Axial Second Moment Of Area Research Articles

Twist-to-Bend Ratios and Safety Factors of Petioles Having Various Geometries, Sizes and Shapes.

From a mechanical viewpoint, petioles of foliage leaves are subject to contradictory mechanical requirements. High flexural rigidity guarantees support of the lamina and low torsional rigidity ensures streamlining of the leaves in wind. This mechanical trade-off between flexural and torsional rigidity is described by the twist-to-bend ratio. The safety factor describes the maximum load capacity. We selected four herbaceous species with different body plans (monocotyledonous, dicotyledonous) and spatial configurations of petiole and lamina (2-dimensional, 3-dimensional) and carried out morphological-anatomical studies, two-point bending tests and torsional tests on the petioles to analyze the influence of geometry, size and shape on their twist-to-bend ratio and safety factor. The monocotyledons studied had significantly higher twist-to-bend ratios (23.7 and 39.2) than the dicotyledons (11.5 and 13.3). High twist-to-bend ratios can be geometry-based, which is true for the U-profile of Hosta x tardiana with a ratio of axial second moment of area to torsion constant of over 1.0. High twist-to-bend ratios can also be material-based, as found for the petioles of Caladium bicolor with a ratio of bending elastic modulus and torsional modulus of 64. The safety factors range between 1.7 and 2.9, meaning that each petiole can support about double to triple the leaf’s weight.

Drooping of Gerbera flower heads: mechanical and structural studies of a well-known phenomenon.

Gerbera, one of the most loved cut flowers, is (in)famous for the drooping of its flower heads under dehydration. This effect has been quantified by analysing both fully turgescent and wilting peduncles of Gerbera jamesonii 'Nuance'. Wilting peduncles display pronounced bending in the region directly below the inflorescence after 24 h of dehydration, while the rest of the peduncle remains upright. Using anatomical measurements and three-point bending tests, we have analysed whether this phenomenon is caused by mechanical and/or geometrical alterations. We have found that both the flexural rigidity and the axial second moment of area are significantly decreased in the apical part of wilting peduncles, whereas the bending elastic modulus shows no significant change. Moreover, cross-sections of wilting peduncles ovalize significantly more than those of turgescent peduncles and exhibit considerable shrinkage of the parenchyma, taking up the majority of the cross-sectional area. Generally, the drooping of wilting Gerbera flowers can be regarded as a temporary instability of a rod-shaped cellular solid caused by anatomical differences (tissue arrangement, existence or the absence of a pith cavity) and geometrical changes (the decrease of axial second moment of area, cross-sectional ovalization, shrinkage of tissues) between the apical and basal regions of their peduncles.

Effect of drought stress on bending stiffness in petioles of Caladium bicolor (Araceae)

Cell turgor plays an important role in the mechanical stability of herbaceous plants. This study on petioles of Caladium bicolor 'Candyland' analyzes the correlation between flexural rigidity and cell turgor. The results offer new insights into the underlying form-structure-function relationship and the dependency of mechanical properties from water availability. Bending modulus E of petioles is calculated from two-point bending tests, taking into account the tapering mode. The corresponding turgor of parenchyma cells during wilting is investigated by pressure probe tests. Wilting petioles show highly significant lower values of E than petioles with sufficient water supply. These differences are also found when comparing well-watered petioles to drought-stressed petioles having parenchyma turgor values in the same range. These results indicate an additional mechanical system sensitive to drought stress. On the basis of analyses of the contribution of different petiolar tissues toward the axial second moment of area and by using experimentally determined and literature values of E for the different tissues, we were able to (1) recalculate E of the intact petiole and to compare it with experimental data and (2) quantitatively estimate the importance of the different tissues for flexural rigidity and E of the petiole. Our results show that the decrease in flexural rigidity of petioles of Caladium bicolor 'Candyland' during wilting results from (1) a water-loss-induced decrease in mechanical efficiency of collenchyma fibers and (2) turgor loss of parenchyma cells.

Development and Growth Form of the Neotropical Liana Croton nuntians: The Effect of Light and Mode of Attachment on the Biomechanics of the Stem

The neotropical liana Croton nuntians (Euphorbiaceae) can occur in a variety of different growth habits. Juvenile freestanding plants are mechanically stable without support and resemble morphologically young trees or shrubs, whereas adult plants are climbers. Ontogenetic variation of bending and torsion properties of different growth phases are analyzed by measurements of flexural stiffness, structural bending modulus, torsional stiffness and structural torsional modulus. Mechanical and anatomical data show two fundamentally different patterns for juvenile freestanding and adult climbing plants. In freestanding plants, mechanical properties and the contribution of cortex, wood, and pith to the stem cross-section vary only little during ontogeny as is typical for semi-self-supporting plants. In contrast, climbing plants become significantly more flexible during ontogeny, as is characteristic for lianas. This is accompanied by a transition to the formation of a less dense wood type with large diameter vessels and an increasing contribution of flexible tissues (less dense wood and cortex) to the cross-sectional area and the axial second moment of area of the stems. Depending on the environmental conditions, freestanding plants can differ considerably in their appearance due to differences in branching system or stem taper. Therefore the influence of light quantity, measured as percentage of canopy opening, on the mechanical properties and the stem anatomy was tested. Freestanding plants grown with strong shade are significantly more stiff in bending compared with plants grown with a moderate light environment.

Stem mechanics, functional anatomy and ecology ofAlnus viridisandAlnus glutinosa

AbstractMechanical properties, the internal stem structure and the wood anatomy of stems ofAlnusglutinosafrom the Black Forest and of two populations ofAlnusviridisfrom the Black Forest and from the Swiss Alps are compared as to their ecological significance. The mechanical bending properties are characterised by the parameters “flexural stiffness” (Estruct·Iax) and “structural modulus of elasticity” (Estruct). The structural modulus of elasticity increases significantly for all three populations during the first years of growth and becomes approximately constant in stems being five years old and older. These ontogenetic variations of the bending mechanical properties are affected by changes of the internal stem structure, i.e. by changes of the amount and distribution of plant tissues with different mechanical properties within the cross section, described by the axial second moment of area (Iax) which weights the position of the tissues by their distance to the neutral plane. The contribution of bending‐stiff wood towards the axial second moment of area increases during the first years of growth whereas the contribution of the more bending flexible bark decreases. With further increasing age, i.e. in five years old and older stems, the distribution of tissues remains approximately constant as do the mechanical properties. Comparing the different species or populations, respectively, the interspecific difference betweenA.glutinosaandA.viridisfrom the same regional origin (Black Forest) are much smaller than the intraspecific differences between the twoA.viridispopulations from the Black Forest and from the Alps. Significant differences between the two species from the Black Forest are only found in the structural modulus of elasticity of 5–7 year old stems. In contrary, in most ontogenetic stages significant differences are found in all tested mechanical and anatomical parameters betweenA.viridisfrom the Black Forest andA.viridisfrom the Alps. The differences may be due to the different environmental conditions in high elevated sites in the Alps compared to lower sites in the Black Forest. However, the observed differences may also originate from a genetic differentiation into two different ecotypes inA. viridis.

The Growth Form of Croton pullei (Euphorbiaceae) ‐ Functional Morphology and Biomechanics of a Neotropical Liana

Abstract: Croton pullei (Euphorbiaceae) is a woody climber of the lowland rainforest in French Guyana and Surinam. During ontogeny, a shift from a juvenile free‐standing growth phase to an older supported growth phase is observed. The following biomechanical parameters were studied: structural Young's modulus, structural torsional modulus, flexural stiffness and bend to twist ratios. Changes in anatomical development were also analysed for different stages of development of C. pullei which differ significantly in their mechanical properties. Free‐standing plants show a nearly constant structural Young's modulus and structural torsional modulus during ontogeny, with flexural stiffness increasing proportionally with the axial second moment of area. These patterns are typical for ?semi‐self‐supporting plants”. In contrast, supported plants show a significant decrease in structural Young's modulus in older stem parts, as well as a decrease in structural torsional modulus. Due to the decrease in structural Young's modulus, flexural stiffness does not increase proportionally with the axial second moment of area. These patterns are typical for non‐self‐supporting lianas. In all supported plants, a sudden transition occurs from early dense wood to a wood type with a much higher proportion of large diameter vessels. In contrast, only the dense wood type is present in all tested free‐standing plants. The data are compared with results from other climbing species of the same study area and discussed with reference to observed features characterizing the growth form and life history of C. pullei.

The mechanics of Norway spruce [ Piceaabies (L.) Karst]: mechanical properties of standing trees from different thinning regimes

In the first part, the authors discuss the mechanical stability of dominant and suppressed Norway spruce trees ( Picea abies) grown under differing thinning regimes with a focus on the stem shape and the mechanical properties of the standing tree. In the second part, they concentrate on the variation of the modulus of elasticity of green wood (MOE fresh) and the density within the stem to analyse the effect of the underlying patterns in variation of these wood properties on the mechanics of the entire stem. At the standing-tree level, the investigations comprised a morphological description and static bending tests on the trees to measure flexural stiffness and structural Young’s modulus of the stem. Discs sampled from different stem heights were examined to distinguish between different tissues (cortex, wood, pith) and wood types (adult, adolescent, juvenile), and to measure quantitatively their contribution to the axial second moment of area of the stem. An analysis of the variation of density and MOE fresh within the stem was carried out on small wood samples cut out of stem sections taken from the same height as the stem discs. The flexural stiffness of the trees is mainly influenced by stem radius, resulting in a higher flexural stiffness for thicker trees. Independently from the stem dimensions, the structural Young’s modulus was slightly higher for the suppressed trees within each site. In an individual tree the structural Young’s modulus decreases with increasing stem height. The cross-sectional analyses of wood type proportions and their characterisation by density and MOE fresh showed that the distribution of adult, adolescent and juvenile wood within the stem affects the structural Young’s modulus and the flexural stiffness of the stem. Finally, a method is presented that allows a re-calculation of the mechanical bending properties of the entire standing trees using MOE and axial second moments of area of the different wood types. Our results suggest that further modelling of mechanical behaviour of trees might be possible taking into account the composite nature of conifer stems.

Biomechanical analysis ofPitus dayi: Early seed plant vegetative morphology and its implications on growth habit

A biomechanical method to distinguish self-supporting and non self-supporting growth habits is applied to exceptionally preserved “twigs” ofPitus dayi Gordon. The analysis investigates whether these isolated stem segments are consistent with a self-supporting tree-like habit as suggested by the stumps, trunks and branches of the genusPitus Witham preserved more commonly in the fossil record. Because of difficulties in accurately identifying certain fossil tissues, three centrisymmetrical models were constructed to test a range of possible tissue combinations over five ontogenetic stages. The results suggest a self-supporting habit with trends in mechanical parameters during ontogeny similar to those of extant, self-supporting plants. Less explicitly constrained to the analysis of habit, the investigation also examines the structural significance of specific tissues during ontogeny as observed from contributions of individual tissues to cross-sectional area, axial second moment of area and flexural stiffness.Pitus dayl produced a physiologically “cheap” primary cauline cortex which was rapidly replaced by the development of a rhytidome. A mechanically significant, cauline hypoderm comprising thickwalled sclerenchymatous tissue is absent. This arrangement differs from other tested Palaeozoic pteridosperms interpreted as semi-self-supporting such asLyginopteris oldhamia andCalamopitys sp. in which the primary cortex is mechanically significant and secondary growth of the wood does not reach mechanically significant thresholds within the primary body.

A biomechanical method to distinguish between self-supporting and non self-supporting fossil plants

A method is described that allows a (semi-)quantitative distinction between extinct self-supporting and non self-supporting plants (i.e. lianas and semi-self-supporting plants). This method is derived from experimental findings in extant plants. In the extant lianas: Aristolochia macrophylla, Clematis vitalba and Fallopia aubertii experimentally derived data of flexural stiffness for older ontogenetic stages are clearly below a neutral line, that is calculated by using as a slope the mean Young's modulus of the youngest ontogenetic stage. Plants of this growth habit show a decrease in Young's modulus during ontogeny. Conversely, the stems of self-supporting woody plants, Syringa vulgaris, Alnus viridis, Alnus glutinosa, show an increase in Young's modulus during ontogeny and for older ontogenetic stages the values of flexural stiffness are clearly above the neutral line. These changes in stem mechanics are caused by changes in anatomy. In lianas the contribution of the peripheral strengthening tissues towards axial second moment of area drastically decreases during ontogeny, whereas in self-supporting taxa, the contribution of the secondary wood considerably increases. These findings make it possible to recalculate the mechanical properties for fossil taxa from which permineralized stems of different ontogenetic stages are preserved. By calculating the axial second moment of area of the different stem tissues and by estimating the Young's modulus of each tissue type with data from similar living tissues, the flexural stiffness and Young's modulus for the fossil stems can be (semi-)quantitatively estimated. For the arborescent lycopsid Diaphorodendron vasculare, the results are very similar to the experimental findings of extant self-supporting trees. In the seed fern Lyginopteris oldhamia, on the contrary, the results exclude a self-supporting growth habit. This taxon was probably a semi-self-supporting plant having either a more lianescent or scrambling growth mode or a habit characterized by individuals providing mutual support in dense monotypic stands. By using a formula that calculates the critical global buckling length for different tapering modes and taking into consideration the stem's weight and any additional load acting at the stem's tip, the theoretical maximum height of trees for different safety factors against global buckling can be estimated. For D. vasculare the calculation points to a maximum height of about 15 m for plants with a basal stem radius of 0.15 m. This corroborates very well, the maximum height inferred by DiMichele (1981).

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

AbstractBiophysical 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.