other statistical properties are briefly discussed, including the dependence of tangent vector projections on contour length. The dependence of a tensor contraction on contour length is derived for an isotropically bending filament with no equilibrium twist.