have found that potato cells can respond very quickly to applied stress. No doubt experienced experimental botanists know of many more examples.

## TISSUES

At the tissue level we are concerned with assemblages of cells which may or may not have an internal pressure above ambient. The mechanics of nonturgor systems are probably moderately well understood (Gibson and Ashby, 1988) but turgor systems continue to present problems (Jeronimidis, this conference) since the response of the turgid cell to external stress depends on its surroundings. A nice example of this is given in the anisotropy of apple flesh, where radial compression (perpendicular to the columnar arrangement of the parenchyma cells) leads to failure in a band across the specimen, while compression in the orthogonal direction leads to shear failure (Khan, 1988). The difference in response is due to the space each cell has available in which to deform. The mechanical properties of leaves and stems have been modeled as a simple sandwich or column of foam with a continuous membrane (epidermis) on the outside (Gibson et al., 1988). This approach is of only limited success since it gives the foam bulk properties which are not derived from the properties of the individual cells (size, wall thickness) and does not take account of turgor. The problems were also sidestepped by Vincent and Jeronimidis (1991), who, trying to model the flowering stem of the dandelion, *Taraxacum officinale*, found they could not cope with the turgor system as an assemblage of cells, but could use the analogy of a metal expanding as it is heated. The gradient in the amount of cell wall material across the stem (1:20, with the largest amount externally) makes the stem equivalent to a bimetallic strip structure which can be solved mathematically.

## STRUCTURES AND STEMS

For many years the perceived wisdom as to the state of stress in an herbaceous stem has been, broadly, that the inner tissues are held in compression and the outer ones in tension. For instance, Strasburger et al. (1903) say that "the rigidity of parenchymatous tissue although to a large extent dependent upon the tension arising from the turgidity of its individual cells, it nevertheless considerably enhanced by the opposing pressure between the inner and outer tissue systems. . . The tension thus arising from the mutual resistance of different tissue systems acts upon the various plant organs like the turgidity of the single cells, and keeps them firm and rigid." The confusion here is due to inadequate distinction between material and structure. It is certainly true that the way in which the cells are packed into the herbaceous stem leads to mutual constraint, which can be released only when the structure of the stem is destroyed, but the
fact that a strip of stem will curl when the constituent cells are turgid is most readily explained by pointing out that the walls of the inner cells are much thinner and therefore will extend farther under the same or similar internal turgor pressure. Thus the inner cells find themselves on the outside of the curl. That this is due simply to the interaction of turgor and the cell walls is shown by strips in which the cells are plasmolysed and have no positive turgor. These strips are straight, showing that under these conditions the strain in the cell walls is constant across the stem (in fact under these conditions the epidermal layer is wrinkled, again suggesting that the cell walls are contracting when turgor is lost). Thus in the intact turgid stem, the cell walls are all at the same state of strain but obviously, therefore, not at the same state of stress since they are of different thicknesses. Therefore a turgor-stiffened system exists in some sort of mechanical equilibrium. If the constraints of the structure are changed (e.g., by cutting it in half and destroying the continuity of the epidermis), then the stresses and strains will be redistributed within the structure and it will assume a new equilibrium shape. Clearly turgor systems are far from understood; we need much more clear thinking if we are to bridge the gap between physiology and mechanical properties.

The decision by the plant to shift from turgor to lignification as the main mechanism for support must involve a number of factors. Although all plants above about 30 cm high seem to have some lignification in the stem, they do not necessarily increase in stiffness. A number of student projects at Reading have shown that in the umbellifers of the genus *Heracleum*, whose stems are made of a number of similar internodes, it is impossible to tell simply by measuring the stiffness of the stem whether the main support is lignification or turgor. Despite the fact that in the one lignified herb which we have analyzed so far (*Nicotiana*) the ring of lignified tissue clearly dominates the stiffening effects of the various tissues, the extra stiffness which lignification gives to woody cells is not required for structural stiffness of the stem. The thought arises that the plant is minimizing the total energy used to maintain its shape. If the plant is to last for a shorter time, low-level expenditure of energy on maintaining turgor is cheaper; if it is to last for a longer time, then it might be worth a short burst at a comparatively high energy level to lignify, and then reduce the energy necessary to maintain turgor in those tissues. Is this what happens?

## EVOLUTION

The morphology of the stems of fossil plants can be used to predict their mechanical properties (Speck *et al.*, 1990). Using the flowering stem of *Taraxacum* as the modern type, Vincent and Jeronimidis (1991) assessed the likelihood that *Rhynia gwynn-vaughni* was truly a land plant. In section, the gradients of cell size and cell wall thickness between the center and the periphery
of the stem are much less pronounced and much more variable in *Rhynia*, which has cell walls which are slightly thicker in the outer layers of cells (3 µm) than the inner (1 µm), but suffers by having a distinct zone (the inner cortex) made of one or more layers of large cells just beneath the outer layer (Edwards, 1986). The calculated distribution of stem stress shows that the maximum stem stresses ascribable to turgor are less than in *Taraxacum*. In *Rhynia* the prestress in the center of the stem is much higher than in *Taraxacum*, which seems rather a bad design since prestress in this position can contribute so little to bending stiffness. Rough calculation shows that the prestress in the dandelion stem is just about sufficient to resist side loads due to wind. With its lower prestress *Rhynia* could not support anything at the tip of its axis which was likely to generate drag forces. It is tempting to suggest that *Rhynia* was actually partly submerged, like many plants found today growing at the edge of lakes and ponds. It is unlikely that they grew totally or continually partially submerged since they have stomata along the entire length of the stem (Kidston and Lang, 1917). However, they are always found densely packed in the chert with their stems laid parallel as if they were growing in clumps. So they might have needed to grow in dense stands simply to make up for the shortcomings in the design of their stem. Since there are so many specimens of *Rhynia* in such good preservation, it would be very interesting to look for evidence of modes of failure of the axis.

## PLANTS AND ANIMALS

The mechanical properties of plants as food are of importance for the animals which feed on plants (Vincent, 1991). The mechanical design of grass has been shown to dictate the way in which animals feed on it (Vincent, 1982, 1983; Wright, 1992) and the ease with which they can extract the nutrients (Lees, 1984; Lees et al., 1981, 1982). This reduces to problems in fracture mechanics at all levels of morphology, ultimately at the cellular level. Very little is known about the mechanical interaction of teeth with plants, and whether it is better for plant cells to resist being broken open by being tough and strong (which is expensive in terms of dry matter in the cell walls) (Choong et al., 1992) or to break open in a brittle ("crisp") manner and hope that the path of the fracture will not branch, thus opening the minimum number of cells and restricting the availability of nutrient to the animal (Vincent and Sibbing, 1991). There is some indication that plants make themselves tough specifically to avoid being eaten (Lucas et al., 1991; Lucas and Corlett, 1991; Wright, 1992).

## DIRECTED FRACTURE

The fracture mechanics of abscission and dehiscence have not been investigated, although a moderate amount is known about the morphology and physiology. There is a fruitful area of work here, which would yield much benefit.
Dehiscence in fruits is, mechanically, of two kinds. The capsule may simply split open to expose the seeds or it may dehisce explosively to broadcast the seeds. In the former the expectation is that the line of dehiscence will be a line of weakness. In the seed pod of rape the middle lamella is dissolved along the joint between the valves and the central diaphragm which supports the seeds. It remains joined in a few places so that a sharp knock will rupture it and send the seeds flying. The energies and forces involved in this mechanism are unknown.

The mechanics of explosive dehiscence have been investigated in the soybean (Weeks *et al.*, 1975). The strength of the dehiscence layer and the force available from the pod were measured by pulling the pod open and then forcing the pod valves together. The force needed to close the pod was assumed to be that available for dehiscence and is directly related to the thickness of the pod (i.e., the elastic prestrain is probably constant and independent of thickness). The force increases greatly as the pod dries out, showing that as the stiffness of the pod wall increases with reduced water content, a constant prestrain results in higher prestress. The strength of the dehiscence layer increases as the water content falls and then becomes more or less constant but very variable, which is typical of brittle fracture, which can be very sensitive to cracks and imperfections. If one wanted to breed a pod which would not dehisce before harvesting, it might be easier to select for a thinner pod wall for a given size of pod, rather than a stronger dehiscence layer (Weeks *et al.*, 1975).

## PLANT MATERIALS

The use of plants as "renewable" sources of materials is gaining political and financial credibility as the concepts of ecology and "Spaceship World" finally filter through to the politicians. The most credible additions to current materials are fibers (from flax, hemp, and such traditional sources as *Urtica*), which will find uses in fiberboard, cementitious and phenolic resin composites, and paper, replacing current wood sources (though this will take up to 50 years, even if the benefits are immediately apparent). Eucalypts are the current best source of quality paper fiber, and there is work in progress to reduce the lignification of these trees transgenically, thus reducing both energy requirements and pollution. If plant fibers are going to be more widely used in materials, then it is important not only to characterize them properly but to persuade those who use them to use the tools of materials science to analyze the properties of the materials into which the fibers are incorporated. For instance, the current units used to describe the mechanical properties of paper are a mess, mixing forces with weights per unit area. They are little better than quality control tests and of little or no use in forecasting the results of specific changes in the components
of the material. Perhaps we should think of some way of ensuring that the
industrial and agricultural aspects of plant mechanics are properly researched.

BIOMIMICRY

If plants are such successful machines, why should we not learn from them?
Relieved of the limitations of processing at ambient temperatures, using readily
available materials, using the weakest of bonds compatible with the necessary
stiffness and strength, we should be able to do better than nature. The trick is
understanding the mechanisms and materials which we are trying to copy. A
successful fiberglass material based on the fracture toughening mechanism of
wood is 50 times tougher, weight for weight, than any other man-made material
(Gordon and Jeronimidis, 1980). Cellular materials offer significant material
efficiencies, and the possibilities of turgor-driven systems have not yet been
explored. The blueprints exist in nature for a variety of hydraulic control and
actuation systems and for an infinite variety of buildings and structures. As
always, the trick is discovering which parts of nature's design to steal and which
bits to leave alone until we are clever enough to understand them.

REFERENCES

Choong, M. F., Lucas, P. W., Ong, J. S. Y., Pereira, B., Tan, H. T. W., and Turner, I. M. (1992). Leaf fracture toughness and sclerophylly, their correlations and ecological implications. *New Phytol.*, **121**:597-610.

Edwards, D. S. (1986). *Aglaophyton major*, a non-vascular land-plant from the Devonian Rhynie chert. *Bot. J. Linn. Soc.* **93**:173-204.

Gibson, L. J., and Ashby, M. F. (1988). *Cellular Solids, Structure and Properties*, Pergamon, Oxford.

Gordon, J. E. and Jeronimidis, G. (1980). Composites with high work of fracture. *Phil. Trans. R. Soc. Lond. A 294*:545-550.

Green, P. B. (1991). Morphogenesis. In Steward, F. C., and Bidwell, R. G. S. (eds.), *Plant Physiology, a Treatise, Vol. X. Growth and Development*, Academic Press, New York, pp. 1-64.

Hush, J. M., and Overall, R. L. (1991). Electrical and mechanical fields orient cortical microtubules in higher plant tissues. *Cell Biol. Int. Rep.* **15**:551-560.

Hush, J. M., Hawes, C. R., and Overall, R. L. (1990). Interphase microtubule re-orientation predicts a new cell polarity in wounded pea roots. *J. Cell. Sci.* **96**:47-61.

Jaffe, M. J., Telewski, F. W., and Cooke, P. W. (1984). Thigmomorphogenesis: On the mechanical properties of mechanically perturbed bean plants. *Physiol. Plant.* **62**:73-78.

Khan, A. A. (1988). *Mechanical and Fracture Properties of Fruit and Vegetables*, Ph.D. thesis, University of Reading, Reading.

Kidston, R., and Lang, W. H. (1917). On Old Red Sandstone plants showing structure, from the Rhynie chert bed, Aberdeenshire. *I. Rhynia gwynne-vaughni* Kidston and Lang. *Trans. R. Soc. Edinb.* **51**:761-784.

Lees, G. L. (1984). Cuticle and cell wall thickness: Relation to mechanical strength of whole leaves and isolated cells from some forage legumes. *Crop Sci.* **24**:1077-1081.

Lees, G. L., Howarth, R. E., Goplen, B. P., and Fesser, A. C. (1981). Mechanical disruption of leaf tissues and cells in some bloat-causing and bloat-safe forage legumes. *Crop Sci.* **21**:444-448.
Lees, F. L., Howarth, R. E., and Goplen, B. P. (1982). Morphological characteristics of leaves from some forage legumes: Relation to digestion and mechanical strength. *Can. J. Bot.* **60**:2126-2132.

Lucas, P. W., and Corlett, R. T. (1991). Quantitative aspects of the relationship between dentitions and diets. In Vincent, J. F. V., and Lillford, P. J. (eds). *Feeding and the Texture of Food*, Soc. Exp. Biol. Semin. Ser. 44, Cambridge University Press, Cambridge.

Lucas, P. W., Choong, M. F., Tan, H. T. W., Turner, I. M., and Berrick, A. J. (1991). The fracture toughness of the leaf of the dicotyledon *Calophyllum inophyllum* L (Guttiferae). *Phil. Trans. R. Soc. Lond.* B **334**:95-106.

Marvin, J. W. (1939). The shape of compressed lead shot and its relation to cell shape. *Am. J. Bot.* **26**:280-287.

Speck, T., Spatz, H. C., and Vogellehner, D. (1990). Contributions to the biomechanics of plants. I. Stabilities of plant stems with strengthening elements of different cross-sections against weight and wind forces. *Bot. Acta* **103**:111-122.

Steucek, G. L., Selker, J. L., and Reif, W. E. (1992). Architecture of the plant shoot apex. *Natural Structures*, Proc. SFB230, Stuttgart, pp. 151-157.

Strasburger, E., Noll, F., Schenck, H., and Schimper, A. F. W. (1903). *A Text-book of Botany* (translated from the German by H. C. Porter, revised by W. H. Lang), Macmillan, London, pp. 166-167.

Thompson, d'A. W. (1917). *On Growth and Form*, Cambridge University Press, Cambridge.

Vincent, J. F. V. (1982). The mechanical design of grass. *J. Mater. Sci.* **17**:856-860.

Vincent, J. F. V. (1983). The influence of water content on the stiffness and fracture properties of grass leaves. *Grass For. Sci.* **38**:107-114.

Vincent, J. F. V. (1991). Plants as food. In Vincent, J. F. V., and Lillford, P. J. (eds)., *Feeding and the Texture of Food*, Soc. Exp. Biol. Semin. Ser. 44, Cambridge University Press, Cambridge.

Vincent, J. F. V., and Jeronimidis, G. (1991). The mechanical design of fossil plants. In Rayner, J. M. V., and Wootton, R. (eds.), *Biomechanics in Evolution*, Soc. Exp. Biol. Semin. Ser. 36, Cambridge University Press, Cambridge.

Vincent, J. F. V., and Sibbing, F. N. (1991). How the grass carp (*Ctenopharyngodon idella*) chooses and chews its good—Some clues. *J. Zool.* **226**:435-444.

Weeks, S. A., Wolford, J. C., and Kleis, R. W. (1975). A tensile testing method for determining the tendency of soybean pods to dehisce. *Trans. A.S.A.E.* **18**:471-474.

Wright, W. (1992). *The Fracture Properties of Grasses and Their Relevance to Feeding in Herbivores*, Ph.D. thesis, University of Reading, Reading.
# On the Research History of Plant Biomechanics<fnref n="1" />

Werner Nachtigall<fnref n="2" />

The research history of plant biomechanics between the period of first analyses and the late thirties of this century is elucidated. Special consideration is given to early ideas, Schwendener and contemporaries, and Rasdorsky and contemporaries. Aspects of bodies of equal resistance, the role of turgescent tissues, stem mechanics and stiffness, dimensional problems, mechanics of "plant buildings," modulus of elasticity, flexibility, and finally, the plant as a compromise construction are discussed. Examples are illustrated by original drawings from the authors cited.

**KEY WORDS:** plant biomechanics; biomechanics; history of botany; research history.

## INTRODUCTION

Investigation of the plant as a biomechanical construction began spectacularly: Simon Schwendener's treatise on *Das mechanische Prinzip im anatomischen Bau der Monocotyledonen* (The Mechanical Principles of the Anatomy of Monocotyledons) (Fig. 1) appeared in 1874. This was the beginning of a new field of research and one it was to influence for the next 25 years. Diverse approaches and notes had been made long before Schwendener, but they were sporadic and unsystematic. He was followed by Rasdorsky, with a fundamental and widespread handling of this subject between 1911 and 1937. Various other morphological and mechanical approaches were made during this period, but they stand more or less apart. The question of automatic regulation under mechanical loading in growing plants was soon taken up.

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²Department of Zoology, Division of Technical Biology and Bionics, University of Saarland, 66041 Saarbrücken, Federal Republic of Germany.</footnote>
<figure><img src="image_169.png" /><figcaption>Fig. 1. Title page from Schwendener's 1874 treatise.</figcaption></figure>

I divide this historical compilation into three sections: (1) Early Ideas, (2) Schwendener and Contemporaries, and (3) Rasdorsky and Contemporaries. Quotations are used to enhance descriptions wherever they seemed appropriate, and for the purpose of continuity some modern concepts have been added to the classical approaches. The period under consideration here begins with the first approaches and continues into the first three decades of this century.

## EARLY IDEAS

### "Plant Skeletons" and Similitude Mechanics

Galilei [1638 (1890)] used the grass stalk to illustrate his ideas on increased bending resistance due to peripheral accumulation of material. He formulated the first ideas on similitude mechanics, comparing a small oak to a large one.
The physicist Hooke (creator of Hooke's law) advised the botanist Grew (1682) in his book on the anatomy of plants. In this case the stiff hollow cylinder with peripheral supporting cords was correctly recognized as an important aspect of a plant's lightweight structure, as opposed to the flexible roots, which have centrally massed stiffening elements. Contemporary quotations, for example, from van Marum (1773) and Senebier (1800), show that these principally correct ideas on construction mechanics were already widespread. In the eighteenth century a discussion began on whether the supporting structures of plants were comparable to the skeletons of animals. Du Monceau (1758) assumed that there is a functional similarity between the wood structure of a tree and the skeleton of an animal. Bonnet (1782) even placed herbaceous plants with respect to their static conception analogous to insects, because both possess external reinforcement. Humboldt (1794), on the other hand, contested that plants possess anything "which may be similar to bones." Although he hindered the correct line of pursuit due to his scientific authority, the idea of a plant possessing a kind of skeleton gradually asserted itself.

## The Role of Turgescence

It is remarkable that, already during these early stages, not just the special strengthening tissues were said to have a skeletal character. In fact, present-day discussions on the significance of internal pressure and the seemingly novel concept "Der Pneu-Bauprinzip des Lebens" [The Pneu-Life's Principle of Construction (Otto, 1978)] just take up the classical ideas again. Halfway through the nineteenth century the significance of turgor as global stiffening in plants was already generally accepted. Thus, according to Sachs (1868), the rigidity or flexibility of a plant is determined by the "global result of the co-ordination between the variable growth in cell walls, their flexibility and elasticity and the turgor." This statement—though using other words—is more or less identical in meaning to the modern definition of a pneu: a flexible membrane enclosing a substance standing under internal pressure.

The combination of turgescent and hard reinforcing materials was recognized more than 100 years ago. So that Schwendener (1874) noted "daß die parenchymatischen Gewebe den Widerstand bedeutend erhöhen" (that parenchyma tissue greatly increases the resistance) during bending in a plant stem attached horizontally at one end and loaded at the other (Fig. 2). He also noted that the sclerenchyma fibers had to be kept apart. This was accomplished by "pressure pads" of highly pneumatic parenchyma lying in between them: "... so leuchtet ein, daß dieselben im Zustand der Tureszenz (spaced out by Schwendener himself!) einen bedeutenden Widerstand gegen Druck darbieten..." (thus it is obvious that they, with their condition of turgidity, offer an important resistance to pressure...; see Studies on Bodies with Equal Resistance and the Role of Turgescent Tissue, below).
<figure><img src="image_170.png" /><figcaption>Fig. 2. Schwendener's (1874) deliberation on the effect of hydraulic stiffening. Top: Unloaded hollow cylinder, filled with compressed air. Bottom: Same as A, with load P(Q hydraulic axial force).</figcaption></figure>

## SCHWENDENER AND CONTEMPORARIES

During the second half of the nineteenth century, the idea that the hollow cylindrical structure of a plant stem or stalk is a principle of lightweight construction, constituting material saving, finally asserted itself. Biologists schooled in physics and even physicists themselves were becoming increasingly interested in the construction of plants. Preliminary ideas on the building of strengthening tissues as a possible reactions to mechanical influences were beginning to form. The philosopher and engineer Spencer (1863/1864) assumed that the “mechanical stress itself” caused the formation of “strong bonds” with which plants “can resist the stresses” (quoted by Rasdorsky 1937). Culmann (1866), the founder of graphical statics, used this very successful method of applying the architecture of bone (the stress trajectories of the femur head) on plants, but apparently he interpreted the growth rings falsely. It was left to Schwendener (1874), a biologist strongly interested in constructional statics, to carry out a comprehensive study (a completely new departure at that time) of “the mechanical principle of anatomical construction” in plants and to raise it to the rank of an independent botanical discipline. Apart from his classical 1874 paper, Schwendener published several papers on plant biomechanics, for example, on elasticity (1878), reinforced plant parts (1882); and microbotany (1887).
## Schwendener's World of Mechanical Concepts

In 1874 Schwendener, stimulated by the use of steel girders in building bridges, published a comprehensive study of statically important structures, in particular, in stalks and blades, but also in leaves and roots of monocotyledons (Fig. 3), and compared them to the mechanical principles of construction in dicotyledons, mosses, ferns, lycopods, and horsetails. First he described "specific mechanical cells" and reported on determining the cross-sectional areas as well as flexibility tests to measure the "load-carrying capacity" (weight per unit cross-sectional area at the breaking limit) and the modulus of elasticity of the inner bark fiber. Xylem, parenchyma, and collenchyma are generally ignored. He assumed (erroneously) that the fibrous strips were I-beams (Fig. 4). [Just how strongly this principle influenced ideas at that time has been shown by studies to demonstrate such I-beams even in lichens (Zukal, 1895) and mosses (Istvanffy, 1896).] However, he interpreted correctly the influence of tensile and compressive stress. He recognized the importance of peripheral accumulation and fusion into rings of the mechanical elements, but also the danger of buckling with too much peripheral emphasis, i.e., a supporting cylinder with too thin walls. He suggested a wall thickness of one-seventh to one-eighth of the diameter as optimal.

## Characterization of Monocotyledons

The blades or stems of monocotyledons were grouped by Schwendener (1874) according to the position and degree of fusion of the fibrous elements. He created no fewer than seven systems with a total of 20 characteristics such as simple rings of fibrous ribbing in *Arum maculatum*, hollow fibrous cylinders with vascular bundles leaning against them as in most Poaceae, etc. Today these allocations, illustrated with extremely schematicized lithographs (creating an impression of suggestive simplification), appear excessively and ineffectively differentiated. If this had to be done again today, one would probably select only four or five clearly defined basic types.

## Studies on Bodies with Equal Resistance and the Role of Turgescent Tissues

Stems (and leaves) are often known as bodies of equal resistance, for which model constructions are presented. This idea and, also, the idea that trees are beams of equal resistance have not been proved experimentally or definitely confirmed (Detlefsen 1884a). Furthermore, this was criticized by Meschayeff (1882) after the studies *Über die Anpassungen zum Aufrechterhalten der*
<figure><img src="image_171.png" /><figcaption>Fig. 3. Schwendener's (1874) plate XI, showing cross sections of a *Fimbristylis spadeacea* (1) leaf and of the stalk of *Cyperus eregius* (2), *Fimbristylis spadicea* (3), *Cyperus conglomeratus* (4), *Hypolytrum argenteum* (5), and *Cyperus* sp. (6).</figcaption></figure>

*Pflanzen* (On the Adaptations of Plants to Stay Upright). One hundred years later, it appears that Schwendener's ideas have been confirmed by recent data (e.g., Silk *et al.*, 1982).

Another approach aspect is whether the turgescence of nonsclerenchyma-
Zur Beurteilung dieses Typus in mechanischer Hinsicht mag die approximative Berechnung eines bestimmten Falles, wie er in Fig. 5 dargestellt ist, als Grundlage dienen. Die Figur ist schematisch gehalten, entspricht aber ungefähr den bei *Scirpus caespitosus* vorkommenden Dimensionsverhältnissen.

Wir denken uns auch hier wieder den Durchmesser des peripherischen Kreises auf 1000 Centim. vergrüssert; die Wanddicke AA, sei 0,17 oder ungefähr 1/6 des ganzen Durchmessers. Für die speziellen Daten genügt folgendes Verfahren. Die Querschnittsfläche der Bastbündel, die übrigens nachträglich durch einfache Division reducirt werden kann, lässt sich mittelst directer Messung am Objecte selbst oder an genaueu Abbildungen ungefähr bestimmen; der Abstand derselben resp. ihrer Schwerpunkte von der neutralen Axe wird dagegen am besten auf der schematischen Figur (hinreichend gross ausgeführt) mit dem Zirkel gemessen. Nachstehend das Ergebniss der Messung und das davon abgeleitete Maass der Biegungsmomente; die Querschnittsflächen sind mit F, die Abstände von der neutralen Axe mit D, die Maasse der Biegungsmomente mit W bezeichnet. Die Epidermis wurde wegen ihrer beträchtlichen Widerstandsfähigkeit mit in Rechnung gebracht.

<figure><img src="image_172.png" /><figcaption>Fig. 5.</figcaption></figure>

1) Die Epidermis, 1 Ctm. dick; $F=3140 \square Ctm.$; $W=392$ Million.

2) Die zwei Bastbündel AA $F=9000$ - $D=470$; $W=1988$ -

3) Die zwei Bastbündel A, A, $F=9000$ - $D=350$; $W=1102$ -

4) Die 4 Bündel BBBB $F=18000$ - $D=240$; $W=1036$ -

5) Die 4 Bündel B, B, B, $F=18000$ - $D=180$; $W=583$ -

6) Die 4 kleinen Bündel aaaa $F=4000$ - $D=330$; $W=436$ -
$61140 \square Ctm.$
$5537$ Million.

<caption>Fig. 4. Schwendener's (1874) calculations of "Maasse der Biegemomente."</caption>

tous tissues (parenchyma, etc.), other than keeping the sclerenchyma fibers apart, plays a measurable part in the bending stiffness of the "whole organ," i.e., the blade of grass (cf. Fig. 3). Comparing the measured downward bending $B_{bl}$ of pieces of blade (which were attached to a rod at one end and bent by their own weight) to the downward bending $B_{scl}$ calculated for a "pure sclerenchymatous system" with the same cross-sectional area, the same second moment of area, and the known modulus of elasticity, a proportionality constant $F$ is obtained; $B_{bl} = F \cdot B_{scl}$. Parenchyma has a positive share in the bending stiffness of a blade at $F < 1$, and the smaller $F$ is compared to 1, the greater the share. Values between 0.94 in *Juncus* and 0.34 in *Molinia coerulea* were obtained, but there were also inexplicable values of $F > 1$, so that Schwendener was unable to solve the question definitely.
Schwendener had already published many of his ideas in 21 essays before his classical treatise (1874), summarizing them again in various later works. Holtermann (1909) issued *Schwendener's Vorlesungen über mechanische Probleme der Botanik* (Schwendener's Lectures on the Mechanical Problems in Botany) (Fig. 5). In 1919 his achievements were summarized by Haberlandt, the famous author of *Physiologische Pflanzenanatomie* (six issues between 1884 and 1924) in a "Gedächtnisrede auf Simon Schwendener" (Commemorative Speech on Simon Schwendener). Schwendener died in 1919 at the age of 91; his final resignation should be taken note of by all laboratory-fixed biologists: "...er hatte kein Interesse, kein Vernügung mehr an einer neuen Beobachtung" (...he was no longer interested in, had no more fun in making new observations) and thus "blieben auch die wissenschaftlichen Anregungen aus, die dem Biologen nur im steten innigen Verkehr mit der unergründlich mannigfaltigen Natur zuströmen" (the scientific stimulus, which flows to the biologist only from a regular and intimate contact with nature in its unfathomable diversity, was not present any more).

<figure><img src="image_173.png" /><figcaption>Fig. 5. Holtermann's (1909) copies of some of Schwendener's tables as a basis for statical considerations. Cross sections. (A) *Cladium mariscus*; (B) *Molinia coerulea* (detail); (C) *Armeria elongata*. "Mechanical tissues," dark area.</figcaption></figure>
Stem Mechanics and Stiffness

Detlefsen (1994a), a high-school teacher in Wismar, was first to criticize profoundly Schwendener's ideas. His work "Über die Biegungselastizität von Pflanzenteilen" (On the Bending Elasticity of Parts of Plants) appeared in two parts and is one of the most important publications within this field to have been made during the nineteenth century. Beginning with a clear definition of the modulus of elasticity and with reference to earlier works (e.g., from Wertheim and Chevandier, 1846), he then discussed the correlation between this modulus and the water content. The tensile tests made by the Schwendener school were rejected as methodically deficient. He then continued with the geometrical distortion of a clearly defined "free of stress, homogeneous, and isotropic straight body," stating that the neutral axis joining the centers of gravity of all cross-sectional areas does not necessarily coincide with the geometrical median line, which gives rise to the relation $r = \Theta EM^{-1}$ ($r$ = bending radius, $\Theta$ = second moment of area, $E$ = modulus of elasticity, $M$ = bending moment). Therefore, $M_1:M_2 = (\Theta E)_1:(\Theta E)_2$ is valid for two bodies of differing rigidity but the same curvature, so that one may use the product $\Theta E$ to characterize the rigidity of a body.

For the first time in botanical literature Detlefsen (1884a) clearly proved that rigidity changes with changes in geometry and absolute size: $d_2^4 = 2$ is valid for a solid rod with a cross section of $d_2$ and rigidity twice that of a rod with a cross section of $d_1 = 1$ cm and of the same material. Thus the diameter $d_2$ is $\sqrt[4]{2} \approx 1.189$ cm. The same rigidity is found in a hollow rod with a wall thickness of 0.095 cm and an internal diameter of 1.189 cm, weighing only 0.414 times as much, etc.

Long before Rasdorsky approached the bending spring concept (1930), Detlefsen (1884) went a step further than Schwendener with the idea of bending elasticity as a constructional goal: "It only requires a little thought to realise that the mechanical efficiency of elastic parts of plants more frequently results from the fact that with less important rigidity they are capable of withstanding considerable bending without damage, rather than preventing any bending from the start due to the type and arrangement of the materials they are made of." These aspects were enhanced by the author with his own calculations (Fig. 6) and measuring techniques and, also, with regard to the different water contents. He concluded his paper with two graphs which show the (nonlinear) decrease in the bending moment (during bending without remaining distortion) of Heilianthus tuberosus and Urtica dioica from the base to the tip. Thus the stiffness and concentration of material (defined as dry weight) gradually decrease from the bottom towards the top, which is more or less equivalent to theoretical, round substitutes with the same stiffness (Fig. 7).

Zimmermann (1884), an admirer of Schwendener, attacked Detlefsen's
<figure><img src="image_174.png" /><figcaption>Fig. 6. Detlefsen's (1884a) demonstration of beam theory.</figcaption></figure>

hypotheses angrily and without reason. Detlefsen (1884b) responded to this briefly, but then, unfortunately, he decided to break off his research ("Following this communication, I do not believe that any one will request me to continue with my studies").

## RASDORSKY AND CONTEMPORARIES

The Russian botanist Wladimir Rasdorsky, writing frequently in German and French, published a series of papers between 1911 and 1937 on the statics of plants, which in principle are still valid today. They deal with questions on
<figure><img src="image_175.png" /><figcaption>Fig. 7. Detlefsen's (1884a) measurements of the cross section of an *Urtica dioica* stem and calculation of the cross section of an equivalent body of rotation. (A) graphs (thin line, object; thick line, equivalent body); (B) table.</figcaption></figure>

research history, for example, the mechanical construction of plants, (1928, 1929), dimensional proportions of blades and stalks, spring-like flexural characteristics (1930), and plant reaction to mechanical loading (1937).

## Dimensional Problems

In the nineteenth century, Sachs (1868), for example, stated that the absolute size of a plant, as far as its organization and construction are concerned, cannot be a matter of indifference to it. However, when comparing plant constructions and high buildings, the laws of similarity were overlooked, even in the twentieth century (Welzien 1926). An unreasonable superiority is given to a plant when one thinks that "the technical achievements of the plant's body are unique of their kind and highly perfected" (Fitting, cited by Strasburger, 1923). "The lightest wrought iron post seems clumsy in comparison to the slim soaring cane of the bamboo" (Schwendener, 1874). "Even the Eiffel tower is clumsy and wide compared to most plant construction" said Francé (1919), who introduced the ideas of technical biology to a larger public with his book *Die technischen Leistungen der Pflanze* (Fig. 8). Comparisons of the slenderness ratio $λ = h/d$ ($h$, height; $d$, mean diameter) are made with the inadmissible proportion $h_1:d_1 = h_2:d_2$. On the other hand, Rasdorsky (1928) recalls the
<figure><img src="image_176.png" /><figcaption>Fig. 8. Francé's (1919) simplified illustrations to the "statics and mechanics of plants." (A) Cross section through a grass stalk. (B) Application of I-beams in the mechanical system of plant; cross section of a rush stalk.</figcaption></figure>

Barba-Kick law on proportional resistance (Kick, 1885), an empirical formula mathematically substantiated, within the limits of elasticity, by Steiner (1884; quoted by Kick, 1885). This law means, among other things, that tower-like bodies of different absolute height can be geometrically similar only when the forces change proportionally to the square of the longitudinal dimensions. However, the body's own weight is already proportional to the cube of its length. It follows that, alone as safety against buckling under self-loading, $d^2 \sim l^3$ and $d \sim l^{3/2} \sim l\sqrt{l}$, and Greenhill (1881) has shown, in continuation of Euler's theories, that self-loading in pines is already relevant for buckling.

Higher plants (and of course technical) constructions must therefore show a lower slenderness ratio, in other words, become more clumsy. Later, this was introduced into the popular literature by Welzien (1927) (Fig. 9). It confirms the actual $\lambda$ values: *Secale cereale* ($h = 2$ m), 500; *Bambusa* sp. (25 m), 100; *Abies nobilis* (70 m), 37.5; *Eucalyptus amygdalina* (128 m), 28; and *Sequoia gigantea* (92 and 100 m), 12.5 and 8.3. This correlation had been formulated by Galilei (1638) by comparing two different-sized oaks: "...una quercia de-gunto braccia alta non potrebbe sostenere i suoi rami sparsi alla similitudine di una di mediocre grandezza."

## Mechanics of "Plant Buildings"

Rasdorsky (1911, 1928, 1929), who refuted much of Schwendener's theories on H-beams (Fig. 10), came to the conclusion, after hearing lectures on reinforced concrete engineering between 1906 and 1907, that a plant should be
<figure><img src="image_177.png" /><figcaption>Fig. 9. Welzien's (1928) explanation of the degree of slenderness, leading back to Rasdorsky. (A) Hypothetical rye stalk, 140 m high, which is statically impossible (B) Halsbrücker Esse (Freiberg, Saxonia), 140 m high. (C) Hypothetical rye stalk, 140 m high, which does not contradict the laws of statics.</figcaption></figure>

<figure><img src="image_178.png" /><figcaption>Fig. 10. Rasdorsky's (1928) drawings of "so-called I-beams" (A, B) and a "real I-beam" (C). (A, B) "Schwendeners I-beams" T-T, T₁-T₁, and T₂-T₂ in *Juncus glaucus*. (C) Normal tensions in a real I-beam with loading at one end (arrow). Above zero line, tension zone; below zero line, compression zone.</figcaption></figure>
taken as a composite construction in which the sclerotic strands and parenchyma correspond to the metal reinforcement and concrete matrix, respectively (Fig. 11). This is the right road to reaching an understanding and clearly demonstrates the important heuristic role of analogy in research, which Helmcke (1972) later demanded most emphatically for biotechnical work: "Accordingly, there is a large degree of analogy in the principles of construction in technical composite construction and plant organisms..." (Rasdorsky, 1929). This idea was presented for the first time by Rasdorsky in 1911 in a section on "the mechanical characteristics of tissues not belonging to those specifically classified as mechanical tissues." Later, other authors will express the same views. During a discussion on leaves, for example, Giesenhagen (1912) observed that their "strengthening tissues formed a grille like the metal gridding of a reinforced concrete floor."

Bach and Baumann (1924) compared the arrangement of tensile fibers in bamboo with "a fortification of the external layer which is most heavily used during bending (similar to reinforced concrete)," and Bower (1930) wrote with reference to reinforced concrete: "Ordinary herbaceous plants are constructed on the same principle. The sclerotic strands correspond to the metal straps, the surrounding parenchyma with its turgescent cells corresponds mechanically to the concrete."

Just what are they, these characteristic and, at the time, so novel peculiarities of the composite building material concrete, "which are basically very simple" (Probst, 1931)?

<figure><img src="image_179.png" /><figcaption>Fig. 11. Rasdorsky's (1928) comparison of the cross sections of a plant and a technical building. (A) *Trichophorum germanicum*, taken from Haberland (1884). (B) Factory stack in Elysabethport.</figcaption></figure>
*Each* elemental area of *each* tissue—besides sclerenchyma, if necessary collenchyma, parenchyma, epidermis, bark—delivers its share, derived from its Young modulus and its distance from the neutral axis, toward bending stiffness. Schwendener's terms "mechanical" and "not mechanical" tissues are not very suitable for the constructional mechanics of a plant's body. "Reinforcement" and "ground mass" or "matrix" would be better terms (Fig. 12). "Even if the weaker tissues... as a result of the lower E-values of their cell wall materials, incur less stress and their constructionally active cross-sectional parts are relatively small (due to thin walls), their total contribution... is not in the least insignificant; because the entire ground mass is extensive..." (Rasdorsky, 1929).

Modulus of Elasticity

Tensile tests with biological materials to determine the strain $\epsilon = e/l$ ($e$ = additional length under stress, $l$ = starting length) and the modulus of elasticity $E = \sigma/\epsilon$ ($\sigma$ = stress = $F/A$; $F$ = tensile force, $A$ = cross-sectional area before loading) were carried out at a very early date. Wertheim and Chevandier (1846) studied the effects of four states of drying on wood from 15 types of trees. Moduli of elasticity between 12,900 N mm$^{-2}$ (pine) and at least 29,700 N

<figure><img src="image_180.png" /><figcaption>Fig. 12. Rasdorsky's (1928) view of a composite beam cross section analogue to Schwendener's cross section of *Scirpus caespitosus*. *T-T* and *T<sub>1</sub>-T<sub>1</sub>*, Schwendener's "I-beams"; stress distribution for "ground mass" (parenchyma, $\sigma_B$) and for "reinforcement" (sclerenchyma, $\sigma_b$).</figcaption></figure>
$mm^{-2}$ (fir) are obtained "for the cell walls alone" after conversion for a moisture content of 23%. A modern comparative value for average conifers amounts to $10,000 N mm^{-2}$. Three methods were used to make these early measurements, which appear quite reliable, in contrast to some later measurements. In longitudinal, radial, and tangential cuts, the tensile forces leading to a lengthening in *Abies pectinata* behave as $1:0.085:0.031$.

## Flexibility

The confusion created by unclear definitions is demonstrated by the discussion on the bending ability of plants at the beginning of the twentieth century. The terms bending and flexibility, bending ability and bending strength, flexural stiffness and bending elasticity, etc., are all muddled and their use is not standardized.

Thus the botanist Schwendener (1874) was accused by Rasdorsky of using his "stereom" as a means of obtaining greater stiffness, i.e., he believed that a plant is well constructed if it bends only slightly or not at all. (Schwendener did not actually maintain this view, but his analogous comparison to earlier railway bridges suggests it. "A maximum sagging of 1/1800-1/5700 of the space is permitted in such bridges; the horizontal extensions caused by storm winds are of the same order.") For comparison, before the First World War, a maximum sagging of 1/2000 to 1/5000 was permitted in fractions of the span of girders in reinforced concrete bridges (Pederij, 1912; Patton, 1915, quoted by Rasdorsky, 1934). On the other hand, Schwendener (1874) wrote, "How conspicuously do the blades of grass play of the air, how lightly do the branches swing on a tree, and how much life seizes the forest when the storm blows through its mighty tops." In his book *A Botanical Journey to the Tropics* (1893), Haberlandt vividly described how far the palms bent over in stormy winds and how they changed their shape in doing so. Detlefsen (1884) complained vehemently about the confusion of terms: "One frequently finds e.g. the expression "bending stiffness" when "flexural elasticity" is meant, a fact which does not simplify comprehension." He went further: "If someone feels the need to express himself vaguely, then he should indicate this with his choice of words, because one writes for people who wish to understand and not for those who admire just what they cannot really understand."

Technicians alone used a clear technical terminology, even when they were analyzing biological materials. Thus Baumann (1913) studied the elasticity and stiffness of bamboo, acacia, ash, and hickory and tested wooden pipes for torsional stiffness, bending strength, and compressive strength. For bending tests, bamboo, for example, was run on two-sided bearings at a distance of 25 times its diameter and loaded centrally. A modulus of elasticity of about $20,000 Nm^{-2}$, a bending stiffness of about $800 Nm^{-2}$, and breakage resulting from bending moments of about 1500 N m were obtained.
Many modern debaters should also take note of the lesson taught by this short historical resumee: Whole libraries have been written for nothing, because scientific analyses were made without standardizing the definitions.

## The Plant as a Compromise Construction

Rasdorsky (1930) clearly defined a compromise construction as follows:

> When stressed by static (bending and buckling) factors stiff materials should be located near the periphery. The feather-like bending in the wind (or from the impact of rain-drops) however, is best provided for by a concentration of this material near the neutral plane or the longitudinal axis of the organ. Due to the composite building nature of plant organs, which dictates a distribution of reinforcement over the total cross section, the two tendencies (centrifugal and centripetal) are set certain limits.

Thus a series of constructional peculiarities, which Schwendener (1884) was unable to bring into accordance with his peripheral pipe systems as "not very rational reinforcing systems," become easily understood.

Haberlandt (1884), still under Schwendener's influence, spoke of beams and tensile fibers in palm leaves and from "irrational constructional peculiarities"—i.e., systems which apparently cannot be understood as "beams or girders"; Schutze (1905) discovered peripheral I-beams in the reinforcement of tropical tree ferns (Fig. 13A). Tschirch (1882) saw an internal and external system of struts (Fig. 13B) in the "long, very thin, but stiff leaves of *Kingia australis*." However, an objective observation shows (Fig. 13) neither I-beams nor "peripheral tendencies" but, rather, an equal distribution of the supporting tissues over the entire cross section (Rasdorsky, 1930).

## Bending Elasticity as a "Constructional Goal"

Compared to a completely stiff stalk, a grass stalk bending in the wind reduces drag and bending moment, thus reducing the danger of breaking. With materials of suitable elasticity and a favorable arrangement of supporting structures, even bending in more complex three-dimensional configurations such as in bushes and trees may constitute the elements of an optimized automatic mechanism, inducing a reduction in area, drag, and moments under wind loading. This again assumes that plants "are able to withstand substantial bending without damage," by means of which "the pressure of the wind on the plant organs above the soil is considerably reduced: thus reducing the area exposed to the wind" (Detlefsen, 1884a). In fact, under high wind speeds *v*, the coefficient of drag $c_D$ and the loaded area $A$ can be reduced by such automatism and, at the same time, the perpendicular distance $a$ from the "basal turning point," so that with $D = c_w A^{1/2} \rho v^2 (\rho$ air density) the drag, $D$, and with $B = a \cdot D$ the bending moment, $B$ are reduced compared to $D$ and $B$ without such automatisms (Fig. 14).
<figure><img src="image_181.png" /><figcaption>Fig. 13. Rasdorsky's copies of figures from the Schwendener school to demonstrate lack of I-beams (A) and tendency to peripheral aggregation of static material (B). (A) Cross section of a tree fern stem (Dicksonia karsteniana). White, parenchyma; black, "sclerenchyma"; dotted, main vascular bundles; a-c, nodes and internodes, α, outer layer; β, fibrillar cell layer with air roots. (B) Schematic cross section of a Kingia australis leaf. gf, vascular bundle; m, mark; p, st, w, "I-beams"; sp, stomata. (A) From Schütze (1928); (B) from Tschirch (1881).</figcaption></figure>

These possibilities were soon recognized: Rasdorsky (1937) saw, in very slender grasses, "even among hollow tube constructions, a high degree of passive withdrawal (due to extensive bending) from the influence of the wind." It has also long been known that, by means of elastic evasion, sedentary seaweeds are geared, in a very extreme manner, fully automatic to more or less pure tensile loading (Oltmanns, 1923). Such aspects have been taken up again in the last decades by the American Zoologist Vogel (1981) and his co-workers. They have shown that a passive automatic adjustment is present not only in algae and trees, but also in sedentary animals such as sea anemones and corals.
<figure><img src="image_182.png" /><figcaption>Fig. 14. Aerodynamic and bending characteristics of a tree. (A) Light wind; (B) very strong wind. See text.</figcaption></figure>

CONCLUSION

"Plant biomechanics" as a scientific discipline certainly goes as far back as Schwendener (1874), who was the first to combine botanical research with technical points of view on a broad, comparative scale. But, as nearly always in "one-man research projects," he was not free of autosuggestion, basing his point of view on an assumed overall existence of "I-beams," so that his scientific building was easily brought to near-collapse by showing that, very often, his I-beams were present only when seen through his eyes.

On the other hand, in Rasdorsky's work on plant statics a wealth of results can be found not only on statics and dynamics, but also on loading, which are, for the first time, biologically and technically satisfying and many of which are still valid today. However, Rasdorsky did not receive only recognition and agreement. Compared to other authors, Schwarz (1930) strangely underrates the Rasdorsky results in a "critical compilation" (how easy it is to veil performance and disguise priorities with tricks in formulation!), thus forcing Rasdorsky (1930) to reply in greater detail and more convincingly. Unlike any other author, Rasdorsky (1911-1937) has summarized, clearly formulated, and studied experimentally so many aspects of plant statics that it is certainly he who has created the essential fundamentals for a modern view of the "construction mechanics
of plants.” However, one cannot imagine Rasdorsky without Schwendener’s courageous beginnings.

## ACKNOWLEDGMENTS

This work was carried out at the Wissenschaftskolleg in Berlin as a preliminary study for plant biomechanics project B2/SFB 230 of the DFG. A short paper on the aspects presented here has already been published (Nachtigall, 1986a). Furthermore, part of the research history of another aspect of this complex, the (once again) much discussed principle of hydraulic formation in a pneu, has been also published (Nachtigall 1986b). A very detailed treatise on the history of plant biomechanics, containing a full bibliography, is in preparation for publication in the near future in a BIONA report (Fischer, Stuttgart).

I wish to thank Ms. Winifred Pattullo for executing the difficult task of translating the original German quotations into English and for the translation of this article in general.

## REFERENCES

* Bach, C., and Baumann, R. (1924). *Elastizität und Festigkeit*, 9 Aufl., Berlin.

* Baumann, R. (1913). Mitteilung über Forschungsarbeiten auf dem Gebiete des Ingenierwesens etc. 131:41-70.

* Bonnet, Ch. (1782). *Contemplation de la nature*, Nouv. ed., Hamburg.

* Bower, F. O. (1930). *Size and Form in Plants*, Macmillan, London.

* Culmann, R. E. (1866). *Die graphische Statik*, Meyer und Zeller, Zürich.

* Detlefsen, E. (1884a). Über die Biegeelastizität der Pflanzen. *Arb. Bot. Inst. Würzburg* 3:144-187, 408-425.

* Detlefsen, E. (1884b). *Bot. Centralbl.* 19:316-319.

* du Monceau, D. (1785). *La Physique des Arbres, etc.* I, Paris.

* Francé, R. H. (1919). *Die technische Leistungen der Pflanzen*, Veit & Co., Leipzig.

* Galilei, G. [1638 (1890)]. *Discorsi e dimonstrazioni matematichi, intorno a due nouvo scienze, Leida*.

* Galilei, G. (1898). *Edizione nazionale VIII*, p. 43.

* Giesenhagen, K. (1912). *Handwörterbuch der Naturwissenschaften*, Fischer, Jena, Vol. 2, pp. 1-35.

* Greenhill, A. G. (1881). Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of green proportions can grow. *Proc. Cam. Phil. Soc.*, 4:65-73.

* Grew, N. (1682). *The Anatomy of Plants etc.*, 2nd ed., Book IV, London, pp. 145-160.

* Haberlandt, G. (1884). *Physiologische Pflanzenanatomie* (1 Aufl.), Engelmann, Leipzig (2 Aufl., 1896; 3 Aufl., 1904; 4 Aufl., 1909; 5 Aufl., 1917; 6 Aufl., 1924).

* Haberlandt, G. (1893). *Eine botanische Tropenreise*, 1 Aufl., Engelmann, Leipzig.

* Helmcke, G. (1972). Cited by Otto (1976), p. 16.

* Holtermann, C. (1909). *Schwendener's Vorlesungen über mechanische Probleme der Botanik*, gehalten an der Universität Berlin, Engelmann, Leipzig.

* Humboldt, A. v. (1794). *Aphorismen aus der chemischen Physiologie der Pflanzen*, Leipzig.

* Istwanff, G. v. (1896). *Jb. Wiss. Bot.* 29:394-402.

* Kick, F. (1885). *Das Gesetz der proportionalen Widerstände*, Leipzig.

* Meschayeff, v. (1882). *Bull. Soc. Nat. Moscou*, No. 4.
Nachtigall, W. (1986a). Pflanzenbiomechanik. Eine forschungsgeschichtliche Studie als Basis für weiterführende Ansätze. In *Nachtigall, W., Wisser, Chr.-M.,* and Wisser, A. (eds.), *Pflanzenbiomechanik (Schwerpunkt Gräser) Konzept SFB 230, Heft 24, Stuttgart, pp. 21–66.

Nachtigall, W. (1986b). Der Pneu-Begriff in der Botanik des 19 und beginnenden 20. Jahrhunderts. Eine wissenschaftliche Analyse des hydraulischen Formhaltungs- und Formbildungskonzepts. *Jb 1985/86 Wissenschaftskolleg, Berlin, pp. 313–327.*

Oltmanns, J. (1923). *Morphologie und Biologie der Algen, 2 Bd., 2 Aufl., Jena.*

Otto, F. (1976). Zum Kolloquium. In Burkhardt, B. (Hrsg.), *Pneus in Natur und Technik, Mitt. d. Inst. f. leichte Flächentragswerke (IL),* Univ. Stuttgart, vol. 9, pp. 11–19.

Otto, F. (1978). Der Pneu-Bauprinzip des Lebens. *Bild d. Wissenschaft 10:124–135.*

Patton, E. O. (1915). *Eiserne Brücken, 1 Bd., Eiserne Balkenträger, 2 Aufl., Kijew, (Russian).*

Pederij (1912). *Handbuch der Eisenbetonbrücken, Petersburg (Russian).*

Probst, E. (1931). *VDI-Nachrichten 11:No. 21.*

Rasdorsky, W. (1911). *Bull. Soc. Nat. Moscou Sect. Biol. 4:351–405.*

Rasdorsky, W. (1928). Über das baumechanische Modell der Pflanzen. *Ber. d. Deutsch. Bot. Ges. 46:48–104.*

Rasdorsky, W. (1929). Über die Baumechanik der Pflanzen (Teil I bis III). *Biologia generalis. Int. Arch. Allgemein. Fragen Lebensforsch. 5:63–94.*

Rasdorsky, W. (1930). Die Lehre von den Biegungsfedern im Dienste der Pflanzenmechanik. *Ber. Deutsch. Bot Ges. 48:253–275.*

Rasdorsky, W. (1937). *Biologia generalis. Int. Arch. Allgemein. Fragen Lebensforsch. XII:359–398.*

Sachs, J. (1868). *Lehrbuch der Botanik (1 Aufl.),* Engelmann, Leipzig (2 Aufl., 1870; 3 Aufl., 1873; 4 Aufl., 1874).

Schütze, M. (1905). *Zur physiologischen Anatomie einiger tropischer Farne, besonders der Baumfarne, Berlin.*

Schwendener, S. (1874). Das mechanische Prinzip im anatomischen Bau der Monocotylen mit vergleichenden Ausblicken auf die übrigen Pflanzenklassen, *Engelmann, Leipzig.*

Schwendener, S. (1878). *Jahresheft Vereins Vaterländ. Nat. Württemberg 34:76–81.*

Schwendener, S. (1882). *Die Schutzscheiden und ihre Verstärkungen, Abh. K. Akad. Wiss., Berlin.*

Schwendener, S. (1887). Über Richtungen und Ziele der mikroskopisch-botanischen Forschung, *Berlin.*

Schwarz, W. (1930). *Beih. z. Bot. Centralbl. 1 Abt. 46:306–338.*

Senebier, J. (1800). *Physiologie vegetale etc. I-V, Geneva.*

Silk, W. K., Wang, L. L., and Cleland, R. E. (1982). Mechanical properties of the rice panicle. *Plant Physiol. 70:46–464.*

Spencer, H. (1863/1864). *The Principles of Biology.*

Strassburger (1923). *Lehrbuch der Botanik für Hochschulen, 15 Aufl., Jena.*

Tschirch, (1882). Beiträge zur Anatomie und dem Einrollmechanismus einiger Grasblätter. *Jb. Wiss. Bot. 13:544–568.*

van Marum, M. (1737). Cited by Rasdorsky (1937) and Senebier (1800), Vol. V, p. 182.

Vogel, S. (1981). *Life in Moving Fluids, Willard Grant Press, Boston, MA.*

Welzien, R. (1926). *Kosmos 44–47.*

Wertheim, and Chevandier (1846). *Poggendorfs Annalen, Ergänz.-Bd. II, p. 481.*

Zimmermann, A. (1884). *Jb. Wiss. Bot. 12:542–577.*

Zukal, H. (1895). *Sitzungsber. Akad. Wiss. Wien 104:1377–1395.*
# Bending Stability of Plant Stems: Ontogenetical, Ecological, and Phylogenetical Aspects<fnref n="1" />

Thomas Speck<fnref n="2" />

Bending mechanical properties, their variation during ontogeny, and the underlying ontogenetic shifts in stem anatomy have been studied for plants of different growth habits. The results found for self-supporting trees and shrubs (Syringa vulgaris, Alnus glutinosa and A. viridis) are significantly different from the findings in non-self-supporting woody lianas (Aristolochia macrophylla, A. cymbifera, A. gigantea, Fallopia aubertii, and Clematis vitalba). In self-supporting woody plants the Young's modulus increases during ontogeny. In non-self-supporting lianas the Young's modulus decreases during ontogeny. These changes in stem mechanics are caused by changes in stem anatomy. In all self-supporting plants the contribution of the secondary wood to the axial second moment of area increases during ontogeny, whereas in lianas the contribution of the peripheral stiffening tissues decreases. In C. recta, a semi-self-supporting species, the Young's modulus is constant during ontogeny. In this plant in all ontogenetic stages a more or less constant contribution of all stem tissues toward the axial second moment of area is found. These findings in extant plants make it possible also to infer the growth habit of fossil plants with permineralized stem remains, by recalculating the mechanical properties of the stems.

**KEY WORDS:** biomechanics; plant stems; self-supporting plants; semi-self-supporting plants; lianas; fossil plants.

## INTRODUCTION

Growth habit is a striking feature of plants and is often taken as an intuitive way for classifying plants in different ‘‘habit groups,’’ as, for example, trees,

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²Botanic Garden of the Albert-Ludwigs-University, Schänzlestr. 1, D-79104 Freiburg, Germany.</footnote>
bushes/shrubs, vines/lianas,herbs,and grasses. This classification is not phylogenetic, but this sometimes intuitive classification combines plants with similar architectural features and also often similar mechanical properties. A tree usually has an upright main trunk with lateral branches (acrotonic growth) and is a tall self-supporting woody plant. A bush, also a self-supporting woody plant, has several more or less equal-sized equivalent stems (basitonic growth) and is typically smaller than a tree. Vines are herbaceous or woody non-self-supporting climbing plants that need an external support to which they are fixed by various means. Woody vines are generally referred to as lianas (Putz and Mooney, 1991). In reality there is a continuum of forms from lianas to trees and it is often difficult to make a clear distinction between self-supporting and non-self-supporting plants by morphological and anatomical methods.

Size and growth habit are related to many properties of plants, important for their fitness and for an understanding of aut- and synecology and for the recognition of selective forces acting on plants. Because size and growth habit of plants are governed by mechanical constraints, quantitative studies in bio-mechanics and functional anatomy of plants with different growth habits assist the interpretation of the ecological significance of stem structure and help to analyze the selective forces acting on plants (Speck et al., 1993, Brüchert et al., 1994). By applying these studies to different ontogenetic stages, variations in biomechanical properties and the underlying changes in stem anatomy can be correlated and compared in self-supporting and non-self-supporting plants (Speck, 1991, 1994). I also summarize a method that allows a (semi-)quantitative distinction between extinct self-supporting and non-self-supporting plants (Speck et al., 1990, 1994; Speck, 1994; Speck and Vogellehner, 1992; Rowe et al., 1993).

## MATERIALS AND METHODS

Plant stems were tested in conventional two-point, three-point, or four-point bending (cf. Bodig and Jayne, 1988; Nachtigall et al., 1988; Vincent, 1990, 1992; Speck, 1991) and in tension. By bending plant stems at different lengths, suitable span-to-radius ratios were determined such that shear forces could be ignored (cf. Kollmann and Côté, 1968; Bodig and Jayne, 1982). Bending forces (F) were increased discretely and the maximum deflection ($x_{max}$) was measured after 60 s. In each bending experiment 6 to 12 discrete force increments were applied. For a force increment from the Hookean part of the force-deflection curve, a regression line can be calculated. The slope of the regression line (b) allows calculation of the flexural stiffness (EI).

$$x_{max} = b \cdot F, \quad E \cdot I = g/b$$

where g (units of m³) is a factor that takes into consideration the geometry of the different bending rigs.
By calculating the mean axial second moment of area (I) of the basal, middle, and apical parts of the tested stem segment, the Young's modulus of the bent stem can be calculated ($E = EI/I$). The allowable variation of the radius along the tested stem length was <5%.

To check the results of the bending experiments, the Young's moduli were determined in tension. For a given species, the Young's moduli measured in tension or bending show the same variations during ontogeny.

Flexural stiffness and Young's modulus may be plotted against the axial second moment of area, which is correlated with the ontogenetic stage of the stems (Figs. 1 and 2).

To discern the different growth habits in the plots of flexural stiffness against the axial second moment of area, the mean value of the Young's moduli of plant stems belonging to the youngest ontogenetic stage ($E_{OS1}$) was used as a slope to calculate a straight line, which I call the neutral line ($NL = E_{OS1} \cdot I$). If the stem diameter increased during ontogeny by a proportionate growth of all stem tissues—that is, if the contribution of the different tissues toward the axial second moment of area were constant during ontogeny—the Young's modulus of the plant stems would also be constant during ontogeny and the data for flexural stiffness would fall on the neutral line. This assumes that the Young's moduli of the different stem tissues are constant for all tested ontogenetic stages. This assumption is acceptable and can be used as a first-order approximation for mature tissues in woody plants, as present in the tested plant stems.

For an analysis of the ontogenetic variations in stem anatomy underlying the observed changes in mechanical properties during ontogeny, a model of a transverse section of the stem is constructed which allows the calculation of the cross-sectional area and the axial second moment of area of the entire cross section and of the various stem tissues (Speck et al., 1990; Rowe et al., 1993). The geometrical parameters for calculation of cross-sectional area and axial second moment of area were measured on transverse sections of fresh material (stained with phloroglucinol/hydrochloric acid). The accuracy of this method has been tested in some transverse sections by using a digitizer pad describing the cross-sectional area and axial second moment of area of the different stem tissues. The results of this lengthy method differ only slightly from the values calculated using the models. Data for fossil plants came from thin sections and acetate peels of permineralized stems; furthermore, photographs from publications were used.

# BENDING MECHANICS AND FUNCTIONAL ANATOMY OF EXTANT PLANTS

## Self-Supporting Trees and Shrubs

More than 250 stems of three woody self-supporting plants [Syringa vulgaris L., Alnus viridis (Chaix) DC., Alnus glutinosa (L.) Gaertn.] were tested in bending experiments. The age of the stems ranges from 1 to 15 years.
The data of flexural stiffness for older ontogenetic stages are clearly above the neutral line (Fig. 1A). This increase in "bending effectivity" during ontogeny is also reflected in the variation of the Young's modulus. In small to medium-sized stems (1 to 3-4 years old) the Young's moduli increase steeply with the axial second moment of area (Fig. 2A). In bigger stems, of self-supporting plants 4-5 to 15 years old, the Young's moduli become more or less constant, varying randomly with large variation (Fig. 2A).

The increase in the Young's modulus during ontogeny is correlated with an increase in wood (Fig. 3). In *Syringa vulgaris*, for example, the contribution of wood to the axial second moment of area increases from 17% in the youngest, 1-year-old axes to 51% in the oldest ontogenetic stage tested (i.e., in 6- to 10-year-old stems). At the same time the contribution of the cortex (including outer bark and phloem) diminishes: in *S. vulgaris* from 75% in the youngest, 1-year-old axes to 49%, and the share of pith decreases from 8% to <0.5% (Table I).

The changes in contribution of the different stem tissues to the cross-sectional area and axial second moment of area occur mainly in stems 1 to 3-4 years old. In these ontogenetic stages the contribution of wood to the axial second moment of area increases rapidly. In older ontogenetic stages the increase in the share of wood to the axial second moment of area is drastically reduced. This means that the contribution of tissues to the axial second moment of area becomes more or less constant for the older ontogenetic stages within the age range tested (i.e., in 5- to 15-year-old stems). This explains why the Young's moduli increase rapidly in woody stems 1 to 3-4 years old. But then in older ontogenetic stages, which have a more or less constant contribution of tissues to the axial second moment of area, the increase in the Young's modulus also slows down extremely, to become more or less constant for 5- to 15-year-old stems.

The kind of variations of bending mechanical properties in bigger stems is possibly different in shrub-like and in tree-like species. In shrub-like species, e.g., *S. vulgaris* and *Alnus viridis*, the Young's moduli become more or less constant for stems older than 4-5 years. In the tree-like *A. glutinosa*, on the other hand, the Young's moduli show a slow increase with increasing stem diameter also in bigger stems. Some first results of bending tests in upright main stems with diameters between 30 and 40 cm yielded Young's moduli between 5.1 and 7.6 GNm⁻² [for the bending equipment used see Wessolly (1989, 1991, 1993) and Sinn and Wessolly (1989)]. The mean Young's modulus for stems of this size range is 6.6 GNm⁻², about 2.4 times larger than that found in 5- to 10-year-old stems with diameters between 1.3 and 2.2 cm (*E*<sub>mean</sub>, 2.8 GNm⁻²; *E*<sub>min</sub>, 1.7 GNm⁻²; *E*<sub>max</sub>, 4.4 GNm⁻²; see Fig. 7). The underlying anatomical reason may be that in trees, as in shrubs, the increase in the relative contribution of wood to the axial second moment of area slows down after a phase of rapid
<figure><img src="image_183.png" /><figcaption>Fig. 1. The flexural stiffness of the plant stems plotted against the axial second moment of area (double logarithmic plot). The neutral line (NL) is calculated from the mean value of the Young's modulus of the youngest ontogenetic stage (OS1; marked by crosses); NL = $E_{OS1} \cdot I$. The ontogenetic stages (OS1 to OS5 for *Syringa vulgaris* and OS1 to OS4 for *Aristolochia macrophylla*) are defined morphologically according to the structure of the bark (Speck, 1991) and correspond to the age of the stems given by the number of annual rings (NAR). (A) In self-supporting woody plants data for older ontogenetic stages are clearly above the NL, as shown here for *S. vulgaris*. OS1, NAR = 1 (youngest annual shoots, $E_{OS1} = 2.85 \text{ GNm}^{-2}$); OS2, NAR = 1 (older annual shoots); OS3 NAR = 2 to 3; OS4, NAR = 4 to 5; OS5, NAR = 6 to 10. (B) In woody lianas data for older ontogenetic stages are below the NL, which is shown here for *A. macrophylla*. OS1, NAR = 1 ($E_{OS1} = 2.47 \text{ GNm}^{-2}$); OS2, NAR = 2; OS3, NAR = 3 to 5; OS4, NAR = 6 to 18. (C) In the semi-self-supporting taxon *Clematis recta* the data are scattered around the NL for all tested ontogenetic stages ($E_{youngest-OS} = 3.89 \text{ GNm}^{-2}$).</figcaption></figure>
<figure><img src="image_184.png" /><figcaption>Fig. 1. Continued.</figcaption></figure>

rise in young ontogenetic stages, but that in trees also in older ontogenetic stages a slow increase in the relative contribution of wood with increasing stem diameter occurs (e.g., in *A. glutinosa*, from 66% in 5- to 10-year-old stems to about 75% in stems 30 to 40 cm in diameter). In shrubs, on the other hand, the relative contribution of tissues to the axial second moment of area remains nearly constant in older ontogenetic stages, perhaps as a consequence of the shrub's restricted ability to develop stems with large diameters.

In addition to changes in the anatomical structure of the stems during ontogeny, (small) variations of the Young's modulus of the different stem tissues during ontogeny are also possible. Other changeable factors include, for example, chemical modifications of cell walls during the formation of heartwood or the functioning of wood as water storage tissue, so altering its mechanical properties. The larger share of late wood sometimes produced in older ontogenetic stages can also change the mechanical properties of the wood cylinder as a whole. These variations seem not to alter the observed overall trend of bending mechanical properties during ontogeny in woody plants. In nonwoody plants the variations of mechanical properties of tissues during ontogeny may be much more distinct, e.g., caused by lignification of the walls of parenchymatous cells (Niklas, 1990, 1992, 1993).

## Non-Self-Supporting Lianas

More than 400 stems of five woody lianas [*Aristolochia macrophylla* Lam., *A. cymbifera* Mart. & Zucc., *A. gigantea* Hook. (=*A. grandiflora* Arruda), *Clematis vitalba* L., *Fallopia aubertii* (L. Henry) Holub] were tested in bending and tension. The age of the stems ranged from 1 to 29 years.
<figure><img src="image_185.png" /><figcaption>Fig. 2. The Young's modulus of the plant stems plotted against the axial second moment of area (double logarithmic plot). The meaning of ontogenetic stages is the same as in Fig. 1. (A) In self-supporting woody plants the Young's moduli increase rapidly during early ontogenetic stages (OS1 to OS3). In older ontogenetic stages (OS4 and OS5) the Young's moduli change very slowly and become more or less constant, at least in shrub-like self-supporting plants, as shown here for *Syringa vulgaris*. (B) In woody lianas, as in *Aristolochia macrophylla*, the Young's moduli decrease with increasing axial second moment of area. The regression line is given by $y = 10^{3.85} \cdot x^{-0.42}$, the correlation coefficient is -0.939. (C) In the semi-self-supporting plant *Clematis recta* the Young's moduli are constant during ontogeny but vary randomly over a large range.</figcaption></figure>
<figure><img src="image_186.png" /><figcaption>Fig. 2. Continued.</figcaption></figure>

The flexural stiffness of older ontogenetic stages is clearly below the neutral line (Fig. 1B), showing that "bending effectivity" decreases drastically during ontogeny. In all five woody lianas, the Young's moduli of stems in the youngest ontogenetic stages are relatively high, after which the Young's moduli decrease during ontogeny to about 25 to 15% of their initial values.

The changes in the mechanical properties during ontogeny are correlated with changes in the amount and cross-sectional shape of the different stem tissues (Fig. 4). The decrease in the Young's modulus during ontogeny is correlated with an increasing fragmentation of the peripheral ring(s) of strengthening tissue which is/are entire in young ontogenetic stages. It is also correlated with a considerable decrease in the contribution of these stiffening tissue(s) to the axial second moment of area. In *A. macrophylla* (Table II), the contribution of the peripheral collenchymatous and sclerenchymatous tissues to the axial second moment of area decreases during ontogeny from 17 to 2% (collenchyma) and from 25 to 2% (sclerenchyma). On the other hand, the contribution to the axial second moment of area of compliant tissues such as parenchymatous cortex and outer bark increases during ontogeny. The contribution of the xylem, which is relatively compliant with large vessels and huge wood rays, also increases significantly (in *A. macrophylla*, from 8 to 29%).

In *A. macrophylla* the secondary phloem is, apart from the assimilate conducting elements, entirely parenchymatous (Figs. 4B and C). In *C. vitalba* and *F. aubertii*, on the contrary, in each annual growth phase in the secondary
<figure><img src="image_187.png" /><figcaption>Fig. 3. Variations in stem anatomy in self-supporting shrubs and trees. The drawings represent transverse sections of different ages; all transverse sections are enlarged to the same size. The original diameter for each transverse section is given in parentheses. Tissues from the center outward: white, pith; hatched, wood; stippled, cortex (including phloem, epidermis, and bark). (A-C) *Syringa vulgaris*: (A) 1-year-old stem (4.0 mm); (B) 3-year-old stem (12.4 mm); (C) 18-year-old stem (38.5 mm). (D-F) *Alnus glutinosa* (stump shoots): (D) 1-year-old stem (4.0 mm); (E) 4-year-old stem (22.1 mm); (F) 8-year-old stem (37.0 mm). (G-I) *A. viridis* (from the Black Forest); (G) 1-year-old stem (2.9 mm, without "wing"); (H) 4-year-old stem (15.0 mm); (I) 12-year-old stem (25.5 mm).</figcaption></figure>

phloem, sclerenchymatous segments are formed (Figs. 4E, F, H, and I). However, these segments are isolated and never fuse to a closed sclerenchymatous ring as in young stems.

In older *C. vitalba* the bark loosens from the stems in strips (Figs. 4E and F). The flexural stiffness for older ontogenetic stages is also below the neutral line and the Young's modulus decreases to 16% of its initial value (Fig. 5). A
<table><caption>**Table I.** *Syringa vulgaris:* Variation of the Percentage Contribution of Different Stem Tissues to the Axial Second Moment of Area During Ontogeny (Mean ± SD)<sup>a</sup></caption><thead><tr><th></th><th>NAR: 1<br>(youngest annual shoots),<br>n = 20</th><th>NAR: 1<br>(older annual shoots),<br>n = 12</th><th>NAR: 2,<br>n = 8</th><th>NAR: 3,<br>n = 12</th><th>NAR: 4-5,<br>n = 14</th><th>NAR: 6-10,<br>n = 25</th></tr></thead><tbody><tr><td>Cortex</td><td>74.6 ± 6.1</td><td>63.7 ± 3.0</td><td>62.0 ± 7.4</td><td>53.4 ± 6.8</td><td>50.5 ± 4.1</td><td>49.2 ± 3.9</td></tr><tr><td>Wood</td><td>17.3 ± 6.6</td><td>29.3 ± 4.1</td><td>34.1 ± 9.8</td><td>45.4 ± 6.9</td><td>49.0 ± 4.0</td><td>50.6 ± 4.0</td></tr><tr><td>Pith</td><td>8.1 ± 2.5</td><td>7.0 ± 2.7</td><td>3.9 ± 3.1</td><td>1.2 ± 0.5</td><td>0.5 ± 0.3</td><td>0.2 ± 0.3</td></tr></tbody></table>

<footnote><sup>a</sup>n, number of analyzed stems; NAR, number of annual rings.</footnote>
<figure><img src="image_188.png" /><figcaption>Fig. 4. Variations in stem anatomy in woody lianas. The drawings represent transverse sections of different ages; all transverse sections are enlarged to the same size. The original diameter for each transverse section is given in parentheses. Tissue types from the center outward: central white area, pith; wood consisting of vessels and tracheids (wedge-shaped area; the vessels are drawn as small circles) and ray parenchyma (white areas between the wedges); white area, parenchymatous cortex and phloem; black area, sclerenchymatous tissue; hatched area, collenchymatous tissue; stippled area, bark; outermost line, epidermis. (A-C) *Aristolochia macrophylla*—this species comprises an entirely parenchymatous secondary phloem: (A) 1-year-old stem (4.8 mm); (B) 4-year-old stem (6.6 mm); (C) 14-year-old stem (29.1 mm). In the other two liana species, *Clematis vitalba* and *Fallopia aubertii*, sclerenchymatous segments are built in the secondary phloem each year. (D-F) *C. vitalba*—in this species in older ontogenetic stages (E and F) the bark loosens from the stems in strips: (D) 1-year-old stem (4.4 mm, mean diameter of the star-shaped transverse section); (E) 6-year-old stem (13.4 mm); (F) 7-year-old stem (24.6 mm). (G-I) *F. aubertii*: (G) young 1-year-old stem (4.3 mm); (H) older 1-year old stem (6.6 mm); (I) 3-year-old stem (11.3 mm).</figcaption></figure>
<table><caption>**Table II.** *Aristolochia macrophylla:* Variation of the Contribution of the Different Stem Tissues to the Axial Second Moment of Area During Ontogeny (Mean ± SD)<sup>a</sup></caption><thead><tr><th></th><th>NAR: 1<br>(n = 7)</th><th>NAR: 2<br>(n = 7)</th><th>NAR: 3-4<br>(n = 7)</th><th>NAR: 5-7<br>(n = 9)</th><th>NAR: 8-18<br>(n = 4)</th></tr></thead><tbody><tr><td>Bark</td><td>5.5 ± 0.4</td><td>17.6 ± 12.2</td><td>20.8 ± 16.3</td><td>26.9 ± 3.3</td><td>18.8 ± 9.2</td></tr><tr><td>Paren. cortex</td><td>42.5 ± 4.5</td><td>40.9 ± 2.4</td><td>39.1 ± 6.4</td><td>46.5 ± 2.6</td><td>48.6 ± 7.9</td></tr><tr><td>Collenchyma</td><td>17.0 ± 1.9</td><td>13.4 ± 5.1</td><td>10.6 ± 4.0</td><td>3.5 ± 0.6</td><td>1.9 ± 0.9</td></tr><tr><td>Sclerenchyma</td><td>24.6 ± 4.0</td><td>17.0 ± 5.2</td><td>12.6 ± 3.2</td><td>4.6 ± 0.3</td><td>2.1 ± 1.6</td></tr><tr><td>Wood</td><td>8.2 ± 0.7</td><td>10.8 ± 0.4</td><td>16.6 ± 4.0</td><td>18.4 ± 2.0</td><td>28.6 ± 3.8</td></tr><tr><td>Pith</td><td>2.2 ± 1.6</td><td>0.4 ± 0.2</td><td>0.3 ± 0.3</td><td>0.05 ± 0.04</td><td>0.01 ± 0.01</td></tr></tbody></table>

<figure><img src="image_189.png" /><figcaption>**Fig. 5.** *Clematis vitalba:* the Young's modulus of the plant stems plotted against the axial second moment of area (double logarithmic plot).</figcaption></figure>

difference from the other lianas is the markedly two-phasic behavior stiffness and Young's modulus. The dotted line in Fig. 5 is a regression over all ontogenetic stages, whereas the three solid lines represent separate regressions for the two phases and the transition zone. These regressions fit the test data much better than the regression line for a continuous decrease in the Young's modulus. The first phase, given by ontogenetic stage 1, is characterized by a high Young's modulus of about 3.45 GNm⁻², which is nearly constant within this stage. The second phase, represented by ontogenetic stage 4, has a low Young's modulus

<footnote><sup>a</sup>n, number of analyzed stems; NAR, number of annual rings.</footnote>
of about 0.55 GNm⁻², showing only a small decrease. The transition zone (ontogenetic stages 2 and 3) is characterized by a rapid decrease (about 84%) in the Young's modulus. This may be because as the bark loosens, parts of the stiffening sclerenchymatous cortex tissues sluff off (Figs. 4E and F). The loose bark contributes to the calculated axial second moment of area of the stem, but its contribution to the flexural stiffness is only small, because of its loose contact with the rest of the stem. This further reduces the Young's modulus of the stems, in addition to the anatomical changes found in the other lianas. A rapid increase in the amount of loose bark is found especially in ontogenetic stages 2 and 3, where the rapid decrease in the Young's modulus occurs.

## Semi-Self-Supporting Plants

Semi-self-supporting plants, which are perhaps best named with the German term "Spreizklimmer", have a growth form that can be described as neither truly self-supporting nor truly lianescent (Schenck, 1912). Plants with this growth habit can live in a more or less liana-like manner with the support of the surrounding vegetation. Furthermore, this life-form also enables growth in dense monotypic stands, where the plants provide mutual support (e.g., *Equisetum giganteum*). For both growth habits, a close interconnection with each other or with the surrounding vegetation is a prerequisite. This interconnection can be achieved by means of fronds or branches. The two growth modes of the semi-self-supporting habit can also be found in the bracken, *Pteridium aquilinum*. In the liana-like growth habit the fronds of *P. aquilinum* can grow up to 6 or 7 m in length (personal observation). In the "normal" self-supporting growth mode—often in dense monotypic stands—the length of fronds is restricted to about 2.5 m.

The mechanical properties of semi-self-supporting plants have been experimentally tested only in the perennial *C. recta*. This plant normally grows with upright axes, but the stems bend in a curved manner if they exceed a given length. It may, however, also grow in a more or less scrambling manner with support from the surrounding vegetation. Stems of the nonscrambling growth habit of *C. recta* show more or less constant Young's moduli during ontogeny, with a large variation, between 2.0 and 6.4 GNm⁻² (Fig. 2C). This huge variation may be because the degree of self-support in the tested stems differs, and the ontogenetic phase from which a given stem has been supported by the surrounding vegetation is different for different stems.

In this taxon there exists neither an increase in the bending effectivity during ontogeny, as found in true self-supporting plants, nor a decrease, as found in lianas. Stems of *C. recta* can be interpreted as being "neutral" during ontogeny as far as the bending mechanical properties are concerned.<fnref n="3" />

<footnote n="3">³ *C. recta* has hollow stems. Values of the axial second moment of area and Young's modulus relate to the massive part of the stem, not to the entire stem with the medullary cavity.</footnote>
This is because there is a proportionate increase in all stem tissues during ontogeny, entailing a more or less constant contribution of all stem tissues to the axial second moment of area. The nearly constant tissue distribution is brought about because the enlargement of the basal internodes is accompanied by the formation of additional entire vascular bundles (Fig. 6). All vascular bundles have a more or less constant tissue composition.

## DISCUSSION

The flexural stiffness and Young's modulus in lianas and in self-supporting plants change in a nearly opposite manner during ontogeny, because of the different selective forces acting on them. The loads caused by wind increase with height. Therefore it is an important selective advantage for self-supporting plants not only to enlarge the volume of the stabilizing tissues in the basal stem parts by secondary growth, but also to optimize the distribution of these tissues with regard to the mechanical properties in bending. This causes an increase in the Young's modulus of the stem as a whole during ontogeny.

For non-self-supporting plants, it is a selective advantage to be relatively stiff when young, until the stems have found a support. In older plants, if the lianas are already attached to their support, it is a selective advantage to be relatively flexible, so that the stems can shed the bending forces to a certain degree.

In some species of liana showing typical bending properties in the young

<figure><img src="image_190.png" /></figure>

Fig. 6. Variations in stem anatomy in the semi-self-supporting plant *Clematis recta*. The drawings represent transverse sections of different internodes of one plant specimen, i.e., of stem segments of different ontogenetic ages. All transverse sections are enlarged to the same size. The original diameter for each transverse section is given in parentheses. Tissue types from the center outward: central white area, pith cavity; vascular bundles arranged as an eustele in the parenchymatous cortex tissue—each bundle comprises xylem (wedge-shaped inner area; the vessels are drawn as small circles), cambium, assimilate conducting phloem (white area), and sclerenchymatous bundle sheath (black area); outermost line of the transverse section, epidermis. Internodes are counted from the stem base upward. (A) Internode 6 (5.2 mm); number of vascular bundles, 24. (B) Internode 3 (6.6 mm); number of vascular bundles, 34. (C) Internode 2 (6.8 mm); number of vascular bundles, 39.
and middle-aged stages, the very basal, old stem parts seem to become more or less self-supporting. This seems to be true for *Wisteria sinensis*. Other lianas, such as many tropical forest lianas, start from a (short) upright stem and become liandescent in a second developmental stage (Putz, 1984, 1985; Caballé, 1986, 1993; Putz and Holbrook, 1991).

This mechanical behavior, also found in other self-supporting woody plants and lianas or liandescent plant parts, seems to be typical for plants of these growth habits. Putz and Holbrook (1991) found, in tropical lianas and tree saplings 2-5 cm in stem diameter (from Luquillo Mountains, Puerto Rico), Young's moduli that are very similar to ours. They reported a mean Young's modulus of $0.61 \pm 0.44 \text{ GNm}^{-2}$ for 12 liana species (ranging from $0.15$ to $1.7 \text{ GNm}^{-2}$), which is very close to the values found in similar-sized stems of the lianas we have tested (*A. macrophylla*, $E_{OS4} = 0.50 \text{ GNm}^{-2}$; *C. vitalba*, $E_{OS4} = 0.56 \text{ GNm}^{-2}$; *F. aubertii*, $E_{OS4} = 0.73 \text{ GNm}^{-2}$). The data for the five tree saplings are, with $E = 4.7 \pm 3.8 \text{ GNm}^{-2}$ (range of data, 1.7 to $12.0 \text{ GNm}^{-2}$), comparable to the values we measured in similar-sized stems of self-supporting woody plants (*S. vulgaris*, $E_{OS4/OS5} = 5.5 \text{ GNm}^{-2}$; *Alnus glutinosa*, $E_{OS5} = 2.7 \text{ GNm}^{-2}$; *A. viridis*, $E_{OS5} = 3.3 \text{ GNm}^{-2}$). Gartner (1991a, b) found, in comparing the mechanical properties of shrub-like and vine-like specimens of *Toxicodendron diversifolium* (T. & G.) Greene, data similar to those of Speck (1991) and the present work. The Young's moduli for different stem diameters of the supported (vine-like) specimens fit very well with the pattern I found (Gartner, 1991a, Figs. 4 and 5). The Young's moduli of the unsupported (shrub-like) specimens resemble more the pattern I found for semi-self-supporting plants than the pattern found in self-supporting plants (cf. Cannell and Morgan, 1987; Cannell *et al.*, 1988). For that reason *T. diversifolium* could perhaps best be compared with the "Spreizklimmer" *Rosa canina* agg. L., a plant for which a liana-like and a shrub-like semi-self-supporting to self-supporting growth mode exists (cf. Haux, 1993).

The results for *C. recta* show that at least this semi-self-supporting plant can be characterized as being neither truly liandescent nor truly self-supporting. However, the question of whether the biomechanically "neutral behavior" is a feature typical of the semi-self-supporting growth habit cannot be answered before other (woody) semi-self-supporting plants have been tested. Recent studies on *R. canina* agg. corroborate these findings for young ontogenetic stages of woody stems from the shrub-like growth mode of this semi-self-supporting plant (cf. Haux, 1994).

These results prove that in extant plants form (growth habit), structure (stem anatomy), and function (mechanical properties) can be correlated in an un-equivocal manner. The experimental findings in extant plants have been used to derive a method that allows a semiquantitative distinction between extinct self-supporting an extinct non-self-supporting plants (i.e., lianas and semi-self-
supporting plants). As methods and results for various fossil plants are described in detail elsewhere (Speck *et al.*, 1990, 1992; Speck and Vogellehner, 1992; Rowe *et al.*, 1993; Speck, 1994; Speck and Rowe 1994), I summarize only some main results.

It is sometimes very problematic to determine size and growth habit in fossil plants, because of (a) the often disarticulated nature of plant fossils and (b) the fact that the mechanical properties of fossil plant material cannot be tested experimentally. Mathematical models of the tissue distribution in the fossil stems make it possible to calculate the axial second moment of area of the various stem tissues and of the entire stem. The Young's moduli of the different fossil tissue types can be estimated quantitatively by recalculation from the most similar extant tissue type for which experimental mechanical data are known. If fossil plant stems are considered as composite structures, the flexural stiffness and Young's modulus of the fossil plant stem can be calculated. Thus, the changes in the bending properties during ontogeny can also be studied in fossil plants and their growth habit can be inferred.

The method was tested by applying it to *Diaphorodendron vascular* (Binney) DiMichele, for which a reliable reconstruction exists (DiMichele, 1981). *D. vascular* is a tree, 8 to 15 m tall, with unbranched main trunk and lateral branches. The calculated mechanical data are entirely consistent with experimental data from extant woody plants (Speck, 1994; Speck *et al.*, 1994). The flexural stiffness for older ontogenetic stages is clearly above the neutral line (Fig. 7A) and the Young's modulus increases during ontogeny to more than 3.5-fold its initial value (Fig. 8A). The results corroborate that this method can be used to identify growth habits in fossil plants with secondary growth.

Calculations for exceptionally well-preserved young stems of *Pitus dayi* Gordon prove that, independent of the tissue type inferred for the cortex/periderm, for which the actual tissue type cannot undoubtedly be determined, the mechanical data are consistent with a self-supporting growth mode (Figs. 7B and 8A) (Speck and Rowe, 1994). The biomechanical analysis of stems of two other pteridosperms, for which either the growth habit was controversially discussed [*Lyginopteris oldhamia* (Binney) Potonie] or a lianescent to self-supporting growth habit was assumed more intuitively for reasons of stem size and anatomy (*Calamopitys* sp.), helps to clear up the actual growth habit of these taxa. The mechanical data calculated for *L. oldhamia* make it very probable that this taxon was a non-self-supporting plant (data on flexural stiffness for older ontogenetic stages below the neutral line in Fig. 7C; decrease in the Young's modulus during ontogeny, Fig. 8B). The calculated Young's moduli of *L. oldhamia* are relatively high compared with the Young's moduli found in extant lianas and decrease to only about 60% of their initial value, so it is probable that *L. oldhamia* was a semi-self-supporting plant rather than a pure liana (Speck, 1994; Speck *et al.*, 1994). Data calculated for a thin stem of *Calamopitys* sp.
<figure><img src="image_191.png" /><figcaption>Fig. 7. The calculated flexural stiffness of fossil plant stems plotted against the axial second moment of area (double logarithmic plot). The neutral line (NL) is calculated from the mean value of the Young's modulus of the youngest ontogenetic stage (OS1; marked by filled squares); NL = E<sub>os1</sub> · I. The ontogenetic stages (OS1 to OS3 for *Diaphorodendron vasculare*, *Lyginopteris oldhamia* and *Pitus dayi*) are defined according to anatomical shifts in stem anatomy (Speck, 1994; Speck and Rowe, 1994). In *Calamopitys* sp. no ontogenetic stages can be defined, as over the length of the studied stem fragment, there is no marked change in the percentage tissue contribution (Rowe *et al.*, 1993). (A) D. vasculare, E<sub>os1</sub> = 1.35 GNm<sup>-2</sup>; (B) P. dayi (calculated under the assumption that cortex tissues are bark-like), E<sub>os1</sub> = 0.38 GNm<sup>-2</sup>. For these two fossil plants the flexural stiffness data for older ontogenetic stages are clearly above the NL, consistent with the assumed self-supporting habit of these taxa. (C) *L. oldhamia*, E<sub>os1</sub> = 4.26 GNm<sup>-2</sup>. Data for older ontogenetic stages are below the NL, making it very probable that *L. oldhamia* is a non-self-supporting plant. (D) *Calamopitys* sp., E<sub>yongest-os</sub> = 3.79 GNm<sup>-2</sup>. The flexural stiffness data are clustered around the NL, i.e., showing an arrangement similar to that found in living semi-self-supporting plants. [A and C from Speck (1994); B from Speck and Rowe (1994); D from Rowe *et al.* (1993).]</figcaption></figure>
<figure><img src="image_192.png" /><figcaption>Fig. 8. Variations of the normalized Young's modulus during ontogeny in self-supporting and non-self-supporting plants. The mean Young's modulus of the youngest ontogenetic stage tested is normalized to 1.0. (A) In all extant and fossil self-supporting taxa a marked increase in the Young's modulus occurs. The black column [Alnus glutinosa (OS6)] represents data for stems 30 to 40 cm in diameter. (B) In all non-self-supporting taxa the Young's modulus either decreases during ontogeny (in plants with a lianescent growth mode) or remains more or less constant during the entire ontogeny (in semi-self-supporting plants).</figcaption></figure>
are consistent with an interpretation, at least for a *Calamopitys* stem of this size,
of a semi-self-supporting plant (Rowe *et al.*, 1993, Speck *et al.*, 1994). As in
the extant semi-self-supporting *Clematis recta*, the data for flexural stiffness are
scattered around the neutral line (Fig. 7D) and the Young's moduli remain more
or less constant during ontogeny.

These few examples of fossil plants show that comparative investigations
of the mechanics of lianas and self-supporting and semi-self-supporting plants
not only yield a better understanding of some features of living plants, but also
yield results important for a better understanding of the structure and function
of fossil plants. In living plants, for example, the methods allow quantitative
conclusions concerning the influence of changes of stem anatomy during ontog-
eny on the mechanical properties. In fossil plants comparatively reliable con-
clusions about whether a given taxon was a self-supporting plant or a liana or
a semi-self-supporting plant are possible. This is of interest not only for recon-
struction of fossil plants but also for questions concerning autecology and sy-
necology of the fossil taxa.

ACKNOWLEDGMENTS

I am grateful for stimulating discussions with Drs. A. Bogenrieder, N. P. Rowe, H.-Ch. Spatz, and D. Vogellehner. I thank N. P. Rowe and H.-Ch. Spatz for critical reading of the manuscript and Dipl-Biol. B. Heneka for preparing some of the drawings.

REFERENCES

Bodig, J.; and Jayne, B. A. (1982). *Mechanics of Wood and Wood Composites*, Van Nostrand Reinhold, New York, Cincinnati, Toronto, London, Melbourne.

Brüchert, F., Bogenrieder, A., and Speck, T. (1994). Anatomischer und biomechanischer Vergleich der Sproßachsen von *Alnus viridis* (Chaix.) DC. aus dem Schwarzwald und den Lechtaler Alpen mit Stockausschlägen von *Alnus glutinosa* (L.) Gaertn. aus dem Schwarzwald im Hindblick auf die Standortsökologie beider Arten. *Ber. Naturf. Ges. Freiburg i. Br.* **81** (in press).

Caballé, G. (1986). *Sur la biologie des lianes ligneuses en forêt gabonaise*, Thèse de Doctorat d'Etat en Sciences, Université de Montpellier 2 Sciences et Techniques du Languedoc.

Caballé, G. (1993). Liana structure, function and selection: A comparative study of xylem cylinders of tropical rainforest species in Africa and America. *Bot. J. Linn. Soc.* **113**:41-60.

Cannell, M. G. R., and Morgan, J. (1987). Young's modulus of sections of living branches and tree trunks. *Tree Physiol.* **3**:355-364.

Cannell, M. G. R., Morgan, J., and Murrey, M. B. (1988). Diameters and dry weights of tree shoots: Effects of Young's modulus, taper, deflection and angle. *Tree Physiol.* **4**:219-231.

Carlquist, S. (1991). Anatomy of vine and liana stems: a review and synthesis. In Putz, F. E., and Mooney, H. A. (eds.), *The Biology of Vines*, Cambridge University Press, Cambridge, pp. 53-71.

DiMichele, W. A. (1981). Arborescent lycopods of Pennsylvanian age coals: *Lepidodendron*, with description of new species. *Palaeontographica* **175**:85-125.

Gartner, B. L. (1991a). Structural stability and architecture of vines vs. shrubs of Poison Oak, *Toxicodendron diversilobum*. *Ecology* **72**:2005-2015.
Gartner, B. L. (1991b). Is the climbing habit of poison oak ecotypic? *Funct. Ecol.* **5**: 696-704.

Haux, M. (1993). *Veränderungen der Anatomie und der biomechanischen Eigenschaften im Laufe der Ontogenese bei Sproßachsen der Gattung Rosa L.*, Staatsexamensarbeit (manuscript).

Kollmann, F. P., and Côté, W. A. (1968): *Principles of Wood Science and Technology*. I. *Solid Wood*, Springer, Berlin, Heidelberg.

Nachtigall, W., Wisser, C.-M., and Wisser, A. (1988). Ein erster Einblick in biomechanische Konstruktionsprinzipien von Gräsern. *Natürliche Konstruktionen, Mitteilungen des SFB 230* 1:59-66.

Niklas, K. J. (1990). *Biomechanics of Psilotum nudum* and some early paleozoic vascular sporo-phytes. *Am. J. Bot.* **77**:590-606.

Niklas, K. J. (1992). *Plant Biomechanics*, University of Chicago Press, Chicago.

Niklas, K. J. (1993). Influence of tissue density-specific mechanical properties on the scaling of plant height. *Ann. Bot.* **73**:173-179.

Putz, F. E. (1984). How trees avoid and shed lianas. *Biotropica* **16**:9-23.

Putz, F. E. (1985). Woody vines and forest management in Malaysia. *Commonwealth Forest Rev.* **64**:369-365.

Putz, F. E., and Holbrook, N. M. (1991). Biomechanical studies in vines. In Putz, F. E., and Mooney, H. A. (eds.), *The Biology of Vines*, Cambridge University Press, Cambridge, pp. 73-97.

Putz, F. E., and Mooney, H. A. (eds.) (1991). *The Biology of Vines*, Cambridge University Press, Cambridge.

Rowe, N. P., Speck, T., and Galtier, J. (1993). Biomechanical analysis of a Palaeozoic gymnosperm stem. *Proc. R. Soc. Lond. B* **252**:19-28.

Schenck, H. (1912). Lianen. In *Handbuch der Naturwissenschaften, 6. Band*, Fischer, Jena, pp. 176-185.

Sinn, G., and Wessolly, L. (1989). Messungen an Bäumen: Ermittlung der Sicherheiten gegen Kippen oder Bruch. *Das Gartenamt* **38**(7):422-430, 38(8):483-489.

Speck, T. (1991). Changes of the bending-mechanics of lianas and self-supporting taxa during ontogeny. *Proc. II. Int. Symp. SFB 230, Mitteilungen des SFB 230* 6:89-95.

Speck, T. (1994). A biomechanical method to distinguish between self-supporting and non self-supporting fossil plants. In Chaloner, W. (ed.), *Fossil Plants as Palaeoenvironmental Indicators*, *Rev. Pal., Pal.* **81**: 65-82.

Speck, T., and Rowe, N. P. (1994). Biomechanical analysis of *Pitus dayi*: Early seed plant vegetative morphology and its implications on growth habit (submitted for publication).

Speck, T., and Vogellehner, D. (1992). Fossile Bäume, Spreizklimmer and Lianen. Versuch einer biomechanischen Analyse der Stammstruktur. *Cour. Forsch.-Inst. Senckenberg* **147**:31-53.

Speck, T., Spatz, H.-C., and Vogellehner, D. (1990). Contribution to the biomechanics of plants. I. Stabilities of plant stems with strengthening elements of different cross-sections against weight and wind forces. *Bot. Acta* **103**:111-122.

Speck, T., Brüchert, F., and Bogenrieder, A. (1993). A biomechanical and ecological comparison of *Alnus glutinosa* with *Alnus viridis*-clans from the Alps and the Black Forest. *J. Exp. Bot.* **44** (Suppl.):50.

Speck, T., Rowe, N., and Vogellehner, D. (1994). Growth habits in plants and their correlation with stem's functional anatomy and biomechanics. II. Fossil plants with secondary growth. *Architecture, structure, mécanique de l'arbre, sixième Séminaire Interne, Montpellier 1994* (in press).

Vincent, J. F. V. (1990). *Structural Biomaterials*, 2nd ed., Princeton University Press, Princeton, NJ.

Vincent, J. F. V. (1992). *Biomechanics—Materials: A Practical Approach*, IRL Press at Oxford University Press, Oxford.

Wessolly, L. (1989). Zwei neue zerstörungsfreie Meßmethoden für Baumuntersuchungen zur Erhaltung der Verkehrssicherheit. *Garten und Landschaft* **89**(9):60-62.

Wessolly, L. (1991). Verfahren zur Bestimmung der Stand- und Bruchsicherheit von Bäumen. *Holz als Roh- Werkstoff* **49**:99-104.

Wessolly, L. (1993). Stand- und Bruchsicherheit von Bäumen. Leistungsfähigkeit und Grenzen der Zugversuche. *Das Gartenamt* **42**:486-491.
# The Biomechanics of Root Anchorage<fnref n="1" />

A. R. Ennos<fnref n="2" />

The mechanics of anchorage has only recently started to be studied, by combining theoretical analysis derived from materials science and foundations technology with experimental investigations in which plants are pulled out of the ground or pushed over. Analysis shows that the mechanical role of roots is restricted to basal regions, the strengthening of which imposes an anchorage cost. Root system morphology is correlated with aboveground morphology: pro-cumbent and climbing plants have fibrous root systems, while upright ones have tap or plate root systems. Experimental studies of anchorage have shown two mechanisms of failure in different plants, which may be related to their pattern of development: In trees and dicots windward roots are pulled out of the ground, while the leeward roots of monocots are pushed farther into it. Despite the recent interest, many aspects of root biomechanics have been largely ignored. Further research in these areas could have both scientific and practical benefits, to help reduce the incidence of lodging in crops and windthrow in trees.

**KEY WORDS:** roots; biomechanics; anchorage; soil.

## INTRODUCTION

Textbooks of botany usually list three functions of roots: to anchor the plant in the ground, to absorb water and nutrients, and to act as a site for food storage. Subsequent discussion, however, usually concentrates on absorption, and apart from brief mentions of the prop roots of maize and mangroves, the mechanical function of roots is ignored. Until recently root anchorage has also been ignored by researchers, partly because it seems intuitively obvious that uprooting of a

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²School of Biological Sciences, Manchester University, Oxford Road, Manchester M13 9PL, England.</footnote>
plant will be resisted by the friction between the roots and the soil; research on such an apparently obvious mechanism has therefore seemed unnecessary. Since roots are underground, they are also both easier to ignore and much more difficult to study than the aboveground parts of plants. Finally, *all* mechanical aspects of plant design have been neglected over the past hundred years. After the excellent mechanical studies of German botanists such as Schwendener, Sachs, Pfeffer, and Metger, much of whose work is summarized by Haberlandt (1914), attention moved to the fields of physiology and biochemistry. Indeed, such has been the swing away from the mechanical sciences in botany that the recent renaissance in plant biomechanics has been inspired largely by zoologists and engineers (Wainwright *et al.*, 1976; Gordon, 1978; Vogel, 1989).

Whatever its cause, the lack of attention and thought has led botanists to hold several misconceptions about root anchorage, which are only now being corrected. Since root systems have very large surface areas for adequate absorption [Dittmer's (1937) rye plants had surface areas of over 200 m²], it has been assumed that they are heavily overdesigned for their anchorage function and that the absorption function alone influences root system architecture. Without clear mechanical models of anchorage it has also been assumed that roots are always loaded in tension. Many experimental studies of lodging in cereal crops have therefore involved tensile tests on the roots (Neenan and Spencer-Smith, 1975) and measurements of the force required to pull individual roots or complete root systems out of the soil (Donovan *et al.*, 1982).

To correct these misconceptions, both theoretical and experimental studies have been performed to investigate the mechanisms by which forces are transmitted from the roots to the soil. At last we are beginning to understand how roots anchor plants and can suggest how and why the anchorage of crop plants (Pinthus, 1973) and forestry trees may be improved.

## THEORETICAL STUDIES

The process of pulling roots from the soil is analogous to the extraction of fibers from a composite material or pulling up foundation piles and can be analyzed in the same way (Ennos, 1989). Because roots are less stiff than the soil in which they are embedded, the top of the root will stretch and shear past the soil if it is pulled up. The soil will fail in shear but, because of its plastic nature, will continue to resist the displacement. Tension will therefore gradually be transmitted from the root to the soil. The greater the upward force applied to the root, the greater the area of soil which will fail and the farther down in the soil the root will be stretched.

The system may fail in one of two ways. If the root is relatively short and strong, the root will be pulled out of the ground, the force required being proportional to its surface area. In contrast, a long, narrow root will break at
the top before lower regions are stretched. In practice unstrengthened roots more
than a few millimeters in length will break, before being pulled out, and at very
low forces (Ennos, 1990).

Design for Resisting Uprooting

These considerations have profound implications for the mechanical design
of root systems. To improve the anchorage performance of a root it must be not
only lengthened but also strengthened, especially at its base, to withstand tensile
forces. The radicles of sunflower seedlings have just such a strengthened basal
region, which is not only increased in thickness by secondary thickening, but
also densely covered with root hairs, which improve the bond between the root
and the soil (Ennos, 1989). These adaptations allow this region to withstand
upward forces of over 0.6 N. Similar strengthening of the stele toward the base
is also found in the seminal roots of wheat (Ennos, 1991a), which must anchor
the seedling against uprooting, and indeed in most other plant species.

The production of strengthening material, which will not improve absorp-
tion, represents a real cost of anchorage. This cost can be minimized by strength-
ening only the basal regions of the roots which have to resist the largest forces,
and by having many narrow roots radiating from the base of the stem rather
than a single thick one (Fig. 1A). Since many thin roots have a larger surface
area than a single root of the same cross-sectional area, they will transfer tension
more rapidly to the soil and need not be strengthened so far down their length
(Ennos, 1993a). However, if there are too many roots, the soil may fail in
tension, and the roots and surrounding soil may be pulled intact from the ground
at a low force. An optimal anchorage system should therefore contain an inter-
mediate number of 10-100 roots (Ennos, 1993a).

<figure><img src="image_193.png" /><figcaption>Fig. 1. Alternative types of anchorage system. A rope-like system (A) efficiently anchors a plant against uprooting, but to resist horizontal forces, rigid elements are also required. Plants may develop a tap root system (B), a system using rigid lateral roots emerging from an expanded stem base (C), or a lateral rosette of stalks (D) to lean against the ground.</figcaption></figure>
Procumbent and climbing plants do indeed have root systems of the form that this model of anchorage predicts. Grasses and sedges have many fibrous roots emerging from the base of the shoot system, while dicots have root axes which branch rapidly near the base (Ennos and Fitter, 1992). However, in all these plants only the basal regions appear to be adapted for anchorage and are usually adequate to resist uprooting forces which exceed the tensile strength of the stem. The large numbers of fine, distal roots are unstrengthened and must have a role purely in absorption.

## Design for Resisting Overturning

Most tall, stiff-stemmed plants are much more likely to be blown sideways by the wind or to be pulled over by their own weight than to be pulled upward. Such plants must therefore have an anchorage system which resists being rotated in the soil, one that, unlike those of climbing plants, contains at least one rigid element. One solution is to produce a stiff "tap root" by secondary thickening of the main root axis (Fig. 1B), to resist toppling directly, like the point of a stake (Ennos and Fitter, 1992; Ennos, 1993a), and to act as the insertion for lateral roots, which can act like guy ropes to stabilize the plant further. A second solution would be to attach relatively stiff roots to the perimeter of a broad stem base (Fig. 1C); the resistances of these roots to bending and to axial motion through the soil should both help stabilize the plant (Ennos, 1991b). A third alternative is for the stem of the plant to support itself by leaning on stiff side shoots while a fibrous root system keeps the base of the stem in the soil (Fig. 1D).

The actual system a plant uses depends on many factors, including its ancestry, ecology, and size. Tap roots are used by the majority of herbaceous dicots and young trees. However, older trees tend to have a more complex plate system (Coutts, 1983; 1986), probably because, with their thick trunks, a tap root system would use too much material (Ennos, 1993a). Many trees actually extend the trunk outward by means of buttresses (Mattheck, 1991; Ennos, 1993b) and anchor the trunk with widely spaced sinker roots. Such a system is also used by the broad-stemmed Himalayan balsam *Impatiens glandulifera* (Ennos et al., 1993a) and by monocots such as wheat and maize, which, being incapable of secondary thickening, cannot produce a tap root. A rosette of stalks is common only in weeds of bare ground, since these stalks would otherwise be shaded out. The stems of these plants are used to hold up flower stalks for better pollination of flowers and distribution of seed.

## EXPERIMENTAL STUDIES OF ANCHORAGE

Despite the success of theoretical studies in explaining the variation in root system morphology between plants of different growth forms, experimental studies of anchorage are also needed to determine exactly how failure occurs. The
first competent experimental study was carried out by Coutts (1983, 1986), who investigated the mechanical resistance of Sitka spruce trees to being overturned by the wind. He developed a practical approach to the study of anchorage which has been used in all subsequent studies.

Most importantly, Coutts emphasized the need actually to observe what is going on underground and developed two methods to do this: He cut vertical trenches in the soil alongside the trunk of his trees to expose a "cross section" of the root system, the morphology of which could be examined and the movements of which could be observed as the tree was being pulled over parallel to the trench. He also inserted microphones into the ground to record when the roots and soil were breaking.

Coutts also monitored the force required to pull his trees over *throughout* the process of uprooting and was able to relate the record of force to the movements and failure of roots and soil which he had observed directly. This allowed him to suggest several components of anchorage. The relative importance of each component could then be determined by carrying out consecutive pulls between which particular components of anchorage were destroyed. Typical experimental procedures involved cutting of roots or trenching around the root system.

Finally, Coutts emphasized the importance of measuring the mechanical strength of both the roots and the soil, as these will influence both the mode of failure and the force required.

To date, published experimental studies of anchorage have been carried out on only five species of mature plants: Sitka spruce (Coutts, 1983, 1986), sunflowers *Helianthus annuus* (Ennos et al., 1993a), Himalayan balsam *Impatiens glandulifera* (Ennos et al., 1993a), wheat *Triticum aestivum* (Ennos, 1991b; Crook and Ennos, 1993), and maize *Zea mays* (Ennos et al., 1993b). Already, however, a pattern is emerging of the manner in which different plant types fail. In trees and dicots, movement of the root system is centered on the *leeward* side of the stem or trunk (Fig. 2A), and the windward roots tend to snap or be pulled out of the soil. Anchorage resistance can be split into several components: the resistance of windward roots to pullout and the soil to tensile failure, the bending resistance of the leeward hinge, and the weight of the root-soil plate or ball. The latter tends to be most important in trees, probably because of scale effects (Ennos, 1993a). In contrast, movement of the root systems of the monocots is centered on the *windward* side of the stem (Fig. 2B) and the cone of roots is pushed down *into* the soil. In these plants the components of anchorage result from the bending resistance of the stiff, basal roots and from the resistance of *compression* of the soil beneath the root cone.

Quite why there are these differences in behavior is unclear. They may occur because the roots of the monocots diverge at a lower angle. They would therefore resist upward forces more strongly but, with their narrower base, be
<figure><img src="image_194.png" /></figure>

<figure><img src="image_195.png" /><figcaption>Fig. 2. The contrasting mechanisms of anchorage failure in (A) the sunflower *Helianthus annuus*, a dicot, and (B) winter wheat *Triticum aestivum*, a monocot. In the sunflower, rotation occurs around the leeward side of the stem and windward roots are levered out of the soil. In wheat, rotation occurs around the *windward* side of the stem and leeward roots are pushed *into* the soil.</figcaption></figure>
pushed down in the soil more easily. The difference in anatomy of dicots and monocots might also have this effect; the roots of dicots and trees are usually solid rods of lignified tissue (though *Impatiens* is an exception). In contrast, monocot roots are stiffened in bending by an external ring of lignified material (Haberlandt, 1914), which results in a rather brittle structure. The lack of secondary thickening in monocots certainly constrains the form of their root systems since they must continually produce ever thicker adventitious roots to provide adequate anchorage as they develop.

## UNSOLVED PROBLEMS

Despite the attention in recent years, work has only scratched the surface of this complex topic. There are many aspects of anchorage that have never been tackled, some of which may be indeed be insoluble because of the great complexity and intraspecific variability of root system form. First, much basic work needs to be performed on the resistance of simple roots, branched roots, and collections of roots to motion through soil. Model tests would undoubtedly give useful information in this area. More studies also need to be carried out on the tensile root systems of climbing and prostrate plants.

The effect of soil strength and texture on anchorage have so far been largely ignored. It has been found that, as expected, failure occurs more distally and at lower forces when the strength of soil is reduced by wetting, but the effect of altering soil texture is more difficult to assess. Comparative studies need to be carried out on models and real plants in sandy and clay soils.

The pattern of development of anchorage systems is also an area which has been little studied. Undoubtedly, the root system becomes stronger throughout the life of a plant to counter the larger forces to which it is subjected, but the exact pattern of development may be critical to its ability to stand up. So far only one study has been carried out in this area, on the development of the anchorage systems of wheat (Crook *et al.*, 1994). It was found that the anchorage system developed its strength slightly in advance of the shoot system, providing a firm foundation for the later growth of the shoot. It is essential to repeat this study on other, wild species.

Little attention has also been paid to the effect of environmental conditions on root growth and anchorage strength. Only two studies have been carried out specifically on the effect of mechanical stimulation on root growth. Gartner (1994) found that shaking the *stems* of tomato plants caused an increased thickening of the base of the tap roots, a response which would have increased the anchorage strength. Working on young Sitka spruce trees subjected to a unidirectional wind, Stokes *et al.*, (1994) found that windward roots exhibited a greater degree of branching, which might improve their effect in preventing windthrow in the direction of the prevailing wind. Many more studies are needed
in this area. It should prove possible to stimulate individual roots of plants grown in nutrient culture or to measure the stresses to which individual roots are subjected with strain gauges. Such an approach is already being taken to examine the development of the buttress and I-shaped roots of trees (Ennos, 1994).

Finally, it is known that soil conditions profoundly affect the primary growth of roots (see Russell, 1977), but the effect on their secondary growth, which will influence anchorage strength, is unknown. Studies in this area are important since they will allow us to determine whether lodging of crop plants may be reduced by altering the management of the soil in which they are growing.

## REFERENCES

* Coutts, M. P. (1983). Root architecture and tree stability. *Plant Soil* **71**:171-188.

* Coutts, M. P. (1986). Components of tree stability in Sitka spruce on peaty gley soil. *Forestry* **59**:173-197.

* Crook, M. J., and Ennos, A. R. (1993). The mechanics of root lodging in winter wheat *Triticum aestivum* L. *J. Exp. Bot*. **44**:1219-1224.

* Crook, M. J., Ennos, A. R., and Sellers, E. (1994). Mechanical development of two winter wheat cultivars. *J. Exp. Bot.* (in press).

* Dittmer, H. J. (1937). A quantitative study of the roots and root hairs of a winter rye plant (Secale cereale). *Am. J. Bot*. **24**:417-420.

* Donovan, L. J., Jui, P., Kloek, M., and Nicholls, C. F. (1982). An improved method of measuring root strength in corn (*Zea mays* L.). *Can. J. Plant Sci.* **62**:223-227.

* Ennos, A. R. (1989). The mechanics of anchorage in seedlings of sunflowers *Helianthus annuus*. *New Phytol.* **113**:185-192.

* Ennos, A. R. (1990). The anchorage of leek seedlings: The effect of root length and soil strength. *Ann. Bot.* **65**:409-416.

* Ennos, A. R. (1991a). The mechanics of anchorage in wheat *Triticum aestivum*. L. I. The anchorage of wheat seedlings. *J. Exp. Bot.* **42**:1601-1606.

* Ennos, A. R. (1991b). The mechanics of anchorage in wheat *Triticum aestivum* L. II. Anchorage of mature wheat against lodging. *J. Exp. Bot.* **42**:1607-1613.

* Ennos, A. R. (1993a). The scaling of root anchorage. *J. Theor. Biol.* **161**:61-75.

* Ennos, A. R. (1993b). The function and formation of buttresses. *Trends Ecol. Evol.* **8**:350-351.

* Ennos, A. R. (1994). Development of buttresses in rainforest trees: The influence of mechanical stress. In Coutts, M. P., and Grace, J. (eds.), *Wind and Wind-Related Damage to Trees*, IUFRO Conference (in press).

* Ennos, A. R., and Fitter, A. H. (1992). Comparative functional morphology of the anchorage systems of annual dicots. *Funct. Ecol.* **6**:71-78.

* Ennos, A. R., Crook, M. J., and Grimshaw, C. (1993a). A comparative study of the anchorage systems of himalayan balsam *Impatiens glandulifera* and mature sunflower. *Helianthus annuus*. *J. Exp. Bot.* **44**:133-146.

* Ennos, A. R., Crook, M. J., and Grimshaw C. (1993b). The anchorage mechanics of maize *Zea mays*. *J. Exp. Bot.* **44**:147-153.

* Gartner, B. L. (1994). Root biomechanics and whole-plant allocation patterns: Responses of tomato to simulated wind. *J. Exp. Bot.* (in press).

* Gordon, J. E. (1978). *Structures or Why Things Don't Fall Down*, Penguin Books, London.

* Haberlandt, G. (1914). *Physiological Plant Anatomy*, Macmillan, London.

* Jeronimidis, G. (1980). Wood, one of mature's challenging composites. In Vincent, J. F. V., and Currey, J. D. (eds.), *The Mechanical Properties of Biological Materials*, S.E.B. Symposium 34, Cambridge University Press, pp. 169-182.

* Neenan, M., and Spencer-Smith, J. L. (1975). An analysis of the problem of lodging with particular reference to wheat and barley. *J. Agr. Sci.* **85**:495-507.
Pinthus, M. J. (1967). Spread of the root system as an indicator for evaluating lodging resistance of wheat. *Crop Sci.* 7:107-110.

Russell, R. S. (1977). *Plant Root Systems: Their Function and Interaction with the Soil*, McGraw-
Hill, London.

Stokes, A., Fitter, A. H., and Coutts, M. P. (1994). Responses of young trees to wind: Effects on root growth. In Coutts, M. P., and Grace, J. (eds.), *Wind and Wind-Related Damage to Trees*, IUFRO Conference (in press).

Vogel, S. (1989). Drag and reconfiguration of broad leaves in high winds. *J. Exp. Bot.* 40:941-949.

Wainwright, S. A., Biggs, W. D., Currey, J. D., and Gosline, J. M. (1976). *Mechanical Design in Organisms*, Wiley, New York.
<figure><img src="image_196.png" /></figure>
# Wind Damage to Forests<fnref n="1" />

B. A. Gardiner<fnref n="2" /> and C. P. Quine<fnref n="2" />

This paper provides a brief summary of research on understanding the mechanics of forest wind damage and the development of predictive and preventive management strategies. It concentrates on work carried out in Great Britain and the experience of British foresters. The first part of the paper details the mechanics of wind damage, how turning moments are produced by the wind and how the tree resists. It also discusses the ways in which trees can fail either by stem breakage or complete overturning. The paper then describes the specific tree and environmental factors that influence the stability of trees and how these are modified by management practices. Finally, an attempt is made to show how it is possible to quantify the probability of wind damage to forests and to develop a national wind risk system.

**KEY WORDS:** wind damage; forest; trees; biomechanics.

## BACKGROUND

This paper discusses research, particularly in Britain, carried out in order to predict and prevent forest damage and illustrates the ways in which biomechanics can help our understanding.

Severe storms such as that on 31 January 1953, which caused substantial damage to forests in Northeast Scotland (Quine, 1989), alerted British foresters to the difficulties of growing exotic conifer plantations in the exposed uplands. The wind climate of these areas is severe and the soils are also generally waterlogged during the winter. Research into site limitations included tree pulling experiments in which mature trees of different size and species, growing on a

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²Forestry Commission, Northern Research Station, Roslin, Midlothian EH25 9SY, Scotland.</footnote>
variety of soils, were winched over and the maximum turning moments required to uproot them recorded. The physical details of the tree were also measured including stem weight, diameter, height, root weight, root depth, and root spread (Fraser and Gardiner, 1967). Over 1000 trees were pulled and the subsequent database provides an essential underpinning to much of our current work.

The results from the tree pulling work were combined with estimates of the national wind climate obtained using tatter flags (Miller *et al.*, 1987) to create a "windthrow hazard classification" (Miller, 1985). This classification combines scores for forest location, elevation, topographic shelter, and soil type to derive an estimate of the height at which trees are likely to begin blowing over and how fast this damage will progress. The system has been widely adopted in Great Britain as a management and investment tool but it is an empirical model which cannot be easily adapted to different situations. It is now being replaced by a physically based model derived from an understanding of the mechanical behavior of trees and the nature of airflow in complex terrain. The availability of this type of model will allow forest managers to make contingency plans and select appropriate preventative techniques.

## MECHANICS OF WIND DAMAGE

Wind damage occurs when the overturning moment due to wind pressure on the canopy can no longer be resisted by the tree and some part of the tree fails. Failure may result in two ways. If the critical stress of the wood is exceeded, wood fibers begin to break or buckle, leading to stem or branch breakage. Alternatively, if the soil/root interface is unable to provide a resistive moment sufficient to counteract the overturning moment, the tree is uprooted and windthrow occurs. Figure 1 illustrates the different components we need to consider when discussing tree mechanics.

The overturning moment (or torque) applied to the tree by the wind is the product of the wind force and the distance between the point at which the force acts and the root hinge about which the tree turns. The total bending moment at the base of the tree is the sum of the force of the wind at each height in the canopy multiplied by that height. For simplicity the force on plantation trees can be regarded as acting at a single point within the canopy (center of pressure in Fig. 1) close to the zero-plane displacement height (d) which is between 70 and 80% of tree height (Gardiner, 1994). Generally, the force on an object due to the wind increases as the square of the windspeed, but because trees are flexible and the canopy streamlines, the wind force on trees increases almost linearly with windspeed (Gardiner *et al.*, 1994). As the tree is bent an additional turning moment develops due to the overhanging mass of the canopy and stem, the magnitude of which is the product of the displaced weight and its horizontal displacement from the root hinge point (see Fig. 1). This additional turning
<figure><img src="image_197.png" /><figcaption>Fig. 1. Components of tree mechanics (after Bell *et al.*, 1991).</figcaption></figure>

moment can be considerable, providing up to 30% of the total overturning moment. If the canopy is wet or covered in snow, the self-weight moment may exceed that due to the wind.

Stem or branch breakage occurs when the wood fibers are no longer able to withstand the compressive or tensile forces exerted on them by the overturning moment. Once a few fibers break or buckle the stress increases (due to reduction in cross-sectional area), additional fibers break, and a catastrophic failure occurs. On other occasions bending leads to stresses normal to the stem surface, causing delamination and rupturing of the wood. This is regarded as a common cause of stem breakage close to the ground following root buttress delamination (Mat-
theck and Bethge, 1990). Different species have different failure modes. Pines are vulnerable to breakage in the crowns at a branch whorl, particularly if they are fast growing at wide spacings, while spruces tend to break in the lowest third of the stem. The shape of the stem determines where the stress is greatest, although there is increasing evidence that trees grow in a manner designed to keep surface stresses as even as possible throughout the stem and roots (Mattheck, 1991). This adaptive growth helps to explain the details of tree shape including stem taper, buttressing, and root morphology. Trees that grow in situations exposed to the wind will be more tapered and invest a greater proportion of their available biomass in root development than those in sheltered situations. This may explain the observed stability of trees at the edge of forests during major storms. The exact biochemical mechanism by which trees modify their growth in response to applied stresses is not completely understood but appears to involve the release of ethylene in those cells that are most highly stressed (Telewski, 1994).

Windthrow occurs when the soil and root system can no longer provide a large enough resistive moment to counteract the turning moment due to the wind. The strength of the tree anchorage depends on the mechanical properties of the roots and soil and on the size and architecture of the root system. Four components of anchorage have been identified (Coutts, 1986). When the tree is displaced by the wind the weight of the roots and adhering soil helps to stop the tree from falling over. The soil underneath and around the root plate has to be physically broken and this contributes a further resistance. The tensile strength of the roots on the windward side of the tree is a third component of the resistive force and the resistance of the roots and soil to bending in the hinge zone provides a fourth component. For small displacements the soil strength is the greatest contributor to the resistive moment but further displacement breaks the soil and fine roots (illustrated by a crack in Fig. 1) and tension in the major roots and soil/root weight becomes more important. The soil/root system is subject to fatiguing by continuous motion and the force required to pull over a tree is reduced by repeated cyclic loading (O'Sullivan and Ritchie, 1993). This suggests that trees may become more vulnerable with successive winter storms, although the magnitude of increase in vulnerability and the ability of the system to recover during subsequent growing seasons require investigation. Although the soil/root system provides the major resistance to overturning, a further contribution is provided by crown contact with neighboring trees, and this can be particularly important in trees grown close together (see Fig. 1).

The resistive moment can be obtained from tree pulling experiments and an estimate then made of the forces required to blow trees over. However, the estimated forces appear to suggest that mean wind speeds of over 50 m s⁻¹ are necessary for windthrow when, typically, trees begin to blow down at just over 20 m s⁻¹. The reasons for this discrepancy are the gusty nature of the wind and
the dynamical behavior of the trees themselves. Trees subjected to wind loading can be regarded as forced damped harmonic oscillators (Gardiner, 1992) and their motion described by the standard equations for such first-order systems (Thompson, 1988). Coherent gusts of wind which form over the canopy intermittently load the tree, causing them to oscillate about their rest position for a few seconds until damping reduces the motion. The sudden impact and higher windspeed during gust passage result in much higher turning moments than those due to the mean wind. Experiments in the field and the wind tunnel have shown that the extreme forces on trees are 8-10 times higher than the mean forces (Gardiner *et al.*, 1994). Resonance at the tree's natural frequency does not appear to be a contributory factor (Gardiner, 1994), although as the windspeed increases gusts become more frequent (Paw U *et al.*, 1992) and they could begin to get in phase with tree motion.

An excellent review of tree mechanics is provided by Wood (1994).

## FACTORS INFLUENCING TREE STABILITY

Stability is a measure of a tree's vulnerability to wind damage. There are two groups of factors that influence tree stability: those that affect the strength of the overturning moment due to the wind and those that affect the resistive moment that the roots can provide.

Any change to canopy or stem shape or to the aerodynamic roughness ($z_0$) of the canopy will change the overturning moment experienced by the tree. Planting density, uniformity of growth, respacing, thinning, and cutting rides or roads all affect the tree shape. Practices which result in a sudden opening of the canopy will increase the loading on the tree and the magnitude of this increase is an extremely strong function of gap size (Stacey *et al.*, 1994). Therefore, trees exposed following thinning or the cutting of a new edge will be more vulnerable than before. However, adaptive growth will lead to recovery in response to the higher wind loading of the exposed position. The duration of this recovery relative to the frequency of damaging winds will be important (Quine, 1994). If root development is not restricted and the canopy can reform, the trees will eventually become as stable as those in an unthinned stand.

Edges, whether adapted to the wind or newly created by clear felling of a neighboring stand, are potential sites for windthrow. Experiments in the wind tunnel (Gardiner *et al.*, 1994) and in the field (Mattheson, 1992) have shown a beneficial effect of shaping the edge to create a more gradual change. This may be carried out by topping edge trees, planting slower-growing species such as broad-leaves along the edge, or having a gradual increase in tree planting density into the stand. Initial planting density or respacing does not appear to affect tree stability provided that root development is not restricted. Even though wind loading increases as the tree spacing is increased (Gardiner *et al.*, 1994), the
tree can compensate for this by enlarging the root system. This compensating growth in the roots may not be possible where the soil is particularly waterlogged or there is furrow plowing, and consequently in these situations stability will be reduced. Wider spacing does increase the stem diameter and taper, which increases the stem strength, and so wider-spaced trees have less chance of stem breakage.

The resistive moment will be reduced by factors which restrict root development. This is particularly true if the growth of the windward roots is affected because of their importance for stability. Plow furrows, drains (see Fig. 1), and damage from vehicles used for extraction can all prevent root spread or cause the roots to branch close to the base of the tree. Similarly, damage to leeward roots will reduce the distance to the root hinge, which reduces the mechanical advantage. Root system asymmetry due to poor planting practice will have similar effects. Root depth is a function of the soil type (for example ironpans restrict root penetration) and the depth of the winter water table. The strength of the soil/root attachment and the soil itself are also strong functions of soil wetness. Ponding, spring lines, or blocked drains can crucially reduce soil strength. Very often the beginning of windthrow is associated with particularly wet areas in the forest and subsequently spreads from that point.

A fuller review of these issues is provided by Coutts *et al.* (1994).

## ASSESSING THE RISK OF DAMAGE

Understanding the mechanics of wind damage allows the vulnerability of a particular tree to be calculated, where vulnerability is defined as the mean windspeed at which the tree will break or be uprooted. Assessing the risk of wind damage involves calculating the probability of this windspeed occurring at that particular site and requires a knowledge of the national wind climate (Quine, 1994), and the modification by topography and surface roughness.

Britain has a severe wind climate compared to the rest of Europe. Strong winds typically result from the passage of Atlantic depressions and only very rarely originate from thunderstorms, tornados, or squalls. Approximately 150 depressions affect Britain every year, crossing from west to east, but they vary in track (and hence area affected) and in intensity (and hence windspeeds experienced). Circulation around a depression is counterclockwise, and the strongest winds are commonly found to the west and south of the center of the low pressure. Most depressions track to the north and west of Britain, and their cumulative influence is responsible for the tendency to stronger winds in northern and western Britain. However, the variability in track can, albeit infrequently, bring strong winds to any part of the country; The general pattern of damage will then bear little relationship to the predictions of a generalized model. The storm of October 1987 is a good example of this variability (Quine, 1989).
Individual lowland forests were extensively damaged, while upland forests, classed as higher risk, remained undamaged. The stochastic nature of storm occurrence means that the level of damage may vary significantly from year to year; a succession of quiet winters can be followed by a sequence of damaging storms. The occurrence of a particularly severe storm in one year does not modify the chance of the same intensity of storm occurring in the following year. Thus, return periods, which are often quoted to illustrate the rarity of storms, represent the long-term average interval between exceedance of specified windspeeds, and not a prescribed cycle.

At the forest scale (kms × kms) the wind climate may vary rapidly, depending on the elevation, local shelter, and presence of surrounding forest. Calculating windspeed variation in complex forested terrain is a difficult task and two approaches have been taken. The first approach is to use computer airflow models in which the equations of fluid flow over digitized topography and surface roughness are calculated (Inglis *et al.*, 1994). The second approach is to use geographic predictors such as elevation, angle to the skyline, slope, aspect, and distance from the coast to calculate the variation in windiness (Hannah *et al.*, 1994). Both approaches have advantages and disadvantages. Airflow models are based on sounder physical principles than geographic predictors and do not require calibration for a particular area. However, the linearized models used to date are unable to deal with flow separation in the lee of hills and valleys even in relatively gentle terrain (slopes <10°). Comparison between field measurement and model prediction shows that the models are conservative, underestimating the speed-up on hills and the speed reduction in their lee. The geographic predictor method is more accurate than airflow models for a particular area, but unfortunately correlations developed in one area may not be applicable elsewhere, and so their general use is less straightforward.

## THE FUTURE

The elements of a new predictive method for assessing the risk of wind damage to British forests are now in place. A detailed understanding of the mechanical behavior of plantation trees along with the data available in the tree-pulling database can be used to calculate the windspeed at which damage may be expected for a particular tree. Such a calculation will be possible for the common plantation species and soil types and will take into account cultivation treatment, management practices, and spacing. The natural variability of trees will be accounted for by calculating the spread in vulnerability of the whole stand. The probability of damaging winds occurring at the precise location of the forest stand will then be calculated using a combination of wind climate information and airflow model or geographic predictor methods to account for local topographic detail. The new risk model will be developed over the next
few years and will replace the existing "windthrow hazard classification." It will be a geographic information system (GIS)-based model so that the necessary forest and soil inventory information, already available to the forester, can be combined with topographic data to produce wind damage probability maps. Furthermore, the interactive nature of the system will allow the consequences of any management operation to be assessed ahead of time.

## REFERENCES

Bell, H. J., Dawson, A. R., Baker, C. J., and Wright, C. J. (1991). Tree stability. In Hodge, S. J. (ed.), *Research for Practical Arboriculture*, HMSO, London.

Coutts, M. P. (1986). Components of tree stability in Sitka spruce on peaty gley soil. *Forestry* **50**:173-197.

Coutts, M. P., Gardiner, B. A., Pyatt, D. G. and Quine, C. P. (1994). *Forests and Wind: Management to Minimise Damage*, Forestry Commission Bulletin, HMSO, London (in press).

Fraser, A. I., and Gardiner, J. B. H. (1967). *Rooting and Stability of Sitka Spruce*, Forestry Commission Bulletin 40, HMSO, London.

Gardiner, B. A. (1992). Mathematical modelling of the static and dynamic characteristics of plantation trees. In Franke, J., and Roeder, A. (eds.), *Mathematical Modelling of Forest Ecosystems*, Sauerländer's Verlag, Frankfurt am Main, pp. 40-61.

Gardiner, B. A. (1994). Wind-tree interactions. In Coutts, M. P., and Grace, J. (eds.), *Wind and Wind Related Damage to Trees*, Cambridge University Press, Cambridge.

Gardiner, B. A., Stacey, G. R., Belcher, R. E., and Wood, C. J. (1994). Field and wind-tunnel assessment of the implications of respacing and thinning on tree stability (submitted for publication).

Hannah, P., Palutikof, J. P., and Quine, C. P. (1994). Predicting windspeeds for forest areas in complex terrain. In Coutts, M. P. and Grace, J. (eds.), *Wind and Wind Related Damage to Trees*, Cambridge University Press, Cambridge.

Inglis, D. W. F., Choularton, T. W., Stromberg, I. M. F., Gardiner, B. A., and Hill, M. (1994). Testing of a linear airflow model for flow over complex terrain and subject to structured, stable stratification. In Coutts, M. P. and Grace, J. (eds.), *Wind and Wind Related Damage to Trees*, Cambridge University Press, Cambridge.

Mattheck, C. (1991). *Trees: The Mechanical Design*, Springer-Verlag, Berlin.

Mattheck, C., and Bethge, K. (1990). Wind breakage of trees initiated by root delamination. *Trees* **4**:225-227.

Mattheson, P. (1992). *Stabilising Effect from Pruning and Topping in Newly Exposed Edges of Old and Middle-aged Norway Spruce*, Internal Report, Forskningscentret for Skov & Landskab. Vejle, Denmark (Danish).

Miller, K. F. (1985). *Windthrow Hazard Classification*, Forestry Commission Leaflet 85, HMSO, London.

Miller, K. F., Quine, C. P., and Hunt, J. (1987). The assessment of wind exposure for forestry in upland Britain. *Forestry* **60**:179-192.

O'Sullivan, M. F., and Ritchie, R. M. (1993). Tree stability in relation to cyclic loading. *Forestry* **66**:69-82.

Paw, U. K. T., Brunet, Y., Collineau, S., Shaw, R. H., Maitani, T., Qui, J., and Hipps, L. (1992). On coherent structures in turbulence above and within agricultural plant canopies. *Agr. For. Meteorol.* **61**:55-68.

Quine, C. P. (1989). Description of the storm and comparison with other storms. In Grayson, A. J. (ed.), *The 1987 Storm: Impact and Responses*, Forestry Commission Bulletin 87, HMSO, London.

Quine, C. P. (1994). Assessing the risk of wind damage to forests: Practice and pitfalls. In Coutts, M. P., and Grace, J. (eds.), *Wind and Wind Related Damage to Trees*, Cambridge University Press, Cambridge.
Stacey, G. R., Belcher, R. E., Wood, C. J., and Gardiner, B. A. (1994). Wind and wind forces in a model spruce forest. *Boundary-Layer Meteorol.* (in press).

Telewski, F. W. (1994). Wind induced physiological and development response in trees. In Coutts, M. P., and Grace, J. (eds.), *Wind and Wind Related Damage to Trees*, Cambridge University Press, Cambridge.

Thompson, W. T. (1988). *Theory of Vibration with Applications*, Allen and Unwin, London.

Wood, C. J. (1994). Understanding wind forces on trees. In Coutts, M. P., and Grace, J. (eds.), *Wind and Wind Related Damage to Trees*, Cambridge University Press, Cambridge.
<figure><img src="image_198.png" /></figure>
# Local Buckling and Other Modes of Failure in Hollow Plant Stems<fnref n="1" />

H.-Ch. Spatz<fnref n="2" /> and T. Speck<fnref n="3" />

Hollow structures such as the stalks of cereals have the advantage that stiffness against bending can be achieved at a relatively low biological cost. However, such structures are endangered by local buckling. An equilibrium approach led to a theoretical description of the process of ovalization for nonisotropic material as found in plants. In particular, the computations allow discussion of three primary causes of failure: (i) failure of the material due to compressive strains in the longitudinal direction, (ii) failure of the material due to strains in the tangential direction, and (iii) failure due to a buckling collapse of the structure. They also show how nodal thickenings stabilize hollow structures. A new experimental device allows observation of the bending behavior of hollow plant stems up to the point of failure of the material or the structuring. Fitting the data with the help of the theory yields a Young's modulus in the transverse direction, which in stalks is difficult to measure directly. A comparison of data for different cereals points to a contribution of the parenchyma to the resistance against local buckling.

**KEY WORDS:** local buckling; nodes; maximal bending moments; testing device; theoretical description.

## INTRODUCTION

Under compressive forces, but especially under bending loads, thin-walled hollow structures, like some hollow plant stems, are in danger of failing by local

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²Institut für Biologie III, Universität Freiburg i.Br., Schänzlestraße 1, D-79104 Freiburg i.Br., Germany.</footnote>

<footnote n="3">³Botanischer Garten, Universität Freiburg i.Br., Schänzlestraße 1, D-79104 Freiburg i.Br., Germany.</footnote>
buckling (Timoshenko and Gere, 1961; Wainwright et al., 1976; Romberger et al., 1993; Niklas, 1992). Theoretical approaches describing the phenomenon of local buckling are rare and confined mainly to local buckling under axial compressive stress (cf. Wainwright et al.; 1976). In engineering sciences most approaches apply only to very thin-walled cylindrical shells of isotropic material (cf. Brazier, 1927; Reissner, 1959; Seide and Weingarten, 1961; Axelrad, 1965; Stephens et al., 1974).

Niklas (1992) gives an excellent account of the problems arising if formulas from engineering sciences derived for hollow tubes built of isotropic material are transferred to highly anisotropic plant materials. Niklas and co-workers also contributed some of the few experimental studies on local buckling processes in hollow plant organs. Niklas (1989) studied local buckling in the horsetail (Equisetum hyemale) and found that in this plant, local buckling occurs mostly in the mechanically weak (meristematic) basal zone of internodes near the nodes. The transverse nodal diaphragms contribute only 2% to the stem's total weight but can increase the modulus of the stem by 17-32% (see also Niklas, 1992). Niklas and O'Rourke (1987) examined the influence of water potential on the mechanical properties of hollow leaves of Allium shoenoprasnum, taking into account the ovalization of the originally ring-shaped cross sections during bending. They also stress the potential stabilizing influence of turgescent pith parenchyma, which can nearly eliminate the danger of local buckling in core-rind structures because of a high stability against compression (Niklas, 1991, 1992; see also Ennos, 1993). Foley (1983) and Lu et al. (1987) mention, in their studies of the mechanical properties of corn stalks (Zea mays), the danger of failing by local buckling in old corn stalks with (partly) decayed central parenchyma. In this developmental stage the corn stalks no longer represent solid stems, as in younger developmental stages, but are essentially slender thin-walled tubes. Vincent (1983) and Vincent and Jeronimidis (1991) investigated the functional anatomy and the mechanical properties of the herbaceous hollow flower stems of the dandelion (Taraxacum officinale) and point to the importance of prestressing in the periphery also of hollow herbaceous stems. They further discuss the problem of local buckling of cells exposed to compression on the outermost leeward side of the bent stem.

Mosbrugger (1990) discusses constructural constraints in fossil trees with (naturally) hollow stems and, especially, the shortcomings in terms of branching frequency and branching angle. Mattheck and co-workers (1992, 1993; Mattheck, 1992) refer mainly, in their studies on hollow plant organs, to trees in which a central cavity respectively a central part of disintegrated, mechanically ineffective wood results from fungus infection. They discuss the problem of safety margins for hollow trees in terms of the ratio of wall thickness to radius and stress the point that different modes of failure may cause local buckling of a hollow structure (Mattheck and Breloer, 1993). Biomechanics and safety mar-
gins of hollow trees have also been studied by Wessolly and co-workers (Herbig *et al.*, 1988; Wessolly, 1988, 1991, 1993; see also Sinn, 1993, 1994; Rinn, 1994).

The theoretical approach developed in Freiburg allows calculation of crit-
ical bending moments for hollow cylinders of nonisotropic material, taking into
consideration different modes of primary failure. Our studies focus mainly on
biomechanics, especially local buckling processes, as related to the anatomy of
grass stalks, but the method can be used to describe the local buckling phenom-
enon in every kind of hollow tube, as, for example, hollow trees (Spatz, 1994).

THEORETICAL CONSIDERATIONS

Equilibrium Approach

As outlined previously (Spatz *et al.*, 1993), buckling of hollow tubes or
plant stems under bending loads (Fig. 1) can be described by an equilibrium
approach (Fig. 2). The considerations are summarized in this contribution. A
more detailed account is given elsewhere (Spatz and Speck, 1994). In essence,
the gain in energy upon releasing strain in the longitudinal direction is equal to
the loss in energy upon deforming the ring-shaped cross section to an elliptical
ring. Using energies (scalsars) greatly simplifies the problem in comparison to
approaches using forces or moments (vectors), which would render the problem
intractable even on large computers.

Upon bending under wind loads (cf. Wainwright *et al.*, 1976; Speck *et al.*, 1990) longitudinal tensile strains develop on the windward side and compressive strains on the leeward side of a plant stem. In hollow structures the walls will tend to move toward the "neutral plane," thus releasing part of the strain (Fig. 1). In our experimental setup care is taken to minimize the bending moment at either end of the plant stem, where it is mounted (see below). The bending line (Fig. 3) for this type of loading can best be described by a third-order spline function. For a constant second moment of area, this function minimizes the strain energy (Böhm and Gose, 1977). For large degrees of ovalization of the cross section, a fourth-order spline function is an even better approximation (Spatz and Speck, 1994). The spline function is used to describe the neutral plane, but also the bending line for the compression and for the tension side, at any degree of ovalization. Using the definitions in Fig. 3 the lengths can be calculated numerically. Knowing the Young's modulus of elasticity in the longitudinal direction $\xi_{long}$, the strain energies can be calculated for any degree of bending and ovalization.

Spatz *et al.* (1993) have shown that the energies for deforming a circular
ring to an elliptical ring can be calculated considering the degree of shearing of
the structure. Both radial and tangential shear strains develop upon ovalization.
<figure><img src="image_199.png" /><figcaption>Fig. 1. Schematic representation of a hollow plant stem with nodes. A wind force $F$ causes a bending moment. This induces stresses in the longitudinal direction. Upon ovalization of the cross section the longitudinal strains are partially relaxed.</figcaption></figure>

If the Young's moduli in the radial and tangential directions are known, the strain energies can be calculated by averaging over the entire ring and in a second step over the entire length of the hollow stem.

The process of buckling is simulated by carrying a given hollow tube through a series of discrete steps of increasing bending until failure either of the material or of the structure is observed (Fig. 4). The degree of bending can be described by the curvature in the vertex (Fig. 3). To obtain the degree of buckling, the bending line of the compression side is changed in small steps to $\Delta s + \Delta\Delta s$, until the sum of all mechanical energies reaches a minimum or

$$dE_{\text{long}} + dE_{\text{deform}} = 0 \qquad (1)$$

Correspondingly the equilibrium is determined for the tension side. Unless the
Equilibrium conditions

$0 = dE_{long}$ [Strains in longitudinal direction]

+ $dE_{shear}$ [Shear strains in radial direction and in tangential direction]

+ $dE_{trans}$ [Change in circumference]

+ $dE_{bend}$ [additional bending of the walls]

$dE_{long} = -\pi * \xi_{long} * \Delta l / l * r * w * dl$

$dE_{shear} = \pi * M * S(a) * l * r * w * dS(a)$

$M = M(\xi_{radial}, \xi_{tangential}, r, w)$

$dE_{trans} = 0,135 * \xi_{tangential} * \Delta u / u * l * w * du$

$dE_{bend} = \pm 3.4 * \xi_{long} * r * w^3 / l^3 * \Delta s * ds$

$l$ = length of bending line; $r$ = radius; $w$ = wall thickness;

$a$ = small axis of the ellipse; $S(a)$ = shear strain;

$u$ = outer circumference; $\Delta s$ = displacement

Fig. 2. The states which a hollow tube undergoes upon bending can be described as equilibrium between energies gained in relaxing longitudinal strains and energies necessary to deform a ring. The first two terms of the equilibrium conditions are dominating (compare Spatz *et al.*, 1993).
3.order spline function

a

<figure><img src="image_200.png" /></figure>

$$B = \Delta s * (1 - 6 * (x/d)^2 + j * 4 * (x/d)^3)$$

$$j = -1 \quad \text{for} \quad -d/2 < x < 0$$

$$j = +1 \quad \text{for} \quad 0 < x < d/2$$

$$\text{Curvature} = 12 * \frac{\Delta s_{\text{neutral}}}{d_{\text{neutral}}^2} \quad \text{for } x = 0$$

b

<figure><img src="image_201.png" /></figure>

$$\% \text{ Buckling} = 100 * \frac{\Delta\Delta s}{(r - w/2)}$$

$$= 100 * \frac{(r - a)}{(r - w/2)}$$

$$= 100 * \frac{(r_{\text{inner}} - a_{\text{inner}})}{r_{\text{inner}}}$$

Fig. 3. (a) The bending line on the compression side can be represented by a third-order spline function. $B(d, \Delta s)$ is the bending line without buckling. Ovalization is described by increasing the displacement to $\Delta s + \Delta\Delta s$ in small steps, leaving $d$ and $\Delta s$ constant. On the extension side buckling is described by decreasing the displacement in small steps. Strains in the longitudinal direction can be calculated by numerical calculation of the length of the bending line for both compression and the extension side. The curvature in the vertex is obtained as the second derivative of an equivalent bending line for the "neutral axis" $B(d_{\text{neutral}}, \Delta s_{\text{neutral}})$. The bending line depicted in this drawing corresponds to the experimental arrangement. (b) An enlarged segment at the vertex showing the definitions of the parameters used. The wall thickness $w$ stands for the thickness of the ring of sclerenchyma.
<figure><img src="image_202.png" /><figcaption>Fig. 4. The relation among bending moment, curvature, and degree of ovalization for a hollow tube of length *l* = 20 cm, radius *r* = 0.2 cm, and wall thickness *w* = 0.01 cm. The Young's modulus in the longitudinal direction is chosen as *ξ*<sub>long</sub> = 10 GN/m². For all computations shown here the Young's moduli in the radial and tangential directions are considered equal: *ξ*<sub>radial</sub> = *ξ*<sub>tangential</sub> = *ξ*<sub>trans</sub>. The numbers in the graphs indicate the ratio of *ξ*<sub>trans</sub>/*ξ*<sub>long</sub>. To account also for high degrees of buckling, a fourth-order spline function has been used (Spatz and Speck, 1994). For small degrees of buckling this is equivalent to the third-order spline function introduced in Fig. 3. The relations between bending moment and curvature are almost perfect parabolic functions.</figcaption></figure>
ratio of length/radius becomes lower than 30:1, the calculations lead to symmetrical results such that the cross section can be regarded as an ellipse. Knowing $\Delta\Delta s$ gives the small axes of the elliptical ring at equilibrium. This determines the axial second moment of area $I$,

$$I = (\pi/4) \cdot \{[a + (w/2)]^3 \cdot [b + (w/2)] - [a - (w/2)]^3 \cdot [b - (w/2)]\} \quad (2)$$

where $a$ is the small axis and $b$ the long axis of the ellipse and $w$ is the wall thickness. The bending moment is related to the curvature in the vertex by the basic equation

$$M = \text{curvature} \cdot I \cdot \xi_{\text{long}} \quad (3)$$

where $M$ is the bending moment at the vertex. $\xi_{\text{long}}$ is the Young's modulus in longitudinal directions.

At equilibrium the degree of buckling is defined as

$$\% \text{ buckling} = 100 \cdot \frac{r - a}{r - (w/2)} \quad (4)$$

where $r$ is the radius of the plant stem or tube before buckling (Fig. 3). The particular form is chosen, such that 100% buckling is reached, when the distal and the proximal wall touch each other, that is, if the structure is fully collapsed.

Figure 4 shows the relations among bending moment, curvature, and degree of buckling for a thin hollow tube. The mechanical behavior depends strongly on the Young's modulus in the radial ($\xi_{\text{radial}}$) and tangential ($\xi_{\text{tangential}}$) directions. Here we assume $\xi_{\text{radial}} = \xi_{\text{tangential}}$ and define it as $\xi_{\text{trans}}$.

## Modes of Failure

Two modes of failure are apparent in Fig. 4. At $\xi_{\text{trans}}/\xi_{\text{long}} = 1$ an almost-linear relationship between bending moment and curvature is observed. The hollow plant stem or tube ovalizes only a little, but upon bending the material reaches a critical compressive strain, chosen here as 1% (Fig. 5a). At $\xi_{\text{trans}}/\xi_{\text{long}} = 0.1$ the tube can be carried to higher curvatures, since ovalization reduces strains in longitudinal direction. But upon further bending the axial second moment of area is reduced to such an extent that the structure reaches the point of mechanical instability and ultimately buckling collapse. For $\xi_{\text{trans}}/\xi_{\text{long}} = 0.01$ mechanical instability is reached at low degrees of curvature.

Two quantities can be defined: the maximal bending moment and the critical curvature. The latter is defined as the curvature leading to critical compression of the material or the curvature reached before the structure collapses. (Since both experiment and calculation use discrete steps, this definition is less arbitrary than it may appear.)

A third type of failure, longitudinal splitting (Fig. 5b), is observed particularly for thick walled tubes of stiff material. Spatz *et al.* (1993) have calculated
<figure><img src="image_203.png" /><figcaption>Fig. 5. Closeup views of two *Arundo donax* internodes during the process of bending, demonstrating two modes of failure. (a) An internode originating from the apical third of a 3.5-m-long stem (length of the internode $l = 8.6$ cm, mean radius $r = 0.66$ cm, mean wall thickness $w = 0.18$). Staining with phloroglucinol/hydrochloric acid reveals moderate lignification of the internodal wall. As shown in the series of drawings, local buckling starts, at relatively little ovalization, with a small indentation slightly right of the vertex on the compression side ($a_3$). The dent increases ($a_5-a_6$), inducing the buckling collapse of the internode ($a_7$). Primary cause of the buckling collapse is failure of the material due to exceeding the critical compressive strain. The ovalization of the hollow stem was relatively small. (b) Failure of a thicker internode of *Arundo donax* from the basal quarter of the same stem (length of the internode $l = 16.6$ cm, mean radius $r = 1.02$ cm, mean wall thickness $w = 0.35$ cm). In this internode the entire wall reveals extremely high lignification. This stiff, thick-walled internode fails by longitudinal splitting at small degrees of ovalization ($b_4-b_5$). Longitudinal splits occur first at the narrow vertex of the ellipse. As a consequence of the splintering of the hollow stem into two shells of semicircular cross section, longitudinal splits occur also in these shells, mainly at the shallow vertex of the deformation ellipse. This is shown in $b_6$, where the buckled internode is shown from above.</figcaption></figure>
the strains in the radial and in the tangential directions which develop if a circular ring is deformed to an elliptical ring. The tangential strains reach their maximum at the point where the long axis meets the ellipse. Figure 6 shows the tangential strain as a function of the wall thickness and various degrees of ovalization. A reasonable estimate for the critical tangential strain in cereals is 2%.

Figure 7 presents the results of the computations for tubes with varying ratios of length/radius and material of different $ξ_{trans}/ξ_{long}$. Longitudinal splitting can reduce the maximum bending moment. In thin-walled tubes, however, such as the hollow stems of cereals, the maximum bending moment is usually reached before the tangential strains are larger than 2%. More salient is the limitation on the degree of curvature (Fig. 7b).

Short stout tubes are more stable than long slender tubes. This is due to the fact that for a ratio of length/radius below 30:1, buckling is no longer symmetrical. It occurs to a much lesser degree on the tension (windward) side than on the compression (leeward) side (Fig. 7c). In this case the cross section resembles a "doughnut" shape rather than an ellipse. At ratios of length/radius between 30:1 and 200:1 the maximal bending moment is nearly constant. At higher ratios we observe the counterintuitive result that for material with $ξ_{trans}/ξ_{long} = 0.1$, the maximal bending moment increases again. For ratios of length/radius > 1000 this is observed even for tubes from a material with $ξ_{trans}/ξ_{long}$

<figure><img src="image_204.png" /><figcaption>Fig. 6. Maximal strains in the tangential direction as a function of the ratio of radius/wall thickness for different degrees of deformation of a circular cross section. The numbers in the graph indicate the degree of ovalization $a/r$.</figcaption></figure>
<figure><img src="image_205.png" /><figcaption>Fig. 7. Maximal bending moment, critical curvature, and buckling as a function of the ratio length/ radius for a hollow tube of radius r = 0.2 cm and wall thickness 0.01 cm. The numbers in the graphs indicate the ratio $\xi_{trans}/\xi_{long}$. The data are normalized with the maximal bending moment and critical curvature of a tube which does not buckle ($\xi_{trans} \gg \xi_{long}$). The filled symbols indicate data for which the criterion for failure was either 1% longitudinal compressive strain or a buckling collapse, which in (c) is symbolized by 100% buckling. The open symbols in (b) indicate data where the additional failure criterion of 2% strain in the tangential direction led to termination of the calculation. For the thin-walled tubes considered, this had little effect on the values for maximal bending moments. (c) In short tubes the degree of buckling is less on the extension side (■) than on the compression side (●).</figcaption></figure>
<figure><img src="image_206.png" /><figcaption>Fig. 7. Continued.</figcaption></figure>

= 0.01. As discussed later the increase is a consequence of the assumption that the bending line on the compression side follows a smooth spline function. For this the relative gain of strain energy in the longitudinal direction decreases with increasing length, such that the equilibrium is reached at lower degrees of ovalization.

Optimal construction has to compromise between high flexural stiffness and elusion of dangerous degrees of buckling. This becomes apparent if structures with equal cross-sectional areas but different ratios of radius/wall thickness are considered (Fig. 8). Upon increasing the radius at the expense of the wall thickness, the maximal bending moment increases. Ultimately, however, ovalization will reduce the mechanical stability again. The ratio of radius/wall thickness, at which the optimum is found, depends on the Young's modulus in the transverse direction.

## The Mechanical Impact of Nodal Thickenings

The approach used allows one to subdivide the length of the hollow tube into segments. No ovalization may occur at the ends of each segment. This is equivalent to introducing nodes into the structure.

Figure 9 shows the maximal bending moments, the critical curvature, and the degree of ovalization for a hollow plant stem of constant length as a function of the length of the segments, i.e., the distance of nodes. The data closely
<figure><img src="image_207.png" /><figcaption>Fig. 8. The maximal bending moment for a hollow tube of length 20 cm and cross-sectional area of $π \cdot 0.4 \text{ mm}^2$ as a function of the ratio of outer radius/wall thickness. The numbers in the graph indicate $\xi_{\text{trans}}/\xi_{\text{long}}$. The data are normalized to the maximal bending moment of a full cylinder of equal cross-sectional area. The criterion for failure is 1% strain in compression, 2% tangential strain, or failure of the structure. Disregard of failure due to tangential strain leads, in a few cases, to higher maximal bending moments. This is indicated by open symbols.</figcaption></figure>
<figure><img src="image_208.png" /><figcaption>Fig. 9. Maximal bending moment, critical curvature, and buckling as a function of the ratio of nodal distance/radius for a tube with length *l* = 2 m, radius *r* = 0.2 cm, and wall thickness *w* = 0.01 cm. Since the bending line is symmetrical, this corresponds to a length of the stalk of grass of 1 m, disregarding tapering. The data are normalized and plotted as in Fig. 7. (c) Buckling on the extension side (■) is less than on the compression side (●) if nodes are located at short distances.</figcaption></figure>
<figure><img src="image_209.png" /><figcaption>Fig. 9. Continued.</figcaption></figure>

resemble those shown in Fig. 7. Again, we observe the counterintuitive increase in the maximal bending moment for a ratio of nodal distance/radius above 200. Most significant is the increase in the maximal bending moment with decreasing nodal distances. As seen in Fig. 9 this results from the fact, discussed before, that the degree of buckling is quite different for the compression side and the tension side.

Stabilization of the structure by nodal thickenings can be effective only for rather thin-walled tubes. Figure 10 shows the maximal bending moments for structures with equal cross-sectional areas as a function of the ratio of radius/wall thickness and different distances of the nodes. If nodes are placed at relatively short distances, the degree of stabilization can become quite remarkable.

## RESULTS

An automated measuring instrument was developed to observe the bending behavior of hollow plant stems up to the point of failure of the material or the structure (Spatz *et al.*, 1993; Spatz and Speck, 1994). A major difficulty in applying forces or bending moments to biological material is damage at the point of attachment. To avoid this problem we constructed a device consisting of a rail and two wagons which can be pulled together by a motor-operated ratchet via a string and pulleys. The wagons carry the mounts for tubes or plant
<figure><img src="image_210.png" /><figcaption>Fig. 10. The maximal bending moment for a hollow tube of length 2 m and cross-sectional area $\pi \cdot 0.4 \text{ mm}^2$ as a function of the ratio of outer radius/wall thickness. The ratio of the Young's moduli $\xi_{\text{trans}}/\xi_{\text{long}} = 0.1$. The numbers in the graph indicate the ratio of nodal distance/radius. The failure criterion is the same as for Fig. 8. Also, the data are normalized and plotted in the same way as for Fig. 8.</figcaption></figure>

stems. The mounts can rotate freely around a horizontal axis, such that bending moments directly at the mounts are avoided. The forces to pull the two wagons together, and thereby bend the object like an arrow, are measured with a force transducer. The position of the two wagons is measured by two displacement detectors. To obtain the height of the vertex, a CCD camera recorder was
mounted on a motor-operated stand, equipped with a displacement detector. A computer program was developed to operate the motor such that the camera was automatically adjusted to the height of the vertex of the bending object. The data force, distance of the two wagons, and height of the vertex were collected every 2 s. From these, the bending moment and the curvature in the vertex were calculated.

Figure 11 shows a corresponding plot for an internodium of *Secale cereale* ssp. *cereale*. The relation between bending moment and curvature is clearly nonlinear. This can be due to two reasons:

(i) a decrease in the second moment of area due to ovalization of the cross section and

(ii) a reduction of the Young's modulus in the longitudinal direction due to exceeding the limits of elastic behavior of the material.

To distinguish between these possibilities, the data were fitted with the help of

<figure><img src="image_211.png" /><figcaption>Fig. 11. The relation between bending moment and curvature as obtained experimentally (filled symbols) for a segment of a stalk of *Secale cereale* ssp. *cereale* of length *l* = 30.7 cm, radius *r* = 0.15 cm, and wall thickness of the sclerenchyma *w* = 0.010 cm. The arrow indicates the point of mechanical failure. The data were fitted up to a curvature of 4.5 m⁻¹ with the help of the theoretical approach (open symbols). From this, the Young's modulus in the longitudinal direction was obtained as *ξ*<sub>long</sub> = 11.3 GN/m² and *ξ*<sub>trans</sub>/ξ<sub>long</sub> as 0.031.</figcaption></figure>
the theory outlined before. Up to a curvature of $4.5 \text{ m}^{-1}$ the fit represents the data quite well, indicating that up to this point ovalization of the cross section accounts for the nonlinearity. If the plant stem is bent further, the bending moments are consistently lower than predicted by the theory, indicating that the material is carried beyond the limits of Hooke's law until it finally fails in compression.

In *Triticum durum* var. *aegyptiacum* the range of non-Hookean behavior is extended to exceptionally high degrees of curvature (Fig. 12) before failure occurs. This correlates with a very large proportion of parenchymatic tissue (Fig. 13d).

To obtain the Young's moduli we take advantage of the fact that the data from the theoretical description can be represented by a second-order polynomial. Knowing the geometrical parameter of the sclerenchyma, it is possible to obtain the Young's modulus in the longitudinal direction from the linear term

<figure><img src="image_212.png" /><figcaption>Fig. 12. The relation between moment and curvature for a segment of *Triticum durum* var. *aegyptiacum* of length *l* = 16.4 cm, radius *r* = 0.12 cm, wall thickness of the sclerenchyma *w* = 0.0055 cm. The data are plotted as in Fig. 11. The simulation with the help of the theoretical approach can represent the experimental data only up to a curvature of 6 m⁻¹. Further increase in curvature at almost-constant bending moments before mechanical failure (arrow) can be attributed to nonelastic behavior of the material.</figcaption></figure>
<figure><img src="image_213.png" /><figcaption>Fig. 13. Cross sections showing parts of the stalk walls of the six cereals studied: (a) *Secale cereale* spp. afghanicum, (b) *Secale cereale* ssp. cereale, (c) *Triticum aestivum* var. lutescens, (d) *Triticum durum* var. aegyptiacum, (e) *Triticum monococcum* var. vulgare, and (f) *Triticum spelta* var. coeruleum. sr, sclerenchyma ring (including epidermis and sometimes also vascular bundles); in some taxa in the outer parts of the sclerenchyma ring areas of photosynthetic tissue are found (ps), pt, parenchymatous tissue; vb, vascular bundles; pc, pith cavity.</figcaption></figure>
of the fit for the data and to extract the Young's modulus in the transverse
direction from the quadratic term.

Table I lists these constants together with the maximal bending moment and the critical curvature for internodal segments of some cereal taxa. Although the mechanical characteristics are not linearly related to the geometrical parameter, presenting averages is not inappropriate since the variations in the geometry are quite small.

The Young's moduli in the longitudinal direction vary only twofold between the varieties investigated. The variation within species is mainly of biological origin. In particular, the ontogenetic state influences the Young's modulus in longitudinal direction. While the ratio of the Young's moduli $\xi_{trans}/\xi_{long}$ is between 0.05 and 0.08 for the majority of the taxa, *Triticum spelta* var. *coeruleum* marks an exception.

DISCUSSION

Hollow plant stems are found in several plant families, for instance, Apiaceae, Poaceae, and Equisetaceae. Such structures realize a high bending stiffness at a relatively low biological cost, but they are endangered by local buckling (Brazier, 1927), particularly under bending loads. In some of the taxa the stem is subdivided into hollow internodes and nodes with a high degree of nodal thickening. These reduce the danger of buckling (Gappoev, 1992; Niklas, 1989, 1992). Hollow structures can be endangered by bending moments exerted by side branches or large leaves. This is presumably the reason why side branches usually include only very small angles to the main stem (Mosbrugger, 1990). Where the angle is nearly 90°, as in the large leaves of *Heracleum mantegazianum*, the nodal thickenings are collar-like and quite massive.

Previous work (Spatz et al., 1990, 1993) showed that the bending behavior of hollow tubes could be described by considering the equilibrium between strain energies in longitudinal direction and deformation energies acting in radial and in tangential direction. Here the numerical values of Young's moduli in radial and in tangential direction are assumed to be equal. This assumption has little significance for the calculation of the maximal bending moment, but it influences the strains developing in the tangential direction upon ovalization and, therefore, the limitations imposed by the danger of longitudinal splitting.

The computations show that the stability of hollow structures against buckling is greatly enhanced by nodal thickenings provided that the ratio of nodal distance/radius is below 30:1. In this range buckling is asymmetrical. The cross section has a "doughnut"-like rather than an elliptical shape. Strains are transferred from the compression side to the tension side, such that the strains on this side can become quite large. We have chosen a value of 2% extension as the failure criterion, compared to 1% in compression.
<table><caption>Table I. Geometrical and Mechanical Data for Intermodia of Cereals (Mean ± SD)</caption><thead><tr><th></th><th>Secale cereale<br>ssp. afghanicum</th><th>Secale cereale<br>ssp. cereale</th><th>Triticum aestivum<br>var. lutescens</th><th>Triticum durum<br>var. aegyptiacum</th><th>Triticum monococcum<br>var. vulgare</th><th>Triticum spelta<br>var. coeruleum</th></tr></thead><tbody><tr><td>Length (cm)</td><td>25.7 ± 2.2</td><td>29.9 ± 4.3</td><td>25.7 ± 2.5</td><td>20.9 ± 2.1</td><td>16.7 ± 4.1</td><td>23.1 ± 5.1</td></tr><tr><td>Mean radius (cm)</td><td>0.174 ± 0.018</td><td>0.173 ± 0.026</td><td>0.19 ± 0.01</td><td>0.130 ± 0.018</td><td>0.118 ± 0.011</td><td>0.155 ± 0.023</td></tr><tr><td>Wall thickness (cm)</td><td>0.0072 ± 0.00066</td><td>0.0092 ± 0.002</td><td>0.0118 ± 0.0019</td><td>0.0076 ± 0.0012</td><td>0.0077 ± 0.0020</td><td>0.0055 ± 0.0006</td></tr><tr><td>of sclerenchyma</td><td>67.9 ± 2.9</td><td>65.7 ± 3.2</td><td>69.7 ± 3.1</td><td>65.4 ± 3.2</td><td>68.6 ± 2.6</td><td>69.4 ± 2.3</td></tr><tr><td>Water content (%)</td><td>0.071 ± 0.005</td><td>0.045 ± 0.006</td><td>0.071 ± 0.020</td><td>0.037 ± 0.018</td><td>0.025 ± 0.005</td><td>0.034 ± 0.015</td></tr><tr><td>Maximal bending<br>moment (um)</td><td>7.23 ± 0.8</td><td>5.19 ± 1.54</td><td>7.27 ± 1.65</td><td>25.9 ± 11.1</td><td>22.5 ± 8.35</td><td>9.07 ± 1.39</td></tr><tr><td>Critical curvature<br>(1/m)</td><td>22.3 ± 3.4</td><td>14.6 ± 3.9</td><td>10.7 ± 2.6</td><td>15.6 ± 4.6</td><td>12.9 ± 5.7</td><td>13.4 ± 4.0</td></tr><tr><td>Young's modulus</td><td>0.08 ± 0.035</td><td>0.053 ± 0.049</td><td>0.054 ± 0.013</td><td>0.07 ± 0.03</td><td>0.071 ± 0.036</td><td>0.17 ± 0.06</td></tr><tr><td>ξ<sub>long</sub> (GN/m²)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ξ<sub>trans</sub>/ξ<sub>long</sub></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Number of<br>measurements</td><td>5</td><td>6</td><td>8</td><td>7</td><td>6</td><td>8</td></tr></tbody></table>
A counterintuitive result is shown in Figs. 7a and 9a. For very thin long tubes the maximal bending moments appear to increase again. This is the result of the assumption of a "smooth" spline function as the bending line on the compression side. A "wavy" bending line would lead to a higher degree of relaxation of strains. This supposition can be tested within the framework of the theoretical approach by introducing "virtual" nodes into the hollow structure. Figure 14 shows the computational results for such an interpretation. The introduction of "virtual" nodes leads to a nearly constant maximal bending moment. For a ratio of distance of virtual nodes/radius above 20:1, the values are lower than those observed without virtual nodes, indicating that wavy bending lines can destabilize the structure. It should be pointed out that we have not yet observed wavy bending lines in our experiments.

It is the aim of our work to understand the mechanical properties from the structural properties of the plant stem, in particular, the distribution of tissues in the cross section. In previous work we observed a correlation between the Young's modulus in the longitudinal direction and the compactness of the sclerenchyma (Spatz et al., 1993). This observation is confirmed here. Secale cer-

<figure><img src="image_214.png" /></figure>

**Fig. 14.** The maximal bending moment as a function of the ratio length/radius calculated for a ratio of Young's moduli $\xi_{trans}/\xi_{long} = 0.1$. The assumption that the bending line on the compression side follows a "smooth" spline function leads to a counterintuitive result. A more reasonable assumption is a "wavy" bending line. In the framework of the theoretical approach this can be demonstrated by introducing "virtual" nodes into the structure. The numbers in the graph indicate the ratio of "nodal distance"/radius used in the computations. A ratio of 20:1 leads to a minimum of the mechanical stability. Wavy bending lines have not yet been observed.
*eale* ssp. *afghanicum*, with the most compact sclerenchyma (Fig. 13a), has the highest Young's modulus in the longitudinal direction, in contrast to *Triticum aestivum* var. *lutescens*, with the least compact sclerenchyma (Fig. 13c).

Spatz *et al.* (1993) suggested that the parenchymatous tissue can contribute to the stiffness in the transverse direction, thus stabilizing the structure against buckling. This supposition is confirmed here in two ways.

(i) The Young's modulus in the transverse direction as obtained from the fit of the nonlinearity of the bending moment-curvature relation implicitly contains the contributions of the parenchyma to the transverse stiffness. The apparent ratios of the Young's moduli $\xi_{trans}/\xi_{long}$ are found in the range of 1:20 to 1:12 with one exception, *Triticum spelta* var. *coeruleum*. If it is assumed that $\xi_{trans}/\xi_{long}$ for sclerenchyma alone is 1:20, as for bamboo (Niklas 1991), then an apparent value of 1:6 is a clear indication of a stabilizing influence of the parenchyma either through sclerenchymatized parenchyma (Spatz *et al.*, 1993) or through turgor pressure.

(ii) *Triticum durum* var. *aegyptiacum* tolerates exceptionally high degrees of curvature (Fig. 12). The portion of the parenchymatic tissue is very large (Fig. 13d). This prevents ovalization to more than 20%. Failure occurs after extensive plastic deformation of the material. High degrees of curvature are also found in *Triticum monococcum* var. *vulgare*. As in *Triticum durum* var. *aegyptiacum* this can be ascribed only partly to the small radii of the internodes tested. Again, a relatively large portion of parenchymatic tissue is characteristic (Fig. 13e), though not as large as in *Triticum durum* var. *aegyptiacum*. Correspondingly, the mechanical behavior is intermediate between that shown in Fig. 11 and that in Fig. 12.

Understanding the mechanical properties of hollow plant stems may help to reduce the enormous economic losses due to lodging of cereal. The base of empirical data, however, is still quite incomplete. Studies of the changes of the mechanical properties during ontogeny are missing. Also, the influence of the degree of hydration or the use of different fertilizers during growth needs to be investigated before generalizations of structure-function relations can be made in such a way as to develop guidelines for plant breeders.

## ACKNOWLEDGMENTS

We are grateful for stimulating discussions with Drs. C. Mattheck, W. Nachtigall, and L. Wessolly. B. Bambai carried out some of the experiments and B. Heneka prepared the drawings.
REFERENCES

Axelrad, E. L. (1965). A more precise estimation of the upper critical bending load of a tube taking into account the geometrical nonlinearity. *Trans. (Iswestia) Acad. Sci. USSR. Sect. Mech. Mech. Eng.* 4:133-139 (Russian).

Böhm, W., and Gose, G. (1977). Einführung in die Methoden der Numerischen Mathematik, Vieweg Verlag, Braunschweig.

Brazier, L. G. (1927). On the flexure of thin cylindrical shells and other "thin" sections. *Proc. Roy. Soc. Lond. Ser. A* **116**:104-114.

Ennos, A. R. (1993). The mechanics of the flower stem of the sedge *Carex acutiformis*. *Ann. Bot.* **72**:123-127.

Foley, D. C. (1983). Mechanical properties of *Zea mays* stems. *Iowa State J. Res.* **58**:235-246.

Gappoev, M. (1992). Statische Interpretation grasartiger Konstruktionen. *Natürliche Konstruktionen-Mitteilungen des SFB 230* 7:73-77.

Herbig, A., Sinn, G., and Wessolly, L. (1988). Zur Standsicherheit von Bäumen im städtischen Bereich Natürliche Konstruktionen. *Leichtbau in Architektur und Natur. Mitteilungen des SFB 230* 1:39-57.

Lu, R.-F., Bartsch, J. A., and Ruina, A. (1987). Structural stability of the corn stalk. *Am. Soc. Agr. Eng. Paper No. 87-6066*:1-16.

Mattheck, C. (1992). *Design in der Natur. Der Baum als Lehrmeister*, Rombach Verlag, Freiburg.

Mattheck, C., Gerhardt, H., and Breloer, H. (1992). VTA: Visual tree defect assessment based on computer simulation of adaptive growth. In Little, E. G. (ed.), *Experimental Mechanics*, Elsevier, Amsterdam.

Mattheck, C., Bethge, K., and Erb, D. (1993). Failure criteria for trees. *Arb. J.* 17:201-209.

Mattheck, C., and Breloer, H. (1993). Der Baumbruch in Mechanik und Rechtsprechung. In *Handbuch der Schadenskunde von Bäumen*, SVK-Verlag, Erndtebrüch.

Mosbrugger, V. (1990). The tree habit in land plants. *Lecture Notes in Earth Sciences* 28, Springer Verlag, Berlin.

Niklas, K. J. (1989). Nodal septa and the rigidity of aerial shoots of *Equisetum hyemale*. *Am. J. Bot.* **76**:521-531.

Niklas, K. J. (1991). Bending stiffness of cylindrical plant organs with a "core-rind" construction: evidence from *Juncus effusus* leaves. *Am. J. Bot.* **78**:561-568.

Niklas, K. J. (1992). *Plant Biomechanics*, University of Chicago Press, Chicago, London.

Niklas, K. J. (1993). Influence of tissue density-specific mechanical properties on the scaling of plant height. *Ann. Bot.* **72**:173-179.

Niklas, K. J., and O'Rourke, T. D. (1987). Flexural rigidity of chive and its response to water potential. *Am. J. Bot.* **74**:1033-1044.

Reissner, E. (1959). On finite bending of pressurized tubes. *Trans. ASME* **81**:386-392.

Rinn, F. (1994). Wie genau kann die Bruchsicherheit eines Baumes ermittelt werden? *Das Gartenamt* **43**:104-108.

Romberger, J. A., Hejnowicz, Z., and Hill, J. F. (1993). *Plant Structure: Function and Development. A Treatise on Anatomy and Vegetative Development, with Special Reference to Woody Plants*, Springer-Verlag, Heidelberg.

Sinn, G. (1993). Grundsätzliches zur Bruchsicherheit von Bäumen. Eine Argumentationshilfe in Schadensfällen. *Das Gartenamt* **42**:387-392.

Sinn, G. (1994). Bruchsicherheit von Bäumen und Restwandstärke geschlossener Stammquerschnitte. *Das Gartenamt* **43**:100-103.

Spatz, H.-Ch., Speck, T., and Vogellenhner, D. (1990). Contributions to the biomechanics of plants. II. Stability against local buckling in hollow plant stems. *Bot. Acta* **103**:123-130.

Spatz, H.-Ch., Boomgarden, Ch., and Speck, T. (1993). Contribution to the biomechanics of plants. III. Experimental and theoretical studies of local buckling. *Bot. Acta* **106**:193-276.

Spatz, H.-Ch. (1994). Ein Kommentar zur mechanischen Stabilität hohler Bäume. *Das Gartenamt* **43**: 92-95.

Spatz, H.-Ch., and Speck, T. (1994). Mechanische Eigenschaften von Hohlrohren am Beispiel von Gräsern. *BIONA-reports* (in press).

Speck, T., Spatz, H.-Ch., and Vogellehner, D. (1990). Contributions to the biomechanics of plants.
I. Stabilities of plant stems with strengthening elements of different cross-sections against weight and wind forces. *Bot. Acta* **103**:111-112.

Stephens, W. B., Starnes, J. H., Jr., and Almroth, B. O. (1974). Collapse of long cylindrical shells under combined bending and pressure loads. In *AIAA/ASME/SAE 15th Structures, Structural Dynamics and Materials Conference*, Las Vegas, pp. 1-8.

Timoshenko, S. P., and Gere, J. M. (1961). *Theory of Elastic Stability*, McGraw-Hill, New York.

Vincent, J. F. V. (1983). *Biomechanics in schools*. *School Sci. Rev.* **43**:648-663.

Vincent, J. F. V., and Jeronimidis, G. (1991). The mechanical design of fossil plants. In: Rayner, J. M. V., and Wootton, R. J. (eds.), *Biomechanics in Evolution*, Cambridge University Press, Cambridge, pp. 21-36.

Wainwright, S. A., Biggs, W. D., Currey, J. D., and Gosline, J. M. (1976). *Mechanical Design in Organisms*, Arnold, London.

Wessolly, L. (1988). Die natürliche Konstruktion Baum als intelligentes statisch-dynamisches System. *Natürliche Konstruktionen. Mitteilungen des SFB 230* **2**:203-212.

Wessolly, L. (1991). *Material- und Struktureigenschaften der Bäume*, Fortschreibung des Stuttgarter Festigkeitskatalogs, Stuttgart.

Wessolly, L. (1993). Stand- und Bruchsicherheit von Bäumen. Leistungsfähigkeit und Grenzen der Zugversuche. *Das Gartenamt* **42**:486-491.
<figure><img src="image_215.png" /></figure>
# Growth and Architecture of the Plant Cell Wall: Biomechanical Problems<fnref n="1" />

Roger Prat<fnref n="2" />,<fnref n="3" />, Michèle Mosiniak<fnref n="2" />,<fnref n="4" /> and Jean-Claude Roland<fnref n="2" />,<fnref n="4" />

This paper deals with the dynamics of the cell wall during plant growth. The study focuses on the extension of mung bean seedlings. First, the parameters of growth were defined (duration, actual localization, polarity, time change of elongation rate, change of osmotic pressure, etc.). Then the specific mechanical properties of the primary cell wall were considered and compared in living and nonliving tissues. Rheology and deformation responses, plasticity, and creep were followed along the growth gradient and when the surface expansion had ceased. Different potential activators (low pH, hormones) were tested. The cell wall texture and its changes were then followed, step-by-step, at the ultrastructural level along the growth gradient; the expanding cell wall appears as a mesophase (liquid crystal-like) of cholesteric type. The helical construction is transient and dispersed when intense surface expansion occurs. Thus, the growing wall changes from order to a dramatic thinning and a complete randomization. Finally, the question of the primary wall dynamics is discussed and the risks of oversimplification when modeling the plastic cellulosic composite within living organs is emphasized.

**KEY WORDS:** biological composite; cellulose; growth; mesophase; mung bean.

## INTRODUCTION

Obviously, the mechanical properties of plants depend mainly on the fact that each constitutive cell is surrounded by a more or less stiff and thick wall. In

<footnote n="1">¹This is the published version of a paper presented at the Plant Biomechanics Congress, Montpellier, France, September 5-9, 1994.</footnote>

<footnote n="2">²University Pierre & Marie Curie, Paris, France.</footnote>

<footnote n="3">³Laboratoire d'enzymologie en milieu structuré, Institut Jacques Monod, tour 43, 2 Place Jussieu, 75 251, Paris cedex 05, France.</footnote>

<footnote n="4">⁴Laboratoire de cytologie expérimentale et morphogenèse végétale, UPMC, bât. N2, 4 Place Jussieu 75 252, Paris cedex 05, France.</footnote>
higher plants, the cell wall is secreted and assembled outside the cytoplasm as a fibrous composite. It is made of ground components (pectic substances, hemicelluloses, etc.), or matrix, in which a frame of microfibrils of cellulose is embedded and strengthens the system.

During growth, the cell wall is in a complex and rather paradoxical situation: It must be both resistant enough to behave as a supporting structure and plastic enough to allow often dramatic surface extension (i.e., an increase of several times the initial surface of the cell wall within a few hours). The cell wall surrounding the young and growing cell is called the primary wall. Its sheets are deposited first against the middle lamella. Growth remains possible because the cellulosic microfibrils keep a certain degree of freedom and can be individually displaced (by sliding and/or reorientation; the point remains moot) within a soft and highly hydrated matrix (the movement is no longer possible in the so-called secondary wall deposited later, after cessation of growth).

It was long considered that the primary wall architecture showed no defined order and was dispersed passively according to the laws of flows of viscous fluids. But with the development of ultrastructural data, the original assumption has been found to be false and it was soon shown that the formation of different textures must be ascribed, at least in part, to the creative power of the plas- malemma, which organizes the cell wall with regard to its future function (Frey- Wyssling and Mühlethaler, 1963). The nature of this morphogenesis and wall ordering is still discussed. Different mechanisms could cooperate at the wall- membrane interface. During recent years, the possibility that the primary wall is, at least in part, precisely and specifically self-assembled in this cell area has emerged. The cell wall could be a "mesophase" or a "mesomorphic phase" somewhat comparable to certain liquid crystals (for reviews see Roland *et al*., 1992; Neville, 1993).

In this paper, we present different mechanical, physiological, and structural data which are integrated during the course of growth of the cell wall and must be taken into account when the living plant is considered.

## THE LOCALIZATION OF CELL GROWTH

In most plant organs, cell growth is localized in minute zones of young levels; it is distributed according to precise gradients which are time dependent (Prat, 1985; Silk, 1992). The most analyzed organs from this point of view were taken from young seedlings in the course of elongation: coleoptiles (maize, etc.), hypocotyls (mung bean, sunflower, etc.), and/or epicotyls (stem of pea, beans, etc.).

The distribution of growth in *Vigna radiata* (mung bean) is shown in Fig. 1. The expansion is highly polarized (= elongation; according to a major axis). The main growth occurs at the top of the organ (just below the apical hook).
<figure><img src="image_216.png" /><figcaption>Fig. 1. Growth gradient of a 4-day-old hypocotyl of Vigna radiata: (A, B) hypocotyl cut in successive 5-mm-long segments; (C) epidermal cell length (mean value) measured for each segment; (D) rate of elongation of successive segments.</figcaption></figure>

The lower and basal regions have lost any expression of cell elongation. The distribution and intensity of elongation of the organ along a growth gradient can be directly correlated with the changes in cell length and local modifications of plasticity.

During a relatively short time [1 or 2 days, depending on the environmental conditions (temperature, light, etc.)], the cells of the subapical zone are able to increase their length 10 times (Fig. 2) or more, before being driven toward lower zones, where their rate of expansion decreases and progressively tends to zero. The intense change in dimensions and size of cells implies both extension (loosening of the preestablished network) and synthesis (apposition and intussusception) of new components. It implies also an increase in the vacuolar volume and a maintenance of the osmotic pressure above a critical threshold by concentration of solutes. A rapid surface increase in the membranes (both of the plasmalemma and of the tonoplast) is also among immediate prerequisite conditions for extension.

Therefore, the regulation of the rate of growth and the arrest of the surface expansion can be due either to variations of pressure, to changes in the plastic properties of the wall itself, or to variations of physiological properties (e.g.,
<figure><img src="image_217.png" /><figcaption>Fig. 2. Diagram of a plant cell before and after growth. The initial length of the cell is about 20 µm, and the final length about 200 µm.</figcaption></figure>

increase in inhibitors, decrease in synthesis and endomembrane flows, change of hormone sensitivity).

THE DRIVING FORCE: TURGOR PRESSURE

Growth occurs only if the cell presents a turgor pressure high enough to ensure cell wall deformation. Turgor pressure of a cell depends on its own water potential and on the water potential of the environment (neighboring cells and external medium); it is relatively easy experimentally to modulate the relative turgor pressure by changing the osmotic pressure of the external medium (Prat *et al.*, 1977, Fig. 3).

For cells whose growth has ceased, variations of the external water potential produce reversible changes of volume (Fig. 3A), which are relevant to the elastic properties of the organ and therefore of the cell wall; if cells are young (Fig. 3B), identical variations more or less block growth. There is an external pressure which prevents any growth, indicating that a minimum turgor pressure is necessary (the "critical turgor pressure"). We can notice that when cells are transferred to a new suitable medium, their growth is restored, but the final size is lower than the final size of the controls (untreated by concentrated medium): That means that there is no "stored growth" (Cleland, 1981).

Kinetics show two components: (1) an elastic deformation comparable to the elasticity of older cells and (2) a plastic deformation (nonreversible) that appears only when the internal pressure is higher than a specific threshold.
<figure><img src="image_218.png" /><figcaption>Fig. 3. Responses of segments excised from the elongating zone (A) and nonelongating zone (B) for different concentrations of mannitol; note that the Y scales in A and B are very different.</figcaption></figure>

## CELL WALL EXPANSION: DISPERSION OF A CHOLESTERIC MESOPHASE

In the growth gradient, tissue differentiation is precocious (Fig. 4) and cell wall thickening begins early. The wall thickness is uneven in different cells: the maximum layering is present at cell corners (Fig. 5) and on certain outer cell facets. The elongating zone is already heterogeneous and complex from the