numerical and experimental validation of unitized beam … · 2018. 10. 27. · model made by an...

NUMERICAL AND EXPERIMENTAL VALIDATION OFUNITIZED BEAM MODEL

Danzi, Francesco∗ , Cestino, Enrico∗ , Frulla, Giacomo∗ , Gibert, James M.∗∗∗ Politecnico di Torino, ∗∗Ray W. Herrick Lab, Purdue University

Keywords: Beam, Homogenization, Optimization, Stiffened structure, Variable Stiffness

Abstract

This paper presents the numerical and experi-mental validation of a variable stiffness box beammodel made by an arrangement of stiffened andunstiffened panels. The derivation of the equiv-alent properties of curvilinear stiffened plates isbriefly summarized. The validity of the equiv-alent continuum plate model is assessed. Thegoverning equations of the variable stiffness box-beam are presented. Once the model is estab-lished, the stiffeners’ path to attain a desiredstatic performance is sought. The optimal config-uration is determined by a topology optimizationproblem where the design variables become theorientation of the curved stiffeners at prescribedpoints. Several analytical examples along withone experiment are presented to show the valid-ity of the model presented herein.

1 Introduction

Aircraft’s contribution to global CO2 emissionhas come under scrutiny since the early 2000s. Inresponse to rising concentrations of green housegas, the 36-State ICAO Council has adopted anew aircraft CO2 emissions standard, which aimsto reduce the impact of aviation’s greenhousegas emissions on the global climate (Annex 16,Volume III). Environmental Responsible Avia-tion (ERA) of NASA’s Fundamental AeronauticProgramme provides guidelines and emission tar-gets for future generation aircraft. Gains in en-ergy efficiency by modifying the wing throughstructural weight reduction and to have higher as-

pect ratios (HAR). The resulting slender, lighterand highly flexible structures are prone to ex-hibit aeroelastic instabilities [1, 2, 3]. Addition-ally anisotropic materials can play a crucial roleenhancing aircraft performance by maintainingrigidity and allowing deformation coupling withno additional penalties on weight. To this end,aeroelastic tailoring is a fundamental tool, as re-ported in the review paper of Jutte and Stanford[4]. The first record of aeroelastic tailoring isdated back in 1949 by Munk[5], where woodenpropeller blade were oriented to create desirabledeformation couplings when operated. In the late1960s, there was a thrust in aeroelastic tailoringresearch, which has continued steadily throughto today. In Weisshaar et al [6], it has beenshown that certain aeroelastic tailoring methodscan modify the wing’s primary stiffness direc-tion, changing the wing’s bending and torsionalstiffness as well as the degree of coupling be-tween the two. Those methods are known asglobal (uniform) aeroelastic tailoring. On theother end, when separate sections of the wingare tailored differently from one another, aeroe-lastic tailoring is applied in a more ”local” man-ner over the wing. These methods are used es-pecially when composite materials are concerned[7, 8, 9, 10, 11].

Potential enabling technologies for passiveaeroelastic tailoring are: Functionally GradedMaterials (FGM) [12], Variable Angle Tow(VAT) [13] and curvilinear stiffeners [14]. Previ-ous research efforts have proposed models withdifferent level of computational complexity todeal with aero-structural design of HAR wing.

1

DANZI, F. , CESTINO, E. , FRULLA, G. , GIBERT, J. M.

While overlooked in most research effort, the ide-alized models have a prominent effect on the fi-nal aircraft; outcomes they provide may influenceconsiderably the entire life-cycle costs [15, 16].These models in fact, represent the physics of theproblem, using a minimum number of degrees offreedom and design variables, rendering the mod-els to have low fidelity that can qualitatively pre-dict the aircraft’s behavior.

In this work, we present a numerical andexperimental validation of a unitized box beammodel subjected to different load conditions. Thebox beam considered has a rectangular cross sec-tion. The upper and lower panels are stiffenedby means of steering stiffeners; the shear websare C spars. The local orientation of the stiff-eners is presumed to vary linearly according toequation proposed by Wu[17] and Wang [18, 19].The stiffened panels are in the concentric or sym-metric configuration. The present paper extendthe current body of research in that it introducescurved stiffeners, previous investigations wererestricted to consider straight but oriented stiff-eners [20, 21]. The aim of the paper is to identifythe stiffeners’ path that maximize the bending-torsion coupling while ensuring a tip deflectionlower than a prescribed limit. Different load con-figurations are considered.

The reminder of the paper is organized as fol-low: section 2 summarizes the derivation of themathematical model adopted herein. Particularly,section 2.1 presents the equivalent plate stiffnessof steering stiffened panels; the equivalent platemodel is validated via Finite Element Analysis(FEA) comparing buckling loads and frequen-cies of simply supported plates obtained with theequivalent model against those obtained with adetailed model. Section 2.2, provides the equa-tions used to calculate the cross sectional stiff-nesses of the beam. Section 3 summarizes themain features of the optimization algorithm im-plemented and presents the mathematical formu-lation of the optimization problem. In section 4the results obtained with the numerical procedureare discussed; a comparison with an experimentis also presented. Finally, in section 5, there areconcluding remarks.

!

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#"

$"

#

#%

#%

&

Fig. 1 : Example of the box-beam cross sectionconsidered in this work.

2 Mathematical model

The derivation of the governing equations ofthe unitized beam starts with the derivation ofthe equivalent continuum plate stiffnesses of thestiffened panels. The latter are used to derive theequivalent beam model by means of the Circum-ferentially Asymmetric Stiffness (CAS) model.For the interest of clarity, an example of the boxbeam’s cross section is depicted in Fig. 1 whereall the relevant dimensions are indicated.

2.1 Derivation of the equivalent plate prop-erties of unitized panel

The structural model is derived on the basis ofthe Reissner-Mindlin plate theory. Consistently,the stiffeners are modeled using the Timoshenkobeam theory. The kinematic equivalence is en-forced within a small repetitive element hence-forth referred to as sub-cell. The strain energydensity equivalence is applied to the repetitivecell, i.e. to the area enclosed between two ad-jacent stiffeners. In the interest of clarity, Fig. 2shows and example of of the repetitive sub-celland cell, respectively.

The local orientation of the stiffeners is pre-sumed to vary linearly as in Wu et al [17] fortow-placed fibers and Wang et al [18, 19] forcurved stiffeners. The stiffeners’ orientation ϑ(x)

2

NUMERICAL AND EXPERIMENTAL VALIDATION OF UNITIZED BEAM MODEL

!

q2

q1

(a)

!"

Straight stiffener

Stiffener path

within the cell

(b)

Fig. 2 : Example of the basic cell Figure (a) (shaded gray area) and sub-cell (b) (shaded red area).

is given as

ϑ(x) = ϑ1 +ϑ2 −ϑ1

bx, (1)

being b the panel length, as shown in Figure2. The stiffeners are presumed being orientedat an angle ϑ(x) with respect to the x-axis ofthe plate. The prismatic rectangular stiffenersare in the symmetric (or concentric) configu-ration and perfectly bonded to the skin panel.The kinematic equivalence is established impos-ing that the strains in any point of the equiva-lent continuum layer are equal to the strains inany point of the stiffeners. The variation of thestrain along the stiffeners width is neglected, con-sistently with the plate’s theory assumption thatMz = 0 . Moreover, since the shear and twistingdeformations act only on one face of the stiffen-ers while on both faces of the differential plateelement, the stiffeners are presumed contributingonly to half of the shear deformation.

In matrix form, the strains of the beam ele-ment rewritten using the usual notation for platetheory (see Reddy [22]) are given as

εb= [E]εP. (2)

Equation 2 relates the generalized strains ofthe Timoshenko beam model to those of the

Reissner-Mindlin plate model. The strain energyfor the Timoshenko beam can be written as inNemeth [23]. Being [Cb] the constitutive ma-trix for the beam and denoting with L j the lengthof the sub-cell, the strain energy for the beam isgiven as

Ub =12

∫L j

εbT [Cb]εbdX (3)

Equivalently, recalling that εTP =

ε(0)xx ,ε

(0)yy ,γ

(0)xy ,κ

(0)xx ,κ

(0)yy ,κ

(0)xy ,γ

(0)yz ,γ

(0)xz , the

strain energy for the beam written in theequivalent plate-strains assumes the followingform

UP =12

∫L j

εPT [CP]εPdX , (4)

where [CP] ∈ R8×8 is given as

[Cp] = [E]T [Cb][E]. (5)

Equation 5 is written in the beam (stiffener) ref-erence frame and must be rotated to align thebeam reference system XY Z to the global refer-ence system xyz (plate system). It follows that

Up =12

∫L j

εpT [Cp]εpdx,

where [Cp] = [T ]T [CP][T ].(6)

3


The transformation T ∈ R8,8 matrix is given as⎡⎣[Tε] [0] [0][0] [Tε] [0][0] [0] [Tt ]

⎤⎦ , (7)

Based on the assumption that the sub-cell is suf-ficiently small, the strain energy can be approxi-mated as follow

Up =L j

2εpT [Cp]εp, (8)

the strain energy density per unit area is given by

Up =UA j

(9)

where A j is the area of the sub-cell. From Fig-ure 2 follows that A j ≈ L jds cos ϑ. The strain en-ergy density for the equivalent continuum layer isgiven as

2U = ns

nsc

∑j=1

εpT [Cp]εpds cos ϑ

=∫ hs

0εT [Q]εdz,

(10)

where ns is the number of stiffeners, nsc is thenumber of sub-cells, and [Q] on the right-handside of the Eq. 10 is the reduced stiffness ma-trix of the equivalent-continuum layer, consistentwith the derivation presented in [24]. Likewise,in the limit for ncell → ∞ one notes the following

limncell→∞

ϑ → ϑ(x),

limncell→∞

2U ⇒∫ hs

0εT [Q(x)]εdz.

(11)

Finally, taking the derivatives of the quadraticforms given in the Equations 10 and 11 respec-tively, yields the stiffness matrices of the equiva-lent continuum layer. It is worth noticing that theequivalent properties obtained using Eq. 11 arethose of a variable stiffness layer, in the follow-ing denoted as EqV S while, Eq. 10, give rise toan equivalent constant stiffness model, denotedas EqH.

In order to assess the validity of the equiva-lent continuum model derived herein are two ex-amples with different topologies of the stiffen-ers. Specifically, the buckling loads and natural

Table 1: Geometric features of the stiffened pan-els under study.

Young’s modulus E 73 GPaPoisson ratio ν 0.3Density ρ 2780 kg/m3

Stiffeners’ width bs 3 mmStiffeners’ spacing ds 100 mmStiffeners’ height hs 20 mmNumber of stiffeners ns 5Plate’s skin thickness hp 3 mmPanel’s width a 500 mmPanel’s length b 800 mm

frequencies of rectangular panels are considered.The panels are simply supported along the edgesand, for the case of buckling loads, are subjectedto uni-axial compression Nx. The results obtainedwith the equivalent models are compared againstthose obtained modeling the skin plate and thestiffeners with shell elements (QUAD4), as de-picted in Fig. 3. The results pertaining the twocases analyzed are reported in Tab. 2 and Tab.3. A good agreement is found among the equiv-alent model and the detailed model. It can benoted from Tables 2 and 3 that the EqV S modelis more accurate than the EqH in either cases.On the basis of the results reported herein, it canbe stated that the stiffness variability introducedby the curved stiffeners cannot be neglected. Amore extensive analysis and discussion, alongwith several other examples, can be found in thefirst author’s PhD dissertation [25].

2.2 Unitized box-beam model

In the following analysis only the flap-torsioncoupling is retained. This assumption derivesfrom the particular arrangement of the panelschosen in this work, i.e. concentric panels withsame topology of the stiffeners for the upper andlower panels. Moreover, the aft and fore panelsare unstiffened C-beams.

The circumferentially asymmetric stiffness

4


(a) (b) (c)

Fig. 3 : Comparison of the buckling modes of a stiffened panel with ϑ1 = −40 and ϑ2 = −14. Fig.(a) is the first buckling mode of the stiffened structure, while Fig. (b) and Fig. (c) are the first bucklingmode for the case of variable (EqV S) and constant (EqH) stiffness respectively.

Table 2: First buckling load [N/mm] of simply supported panels with curved stiffeners subjected touni-axial compression Nx.

TopologyStiffened Eq. V.S. ErrR% Eq. H. ErrR%

θ1 θ2

a −40 −14 293.2 300.0 2.3 338.19 15.3b 10 34 292.7 293.2 0.2 313.5 7.1

Table 3: First frequencies [Hz] of a simply supported panels with curved stiffeners.

TopologyStiffened Eq. V.S. ErrR% Eq. H. ErrR%

θ1 θ2

a −40 −14 114.25 115.01 0.7 118.65 3.8b 10 34 103.91 108.78 -4.7 112.46 7.6

coefficients are given as follows

GJ =4Ω2∮

1/A∗66ds

+4∮

D∗66ds (12a)

k = 2Ω

∮ (A∗

16/A∗66)

zds∮1/A∗

66ds−2

∮D∗

16

(dyds

)ds (12b)

EI2 =∮

z2(

A∗11 −

A∗216

A∗66

)ds+

[∮A∗

16/A∗66zds

]2∮1/A∗

66ds

+∮

D∗11

(dyds

)2

ds

(12c)

EI3 =∮

y2(

A∗11 −

A∗216

A∗66

)ds+

[∮A∗

16/A∗66yds

]2∮1/A∗

66ds

+∮

D∗11

(dzds

)2

ds

(12d)

Retaining only the membrane contributions tothe stiffnesses, the above equations agree withthose given in [26, 27], while, the expression asreported in Eqns 12 are in agreement with [20]where also the bending contributions to the stiff-ness were considered. It is worth mentioningthat, despite the integral reported in the Eqns 12,they are limited to the area enclosed by the mid-line of the thin walled beam, also the contribu-

5


tions of the spar’s flanges has been considered tocompute the overall beam stiffness.

In Figure 4 are reported the effective beamproperties GJt ,k,EI2 with respect to the angles oforientation of the stiffeners ϑ1,ϑ2. For the sakeof simplicity we neglected the variation of thestiffness coefficients with respect to the beam ab-scissa that is, the expression of the effective beamproperties are calculated using the homogenizedcoefficients (see Eq. 10). It is seen that the max-imum coupling k can be obtained with straightstiffeners oriented at ϑ = 27.5. It is worth not-ing that the maximum bending stiffness is givenfor stiffeners oriented at zero while, orienting thestiffeners at ϑ = 45 ensures the maximum tor-sional stiffness GJt . Figure 5 illustrates the varia-tion of the effective beam stiffnesses with respectto the beam abscissa x when the upper and lowerpanels have curved stiffeners with ϑ1 = 0,ϑ2 =25; all the others geometric parameters for theunitized beam are listed in Table 4. The solidlines are the variable stiffness properties whilethe dashed lines are for the homogenized beamstiffnesses. Figure 6 shows the attainable effec-tive beam stiffnesses when ϑ2 varies within theentire range of possible orientations while ϑ1 ischanged parametrically1.

The governing equations for the flexural-flexural-torsional, variable stiffness beam aregiven as follow .

− (GJ)′ ϕ′−GJϕ′′+ k′w′′+ kw′′′ = qϕ, (13a)

−k′′ϕ′−2k′ϕ′′− kϕ′′′+(EI2)

′′ w′′+

2(EI2)′ w′′′+EI2wIV = qw,

(13b)

(EI3)′′ v′′+2(EI3)

′ v′′′+EI3vIV = qv. (13c)

Note that the case of constant stiffness can be re-covered from Eqns 13 neglecting the derivativesof the stiffness’ coefficients; moreover, the caseof isotropic structure can be obtained by assum-ing the coupling term k equals zero.

1The curves reported in Figure 6 are the contour plotsof the envelopes shown in Figure 4

3 Problem formulation

The topology optimization problem is formulatedsuch that the design variables are the stiffeners’orientations at prescribed control points. All theother geometric parameters are fixed (see Table4). The optimization problem for the static casescan be formulated as follows

max12

∫ L

0

(M2

y GJt

GJtEI2 − k2

)dx (14)

subject to: Ku = q

ϑi+11 = ϑ

i2

−45 ≤ ϑi ≤ 45

ϕtip ≥ ϕ0

wtip ≤ w0

where u is the vector of the generalized displace-ment u = [ρx ρy ρz]

T , K is the stiffness matrix,ϑi is the angle of orientation of the stiffeners atcontrol point i, and q = [Mt M f lap Mlag]

T is thegeneralized force vector. The stiffness matrix isgiven as follow

K =

⎡⎣GJt K 0k EI2 00 0 EI3

⎤⎦In the case of planar deformation (i.e. ρz = 0),the strain energy (compliance) is given as follow

12

∫ L

0

(M2

y GJt

GJtEI2 − k2

)dx. (15)

The optimization is carried out by means ofthe Studp GA, i.e. a population based algorithmwhich implement the breeding farm paradigm toenhance the algorithm exploration and exploita-tion. A stress parameter η of 0.4 and a probabilityof extinction of 0.8 have been selected; the readercan refer to Danzi et al [28, 29] for further details.A population of 20 chromosomes is used. Stan-dard mutations has been implemented; the crossover probability is set to 1. The allowable orien-tations range from ϑ = −45 to ϑ = 45 with aset space of 2.5.

6


4.5

40

5

20 40

GJt

109

2

200

1

5.5

0-20-20

-40 -40

(a)

45

30

15

2

0-2

-45-15-30

1

-150 -30

15

0

109

30

k

-4545

2

(b)

1

1.2

45

1.4

1.6

EI2

1010

30

1.8

150

2

45-15 3015-30

1

0-15-45 -30-45

(c)

Fig. 4 : Effective beam properties with respect to the stiffeners’ orientations ϑ1,ϑ2. The envelopes areobtained considering the homogenized properties.

0 275 550 825 1100

x

mm

0

0.5

1

1.5

2

( )

Nmm2

1010

GJt

k

EI2

Fig. 5 : Effective beam stiffnesses with respectto the beam abscissa x for the case of curvilinearstiffeners ϑ1 = 0,ϑ2 = 25. Solid lines representthe variable stiffnesses while the dashed lines arethe homogenized stiffnesses.

4 Results

The wing box under study is made by Al 6060Aluminum alloy (E = 58000 MPa, v = 0.33 andρ = 2780 kg/m3). With reference to Fig. 1, allthe other geometrical parameters, i.e. aft andfore panels thicknesses tw, plate’s skin thicknesshp, stiffeners width bs and height hs, cross sec-tional width b and height of the shear webs hware fixed. For the sake of clarity, the corre-sponding values are listed in Table 4 . Threestatic cases are considered, namely: (a) tip con-centrated load, (b) uniform distributed load and(c) triangular load. For all the cases consid-ered, only qw is applied. The load is appliedat the cross-sectional centroid. The followingpairs of tip displacements w0 and rotations ϕ0are given for the three problems in hand, namely:concentrated load 14 mm, 0.287, uniformlydistributed load 4.6 mm, 0.08 and triangularload 1.1 mm, 0.03.

The first load case is the same used in Ces-tino [20, 21] considered herein as benchmark tovalidate the mathematical model derived. Thebeam is clamped at one end and subjected to aconcentrated load F = 41.37 kg applied with anoffset of 40 mm with respect to the beam tip.Two transducers are placed at the beam’s tip inorder to measure the deflection and rotation re-spectively, as shown in Figure 7. In Figure 8 isreported the comparison between the stiffnessespredicted with the present model and those ob-tained as in [20]. A small deviation with respect

7


0 15 30 45

2 [deg]

4.5

5

5.5

GJt

[Nmm2]

109

1=0°

1=22.5°

1=45°

(a)

0 15 30 45

2 [deg]

0

2

4

6

8

10

12

14

k

[Nmm2]

108

1=0°

1=22.5°

1=45°

(b)

0 15 30 45

2 [deg]

1.1

1.2

1.3

1.4

1.5

1.6

1.7

EI2

[Nmm2]

1010

1=0°

1=22.5°

1=45°

(c)

Fig. 6 : Effective beam properties with respect to the stiffeners’ orientation ϑ2 at fixed ϑ1. Solid linesare for ϑ1 = 0, dashed lines are for ϑ1 = 22.5 and the marker are for ϑ1 = 45. The envelopes areobtained considering the homogenized properties.

Table 4: Geometric features of the wing box un-der study.

Stiffeners’ width bs 3 mmStiffeners’ spacing ds 10 mmStiffeners’ height hs 4 mmNumber of stiffeners ns 6Plate’s skin thickness hp 2 mmBeam’s length L 1100 mmSpar caps length Lw 20 mmSpar height hw 40 mmSpar’s thickness tw 2 mm

to the stiffnesses computed as in [20] is appre-ciable; the deviation is due to the model adoptedto calculate the bending stiffnesses. Indeed, Ces-tino and Frulla considered the bending stiffnessof the stiffened plate being calculated as for solidpanels, that is [D] = [Q](h3

+ − h3−). Danzi [25]

noted that this lead to a discrepancy with respectto the stiffnesses attainable using Nemeth’s for-mulation [23], especially for thicker stiffeners. Inthis case, however, the stiffeners’ height is com-parable with that of the skin plate therefore theaforementioned discrepancy is small. Figure 9reports the bending displacement and the rotationof the beam measured at the beam’s tip. Differ-ent models, namely: (a) solid FE mode, (b) the-oretical derivation as in Cestino [20], (c) exper-iment and (d) present derivation for the case of

straight stiffeners oriented at 25 are comparedagainst each other.

Fig. 7 : Experimental setup for the first load casewith straight stiffeners oriented at 25.

The beam tip deflection and rotation forthe different load cases are listed in Table 5.The last column of Table 5 exemplifies theoptimized topology for the stiffeners path. It isworth noting that, differently to Cestino [20],the optimum orientation of the stiffeners for thecase of concentrated load is at 27.5 rather than27.5. The results are in agreement with thoseobtained in [21] where a topology optimizationwas performed using the SIMP algorithm.Examining the case of uniformly distributed

8


Fig. 8 : Comparison of the stiffnesses obtainedwith the presented model and the model adoptedin [20].

Fig. 9 : Comparison of the mean tip displacementand tip rotation obtained with different modelsand experiments.

load, the optimization leads to curved stiffenerswith ϑ1 = 20 and ϑ2 = 22.5. It is seen thatthe problem is multi-modal, particularly, sincethe homogenized model have been consideredherein, the solution ϑ1 = 20 and ϑ2 = 22.5

is equivalent to the solution ϑ1 = 22.5 andϑ2 = 20. The same behavior has been observedfor the case of triangular distributed load, in thiscase the optimum orientations are ϑ1 = 5 andϑ2 = 12.5.

5 Conclusions

In this work, a Strain-Energy-based equivalencemethod has been adopted to derive the equivalentcontinuum model of curvilinear stiffened plates.It is shown that the derivation gives rise to twoequivalent plate models, i.e. equivalent constantstiffness and equivalent variable stiffness. Thetwo models have been assessed comparing buck-ling loads and frequencies of vibrations of sim-ply supported, rectangular plate. The compari-son were made by means of FEA with respectto the buckling loads and frequencies obtainedconsidering the stiffened structure. It is shownthat the variable stiffness model gives better ac-curacy with respect to the constant model andthus should be preferred to the constant stiffnessone.

A unitized box-beam made by an arrange-ment of stiffened and unstiffened panels has beenthen considered. By means of CAS model, the ef-fective beam cross-sectional properties have beencalculated. A parametric analysis has been per-formed in order to investigate the effect of thestiffeners orientations on to the apparent beamstiffnesses. A comparison of the effective beamstiffnesses with variable and constant stiffnessesis presented for one case aiming to highlight thedifferences between the two models. It has beenobserved that for the cases analyzed herein thedifference between the stiffnesses attainable withthe two models is small.

The effect of the applied load to the stiffen-ers’ path is investigated. Particularly solutionswhich maximize the compliance (strain energy)given a set of requirements in term of maxi-

9


Table 5: Results of the topology optimization for the static cases.

Load ϑi[deg] w[mm] ϕ[deg] Topology1

qw = δ(L)Fϑ1 = 27.5ϑ2 = 27.5

-13.89 -0.29

qw = FL

(!) ϑ1 = 20ϑ2 = 22.5

-4.548 -0.801

qw = FL

(1− x

L

)

(!) ϑ1 = 5ϑ2 = 12.5

-1.0994 -0.0317

1 The beams are drawn out of scale.

mum tip deflection and minimum tip rotation aresought. The optimization has been performed us-ing the Studp GA. The optimization problem hasbeen written as topology optimization where thedesign variables were the orientations of the stiff-eners at the root and tip of the beam respectively.A good agreement between numerical and exper-imental results is found. The results obtained arein agreement with results in published literature.

In conclusion, the authors envision that thelow fidelity model presented here will be use-ful in the preliminary design stages of an air-craft. Indeed, it has been shown that despite us-ing a limited set of degrees of freedom and aminimum number of design variables, the modeland subsequent analysis can qualitatively predictthe behavior of complex structures. It is use-ful in obtaining insights for the successive de-sign stages. Moreover, it offers the advantage ofbeing computationally efficient, the opposite ofcumbersome Finite-Element-based optimizationusing commercial software.

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6 Contact Author Email Address

The corresponding author can be con-tacted at the following email addresses:[email protected] or [email protected]

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