Plant Physiol. (1982) 70, 460-4640032-0889/82/70/0460/05/$00.50/0

Mechanical Properties of the Rice Panicle'Received for publication July 8, 1981 and in revised form January 12, 1982

WENDY KUHN SILK, LILY L. WANG,2 AND ROBERT E. CLELANDDepartment ofLand, Air and Water Resources (W.K S.) and Department of Chemical Engineering (L.L. W),University of California, Davis, California 95616; and Department ofBotany, University of Washington,Seattle, Washington 98195 (R.E. C.)

ABSTRACT

Curvature, bending moment, and second moment of stem cross-sectionalarea were evaluated from photographic data and used to compute flexuralrigidity and Young's modulus in the panicle rachis of rice, Oryza sativa L.'M-101.' Flexural rigidity C, and its components E, Young's modulus, and1, the moment of inertia of the area about the neutral axis, were evaluated1.5 cm (tip), 9.5 cm (mid), and 16.5 cm (base) from the tip of the paniclerachis. In dynes per square centimeter, C increases from 1.1 x 103 nearthe tip to 1.09 x 104 in the middle to 5.35 x 104 in the basal region of therachis. Of the components of C, the I changes have the larger effect,increasing from 2.12 x 10-7 centimeters4 near the tip to 8.21 x 10-7centimeters' in mid regions to 6.0 x 10-6 centimeters4 in the basal regions.Young's modulus increases from 4.8 x 10' dynes per square centimeternear the tip to 1.4 x 1010 dynes per square centimeter in mid regions thenfalls to 7.4 x 10i dynes per square centimeter near the base of the mainstem. Values of Young's modulus from Instron experiments were in satis-factory agreement with values calculated from the beam bending equation.Flexural rigidity in the curved region of the panicle proved independent ofpanicle load, indicating that the dissected panicle rachis behaves in somerespects as a tapered loaded beam.

When the rice panicle emerges from its sheathing leaf, it isvertical. As grains fill, the panicle droops so that it is bent in aplane curve with the tip often deflected past the horizontal planewhich contains the insertion of the basal branches. Removal ofthe ripe grains leads to restoration of the vertical orientation. Thissuggests the beam bending equation as a model relating grainweights to shape changes during grain ripening. According to theengineering theory of bending (4), shape (local curvature) isproportional to bending moment

K= M/C (1)

where K equals curvature, M is the bending moment, and C is acoefficient, the flexural rigidity, which describes the resistance tobending. In a homogeneous beam, the flexural rigidity is theproduct of E, Young's modulus (the ratio of stress to a longitudinalstrain produced by the stress) and I, the moment of inertia of thearea about the neutral axis

C= Ex I (2)

Supported by grants from the California Agricultural ExperimentStation and PCM 78-23-710 from the National Science Foundation to W.K. Silk and Contract DE-AM06-76 RL02225 from the Department ofEnergy to R. E. Cleland.

2 Present address: Clorox Technical Center, Pleasanton, CA 94556.

This paper describes the evaluation of the components of equa-tions I and 2 in the rice panicle. A panicle rachis was loaded indifferent ways to test that flexural rigidity is independent of load.As an external test of the suitability of the bending equation,Young's modulus obtained by calculation from values of curva-ture, bending moment, and moment of inertia are plotted withYoung's modulus obtained directly from Instron tests.

MATERIALS AND METHODS

The Rice Panicle as a Loaded Beam. Mechanically, the intactrice panicle is a complex structure. Eight to ten branches, eachbearing grains, are borne on the rachis (main axis). The branchesare wiry and twist to some extent around the stem. Some appearto support the rachis, increasing its rigidity and decreasing localcurvature, while others hang below the stem and may act asconcentrated loads. To study the mechanical properties of therachis, the structure was simplified by removing branches withsharp scissors until only the most apical branch remained. Bendingmoments and curvatures were determined on the singly branchedrachis, as described below. Then the branch was removed, and themeasurements were repeated on the rachis with its ten grains.The beam model is sketched in Figure 1. Ripening grains act as

loads to bend the panicle rachis. A single rice grain attached tothe rachis at d produces a bending moment at s, where both d ands are measured from the tip. The bending moment has a direction,which indicates the direction of rotation, and a magnitude, whichmeasures the tendency of the force to make the rigid grain rotateabout a fixed axis through s perpendicular to the plane of thecurve. The magnitude of the bending moment is the product ofthe force exerted by the grain and the moment arm between d ands. The force equals grain mass times the gravitational coefficient,while the moment arm is given by [x(s) - x(d)], the horizontaldistance between the grain and the position of interest. (Thenotation x(-) represents the component of (.) along the x-axis.)The moment due to the grain at d tends to produce a counterclock-wise rotation ofthe grain about the axis through s. This is balancedby an internal moment tending to produce a clockwise rotation.The external and internal moments result in the observed equilib-rium.As it bends with curvature K (Fig. IC), the rachis is stretched

through its upper half and compressed through its lower half. Asurface of zero strain, the neutral surface, extends through thepanicle center. Magnitude of the strain (relative extension) in-creases with distance from the neutral surface so that maximumextension occurs at the upper edge, and greatest compressionalong the lower edge. Distances d and s are measured along theneutral axis.

Experimental and Numerical Methods. Caryopses of Oryzasativa L. 'M-101' were obtained from N. Rutgers of the Depart-ment of Agronomy and Range Science, (University of Californiaat Davis), and were planted in a mixture of Yolo clay loam andpotting soil in 5 gallon, plastic pots. After emergence, plants were

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MECHANICAL PROPERTIES OF THE RICE RACHIS

A

CM

F: mg

[x(s)-x(d)] = moment armof F

C

Neutral surface

FIG. 1. Diagrams of tapered beam model of the panicle rachis. A,single grain located d cm from panicle tip produces a bending moment at

location s distal to the grain. B, free body diagram of the s cm of A showsforce, F, exerted by the grain, moment arm of the force, and the compen-

sating internal bending moment about an axis through s and perpendicularto the page. C, rachis cross-section in perspective drawing shows R, theradius of curvature (the reciprocal of K of equationI), the neutral surfacewhich is free of strain during bending, and the neutral axis which runs

through the center of the rachis. During bending, the tissue above theneutral surface extends and is under tension, while tissue below the neutralsurface shortens and is under compression.

thinned to five per pot. The rice was grown in the departmentalgreenhouse and was watered and fed with half-strength Hoaglandsolution several times weekly. Panicles emerged about 14 weeksafter planting. Grains filled approximately two weeks after panicleemergence.The intact and dissected panicles were photographed with 35-

mm film and photographs were printed on 8 x 11 inch paper toproduce a total magnification of approximately 1:1. A plumb boband a scale were included in all photographs. After photographyof the dissected rachis, grains on the branch and rachis were

removed, identified on a panicle sketch, and weighed. Fresh cross-

sections ofthe rachis were photographed with an Olympus viewingtube and automatic camera mounted on a dissecting microscope.

Curvature, bending moment, and moment of inertia determi-nations involved digitizing curves from the photographs. A dividerset at 2 mm was used to inscribe equally spaced points along thephotographs of the rachis. The x, y coordinates of the equallyspaced points were recorded with a Tektronix 4956 digitizerinterfaced with a 4052 desktop computer. Care was taken to lineup the vertical plumb bob in the photograph with the ordinate ofthe digitizer. The location and weight of each grain on the rachisand the apical branch were recorded manually. Bending moment

at s, M(s), was assumed positive and was computed as:

M(s) = I F(i)x [x(i) x(s)] (3)i

where x(i) represents the x component of the location of the ith

grain, and the force exerted by the i1h grain, F(i) was evaluated as

the mass of the ith grain times the gravitational coefficient.Curvature was evaluated using a modification of the method

described by Silk and Erickson (9). The distance between pointswas used as the parameter s in the parametric form of the equationfor curvature:

(dxi (d y -(dy\ (d x

dsds)kds2 ds)(dsdK(s)= [(ds) (ds)2dy

[ds7 dTSiNine-point differentiating formulae were used on alternate points.

Under the assumption that the rachis was circular in cross-section,moment of inertia, fy2dA, was computed as half the polar momentof inertia evaluated from digitized micrographs. Elliptic cross-

sections (defined as having I about the x-axis more than 10%different from I about the y-axis) occurred near branch insertion

points but were not used in the analyses.For Instron analysis, care was taken to maintain tissue in a

moist environment during transit. The rachis base was wrappedin a wet sponge and cut with sharp scissors, and the panicle wasimmediately placed on wet paper towels. Tip, mid, and basesegments of 1.5 cm length were cut with razor blades and quicklyplaced on moist pieces of sponge in marked plastic bags whichwere sealed with a commercial 'seal-o-pack' heating device. Formeasurement of load bearing tissue area, fresh cross-sections werecut from the rachis just basal to the Instron segments and werephotographed through a dissecting microscope. Segments werestill moist when they arrived in Seattle after 24 h Express Mail.

Young's modulus (E) was determined for rice panicles in thefollowing way, using an Instron extensometer (3). Preliminarytrails confirmed that fresh tissue gave extension-force curves sim-ilar to methanol-boiled tissue. Panicle sections were boiled 5 minin methanol, rehydrated, placed between the clamps of an InstronTM extensometer (5 mm between clamps), and subjected to twoextensions to 0.5, 1, or 2 kg loads. The slope of the second, elasticextension was determined (percent extension/load). Because atthe high loads used here some movement occurs in the leather-faced clamps, a control extension was run with a steel pin, and theslopes from this extension were subtracted from that of the panicle.The corrected slope was then used, with cross-sectional areasdigitized from photographs, to calculate Young's Modulus.

RESULTS

Curvature. Curvature of intact panicles was first evident at thetip. A panicle photographed on August 22 had K = 0.08 cm-' inthe apical 4 cm (Fig. 2A). Four days later, the curvature maximumwas greater (Kmax = 0.31 cm-') and was located 5.3 cm from thetip (Fig. 2B). The region of significant curvature (K > 0.05)extended to 12 cm (Fig. 2B). By September 2, the grains were fullyripe. Maximum curvature had not changed (Kmax = 0.32 cm-but the length of the curved region had increased to 17 cm (Fig.2D).Removal of branches increased the maximum curvature while

decreasing the length of the curved region, especially when therachis was held vertical (Fig. 3, rows 1 and 2). These effects areprobably due to the increase in moment arm which results fromremoval of the loaded branches. Maximum curvature for thepanicle with the apical branch remaining is Kmax = 0.41 cm-'ats = 6.6 cm. When the last branch is removed so that only thegrains attached to the rachis act as loads, maximum curvature ofthe vertically held rachis is Kmau, = 0.60 cm-'at s = 6.1 cm. Whenthe unbranched rachis is held at 450 to the vertical, the curvaturemaximum is K = 0.49 cm-' at s = 6.5 cm.Bending Moments in the Dissected Panicle Rachis. Bending

moment increases with distance from the tip due both to increasein grains encountered and increase in moment arm. These effectscan be distinguished in the different loadings shown in Figure 3,rows 1 and 3. The abrupt increase in bending moment at s = 7 cmin the branched panicle (Fig. 3A), is due to the extra load whichbegins to act at the point of insertion of the branch on the mainteam. The moment arm effect is clearly visible in a contrastbetween the vertical and inclined orientations. Column B showsa bending moment which approaches a constant value in thevertical region of the rachis, while column C shows a bendingmoment which continues to increase as a consequence of theincrease in moment arm with distance from panicle tip.

Flexural Rigidity, Young's Modulus, and Moment of Inertia inthe Panicle Rachis. Flexural rigidity can be computed as MIKwherever K is large enough to be significant. Figure 3, row 4,shows flexural rigidity as a function of position for three differentloadings of the same panicle. Theoretically, the flexural rigidityshould be a property of the panicle and hence should not varywith panicle loading. Figure 3 confirms that local flexural rigidity

461

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Plant Physiol. Vol. 70, 1982

.4

a X.II

I

FIG. 2. Curvature of an intact panicle rachis during grain ripening. Photographs are above corresponding plots of curvature versus distance from tip.Date of each photograph is indicated.

of the rachis is independent of panicle orientation and presence ofthe apical branch. Thus, Figure 3 represents an internal test of thesuitability of the numerical methods used for curvature and bend-ing moment and also of equation I as a model for panicle shape.In all three figures, flexural rigidity is approximately 3 x 103 dynescm-2 until s = 7 cm where it begins to rise steeply to 1.7 x 104dynes cm-2 at s = 10 cm.Moment of inertia (in cm4) of the area about the neutral axis

increases from 2.1 x 10-7 at the tip to 6 x 10-6 in the basal regionof the rachis (Fig. 4). The 30-fold increase is not unexpectedbecause moment of inertia varies with the fourth power of thestem radius. The branch taper appears to account for the largeincrease in flexural rigidity with distance.Young's modulus was evaluated both with Instron measure-

ments (Table I) and by calculation as C/I (Fig. 5). There was nosignificant difference in Young's modulus between younger,straight (but fully elongated) panicles and mature, bent panicles(Table II). The values are approximately twice as great in the midregion as at the tip but are smaller again in the hollow base region(Table I). The magnitude of the changes in Young's modulus aresmall compared to the increase in moment of inertia with position.The values calculated from the bending equation are in satisfac-tory agreement with the Instron measurements (Fig. 4).

Stress in the Intact Panicle Rachis. The physical complexity ofthe intact panicle makes it difficult to model, but branch removalsimplifies the structure to permit calculation of the mechanicalproperties of the rachis. It is also possible to calculate an effectivelongitudinal stress, ao,, in the fully loaded rachis. The formulausually used for this calculation is:

as= Mh/I (4)

where h is the distance from the neutral surface to the point inquestion (4). (Note that tissue above the neutral surface is intension, a positive pressure by our convention, while tissue belowthe neutral surface is in compression.) Calculation of the bendingmoment is not feasible in the intact panicle. Curvature measure-ments are straightforward, however (Fig. 2); and, by combiningequation 4 with equation 1, one sees that stress can be calculatedas a,, = EKh. Table III shows effective longitudinal stress due toself loading at the tip, mid, and basal regions of the intact, fullyripened panicle. The stress is computed for the outside of the stem

and thus represents the maximum stress in the panicle at the threepoints. Stress is greatest in the middle region.

DISCUSSION

It is interesting to speculate on the usefulness to the plant of themechanical properties described here. The lower flexural rigidityof the tip permits the panicle to bend and thereby to reduce themoment arm produced by the ripe grains. The higher flexuralrigidity at the base of the panicle lessens the curvature there andthereby lowers the stress experienced by this region. Both of theseeffects probably lower the maximum stress in the panicle rachis.Of the components of flexural rigidity, one, Young's modulus,

depends on the composition of the plant tissue, while the other,moment of inertia, depends on the configuration, i.e. the geomet-rical arrangement of the existing material. This work shows thatthe rice rachis modifies flexural rigidity morphologically, via taper,rather than via changes in tissue composition (Figs. 4 and 5). It isworth noting that the hollow center which develops beyond s =10 cm is a particularly efficient structure for conferring flexuralrigidity because it permits a large flexural rigidity for a smallamount of plant tissue. Since moment of inertia increases with thefourth power of the radius, a hollow tube is more resistant tobending than is a solid rod containing the same volume ofmaterial.To a first approximation, the panicle rachis behaves like a

tapered beam (Figs. 3 and 5). Our model considers the variationwith s of Young's modulus and panicle geometry, including thehollow center encountered beyond s = 10 cm, but ignores cross-sectional heterogeneity within the living tissue. Stem anatomymight profitably be considered in more refined studies. Scleren-chyma along the outer perimeter of the stem and vascular bundleswithin the tissues should have elastic moduli different from thecortical cells, (particularly the central cells which are frayed nearthe hollow region). The presence of relatively rigid bundle platesat the nodes in the hollow stem sections (5) is another anatomicalinhomogeneity. These structural inhomogeneities may account forthe discontinuities in the curvature plots (Fig. 2) and the existenceof a preferred angle of bending. Panicles manually bent in a planedifferent from that encountered naturally do not droop as far asthose naturally bent. This is not predicted from the simple beammodel.

462 SILK ET AL.

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MECHANICAL PROPERTIES OF THE RICE RACHIS

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FIG. 3. Evaluation of terms of equation I for three different loadings of a dissected panicle rachis. Top row shows photographs of rachis held verticalwith a terminal branch (A), debranched rachis held vertical (B), and debranched rachis held at 450 (C). Row 2 shows plots of curvature versus distancefor the corresponding photographs. Row 3 shows corresponding plots of bending moment versus distance. Bottom row shows flexural rigidity, computedas M(s)/K(s) in the region of significant curvature. Although the rachis takes different shapes (curvature plots) under the different loadings (bendingmoment plots), its flexural rigidity is similar for three loadings.

The approach taken here owes much to the work of Archer andWilson (1, 2) who have studied the mechanics of the compressionwood response. In preliminary analyses, they calculated the effec-tive elastic modulus in stem leaders of white pine; and in laterstudies they determined the distribution of bending moment,moment of inertia, and strain in the bent leaders. Other workershave found the physiological significance ofmechanical propertiesand have used mechanical analyses to reveal design principles.Leiser and Kemper (6, 7) modeled the tree trunk as a taperedbeam and showed that the most stable structure, ie. the trunk with

lowest maximum stress under wind loading, has a particular taper.This work had implications for nursery management of saplings.McMahon and Kronauer (8) suggested that tree branches mightbe designed to be beams of greatest lateral extent. The mathemat-ical formulation of this hypothesis gave a solution which impliedthat branches would exhibit elastic similarity and that branchdiameter would increase with length raised to the 3/2 power. Thisprediction was verified empirically for cherry, oak, and pine aswas another prediction of the model that the frequency of vibra-tion would be inversely proportional to the square root of the

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Table I. Instron Determinations of Young's Modulus in Panicle Rachis

Position Developmental Stagefrom Tip Straight with grains unripe Bent with ripe grains

cm dynes cm-21.5-2 (5.35 + 1.54) x 109 (4.35 ± 1.39) x 1099.5-10 (1.41 ± 0.73) x 10'0 (1.44 ± 0.57) x 101015.5-16 (6.80 ± 2.14) x 109 (8.08 ± 2.00) x 109

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FIG. 5. Values of Young's modulus. Logarithm to the base 10 of E(expressed originally in dynes cm-2) is plotted against cm of distance fromtip. (*), Instron determinations; (l), calculated as E = MIKI.

Plant Physiol. Vol. 70, 1982

Table II. Analysis of Variance of Young's Modulus with Position andDevelopmental Stage (Straight versus Bent)

Source of Variation F-Value Degrees of SignificanceFreedom Level

Position 14.4 2 *aDevelopmental stage 0.02 1 NS

ap < 0.01.

Table III. Estimatesfor a8, Bending Stress at the Outer Edge of theRachis in the Fully Loaded Panicle in Absence of Wind

s from Tip a,cm dynes cm-21.75 1.73 x l079.75 8.29 x 10715.75 4.72 x 107

branch length.It is instructive to compare the mechanical properties of the

panicle rachis to those of other materials. The elastic modulus of2 to 6 x 10'0 dynes cm-2 calculated for white pine branches (2)shows the rice rachis is approximately 4 times as extensible, orone-fourth as rigid, as pine wood. Other compansons can be madeto nylon, with E = 2.8 x 1010 dynes cm-2, and to steel, with E=2.1 x 1012 dynes cm-2. It is interesting that saplings under windloading would have bending stress, as,, as great as 4 to 8 x 108dynes cm-2 (7), approximately 10 times the as of the self-loadedrice panicle.The measurements described in this paper reveal the mechanical

properties of the panicle rachis. The principles of mechanicaldesign in the intact panicle, the existing developmental strategy,and the possibilities for optimal mechanical design remain to bedetermined.

Acknowledgment. We thank Joseph Keller for an instructive conversation andRobert R. Archer for helpful criticism of our manuscript.

LITERATURE CITED

1. ARCHER RR, BF WILSON 1970 Mechanics of the compression wood response. I.Preliminary analyses. Plant Physiol 46: 550-556

2. ARCHER RR, BF WILSON 1973 Mechanics of the compression wood response. II.On the location action and distribution of compression wood formation. PlantPhysiol 51: 777-782

3. CLELAND RE 1967 Extensibility of isolated cell walls: measurement and changesduring cell elongation. Planta 74: 197-209

4. CRANDALL SH, MC DAHL, TJ LARDNER 1972 An Introduction to the Mechanicsof Solids. McGraw-Hill, New York

5. KAUFMAN PB 1954 Development of the shoot of Oryza sativa L. and thecomparative structure of 2,4-D treated plants. PhD dissertation. University ofCalifornia, Davis

6. LEISER AT, JD KEMPER 1968 A theoretical analysis of a critical height of stakinglandscape trees. J Am Soc Hortic Sci 92: 713-720

7. LEISER AT, JD KEMPER 1973 Analysis of stress distribution in the sapling treetrunk. J Am Soc Hortic Sci 98: 164-170

8. MCMAHON TA, RE KRONAUER 1976 Tree structures: deducing the principle ofmechanical design. J Theor Biol 59: 001-024

9. SILK WK, RO ERICKSON 1978 Kinematics of hypocotyl curvature. Am J Bot 65:310-314

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