American Journal of Botany 86(6): 767±775. 1999.

BIOMECHANICS OF THE COLUMNAR PRINGLEI1

KARL J. NIKLAS,2 FRANCISCO MOLINA-FREANER,3 AND CLARA TINOCO-OJANGUREN3

2Section of Biology, Cornell University, Ithaca, New York 14853; and 3Instituto de Ecologia UNAM, Apartado Postal 1354, Hermosillo, Sonora CP 83000, Mexico

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

Key words: biomechanics; Cactaceae; mechanical stability; plants; stems; tissue density; tissue stiffness; wood; Young's modulus.

The relationship between the mechanical and anatom- The stems of columnar cacti are ideal organs with ical properties of plant tissues has been the subject of which to empirically study the relationship between anat- considerable speculation because it is evident that, aside omy and tissue stiffness because of their tall, compara- from their physiological functions, every tissue type con- tively sparsely branched and woody growth habit, the tributes in some way to the mechanical behavior of or- persistence of pith and cortex in older portions of stems, gans (Schwendener, 1874; Carlquist, 1961, 1969, 1975; and the presence of vascular strands that, despite the ac- Wainwright et al., 1976; Niklas, 1992; Speck, 1994; cumulation of secondary tissues, remain slender and Spatz et al., 1995). A number of factors in¯uence the largely unconnected to neighboring strands (Gibson, mechanical properties of each tissue type, but prior stud- 1978; Mauseth, 1988; Mauseth and Plemons, 1995; Mau- ies suggest that stiffness is positively correlated with the seth et al., 1995). These and other features provide an volume fraction of cell wall materials (and thus speci®c opportunity to remove nearly untapered cylindrical sam- gravity and density), especially among the secondary tis- ples of xylem and other tissue types from different lo- sues (Forsaith, 1926; Record, 1934; Seibt, 1964; see cations along the length of stems, test these samples in Esau, 1967), and with water content and thus turgor, es- bending to determine their stiffness, assess whether lon- pecially among the primary tissues (Falk, Hertz, and Vir- gitudinal gradients in this property exist, and evaluate gin, 1958; Niklas, 1992). However, comparatively few whether observed gradients correlate with anatomical studies have addressed these relationships experimentally variations. in quantitative terms. In this paper we present the ®rst phase of this research agenda by reporting the longitudinal variations in the 1 Manuscript received 21 August 1998; revision accepted 21 Decem- stiffness of tissue types surgically removed from different ber 1998. The authors thank Prof. James D. Mauseth (University of Texas) who locations along the length of Pachycereus pringlei stems. acted as Editor-in-Chief during the review process; two anonymous re- This species is the most massive plant in the Sonoran viewers who made constructive recommendations to improve an earlier Desert, reaching heights of 15±20 m and producing stems draft; the owners of El Sacri®cio who provided access to their property; up to 1.5 m in diameter (Turner, Bawers, and Burgess, Ivan Romo for logistical support; Conrado Velenzuela, Oscar Gutierrez, 1995). It was selected for study because of the availabil- Mauricio Cervantes, Grethel Ramirez, Martin Villegas, and Daniel Mo- rales for assistance in the ®eld. Field work was supported by funds from ity of specimens differing in size and thus presumably the operating budget of the Instituto de Ecologia UNAM to FMF and age. It was also selected because of its growth habit, CTO. which permits a comparatively straightforward biome- 767 768 AMERICAN JOURNAL OF BOTANY [Vol. 86 chanical interpretation. The cactus ranges from sea level study from this locality because of their size and healthy appearance. to 950 m and grows mainly in areas dominated by warm- These plants measured 3.69 and 5.89 m in height and are denoted season rainfall, with the exception of central Baja Cali- throughout this paper as plants 1 and 2, respectively (Fig. 1A, B). Ad- fornia where it can be found in areas of mainly winter ditional specimens were examined during the course of our ®eld inves- rainfall (Shreve, 1964; Turner, Bowers, and Burgess, tigations to determine patterns of self loading, especially on lateral 1995). Since individual stems are unbranched and since branches. plants are severely damaged or killed by frost, the bio- The stems of plants 1 and 2 were sectioned to obtain representative mechanics of P. pringlei is not likely to be adapted to transections (Fig. 1C) measuring a few centimetres in length and seg- transient snow or ice loadings. Finally, although the xy- ments of equivalent lengths (1.1 and 1.18 m for plants 1 and 2, respec- lem strands of old plants become interconnected near the tively) for mechanical study (Fig. 1D±F). Three segments comprising base of stems, they are only modestly laterally intercon- the stem of plant 1 were designated ``bottom,'' ``middle,'' and ``top''; nected to adjoining strands along much of the length of ®ve segments from plant 2 were assigned letters A to E in an acropetal even very massive and tall stems. These xylem strands direction starting from the base of the stem. In addition to these stems, a representative curved lateral branch on plant 2 measuring 2.12 m in are thus easily removed from ground tissues, and seg- length was studied. This branch was cut into two segments designated ments differing in position with respect to stem height ``B'' and ``T'' (for bottom and top), each measuring ϳ0.93 m in length. can be tested in bending to determine how stiffness varies Stem segments were dissected either in the ®eld or laboratory to along stem length. remove their xylem strands, which contained secondary tissues in all In this paper we show that the stiffness of the xylem cases, from surrounding ground tissues. During this process, the smaller increases in a basipetal direction toward the base of lateral strands that interconnected the main strands were purposely bro- young and old stems but sharply decreases ϳ1 m above ken and removed to produce beam-like specimens for mechanical tests ground level to a level comparable to that found just be- (see Fig. 1D±E). On average, 12 of these specimens in excess of 1 m low the stem tip. In terms of their per unit volume con- long were successfully removed from each of the three stem segments tribution to the ability of stems to resist bending forces, of plant 1 and tested in bending. Between 7 and 15 beam-like specimens the xylem strands of P. pringlei are ill equipped to cope in excess of 1 m long were removed from each of the ®ve stem seg- with the potentially large bending forces that can occur ments of plant 2 and tested in bending. A smaller number of vascular in the base of stems. However, we demonstrate that the bundle specimens was sampled from segment A of plant 2 because the geometric contribution made by the xylem to the ability xylem strands in the base of this segment were laterally fused together of stems to resist bending forces increases sharply at the in various combinations such that a beam-like sample for each xylem base of plants. Consequently, the amount of xylem at the strand could not be obtained. Finally, between 9 and 15 xylem strand base of stems more than compensates for its low stiffness. specimens were removed from the lateral branch of plant 2 re¯ecting We also demonstrate that the stem ribs running much of the fact that some of the branch strands broke during the process of the length of even old and tall stems contribute signi®- dislodging the branch from its main stem. We present the data from the cantly to mechanical stability, by virtue of their location strands that were successfully removed and tested from this branch even and the stiffness of their tissues. The mechanical princi- though they reveal no statistically legitimate trend in tissue stiffness. ples underlying the architecture of P. pringlei and pre- Portions of the stem ribs (consisting of epi- and hypodermal tissues), sumably other columnar cactus species are reviewed and pith, and inner cortex were removed from plant 2 to determine their discussed in light of the data presented here. The rela- stiffness. Three stem ribs with triangular cross sections were removed tionship between tissue stiffness and anatomy is dis- with the aid of a sharp knife from each of segments B±E (segment A cussed in terms of our preliminary ®ndings, but will be had a well±de®ned periderm and lacked evident external ribs). Three treated in greater detail in subsequent publications. pith samples were removed with the aid of a cork borer or a sharp knife from the base of each of segments B±E; pith samples were also removed from the distal part of segment A (no pith was found at the base of this MATERIALS AND METHODS stem segment). Three cortex samples were removed from segments A This study was carried out in the coast of the state of Sonora, Mexico, and B and successfully tested in bending; cortical samples removed an area that belongs to the Central Gulf Coast vegetational subdivision from segments C±D could not be tested in bending because they sagged of the Sonoran Desert (Shreve, 1964; Felger and Moser, 1985). The under their own weight when suspended horizontally. study site was located on a west-facing bajada of the Sierra Seri. Plants Bending tests were used to determine the material stiffness (Young's of Pachycereus pringlei (S. Watson) Britton & Rose were studied at modulus) E and the ¯exural rigidity EI of tissue samples (Fig. 1F). Rancho El Sacri®cio (29Њ05.82Ј N, 112Њ08.00Ј W). The population den- (Young's modulus is a measure of the ability of a material to resist sity at the study site was 58.0 Ϯ 10.8 plants/ha (mean Ϯ SD). Individ- bending; ¯exural rigidity is a measure of a structure's ability to resist uals in this population ranged from 0.012 to 1.02 m in basal diameter bending. The latter is important because the second moment of area I and from 0.086 to 12.62 m in height. Young plants had a single vertical is a measure of the contribution made by the transverse geometry and stem with 10±15 vertical ribs that can expand and contract depending size of a material to the ability of a structure to resist bending. Mea- on the availability of water (Moran, 1968). Older plants, which may be suring E and I is necessary therefore because stems can rely on either 300 yr old, had lateral stems that rise near the trunk base at acute angles the stiffness or the quantity of different tissues for mechanical support.) and may surpass the main stem in height. Two plants were selected for Tissue samples from each stem segment were placed between two ver-

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Fig. 1. Pachycereus pringlei growth habit and anatomy. (A) Plant 1 measuring 3.69 m high from ground level. (B) Plant 2 measuring 5.9 m high from ground level. (C) Representative transverse cross sections from plant 1 showing ¯uted geometry and xylem strands (section from base of stem, upper left; section from upper 1 m of stem, lower right). (D) Xylem strand segments (1.1 m long) dissected from bottom (in foreground), middle, and upper third (in background) of plant 1. (E) Appearance of xylem strands before interconnecting tissue strands are removed for mechanical testing. (F) Xylem strand segment tested in bending. See text for further details. June 1999] NIKLAS ET AL.ÐCACTUS BIOMECHANICS 769 770 AMERICAN JOURNAL OF BOTANY [Vol. 86 tical supports and then loaded by placing bags of sand varying in weight attached at their mid-lengths (Fig. 1F). A horizontally oriented needle was sighted against a metric ruler to measure the mid-length vertical de¯ection ␦ resulting from the externally applied load. The Young's modulus of each sample was computed from the formula E ϭ Pl3/48␦I, where P is the mass-force of the load and l is the free length of the sample between the two vertical supports. Second moments of area were computed on the basis of morpho- metric measurements of transections taken at the bottom, middle, and upper thirds of each sample. Different formulas for I were used de- pending on the transverse geometry of tissue samples and on the ori- entation of the sample's cross section with respect to the plane of bend- ing. The transverse geometry of the xylem strands varied as a function of location with respect to the stem tip (Fig. 1D and F); most of the strands examined had an elliptical cross section near the stem tip and irregular cross section near the stem base. Different formulas for I were also necessary because each vascular tissue sample was tested in bend- ing by loading it in the radial and tangential direction with respect to stem transverse anatomy to determine the radial and tangential stiffness

(designated as ER and ET, respectively) and because the geometry and dimensions of strands viewed in these opposing directions differed. Fi- nally, we note that each stem rib was tested in bending such that its epidermis was located on the side of the specimen experiencing tension. This orientation was selected because it crudely conformed to when stems dilate due to water storage and place their epidermis in tension. The reverse orientation with regard to bending forces would also have underestimated the stiffness of this portion of stems. The transverse geometry and size of tissues were not uniform along the length of some vascular bundle segments (Fig. 1F). For these seg- ments, the relationship E ϭ Pl 3/48␦I was invalid because this formula is predicated on the assumption that beams have uniform I. For these ``irregular'' bundle segments, which occurred near the stem base, we used another formula adapted for ¯exed conical beams (see Niklas, 1992, p. 336±337). The geometry of each conical beam was determined on the basis of three sets of morphometric measurements taken at the middle and ends of these irregular strands. Sensitivity analyses indicated that the alternative formula consistently overestimated E. We consider this error unimportant because it is biased in favor of a conservative interpretation of the sharp drop in E observed at the base of stems for each stem (see Results). ANOVA and Model Type I and II regression analyses were used to determine whether differences in the E or EI of tissues varied signi®- cantly and predictably as a function of distance from shoot tips. Statis- tical comparisons were also drawn among the different tissue types to Fig. 2. Tissue stiffness (Young's modulus E) of xylem strand seg- estimate their respective contributions to the ability of intact stems to ments removed from plants 1 and 2 (see Fig. 1) plotted against relative cope with their static (self-weight) loadings. All statistical analyses were location along stem length. (A) Young's modulus (measured in the tan- performed using the software package JMP᭧ (SAS Institute Inc.) using gential direction with respect to stem ET) of individual xylem strand a Power Macintosh 8100/80. segments removed from plant 2 (arbitrary segment numbers shown in upper right of ®gure). (B) Mean tangential and radial stiffness (ET and ER, respectively) of xylem strand segments removed from the main stem RESULTS and one curved lateral branch removed from plant 2. (C) Mean tangen- tial and radial stiffness (ET and ER, respectively) of xylem strand seg- A pronounced and consistent pattern of longitudinal ments removed from the main stem of plant 1. Dark vertical lines are variation was observed in the stiffness of xylem strand the standard errors of means; thin vertical lines are standard deviations segments, viz., tissue stiffness increased in a basipetal of means. direction from the tip to the base of each strand and then abruptly decreased at the base of both plants where large trend within each stem; Fig. 2A), or when the mean stiff- quantities of secondary xylem had accumulated. The ness of all samples drawn from the same stem segment magnitude of the stiffness measured for xylem strands was plotted as a function of distance from the stem tip removed from the base of the main stems of plants 1 and (to determine whether differences among stem segment 2 was statistically indistinguishable from that measured stiffness were statistically meaningful; Fig. 2B). for segments of the xylem strands removed from the bot- Even though the xylem was mechanically anisotropic, tom of the lateral branch of plant 2. This general trend its stiffness measured in the radial and tangential direc- was evident when either the stiffness of individual xylem tions with respect to stem length was highly correlated strands was plotted as a function of distance from the (r2 ϭ 0.59, N ϭ 122). The mechanical anisotropy of the stem tip (to assess the reproducibility of the longitudinal xylem was evident from comparisons among the means June 1999] NIKLAS ET AL.ÐCACTUS BIOMECHANICS 771

Fig. 3. Tangential stiffness ET plotted against radial stiffness ER of xylem strand segments dissected from plants 1 and 2. Dashed line shows the isometric relationship (ET ϭ ER); solid line is ordinary least squares regression curve (see formula at lower right).

(and the standard errors) of ET and ER (Fig. 2B, C) and from Model Type I and II regression analyses of ET against ER, which gave slopes Ͻ1 (Fig. 3). Because val- ues for ET and ER were calculated independently of the transverse geometry of tissue samples, the anisotropy of Fig. 4. Mean tangential and radial ¯exural rigidity (EIT and EIR, the xylem must be a manifestation of the effect of ana- respectively) of xylem strand segments removed from plants 1 and 2 tomical features on the ability of samples to resist bend- (B and A, respectively) plotted against relative location of segment with respect to stem length (see Fig. 2 for E and E data). Dark vertical ing forces. Preliminary anatomical studies revealed that T R lines are standard errors of means; thin vertical lines are standard de- xylem strands have large living rays that, together with viations of means. regions of axial cell types, give the wood a banded ap- pearance when seen in tangential and transverse planes of section (see Discussion for further comments). The ¯exural rigidity of the strands increased from the In terms of other tissues and portions of the stems, a stem tip to ground level as a consequence of the amor- signi®cant difference in cortex stiffness was observed be- tization of secondary xylem and the attending increase in tween samples successfully removed and tested in bend- its second moment of area I (Fig. 4). The absolute mag- ing from stem segments A and B (5.30 and 0.549 GN/ nitudes of EI at the base of plants 1 and 2 differed sig- m2, respectively). The stiffness of the cortex in the more ni®cantly, presumably because of differences in the age distal segments of the stem was assumed to be K0.549 of plants and thus the amount of secondary xylem that GN/m2 because these samples failed to support their own had accumulated in the stem bases. This was con®rmed weight when suspended horizontally. A statistically sig- by visual inspection of representative transverse sections ni®cant basipetal increase in pith stiffness was observed taken at different locations along the lengths of stems. (Fig. 5A). Although no strong statistical trend was evi- Differences in the EI of individual xylem strands were dent for longitudinal variations in the bulk stiffness of observed and were correlated with the position of strands stem rib tissues (Fig. 5B), the stem ribs removed from with respect to stem ¯exure and thus bending stresses. the upper two-®fths of stems were stiffer than their cor- This was most evident in the curved lateral branches of responding xylem strands (Fig. 6). additionally examined plants where tensile and compres- sive bending stresses reach their maximum intensities No statistically signi®cant correlation was observed along the concave (adaxial) and convex (abaxial) surfaces between tissue stiffness and bulk density. The general of stems, respectively. For example, the ¯exural rigidity trends observed for longitudinal variations in tissue bulk of the strands removed from the abaxial surface of the density and stiffness, however, were reversed for the vas- curved branch from plant 1 was signi®cantly greater than cular and stem rib tissues (Fig. 7). Speci®cally, tissue that of the strands removed from the adaxial surface of samples removed from the top of stems were, on average, the stem because the abaxial strands were more massive more dense but less stiff than anywhere along the length in cross section (and thus had larger second moments of of stems, while the bulk density of vascular tissues in- area) than the adaxial strands. Even though no statisti- creased at the stem base, where tissue stiffness decreased cally signi®cant difference in the material stiffness of the sharply (compare Figs. 2 and 7). The signi®cance of these abaxial and adaxial strands was observed, we concluded observations is unclear, but we note that bulk tissue den- that the differences in the accumulation of secondary xy- sity was measured for non-aspirated samples as the quo- lem between the opposing adaxial and abaxial strands tient of wet weight and volume, and thus do not provide were a function of the differential distribution of bending a direct measurement of the volume fraction of cell wall stresses. materials in these samples (see Discussion). 772 AMERICAN JOURNAL OF BOTANY [Vol. 86

Fig. 6. Histogram comparisons among the mean stiffness of xylem, stem rib, and pith tissue samples removed from plant 2.

chanical support, yet are living and thus photosyntheti- cally competent. Our data show that the bulk stiffness of P. pringlei stem rib tissues is equivalent to that of the vascular tissues. The peripheral stem tissues of this and presumably similar species can thus serve as an important stiffening agent especially in the upper portions of tall and presumably old plants where the stiffness of the rib tissues can exceed the stiffness of the vascular tissues by nearly an order of magnitude. The fact that this ¯uted

Fig. 5. Young's modulus E of pith and stem rib tissue samples (A and B, respectively) from plant 2 plotted against relative location with respect to stem length. Dark vertical lines are standard errors of means; thin vertical lines are standard deviations of means.

DISCUSSION In addition to providing mechanical support, the ele- vated stems of columnar cacti like P. pringlei serve as photosynthetic and water storage organs (Gibson and No- bel, 1986; Mauseth, 1988; Nobel and Meyer, 1991). Thus, the morphology and anatomy of these stems should not be interpreted exclusively in the context of a single biological function but are more pro®tably discussed in terms of how they successfully cope with performing all three functions simultaneously. For example, engineering shows that the best location for the principal stiffening agent in any vertical support member is at the surface because bending and twisting stresses reach their maxi- mum intensities at this location (Timoshenko and Gere, 1961). The optimal location for photosynthetic tissues is likewise at the surface of the columnar stems lacking foliage leaves because this location favors gas exchange and light interception. Since the stiffest known plant tis- Fig. 7. Bulk tissue density ␳ of samples removed from plants 1 and sues are ill equipped for photosynthesis (sclerenchyma 2 plotted against the relative location of samples with respect to stem and wood), it is clear that the requirements for mechan- length. (A) Bulk density of xylem and stem rib tissue samples removed ical support and photosynthesis must be reconciled. In from plant 2. Solid horizontal line denotes mean bulk density of pith samples. (B) Bulk density of xylem tissue samples removed from plant the case of columnar cacti, this reconciliation is achieved 1 (no stem rib tissues were examined for this plant). Dark vertical lines in the form of a ¯uted transverse stem geometry whose are standard errors of means; thin vertical lines are standard deviations peripheral tissues are suf®ciently stiff to provide me- of means. June 1999] NIKLAS ET AL.ÐCACTUS BIOMECHANICS 773

average radius of the triangular sections and that of the polygon section, respectively.) We now denote the unit radius of the circular cross section with area AP as ri and the unit radius of the stem as a whole (due to the average radial thickness added by the stem ribs) as ro. Solving for the ¯exural rigidity contributed by the polygon gives 4 0.7854r i EP , where EP is the bulk material stiffness of the tissues comprising this portion of the stem. Solving for the ¯exural rigidity contributed by all of the triangular 44 ribs gives 0.7854(rroi ± )ER , where ER is the bulk ma- terial stiffness of the stem rib tissues. Setting these two formulas equal to one another reveals when the stem ribs contribute as much to the ¯exural rigidity of the stem as the central portion of the stem, and solving this identity for the dimensionless quotient h/b gives the formula h 11/2 180Њ ϭ 0.5 ϩ 1 Ϫ 1 cot , b []΂΃␥ ΂n ΃ where h/b is the stem rib aspect ratio (the quotient of the rib radial and tangential dimensions) and ␥ϭER /EP (the dimensionless quotient of the bulk stiffness of rib tissues and the bulk stiffness of the remaining stem tissues). The Fig. 8. A model for the geometric contribution of the stem rib tis- demonstration that the ribs are structurally important is sues to the ability of a stem as a whole to resist bending forces. (A) The cross section of a ¯uted columnar cactus stem is geometrically completed by plotting the aspect ratio h/b against the approximated by a series of n number of triangles (each with radial and number of ribs n on a stem for different values of ␥ϭ tangential dimensions h and b, respectively, and transverse area Ar) sur- ER /EP. This graphic device allows us to predict the rib rounding a regular polygon with n sides of length b and transverse area aspect ratio for any stem with n number of ribs that ob- AP (diagram at left). The average radius of the series of triangles equals tains a ¯exural rigidity equal to that of the remaining r ± r , where r is the average radius of the polygonal portion of the o i i portion of the stem (Fig. 8B). In the case of plant 2, n ϭ stem and ro is the average radius of the stem as a whole. Thus, the second moment of areas of the polygon and the triangular series equal 15 and, for the upper third of stem length, ER ഡ EP (i.e., 444 1). Inserting these values into the formula gives h/b 0.785rrrioi and 0.785 ( ± ), respectively. (B) The stem rib aspect ratio ␥ϭ h/b that holds for the condition that the second moment of area of the ϭ 0.974 in contrast to h/b ϭ 1.02 Ϯ 0.5, which was triangular stem ribs equals that of the remaining portion of the stem observed based on 30 measurements of 15 ribs. Taken at plotted against the number of stem ribs n for different values of ␥ϭ face value, the agreement between the predicted and ob- E /E , where E is the stiffness of the stem rib tissues and E is the R P R P served rib aspect ratio argues that, by virtue of their lo- stiffness of the rest of the stem. The thin horizontal line denotes the predicted value of h/b for a stem with 15 ribs (shown by a vertical line). cation as well as their bulk tissue stiffness, the ribs of P. See text for further details. pringlei contribute signi®cantly to the ability of stems to cope with bending forces. Our formula does not take into account the conse- geometry can simultaneously cope with stem dilation quences of growth dynamics or changes in stem water when water is stored in stem ground tissues must not content, either of which can in¯uence the stem rib aspect escape attention (Moran, 1968; Mauseth, 1988). ratio. The accumulation of secondary tissues in older por- Owing to their location, the geometric contribution (the tions of stems or transient changes in stem turgor are second moment of area) made by stem ribs to the overall expected to decrease the aspect ratios of ribs, which, on ¯exural rigidity of stems is substantial and is shown eas- average, decrease toward the stem base as the primary ily with the aid of a few simple calculations. Each ¯uted tissues comprising the ribs are replaced by periderm. In transverse stem geometry (Fig. 8A) can be approximated this regard, our measurements indicate that the periderm by a series of triangles (each representing the transverse at the base of old stems is as stiff or stiffer than the wood geometry of a stem rib) with radial and tangential thick- it surrounds, such that our formula predicts h/b K 0.05. ness h and b surrounding a regular polygon with n sides This value corresponds to the trivially corrugated stem (representing the remaining portions of the stem cross surface that is typically seen at the base of old P. pringlei section). Noting that the number of triangles and their stems. tangential thickness de®ne the number and length of the Despite the mechanical importance of the stem ribs in polygon's sides, we see that the total transverse area of the younger portions of stems, the longitudinal variations all the triangles (i.e., the sum of all the rib cross sections) in the material stiffness and cross-sectional area of the AR equals nbh/2 and that the transverse area of the poly- xylem cannot be neglected. Despite the sharp drop in the 2 gon AP equals (nb /4) cot (180Њ/n). It also follows that stiffness of the xylem at the base of stems, the geometric the second moment of area of all the triangles IR and the contribution made by this tissue (i.e., the axial second second moment of the polygonal portion of the stem IP moment of area of the xylem) to the ¯exural stiffness of must equal the second moments of area of circular cross the stem as a whole increases signi®cantly toward the sections with equivalent cross sectional areas AR and AP, stem base because of the basipetal increase in the xylem respectively. (Note that this is equivalent to taking the transverse area. Speci®cally, engineering theory shows 774 AMERICAN JOURNAL OF BOTANY [Vol. 86 that this geometric contribution increases as a function of the fourth power of the radial thickness of the xylem (see Wainwright et al., 1976; Niklas, 1992). Thus what the xylem may lack in material stiffness is more than com- pensated for by its basipetal increase in cross-sectional area. The xylem is also mechanically anisotropic since it is much stiffer when bent in the radial than in the tan- gential direction with respect to stem length. This an- isotropy may be functionally important since the xylem strands are arranged in a circle when seen in stem tran- section, such that an almost equivalent number will bend in the radial and in the tangential directions regardless of the direction of a bending load with respect to the stem as a whole. Under any circumstances, the mechanical anisotropy of the xylem is entirely compatible with the anatomy of this tissue. Preliminary anatomical studies reveal that P. prin- gleii xylem consists of very tall rays composed of living thin-walled cells that, when viewed in either the tangen- tial or transverse plane of section, are approximately as wide as the intervening layers of axial cell types. The xylem strand of P. pringleii can be thus crudely approx- imated as a beam composed of alternating vertical plate- like layers of two materials differing in stiffness (Fig. 9A). Designating the stiffness of these two materials as E1 and E2, it follows from the basic theory of composite materials (Hollister and Thomas, 1966; Wainwright et al., 1976; Niklas, 1992) that the bulk stiffness of the strand as a whole measured in the radial and tangential direc- tions ER and ET are given by the formulas ER ϭ (E1V1 ϩ E2 V2) and 1/ET ϭ (V1/E1) ϩ (V2/E2), respectively, where V1 and V2 are the volume fractions of the two component materials such that V1 ϩ V2 ϭ 1. Taking the quotient of the radial and tangential stiffness, we see that ER/ET ϭ (␣ V1 ϪV1 ϩ 1)[(V1/␣) Ϫ V1 ϩ 1], where ␣ϭE1 / E2. Fig. 9. A beam composed of two materials differing in stiffness (E1 and E ) arranged as vertical plates (A) is always as stiff or stiffer in the Plotting the quotient ER/ET against the volume fraction of 2 radial than in the tangential direction (E Ն E ) (B). See text for further either of the two materials (i.e., V ϭ V or V ) shows that R T 1 2 details. ER Ն ET regardless of the numerical value of E1/E2. That is, the stiffness of the composite material (i.e., the bulk stiffness of the vascular strand) measured in the radial stems can grow 11±13 m high, thicker walled libriform direction must always equal or exceed the stiffness mea- ®bers and ray cells occur in the comparatively slender sured in the tangential direction (Fig. 9B). wood column nearer the summit than at the base of stems Currently, we have no empirical basis to explain why (Carlquist, 1969, cf. ®gs. 41±42). The anatomical trends the stiffness of the xylem sharply decreases at the base observed for this lobelioid species show that the volume of stems. Indeed, we expected the stiffness of this tissue fraction of cell wall materials (and perhaps the stiffness) to increase basipetally just as we anticipated a strong and of secondary vascular tissues is not invariably correlated positive correlation between tissue stiffness and bulk den- with the age and location of the cambium with respect to sity, whereas none was found. The absence of a corre- the stem apex. It is clear that detailed anatomical studies lation between tissue stiffness and density may be a result are required to determine why the stiffness of P. pringlei of measuring the latter using non-aspirated tissue sam- vascular tissues changes abruptly near the stem base. ples. (An inverse relationship between tissue density and These studies are currently underway and will be report- stiffness can result if wood progressively accumulates ed in subsequent publications. embolized vessels, even if stiffness correlates well with Comparatively few studies have devoted attention to the volume fraction of cell wall materials.) A dispropor- the biomechanics of cacti (see Nobel and Meyer, 1991; tionate accumulation of living secondary tissues with thin Niklas and Buchmann, 1994; Cornejo and Simpson, non-ligni®ed walls near the stem base could also account 1997; Molina-Freaner, Tinoco-Ojanguren, and Niklas, for the trends observed for vascular tissue stiffness and 1998). But there is little doubt that these organisms pro- density. In this regard, we note that anatomical studies vide exciting opportunities to study how their manifold indicate that the wall thickness of secondary xylem cell biological functions are anatomically resolved in what are types produced at the stem base of some dicot species generally considered dif®cult environmental conditions. can be appreciably thinner than elsewhere along the The cacti also provide numerous opportunities to identify length of stems. For example, in Cyanea leptostegia, an the extent to which anatomical features correlate with tis- unbranched palmiform lobelioid from Hawaii, whose sue stiffness and strength and how these mechanical June 1999] NIKLAS ET AL.ÐCACTUS BIOMECHANICS 775 properties are affected by ecological specializationÐa re- ÐÐÐ, Y. UOZUMI,B.J.PLEMONS, AND J. V. LANDRUM. 1995. 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