Plant Physiol. (1973) 51, 777-782

Mechanics of the Compression Wood ResponseII. ON THE LOCATION, ACTION, AND DISTRIBUTION OF COMPRESSION WOOD FORMATION'

Received for publication September 28, 1972

ROBERT R. ARCHER AND BRAYTON F. WILSONDepartments of Civil Engineering and of Forestry and Wildlife Management, University of Massachusetts,Amherst, Massachusetts 01002

ABSTRACT

A new method for simulation of cross-sectional growth pro-vided detailed information on the location of normal woodand compression wood increments in two tilted white pine(Pinus strobus L.) leaders. These data were combined withdata on stiffness, slope, and curvature changes over a 16-weekperiod to make the mechanical analysis. The location of com-pression wood changed from the under side to a flank sideand then to the upper side of the leader as the geotropicstimulus decreased, owing to compression wood action. Itslocation shifted back to a flank side when the direction ofmovement of the leader reversed. A model for this action, basedon elongation strains, was developed and predicted the ob-served curvature changes with elongation strains of 0.3 to0.5%, or a maximal compressive stress of 60 to 300 kilogramsper square centimeter. After tilting, new wood formation wasdistributed so as to maintain consistent strain levels along theleaders in bending under gravitational loads. The computedeffective elastic moduli were about the same for the twoleaders throughout the season.

done here. However, several extensions and improvements weremade.

First, the previous model, which was based on a single "effec-tive" modulus of elasticity, was replaced by a composite materialmodel which allowed for two different moduli, EN and EC forthe "normal" wood and compression wood materials. In orderto get more precise estimates of the development of the patternsof NW and CW over the experimental period, we developed anew method which uses "wedges" of each cross section boundedby pairs of rays originating at the pith center and terminaingat the outside of the bark (Figs. 1 and 2). Basically, each wedgeis divided into subunits representing the daily increments ofwood during the experiment. Each subunit is further subdividedfor mapping the location of CW. When all the wedges are com-bined, it is possible to simulate the daily growth of a cross sec-tion. The details of this technique are given in Appendix A.Our measurements showed that there was no significant changein bark thickness during the experiment.

Determination of Zero Gravity Shapes. In a slight modificationof the procedure given before (1), we now define

GK(S) = f MK(SI) ds, (1)

A previous paper (1) presented analytic techniques for re-covering basic geometric and mechanical data for the bendingof a tilted leader. Working with field photographs of two whitepine (Pinus strobus L.) leaders, one tilted 300 and the other 900,cross-sectional data taken during the experimental period, andwith detailed cross-sectional data obtained after the end of theexperiment, it was possible to compute the shapes and stiffnessesduring the righting of the leaders by compression wood action.

In the present paper we examine (a) the correlation betweenorientation to the vertical (geotropic stimulus) and the timingand location of CW' formation; (b) a possible mechanical modelfor the righting of a stem by CW action which changes localcurvatures along the stem; (c) the distribution of mechanicalstrains along the stem during the experiment.

MATERIALS AND METHODS

Composite Material Model. The mechanical model and relatedanalysis presented in Reference 1 formed the basis for the work

1 This work was supported by National Science FoundationGrant GK-3 1490 and Massachusetts Experiment Station ProjectMcIntire-Stennis 10.

'Abbreviations: CW: Compression wood; NW: normal wood.

where the subscript K corresponds to test K. MK is the bendingmoment, s the arc length measured along the stem, and I thecomposite moment of inertia defined by

I = IN + RIC (2)

where R = Ec/EN and EC and EN are the moduli of elasticityin the CW and NW zones, respectively.Now taking pairs of test results with K = 1 denoting the natu-

ral leader without additional loads and K = I denoting thestem with particular extra loads, one computes

8R[GI - G1] ds,

ENl eLrSR

[A401 - AO,] ds,SL

where the difference in slope angles is given byA'K = OK (S) - OK (SO)

(3)

(4)

and SL and sR are chosen so that the curvatures d4Ol/ds are "wellseparated" over the interval SL < s < sR. As presented previ-ously (2), these curvatures are computed from the x - y co-ordinates of the centerline of the leader for the various loadcases. Near the base of the leader curvatures are small and nearthe tip moments are small, so in general SL and SR were chosen tocover the middle 60 to 80% of the stem.

777

778 ~~~~~~ARCHER AND WILSON PatPyil o.5,17

FIG. 1. Cross section 58.4 cm from the base of the leader tilted300. The 36 rays used for analysis are shown. Ray 1 points to theunder side along the n axis. The area of CW is stippled. The dottedline marks the estimated location of the cambium at the start ofthe experiment (RR' in Fig. 2).

RESULTS AND DISCUSSION

Compression Wood Formation Related to the Geotropic Stimu-lus. As the leader in the 3Q0 case bent upwards, the geotropicstimulus (measured as the sine of the angle from vertical con-verted to per cent, with vertical at 0% and horizontal at 100%)decreased and the location of compression wood in cross sec-tions of the leader changed. Four successive stages could be dis-tinguished on the basis of the location of CW. These stages areillustrated for a single cross section in Figure 3 and summarizedfor the entire 30' leader in Figure 4. The stages occurred in thefollowing time sequence: (a) CW formation on the lower side;(b) intermittent CW on a flank side or continuous CW on aflank side; (c) CW formation on the upper side; (d) CW forma-tion again on a flank side. The progression through these stageswas least at the base of the leader, which had the least change ingeotropic stimulus, and showed only stage 1, while the tip of theleader, which had the greatest change in stimulus, progressedthrough all four stages.The transition between stages 1 and 2, and between stages 2

and 3, appeared to 1te related either to the amount of change ingeotropic stimulus or to reaching a certain threshold of geo-tropic stimulus. The end of stage 1, the cessation of CW forma-tion on the under side, occurred all along the leader when thegeotropic stimulus had decreased to 27 to 36% of the maximumand when the respective parts of the leader had reached a stimu-lus level of 24 to 35%. The basal portion of the leader had notstopped forming CW on the under side at the end of the experi-ment. The biggest change was 33% of maximal stimulus althoughit had reached a stimulus level of 18%, 6% less than where stage1 stopped on the rest of the leader. This observation suggeststhat the basal portion continued to form CW on the underside because there was not enough change in geotropic stimulus.The beginning of stage 3, CW formation on the upper side,

occurred when the leader was still 7 to 120 from the vertical.Therefore, it did not occur because the leader passed verticaland the previously upper side became the lower side. Stage 3began when the geotropic stimulus had decreased by 14 to 22%from where stage 1 ended, or a total reduction of 43 to 52%from maximum to a stimulus level of 12 to 25%.

FiG. 2. Diagram of the method for simulating the growth of thewedge between rays 1 and 2 in Figure 1. a: The configuration at thestart of the experiment. PP' is the pith boundary, AA' the first an-nual ring, CC' the cambium, BB' the outer boundary of the bark.b: Configuration at about day 36. RR' marks the inner boundaryof the region of CW (stippled). c: Configuration at the end of theexperiment. WW' marks the outer boundary of the CW. d: Sub-divisions for mapping of CW in b. CC' and DD' mark the limits ofthe strip of area, EFGH marks the subarea that can be scored aseither NW or CW.

It will be noted that a value for R = Ec/EN must be assumedin order to compute the estimate EN, .

Since the individual load cases are independent, a single esti-mate for EN is obtained from

M

EN EN1= (5)EN=(M - 1)

where (M - 1) is the number of loading cases.The zero gravity shape follows as

VI'S) = 01(s) - M(SD ds, (6)

110.

90.

70,

50-

30,

10.

10

O2- 30 40 5O 510GEOTROPICISTIMULUS (%) - -

z

Ii _ _~~~~~-

_~~~~U

19 28RAY NUMBER

laa

36 I0

FIG. 3. The changing geotropic stimulus and CW location overtime in a section 58.4 cm from the base of the leader tilted 300.Extent of the CW increment, shown by horizontal bars, is estimatedfor every 2-day period during the experiment. Position of the raysis shown in Figure 1 for the same section. See Reference 2 forsimilar diagrams of other sections.

778 Plant Physiol. Vol. 51, 1973

, ,ib

COMPRESSION WOOD RESPONSE. II

The top 25 cm of the leader, the portion that formed CW onthe upper side, changed back to forming CW on a flank side,stage 4, at approximately 80 days after the beginning of the ex-periment (Fig. 3). At about that time this entire portion reversedits direction of movement as the action of CW on the upper sideexceeded that on the lower side (Fig. 4). This final stage in theshifting of CW around the leader seemed to be clearly related tothe change in direction of movement; it seemed to be unrelatedto the level of stimulus or to the amount of change in stimulus.The formation of CW on the flank side of the leader was

unexpected because we did not observe lateral or torsionalmovements by the leader as it was bent upwards. Major move-ments would have been observed, but movements of the mainpart of the stem below the point of attachment could havecaused minor lateral movements. It is not clear to us why duringphase 2 only NW was formed from 15 to 30 cm, intermittentNW and CW from 30 to 47.5 cm, and all CW from 47.5 cm to72.5 cm. Again, these longitudinal differences may be due toundetected lateral or torsional movements that were greatestnear the end of the leader.The arc of CW seen in transverse section was greatest (150-

1600) when it was formed on the under side, less (average of1100) when formed on the upper side, and least (800) whenformed on a flank side. CW that formed on the under side ofthe leader which was tipped to 900 from the vertical also coveredan arc of 150 to 1600. Although this leader underwent considera-ble upward curvature owing to CW action, the maximal reduc-tion in geotropic stimulus was only 147c, and CW was formedcontinuously on the under side of the entire leader.

Clearly, CW does not merely form on the under side of a lean-ing stem. In fact, the leader tilted 300 seems to have developed an"equilibrium position" comparable to that in branches (5, 7).CW formation on the under and upper sides seems to changelocation so as to bring the stem into a position where the geo-tropic stimulus is about 25% (150) near the tip to 20% (120)toward the base. Formation of CW on a flank side during phase2 occurs after vertical correction has occurred from CW under-neath the leader, and it stops when CW begins to form on theupper side. Little (5) described a similar phenomenon wherebranches which had been bent first into a horizontal loop andthen downward formed CW on the concave flank side of thebranch only after the branch had moved upwards to its "equi-librium position" through formation of CW on the under side.Vertical orientation seems to take precedence over horizontalorientation.

Thus, tipping the main stem has to some degree made itbilaterally symmetrical like a branch. However, stage 5, theswitching of location from upper to a flank side with the changein direction of movement of the leader, has not been observedfor branches. This switch may represent the loss of the initial"equilibrium position." In any case, stages 2 and 4 both involveCW formation on a flank side after cessation of CW formationon the upper or lower sides. The stimulus for CW formation on aflank side could have been of the same type for both stages.The angular extent of the arcs of CW in branches of white

pine has been reported at 1460 (8); on the under side of ourtilted stems it was 150 to 1600. Some consistent mechanism mustregulate the extent of these arcs. The increase in extent of thearcs is not as important mechanically as it might seem, becausethe contribution of the first moment of the added CW to changein curvature is approximately 1.28r2t, 1.83r2t, and 2.Or2t for 80,160, and 1800 arcs, respectively (taking r and t as the nominalradius and thickness of the arc). Thus, doubling the arc from800 to 1600 only increases the contribution to curvature changefrom 64 to 92% of maximum.

Compression Wood Action. A theory for the mechanical ac-tion of CW was constructed which assumed that CW fibers

(n4mco

0

IL

z4

1--C,,S

10 0 10 20 30 40 50 60 70GEOTROPIC STIMULUS (%)

FIG. 4. Geotropic stimulus related to location of CW incrementson the sides of the leader tilted 300. Heavy lines mark the begin-ning (right) and end (left) of the experiment. Arrows mark the di-rection of the changing stimulus over time as CW action reorientsthe leader.

develop an elongation strain, Ec. This elongation strain can beanalyzed using methods similar to those used for thermal straineffects (Ref. 3, p. 287), i.e.,

e= + sc

where e is the actual mechanical strain, o- the stress in the fibersand Ec the elastic modulus for CW. If a fiber is unconstrained bymaterial attached to it, then the actual mechanical strain e = cand oa = 0. If the fiber is fully constrained then E = 0 and themaximal compressive stress of o-m,l, = -ECe, would result.It will be seen that for the present experiment values of E in therange of 0.3 to 0.5% were computed for Ec from 2 to 6 x 104kg/cm2. Thus o-m. ranging from 60 to 300 kg/cm2 is availableas a mechanical action for changing the curvature. These valuesare within the range of others reported in the literature (7, 9).The actual compressive stress which develops in each fiber isfound only by accounting for the complex interaction of thiselongation strain and the bending resistance of the cross section.The details of this theory are presented in Appendix B.

Every increment of CW develops an elongation strain whichproduces a curvature (Equation B-8). Both leaders sagged duringthe initial week or two of the experiment, presumably beforeCW was mechanically active. Therefore, we started the modelfor curvature recovery after the initial sag period. We found thatthe model was very sensitive to the timing of strain development;assuming that strains developed within a 1-week period producedunsatisfactory results. The best results were achieved by simu-lating an increasing strain from 0% to 10, 60, 90, and finally100% of the total strain over a 4-week period. We assume thispattern of strain development is related to the course of second-

Plant Physiol. Vol. 51, 1973 779

ARCHER AND WILSON

0 15 30 45 60 75 90 105DAYS

FIG. 5. Curvature recovery of sections 43.2 and 53.3 cm frombase of the leaders tilted 90° and 300, respectively, over the courseof the experiment. Measured curvatures (heavy lines) with self-weight influence removed are compared to those predicted by themodel for CW action (dashed lines) for two levels of CW elongationstrain (Ec EN). The upper and lower predicted curves are, re-spectively, for ec of 0.5 and 0.4% for the 30° leader, and of 0.4 and0.35% for the 900 leader.

ary cell wall deposition in the CW. Because new increments ofCW and NW were being added throughout the 4-week period,the restraint of the rest of the section had to be changed continu-ously.The curvature changes predicted by the model (Appendix B)

at a given point on each of the experimental leaders are com-pared to the actual curvatures in Figure 5. For both cases,where EC = EN, potential strains (Ec) of 0.3 to 0.5%vc gave areasonable correspondence with the actual data. Similar re-sults, presented in detail elsewhere (2), were found for four otherlocations on each leader. The oscillations in the actual curvaturedata (Fig. 5) are due to the difficulty of obtaining the requireddata by numerical methods from the points taken from fieldphotographs.These results give strong support to the validity of the mechani-

cal model based on elongation strains. Because of the taper ofthe leaders and the variable amounts of CW, the values used tocompute curvature changes varied by factors of 4 to 5 bothalong the leader and with time, and yet strains in the samerange gave good predictions. The model also successfully pre-dicted curvature reversal at points where CW formed on theupper side of the 30° leader (Fig. 5). Preliminary results measuringstrain relief of CW gave strains of 0.2 to 0.3%cO (A. C. Page,Forestry Department, University of Massachusetts, unpublisheddata) that are close to the strains we found by fitting our modelto the experimental recovery curvatures.

Strain Variation during Growth. When the leaders were firsttilted from the vertical, a state of strain caused by the weight ofthe leader and the terminal group of buds was set up. The peakstrains can be found from

Ml(IB,T - n)eI-, T = - Ex

ment to the moment of inertia, both of which have considerablevariation in both position along the leader and time (Figs. 6and 7). The increase in bending moment with time at the tip isnearly 15-fold while at the middle and base it is only 3.1 and 2.5,respectively. The moment of inertia curves are very sensitive tothe distribution of the new increments to the xylem over thecross section since the higher moments of area (both first andsecond) are involved in its calculation.

Before the experiment the leader had grown for 1 year in avertical position. The flatness of the strain curve marked 1would lend support to the constant strain hypothesis for stemgrowth (4), assuming that intermittent wind loads flex the leaderin bending approximately as gravity loads did just after tilting.Because of the taper the gravity loads do increase slightly towardthe base, but so do the wind loads owing to the wider profilepresented to the wind.Using this same interpretation after tilting, following 4 weeks

of growth the level of strain has been reduced by a factor of 2while still remaining very flat along the length of the leader(Fig. 8). The continued flatness of the strain curve seems con-sistent with the operation of a mechanism which distributesxylem increments so as to maintain a constant strain, in otherwords, a stimulation of growth at points along the stem so as toincrease the stiffness and thus the resistance to the bendingmoments and therefore to reduce strains. The leaders were free

12x

9zLu

(906-

z 3zLLI m

,E0

-

a:LUizLL0

zLL20

-0,E0

zZcIa:Ho(7)

where M, (s) = bending moment due to gravity loads (withoutadditional loads); h(s) = neutral axis in bending; I(s)EN =

bending stiffness; and nB,T = distance along normal to the bot-tom and top fibers of the section.The strain results are shown in Figure 8 for the leader which

was tilted initially to 90° from the vertical. The reasonably con-stant level of strain over most of the length of the leader is note-worthy, since e is computed from the ratio of the bending mo-

10

5

0

0 10 20 30 40 50 60DISTANCE FROM BASE (cm)

FIGs. 6 to 8. Distribution of the bending moment (Fig. 6), mo-ment of inertia (Fig. 7), and mechanical strain (Fig. 8) along theleader tilted 900 at different times during the experiment. Curves1 to 5 are at 0, 27, 63, 91, and lIt days, respectively, after thestart of the experiment. The strains in Figure 8 were computed atthe top- and bottom-most xylem element of each cross section.

2- 8

1 - -

TOPO- 7 . .

BOTTOM

--2 -I

780 Plant Physiol. Vol. 51, 1973

6

COMPRESSION WOOD RESPONSE. II

- 6- !\! 90°E

o50'

0 15 30 45 60 75 90 165DAYS

FIG. 9. Effective elastic moduli of the two leaders throughoutthe experiment (Ec = Ev). Vertical bars give the range of E'scalculated for different loading experiments, and the heavy lineconnects the average moduli.

to flex in the wind, so both the gravity loads and added windloads (both leading to similar strain patterns) could have beenacting.The results for the 30° leader are similar (2) until the tip por-

tion has nearly reached vertical and the bending moment issharply reduced.

Elastic Moduli. Since a value for the ratio R = ECI/EN mustbe assumed in order to calculate EK (equation 5), the range ofvalues R = 0.0, 0.5, 1.0, 1.5, and 2.0 were used. Pillow andLuxford (6) found values for R in static bending (tested in greencondition) between 0.51 and 0.92 for six different species ofconifers. We did not find any indication from our analysis thatone could choose between the cases 0.5, 1.0, or 1.5. As R in-creased, EN decreased so as to give about the same effectivemodulus. In the present work only EN = EC (R = 1) resultsare given. (See Ref. 2 for other cases.) Comparison of the newresults (Fig. 9) with the earlier results (Fig. 4) show the samebasic trend, with the same "hump" in the 900 results over thefirst month, but the new results for the two cases are in betteragreement. Considering the range of loading and large changesin the natural shape of the two leaders, the variation of the elasticmodulus is not surprising.

CONCLUSION

CW formation is a remarkably sensitive physiological-mechani-cal response to a gravitational stimulus. The physiological sys-tem that regulates the location of CW appears to react like acomplex switch to changes in the magnitude of the gravitationalforce acting as the geotropic stimulus; there is not just a simpleformation ofCW on the under side of a stem under all conditions.As the CW response continues, the stem thickens owing to CWand NW formation, and the curvature changes from the elonga-tion strains of the CW. Thus, all the mechanical parametersthat determine the amount of inertia and resistance to bendingby CW action are changing at the same time that the geotropicstimulus and the strains at the outside of the stem change.

Because of the sensitivity of the CW response to changes ingeotropic stimulus, stiffness, diameter, CW location, and stemcurvature, we conclude that an acceptable analysis requiresdetailed cross-sectional information along the stem, data onstiffness, and centerline information for slope and curvature cal-culations. These parameters are so interdependent in the me-chanical-mathematical models used that they must all be meas-ured for a proper dynamic analysis. Conifers seem to have evolveda superbly efficient set of mechanisms to maintain a certainposition through the action of CW while at the same time main-taining a stem form with uniform strains in bending.

Acknowledgment-We thank 'Mr. R. S. Deshmukh for assistance in dataprocessing.

LITERATURE CITED

1. ARCHER, R. R. AND B. F. WILSON. 1970. MIechanics of the compression woodresponse. I. Preliminary analyses. Plant Phlysiol. 46: 550-556.

2. ARCHER, R. R., B. F. WVILSON, AND R. S. DESHNIMKH. 1972. Computer sinmula-tion of the compression wood response. TR No. EMI72-1, Civil EngineeringDepartment, University of MIassachusetts, Amherst.

3. CRANDALL, S. H., N. C. DAHL, AND T. J. LARD-NER. 1972. Introduction to theM\echanics of Solids, Ed. 2. MIcGraw-Hill, New York.

4. LARSON, P. R. 1963. Stem form development of forest trees. Forest Sci.MIonograph 5.

5. LITTLE, C. H. A. 1967. Some aspects of apical dominance in Pinus strobusL. Ph.D. dissertation. Yale University, New Haven.

6. PILLOW, MI. Y. AND R. F. LUXFORD. 1937. Structure, occurrence, and propertiesof compression wood. U.S.D.A. Tech. Bull. No. 546.

7. WESTIN-G, A. H. 1965. Formation and function of compression wood ingymnosperms. Bot. Rev. 31: 381-480.

8. 'VESTIN-G, A. H. 1965. Compression wood in the regulation of branch angle ingymnosperms. Bull. Torrev Bot. Club 92: 62-66.

9. WESTIN-C, A. H. 1968. Formation and function of compression wood ingymniiosperms. Bot. Rev. 34: 51-78.

APPENDIX A

SIMULATION OF A GROWING SECTION USING A RAY-ORIENTEDDATA POINT SET

In the present work we require a much more detailed andspecific account of the amount, type, and site of each incrementof new wood than previously (1). A procedure which organizedthe section data by radial files turned out to be convenient forthe computer simulation as well as representing a good approxi-mation to the actual radial growth process.For each cross section 36 radial lines were drawn at an angu-

lar spacing of roughly 100 (Fig. 1). On each ray the followingpoints were converted to machine-readable form using a Grapha-con tablet on line to a PDP-15 computer; the pith center (fororientation); the pith boundary; the boundary of the first annualring; pairs of CW boundary points (if any) for entry and exitfrom a region of CW; a pair of bark boundary points.For each cross section the x and y coordinates of about 230

to 250 points were stored on tape. Thus the growth simulationfor the two leaders drew on a set of roughly 30,000 three-digitnumbers which represented the cross-sectional geometry inmachine-readable form for the 27 sections for the leader tilted900 and 30 sections for the one tilted 300.The necessary information about each new wood increment

in each wedge, i.e., its contribution to the area, first momentabout the axes, etc. is obtained by constructing a thin strip ofarea (Fig. 2d) corresponding to a prescribed fraction At (tmeasures the days elapsed in the experiment). The movementof CC' from RR' to its final location is assumed to be linearin time (1). The location of RR' is set along other rays, aroundthe section, equal to the distance from AA' to the first CW on theunder side of the section. Since we have the coordinates of theboundary lines of the CW zone for each wedge, it is possible toassign each subarea (Fig. 2d) to either the CW set or NW set.By this method the entire xylem which grew during the experi-ment is broken down into small areas of specified size. In thepresent work we used areas corresponding to about 2% of thetotal xylem ray size in radial extent and about 20 of circumfer-ential arc in the tangential direction. This gave about 10,000area units over the entire xylem. For construction of "bar"graphs (Fig. 3) the strips of area extending around the entire360° of the circumference are drawn horizontally with the CWportions as solid lines.The present work uses a composite-material model (Ref. 3,

p. 495) in which Ec denotes the modulus for the CW and EN theNW. In order to locate the neutral axis, -n, we write the axialstrain as

Plant Physiol. Vol. 51, 1973 781

ARCHER AND WILSON

()t-ni)e = -

p0 = + ec

EC

where1 do d4' (A-2)p ds ds

The force balance along the axis of the stem gives

0 = f £dA =IE (n - n)dA + E f (n - i)dA (A-3)AT P AN P AC

where co is the axial stress and ANandAC denote the areas of thexylem consisting of NW and CW, respectively. It follows that

_ SN + RScn =

ANv + RAC

where

SN,C = CAN ,C

(A-4)

where a- is the axial stress in the fiber and e is the actual mechani-cal strain.The analysis which follows will yield a local change in curva-

ture of the stem and a distribution of self-equilibriating residualstresses across the section due to each new increment of CW.

Strain Distribution. Let 1/po be the curvature of the stem at agiven time t, then due to a distribution of elongation strains E,in the CW fibers acting over a time At, a new curvature 1 Ipwill result. If we assume the associated strain distribution to belinear across the section we get

(n - ) = C ()I-'(p - po)

(B-2)

where AC = (p - po)-' is the change in curvature due to theCW action and fi denoted the distance to the neutral axis of the

(A-5) strain distribution.Force Balance. In order to use the requirement that there be

no net axial force on any section, we integrate (B-1) to get

iidA

AN,= (A-6)dAlNj, C

andR = Ec/EN

The moment-curvature relation is found from

M= I (n-n)dAAT

JEN (n .F)2dA+ Ec(nE-C )dAAN P AC P

or

M =- [fN + RIcIp

wherefN = IN - 2_ZSN + rFiAN

fc = Ic - 2flSc + h2Acand

IN,C= f n2dAAN, C

(A-7)

- [EC (n - fi)dA + Ec (I i)dA

C LA C

and drop the first integral on the right side. Also the strain(A-8) elongation E, is assumed to act only over the small increment

AA, of new CW and the integral is approximately equal toEc,/,AAC .

(A-9) A similar argument making use of

a(n - i)dA = 0

(A-10)A,

yields

(A-1i1) -AC EN (n- f)ldA + Ec (n-fi)2dAN AC(A-12)

(A-13)

All of the integrals indicated above can be calculated approxi-mately by sums taken over the small fundamental areas.The basic relation connecting the moment to the curvature

response takes the form

d-O d*+

M

dsq ds EN(UA + RIc)

and the axial strain becomes

M(n -)EN(fN+ Rfc)

= ECC [LAC (n-n)dA] (B-4)A^C

Solving (B-3) and (B-4) for fi gives

- AAC(IN + RIC) -ASC(SN + RSC)AAC(SN + RSC) -ASc(AN + RAC) (B-)

where the R, AN, AC, SN, SC, IN, and Ic are as defined in(A-5,6,7) and (A-13) and

(A-14)AAC=| dA

S A,c

ASC = nd4AC

(A-15)

APPENDIX B

MECHANICAL ACTION OF THE COMPRESSION WOOD

Assuming that CW fibers undergo an elongation strain Ec,

Using n from (B-5) we obtain AC from (B-3) and finally thenew curvature 1/p in the form

1/p = l/po + ReCAA) (B8)fi(AN + RA,) (SN + RS,)

(A-1) (B-1)

(B-3)

(B-6)

(B-7)

782 Plant Physiol. Vol. 51, 1973

ly% .\