Size optimization of a cantilever beam under deformation...

Struct Multidisc Optim (2009) 39:327–336DOI 10.1007/s00158-008-0342-4

BIOMECHANICAL APPLICATION

Size optimization of a cantilever beam underdeformation-dependent loads with applicationto wheat stalks

P. Sivanagendra · G. K. Ananthasuresh

Received: 8 January 2008 / Revised: 27 September 2008 / Accepted: 9 November 2008 / Published online: 14 January 2009© Springer-Verlag 2009

Abstract In this paper, we consider the optimizationof the cross-section profile of a cantilever beam underdeformation-dependent loads. Such loads are encoun-tered in plants and trees, cereal crop plants such aswheat and corn in particular. The wind loads acting onthe grain-bearing spike of a wheat stalk vary with theorientation of the spike as the stalk bends; this bendingand the ensuing change in orientation depend on thedeformation of the plant under the same load. Theuprooting of the wheat stalks under wind loads is an un-resolved problem in genetically modified dwarf wheatstalks. Although it was thought that the dwarf varietieswould acquire increased resistance to uprooting, it wasfound that the dwarf wheat plants selectively decreasedthe Young’s modulus in order to be compliant. Themotivation of this study is to investigate why wheatplants prefer compliant stems. We analyze this by seek-ing an optimal shape of the wheat plant’s stem, whichis modeled as a cantilever beam, by taking the largedeflection of the stem into account with the help of co-rotational finite element beam modeling. The criteria

This work was presented at the 7th World Congress onStructural and Multidisciplinary Optimization (WCSMO-7),May 21–25, 2007, Seoul, Korea.

P. Sivanagendra · G. K. Ananthasuresh (B)Mechanical Engineering Department,Indian Institute of Science,Bangalore 560012, Indiae-mail: [email protected]

P. Sivanagendrae-mail: [email protected]

considered here include minimum moment at the fixedground support, adequate stiffness and strength, andthe volume of material. The result reported here is anexample of flexibility, rather than stiffness, leading toincreased strength.

Keywords Deformation-dependent loads · Sizeoptimization · Dwarf wheat stalks · Compliant design

List of symbolsE Young’s modulus of materialF Aerodynamic loadCD Drag coefficientCL Lift coefficientFD Drag forceFL Lift forceρ Mass densityA Area of cross-section of the beamAf Frontal areaV Air-flow velocityφ Angle of the rotated spike with the verticalα Rigid body rotation of the beam elementPg Degrees of freedom in the global coordinate

systemPl Degrees of freedom in the local coordinate

systemfint Internal force vectorfext External force vectorKt_int Global tangent stiffness matrix due to internal

forcesKt_ext Global tangent stiffness matrix due to external

forcesU0 Global reference displacement vector�U Global incremental displacement vectorε Strain in the beam element

328 P. Sivanagendra, G.K. Ananthasuresh

σ Stress in the beam elementMR Ground reaction momentdl Lower bound on the diameter of the stemdu Upper bound on the diameter of the stemd(x) Diameter of the stem along its axisS Permissible stress� Permissible lateral deflectionL LengthL0 Initial lengthu Nodal axial deflection in the global coordinate

systemw Nodal transverse deflection in the global coor-

dinate systemθ Nodal rotation of the element in the global

coordinate systemu Nodal axial deflection in the local coordinate

systemθ Nodal rotation in the local coordinate systemV∗ Permissible volume of the stem

1 Introduction

Wind-related crop damage is a major obstacle to cerealproduction (mainly in wheat) that costs several billiondollars per year (Farquhar et al. 2002). Mechanicaldamage to the wheat plant is due to wind-induced stress(Norman 1986), resonance (Miller 2005), anchoragerotation, and buckling under its own weight (Farquharet al. 2002). These biomechanical factors play an im-portant role in the growth and endurance of wheatplants. Hence, the plant biomechanics continues to bean important area of study. The “reduced height” Rhtgenes, known as the dwarfing genes, have been usedto make the plants grow short with a view to mitigatethe aforementioned problems and thereby increase theyield (Borojevic 2005). A study that measured the elas-tic moduli of the dwarf variety of wheat plants foundthat dwarf plants reduced their effective moduli se-lectively (Farquhar and Meyer-Phillips 2001; Farquharet al. 2004). This can be interpreted as the plant’spreference to be compliant (i.e., elastically flexible)even when they are forcibly made to grow shorter thantheir normal height through genetic modification. Bymeans of a simple optimization problem, we argue inthis paper that compliance is indeed preferred undersome assumed performance criteria for wheat plants.

While stress, resonance, and buckling are familiarterms to engineers, “anchorage rotation” is a new term.It refers to one of the main modes of failure whereinthe plant is uprooted at the ground anchor. Agriculturalscientists call this failure dislodging. If we model the

vertically growing plant’s stem as a cantilever beam,then the anchorage rotation is a consequence of thereaction moment at the ground anchor. Minimizing thismoment helps prevent anchorage rotation. Adequatestrength and stiffness are also additional criteria whilethe mass of the material of the stem is limited. The massof the material for the plant stem is a limited resourcebecause there is competition between the growing spikewith grains and the rest of the plant. Thus, optimizationis inherent in plants’ stems to satisfy all these criteria.

From the viewpoint of structural optimization, theoptimization of the plant stem of wheat is an interestingproblem in two ways. First, the wind loads acting on theplant stems are deformation-dependent. This is becausesignificant wind loads act primarily on the grain/flowerbearing spike at the tip. This load changes as the ori-entation of the spike changes due to the deformationcaused by the wind load itself (see Fig. 1). So, the loadhere is dependent on the deformation—a situation thatis considered in fluid-structure interaction problemsinvolving large displacements (Moller 2002).

The second interesting aspect of this problem is thatthere is a conflict between compliant and stiff designsin plant stems. If we consider that the aerodynamicload on the spike can be given by the following formula(Hoerner 1965), we can see that the increased defor-mation helps reduce the tip load, which can be approx-imated as

F = 1

2CDρ AfV2 = F0 cos φ (1)

(a) (b)

Spike

0 cosF F φ=

0Fφ

Fig. 1 Deformation-dependent loads on plant stems: a un-deformed state, b deformed state

Size optimization of a cantilever beam under deformation . . . 329

where F is the aerodynamic load on an ellipticallyshaped object oriented at an angle φ with the directionperpendicular to that of the air-flow velocity, V. Here,the drag coefficient CD, frontal area Af, mass densityρ, and V are combined into a constant term F0. Thecos φ term includes the effect of the orientation. Whenthe plant deforms and bends more, not only the tip loadreduces but also the moment arm relative to the groundanchor reduces. Consequently, the reaction moment atthe ground anchor reduces. Thus, there are two moti-vations for preferring compliant (i.e., flexible) stems.But excessive deformation due to large flexibility hasother problems. These are: breaking of the stem dueto excessive stress; excessive deformation that makesthe plant interfere with its neighbors; resonant behaviorunder varying wind loads; and finally buckling underthe self-weight. Thus, plants seem to seek an optimalconfiguration for adequate stiffness, strength againstbuckling and material failure, and resonance while pre-venting anchorage rotation due to the reaction momentfelt at the ground anchor. There is also the limitedresource condition of available mass as the plant growsand sustains itself.

In order to investigate whether plants prefer com-pliant or stiff stems in the aforementioned con-flicting requirements, in this paper we consider thesize-optimization of a plant’s stem by modeling it asa cantilever beam. We include the large deformationeffects. As noted above, the load depends on the defor-mation which is taken into account in the geometricallynonlinear finite element analysis with co-rotationalbeam elements (Crisfield 1991). Due to this kind ofloading pattern, there will be an additional contributionto the tangent stiffness matrix due to the external load(Sivanagendra 2006). For simplicity and the lack ofinformation, we consider only linearly elastic materialbehavior. The properties of wheat plants’ stems aretaken from Farquhar et al. (2002). In addition to theload on the spike, there exists distributed load all alongthe stem and leaves but it is considered to be negligiblewhen compared with the load on the spike.

The problem is posed in different ways so that weunderstand the criteria that resolve the conflict betweencompliant and stiff designs for the plant’s stem. Preven-tion of the anchorage rotation is taken as the primarycriterion. Stiffness and strength are taken as secondarycriteria. Buckling and resonance are not considered inthis paper although they are important in the overallbehavior of the plant in its operating environment.

The remainder of the paper is organized as follows.In the next section, we include the details of how wedealt with the large-displacement analysis of the beamunder the deformation-dependent loading. Following

this, we present different formulations of the optimiza-tion problem and the results. Discussion and concludingremarks are in the last section of the paper.

2 Large displacement analysiswith deformation-dependent loading

We use large-displacement co-rotational finite elementformulation (Crisfield 1991) for the analysis of 2Dbeams using two-noded elements. We consider a beamelement undergoing large displacement and rotationas shown in Fig. 2. Let Pg = {u1 w1 θ1 u2 w2 θ2}T be thecolumn vector that denotes the displacements in the Xand Y directions and rotation about the Z axis at thefirst and second nodes in the chosen global coordinatesystem, XYZ. In the co-rotational formulation, we needto subtract the rigid body rotation α of the element.After we do that, the local degrees of freedom of thebeam element are expressed as Pl = {

u θ1 θ2}T

where

u = change in the length of the beam = L − L0

θ1 = θ1 − α

θ2 = θ2 − α (2)

where the expressions for L0, L and α are obtainedusing the global coordinates {x1, z1, x2, z2} of the beamelement and its global degrees of freedom, Pg ={u1 w1 θ1 u2 w2 θ2}T. By using the following shape func-tions to interpolate the axial and transverse displace-ments with the local nodal variables of (2), we canwrite the displacement u1 at any point with coordinates

Fig. 2 Beam kinematics in co-rotational formulation


x and z in the axial and transverse directions respec-tively in the local coordinate system, xyz.

una = xL0

u = ζu

wl = L0{ζ (1 − ζ )2 θ1 + ζ 2 (1 − ζ ) θ2

}

ul = una − zdwl

dx(3)

We use a shallow arch based modified strain measurethat avoids membrane locking (Battini 2002), which isshown below.

ε = 1

L

L∫

0

{∂ul

∂x+ 1

2

(∂wl

∂x

)2}

dx − z∂2wl

∂x2(4)

The local and then the global tangent stiffness matri-ces are derived using the general method described inCrisfield (1991). By using the strain measure of (4), wepresent the essential details of the derivation given inSivanagendra (2006) as described below.

By substituting for the interpolated displacementsfrom (3) into (4), we write the strain as follows.

ε = uLo

+ 1

15

(

θ21 − θ1θ2

2+ θ

22

)

+ zL

((4 − 6ζ ) θ1 + (2 − 6ζ ) θ2

)(5)

We assume that the material is linearly elastic andwrite the stress as shown below.

σ = Eε (6)

where E is the Young’s modulus of the material.The internal virtual work, IVW, in the beam element

can be written by using the stress and strain from (6)and (5).

IVW =∫

V

σ δε dV (7)

By defining the reaction forces and moments inthe local coordinate system, Fl = {N, M1, M2}, we canwrite an alternate expression for IVW.

IVW = FTl δPl = {

N M1 M2}T

δPl (8)

By equating (7) and (8), we can obtain the expres-sions for the internal reactions as follows.

N = EA

{u

L0+ 1

15

(

θ21 − θ1θ2

2+ θ

22

)}

(9a)

M1 = EAL0

{u

L0+ 1

2

(

θ21 − θ1θ2

2+ θ

22

)}

×(

2

15θ1 − 1

30θ2

)+ EI

L0

(4θ1 + 2θ2

)(9b)

M2 = EAL0

{u

L0+ 1

15

(

θ21 − θ1θ2

2+ θ

22

)}

×(

2

15θ1 − 1

30θ2

)+ EI

L0

(2θ1 + 4θ2

)(9c)

where I is the moment of inertia of the beam elementthat arises in the volume integration in (7). In order tocompute the internal force vector in the global coor-dinate system, we use the following equality betweeninternal work expressions computed in the local andglobal coordinate systems.

δPTl Fl = δPT

g Fg (10)

By referring to Fig. 2 and differentiating (2), we canobtain the transformation Jacobian between the localand global displacements.

δPl = BδPg (11)

with

B =⎡

⎣−c −s 0 c s 0

−s/L c/L 1 s/L −c/L 0−s/L c/L 0 s/L −c/L 1

⎤

⎦ (12a)

and

L =√

(x2 + u2 − x1 − u1)2 + (z2 + w2 − z1 − w1)

2

c = x2 + u2 − x1 − u1

L

s = z2 + w2 − z1 − w1

L(12b)

From (10) and (11), we can write Fg as follows.

Fg = BTFl (13)

In order to obtain the tangent stiffness matrix in theglobal coordinate system, we take the first variation ofFg with respect to the global displacement vector.

δFg = KgδPg = BTδFl + Nδb1 + M1δb2 + M2δb3 (14)

where bi (i = 1, 2, 3) are the columns of B. The first ofas yet unknown terms on the right hand side of (14) canbe obtained as follows.

δFl = KlδPl = KlBδPg (15)


where Kl can be obtained by writing the internal strainenergy using (5) and (6) and taking its derivative withrespect to the local displacement vector, which yields:

Kl =⎡

⎢⎣

K11l K12

l K13l

Sym K22l K23

l

Sym Sym K33l

⎤

⎥⎦ (16)

where

K11l = EA

L0(17a)

K12l = EA

(2θ1

15− θ2

30

)

(17b)

K13l = EA

(2θ2

15− θ1

30

)

(17c)

K22l = EAL0

(2θ1

15− 2θ2

30

)

+ 2

15EAL0

(u

L0+ θ

21

15− θ1θ̄2

30+ θ

22

15

)

+ 4EIL0

(17d)

K23l = EAL0

(2θ2

15− θ1

30

)(2θ1

15− θ2

30

)

− 1

30EAL0

(u

L0+ θ

21

15− θ1θ2

30+ θ

22

15

)

+ 2EIL0

(17e)

K331 = EAL0

(2θ2

15− θ1

30

)2

+ 2

15EAL0

(u

L0+ θ

21

15− θ1θ2

30+ θ

22

15

)

+ 4EIL0

(17f)

where A is the area of cross-section of the beam ele-ment. The second to fourth unknown terms on the righthand side of (14) are obtained by differentiating thecolumns of A with respect to the displacements using(12a) and (12b). Thus, we can compute the elementlevel tangent stiffness matrix in the global stiffnessmatrix from (14) and then the complete matrix forthe entire structure by assembling all the elementalmatrices into a global one.

The tangent stiffness matrix derived above is neededwhen the static equilibrium equation is linearized forsolving it using the incremental-iterative method. Thedifference between the normal procedure and that ofthe deformation-dependent external load can be seenbelow.

fint = fext (18a)

fint (U0) + Kt_int�U = fext (U0) + Kt_ext�U (18b)

where fint is the internal force, fext the external force,Kt_int and Kt_ext the tangent stiffness matrices of theinternal and external forces respectively, U0 a referencedisplacement, and �U the incremental/iterative dis-placement change. The re-arrangement of (18b) yields:

�U = [Kt_int − Kt_ext

]−1 {fext (U0) − fint (U0)} (19)

This equation is used to iteratively compute thechange in the displacements until convergence.

The external force for a two-noded beam element,which is modeled as a slender cylinder, due to dragand lift forces is shown in Fig. 3 and (20b) below. Anapproximate version of this was given in (1).

fe_ext = {0 0 0 FD FL 0}T (20a)

where

FL = 1

2CLρ A f V2 and FD = 1

2CDρ A f V2 (20b)

CD = 1.1 cos3 φ + 0.02

CL = 1.1 cos3 φ sin φ (20c)

where φ is as shown in Fig. 3, ρ is the mass density, andV is the speed of the wind. The tangent stiffness matrixof the external force on an element can be obtained bytaking the derivative of the force with respect to thenodal displacement vector Ue. Because the wind loadacts on only the last finite element and that too on thefar-end node, the 6 × 6 matrix Ke_ext is computed bydifferentiating (20a) with respect to the displacementvector of the tip element. For this, we note that angleφ is equal to the rigid-body rotation of the tip element,α, shown in Fig. 2. Since α, as can be seen in Fig. 2,

VLF

DF

L

d

φ

Fig. 3 Drag and lift forces on a cylinder subjected to steady wind


depends on the x and z displacements of the two nodesof the element (i.e., u1, w1, u2, and w2), only thoseelements of Ke_ext that correspond to u1, w1, u2, and w2

are non-zero.To validate the numerical solution technique of

the deformation-dependent load, we bench-marked itagainst the commercial finite element solver, ABAQUS(www.simulia.com). Deformation-dependent load is oftwo types:

(a) the first is the “follower load” problem whereonly the direction of the load changes with theorientation of the element on which the load isacting. A classic example of this is a cantileverbeam with a tip load that is always normal to thebeam as the beam bends.

(b) the second is a situation where the magnitude ofthe load depends on the deformation.

Here, we have a general situation where both thedirection and magnitude are changing. Most commer-cial finite element software can deal with only the firsttype, i.e., the follower load case. Hence, we solve thisproblem in order to benchmark our analysis results.In Fig. 4, we can see the good agreement between theresults of ABAQUS and our implementation (labeledFEM in the figure) for the follower load case.

In lieu of verification of the general deformation-dependent load case using another software, we con-ducted an experiment in a wind tunnel (see Figs. 5 and6). A cylindrical piece of foam was used to model thegrain-bearing spike of the wheat plant and a steel wirewas used to model the stem. The wire was mounted

Fig. 4 Comparison of the tip displacements (U for horizontal andW for vertical) obtained with ABAQUS and our finite elementcodes for the cantilever with a follower tip load

Fig. 5 Experimental setup to induce deformation-dependentloads on a cantilever beam as a model of a wheat plant with agrain-bearing spike subjected to wind loads in a wind tunnel

like a cantilever beam by fixing it to a stand. A pitotstatic tube was placed to measure the wind speed. Thedisplacement of the wire (70 mm long and 2 mm indiameter) and the foam-cylinder (20 mm diameter and25 mm tall) were measured using image processing ofthe photograph captured using a digital camera. Theimage of the deflected wire is shown in Fig. 6 and thecomparison between simulation and the experiment isshown in Table 1. The relative error is less than 10%barring one case of 18% relative error for a quantitywhich is very low. We attribute the errors to the mea-surement error in capturing the displacements and pos-sibly due to misalignments in mounting the wire insidethe wind tunnel. This simple experiment confirms thatthe numerical modeling of the deformation-dependentloading is reasonably accurate.

u

w

Fig. 6 Undeformed (a) and deformed (b) configurations of themodel of the wheat plant with a grain-bearing spike in a windtunnel

http://www.simulia.com


Table 1 Comparison of thedisplacements of thecantilever beam subjectedto deformation-dependentwind loads obtainedexperimentally and using thefinite element analysis

No. Velocity FEM solution Experimental % Error(m/s) u (mm), w (mm) u (mm), w (mm) u, w

1 13.35 2.32, 0.06 2.10, 0.05 −9.50, −18.032 15.09 2.96, 0.10 2.78, 0.09 −6.08, −6.063 18.30 4.25, 0.21 4.02, 0.23 −5.40, 9.764 24.24 6.94, 0.55 6.83, 0.58 −1.59, 5.845 27.81 8.61, 0.85 8.64, 0.90 0.35, 6.386 28.66 9.01, 0.93 9.28, 1.01 2.99, 9.19

3 Optimization problem formulation and solutions

Because uprooting at the ground anchor due to a largereaction moment is the most significant failure mode inwheat plants (Farquhar et al. 2002), we first pose thefollowing problem.

Minimized(x)

Ground reaction = MR

Subject toEquilibrium equationVolume constraintdl ≤ d (x) ≤ du (P1)

where circular cross-section is assumed with d(x) asthe diameter along the length of the beam.1 But thisis an inappropriately posed problem because the stembecomes as flexible as it can be (limited only by thelower bound dl) in order to deform to achieve θ → π/2to make the force zero as per (1). Clearly, the stress inthat case would exceed the strength and the stiffnessis compromised too. The strength consideration can beadded in two ways. One method is shown below.

Minimized(x)

MR

Subject toEquilibrium equationsmax

x(σ (x)) ≤ S

dl ≤ d (x) ≤ du (P2)

where σ is the stress and S the permissible strength.The problem was solved using fmincon routine in theoptimization tool-box of Matlab. The gradients werecomputed using the finite difference method ratherthan analytically. Since the number of variables is small,it did not drastically affect the computation time. We

1Since the plant, as reported in the literature, selectively changesthe material stiffness property, it would be more appropriate tomake the Young’s modulus as the design variable. But we chosethe cross-section here as it is treated as a “size-optimization”problem.

note that analytical computation of the gradients ispossible but it would be slightly more cumbersomebecause of the deformation-dependent load. A solutionobtained using this formulation is shown in Fig. 7 alongwith the data used.

Alternatively, the strength criterion can be posed asthe primary criterion while trying to lower the reactionmoment at the anchor indirectly. This is because thestress is maximum at the ground anchor and is pro-portional to the reaction moment. This leads us to theformulation shown below.

Minimized(x)

{S − max

x(σ (x))

}2

Subject toEquilibrium equations

L∫

0

0.25πd2dx ≤ V∗

dl ≤ d (x) ≤ du (P3)

where V∗ is the permitted maximum volume, which istaken as one third the volume obtained if the upperbound were reached on the diameter everywhere inthe beam. Figure 8 summarizes the results of problem(P3) for different values of S using the same data aswas used for problem (P2). It can be noticed in Fig. 8that as the value of S increases there is a tendencytowards compliant designs. In all cases, the reactionmoment was less than F0L and it gets better (i.e.,lower) with increasing values of S. This is in agreement

Young’s modulus = E = 0.5 GPa Height of the plant = L = 1m

Tip load = F0 = 10 N

Lower bound = dl = 2 mm

Upper bound = du = 6 mm

Strength = S = 15 MPa

Fig. 7 Data and the optimum diameter profile of the verticalbeam (i.e., plant’s stem) for problem (P2)


(a) (b) (c) (d) (e)

Fig. 8 Optimum cross-sections of problem (P3). a S = 2 MPa,tip-deflection = 0.0590 m, b S = 3 MPa, tip-deflection = 0.0591 m,c S = 6 MPa, tip-deflection = 0.0597 m, d S = 12 MPa, tip-deflection = 0.0895 m, e S = 15 MPa, tip-deflection = 0.1146 m.Volume constraint was active only in case (a)

with the intuition that higher strength permits largerdeflection and larger deflection reduces the tip load andthe moment arm about the ground anchor. The areaof cross-section profile increasingly tapers upwards andbecomes narrower which is an indication of increasedflexibility. Similar runs for different values of S canbe done for problem (P2) to see that the same resultscould be observed. It can be noticed that the obtainedcross-section profile of Fig. 7 (that had a coarse meshbut the same height as that of Fig. 8a–e) is aboutthe same as those of Fig. 8a–c. But the profiles inFig. 8d–e are substantially different. Note in particularthat the maximum stress is the same in the profiles ofFigs. 7 and 8e. This indicates that problems (P2) and(P3) are different and that the formulation influencesthe obtained cross-section profile. The main differencelies in the fact that in problem (P2) not only the pre-scribed stress value is met but also the ground reactionmoment is minimized. On the other hand, in (P3), onlythe stress value is met. This difference can be under-stood by noting that the maximum stress is dependenton the bending moment as well as the moment of inertia(i.e., the cross-section geometry). Hence, the problem(P2) is more appropriate than (P3).

Fig. 9 Optimal cross-sectionprofiles for problem (P4)a without the strengthconstraint using �/L = 0.2b with the strength constraint,S = 15 MPa and the stiffnessconstraint, �/L = 0.2.dl = 3 mm, du = 10 mm

(a) (b)

Fig. 10 Optimal cross-sectionprofile for problem (P4)wherein the deformation-dependent load is modeledusing (7b) with S = 15 MPa,�/L = 0.3, dl = 2.5 mm,du = 10 mm

The horizontal deflection in the last case with S =15 MPa is about 11.5% of the height of the plant, whichis high. Furthermore, the solution shown in Fig. 8e hada horizontal deflection of more than 33% of the heightof the plant. Thus, it is important to impose an upperbound on the deflection to prevent plants from inter-fering with their neighbors. This indirectly takes careof the stiffness requirement, which is also important forpreventing the mechanical damage to the plant. This isincluded in the following problem statement.

Minimized(x)

MR

Subject toEquilibrium equationsmax

x(σ (x)) ≤ S

ux=L ≤ �

dl ≤ d (x) ≤ du (P4)

Figure 9a–b show the sample results of Problem (P4)without and with strength constraint respectively. Itcan be seen that the optimum profiles are those thatlimit the deflection to the upper portion so that it candeform and lower the tip-load as per (1) while retain-ing stiffness and ensuring that the reaction moment isminimized. It should also be noticed that many ele-ments have reached the upper or lower bounds in bothcases. A more accurate modeling of the deformation-dependent load was given in (20b–c). When this is usedinstead of the approximate load of (1), we obtainedthe result shown in Fig. 10. In this too, most of theelements reached their upper or lower bounds. Thesetwo cases indicate that the optimized profile of thecross-section achieves stiffness (i.e., horizontal tip-loaddeflection) by effectively decreasing the length of thestem. This is done by pushing the cross-sections of thelower elements to their upper bound.

3.1 Discussion

Given the fact that the load and the moment armreduce with increasing deflection—which makes theground reaction small—intuitively we could guess thatplants prefer compliant stems. As noted in the introduc-tion, this issue was taken up for an optimization study


because agricultural scientists and genetic engineershave chosen to make the plants grow shorter than theirnormal height. Engineering view tells us that reducedlength implies increased bending stiffness; indeed, if thelength is decreased by a factor f , the stiffness increasesby a factor of f 3 for transverse bending loads. Stiffstems have more drag (as per (1) and (20b–c)) andlarge moment arm for the ground reaction moment.Thus, growing them to a reduced height appears tobe counter-productive to reducing the ground reactionmoment. Thus, it is probably not surprising that theeffective elastic modulus decreased in the geneticallymodified dwarf variety wheat plants. This seems toindicate that the plant found its way to be compliant.This study, through optimization, confirms the intuitionthat plants do prefer compliant stems when there isno limit on the horizontal tip deflection. This was seenin Fig. 7 where the tip deflection was as much as onethird of the stem’s height. When the tip deflection isconstrained, the optimal stem becomes as flexible asit can be within the limits of that constraint. Anotherimportant observation is that the optimal stem profilesare rather straightforward; that is, they taper from theground to the tip, their significant portion is of uniformcross-section, and many elements reach their lower orupper bounds. This is consistent with the cross-sectionprofiles of the wheat and other cereal crop plants ingeneral.

In this work, as noted in a footnote earlier in thepaper, we could have taken Young’s modulus as the op-timization variable rather than the area of cross-section.This is because the dwarf plants selected changed theirmodulus. The results of that will be slightly differentbecause the bending stiffness is linearly proportionalto the Young’s modulus where as the bending stiff-ness (mainly moment of inertia) is proportional to thefourth power of the diameter. However, the trendsin the performance are not likely to be different. Byperformance, here, we mean that compliant stems arepreferred to stiff stems. Thus, the optimization resultsof the paper support the somewhat counter-intuitiveidea that reducing the height of the stem does notreduce the reaction moment to prevent the anchoragerotation. The lack of quantitative comparison betweenthe selectively lowered modulus notwithstanding, theresults of this preliminary work suggest that the moti-vation behind dwarfing of cereal plants should be re-examined from the biomechanics perspective. Furtherstudies should consider buckling under self-weight andresonant frequency and look for quantitative data forcomparing with the results of the optimization with.Such data is scarce but exists (Crook and Ennos 1996).The current work, in that sense, is not exhaustive but

serves as an example of structural optimization underdeformation-dependent load where compliant designsare preferred over stiff designs.

4 Closure

Biomechanical factors are very important for cerealcrop plants such as wheat to prevent uprooting due toground reaction moment, stress, buckling, etc., whenthey are subject to wind loads. In this paper, we con-sider only the steady wind loads. As explained in thispaper, the determination of the cross-section profile ofthe plant stem (modeled here as a cantilever beam)leads to a novel structural optimization problem withdeformation-dependent loads. An interesting aspectis formulating different optimization problems to seewhy plants prefer compliant stems as was noticed inthe genetically modified dwarf variety (Rht) wheatplants. The results of optimization confirm the counter-intuitive idea that compliant designs are preferred overstiff designs in reducing the ground reaction moment.Competing effects of minimizing reaction moment atthe ground anchor, maximum stress and adequate stiff-ness are considered here. Buckling and dynamic be-havior need to be considered in future work alongwith quantitative comparison with experimental dataavailable in plant biomechanics literature.

Acknowledgements We thank Prof. Raghuraman Govardhanfor pointing us to Hoerner’s book on fluid dynamic drag andproviding the wind tunnel facility for conducting the experimentreported in this paper.

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