MASTER'S THESIS - Fakultet strojarstva i brodogradnje · PDF fileSection 11 2.2.3.1.3. ... Figure 13 Example of a FEMAP interactive form for the definition of tapered beam ... Maja

UNIVERSITY OF ZAGREB

FACULTY OF MECHANICAL ENGINEERING

AND NAVAL ARCHITECTURE

MASTER'S THESIS

Maja Hećimović

Zagreb, 2014.

UNIVERSITY OF ZAGREB

FACULTY OF MECHANICAL ENGINEERING AND NAVAL

ARCHITECTURE

STUDY OF THE MORPHING LEADING EDGE

ON THE WHOLE WING STRUCTURAL

CHARACTERISTICS

Supervisors: Student:

Prof. dr. sc. Vedran Žanić, dipl. ing.

Prof. dr. sc. Shijun Guo Maja Hećimović

Zagreb, 2014.

I hereby declare that this thesis is entirely the results of my own work except where otherwise

indicated.

I would like to give my gratitude to my supervisor, dr.sc. Pero Prebeg for being always ready

to help and discuss on topics regarding this thesis.

Also, I would like to thank to my family, especially my sister Marina, for always being there.

Last but not the least I would like to thank Thomas Sheppard and Damir Zahirović. for

English grammar help and few technical tips.

Maja Hećimović

Maja Hećimović Master’s Thesis

Faculty of Mechanical Engineering and Naval Architecture I

1 Contents

2 INTRODUCTION .............................................................................................................. 1

2.1 Aims and objectives..................................................................................................... 2

3 CALCULATION OF STIFFNESS CHARACTERISTICS ............................................... 4

2.1. Relevant cross-sections.................................................................................................... 4

2.1.1. Determination of relevant point coordinates ............................................................ 7

2.2. TORO program ............................................................................................................ 8

2.2.1. TORO input files ...................................................................................................... 8

2.2.2. Cross-section visualization ...................................................................................... 9

2.2.3. TORO Calculation ................................................................................................. 10

2.2.3.1. Method for generation of the primary response fields in bending and restrained

torsion of thin-walled structures [7] ......................................................................................... 10

2.2.3.1.1. Modeling philosophy for Primary Response in Concept Design ................... 10

2.2.3.1.2. Calculation of Response for a Transverse Strip with a Complex Cross

Section 11

2.2.3.1.3. Cross-Sectional Shear Stress Distribution Due To Bending ....................... 13

2.2.3.1.4. Corrected Normal Stresses due to the Influence of Shear (Shear Lag) 14

2.2.3.1.5. Calculation of Warping and Primary Shear Stresses due to Pure

Torsion 14

2.2.3.1.6. Calculation of Torsional and Warping Stiffness of Thin-Walled

Structures 15

2.2.3.1.7. Normal and Secondary Shear Stresses due to Restrained Warping .... 16

2.2.4. TORO output files .................................................................................................. 18

3. TAPERED BEAM FEM MODELS OF SADE WING .................................................... 20

3.1. Definition of cross-section properties for tapered element ....................................... 20

3.2. Geometry ................................................................................................................... 21

3.3. Mesh .......................................................................................................................... 21

3.4. Constraints ................................................................................................................. 22

3.5. Loads ......................................................................................................................... 23

3.5.1. Pure bending load case ........................................................................................... 24

3.5.1.1. Results ................................................................................................................ 26

3.5.2. Pure torsion load case ............................................................................................ 27

3.5.2.1. Results ................................................................................................................ 28

3.5.3. Combined torsion and bending load case .............................................................. 29


Faculty of Mechanical Engineering and Naval Architecture II

3.5.3.1. Results ................................................................................................................ 31

4 SHELL MODELS ................................................................................................................. 32

4.6. Geometry ................................................................................................................... 32

4.7. Mesh .......................................................................................................................... 33

4.8. Constraints ................................................................................................................. 34

4.9. Loads ......................................................................................................................... 35

4.9.1. Definition of nodes ................................................................................................ 35

4.9.2. Pure bending load case ........................................................................................... 36

4.9.2.1. Results ................................................................................................................ 39

4.9.3. Pure torsion load case ............................................................................................ 40

4.9.3.1. Results ................................................................................................................ 42

4.9.4. Combined bending and torsion load case .............................................................. 43

4.9.4.1. Results ................................................................................................................ 45

5 COMPARISON OF THE STRUCTURAL DISPLACEMENTS BETWEEN BEAM

(BWBLE,BWB) AND SHELL (SWBLE,SWB) MODELS .................................................... 47

5.1. Pure Bending ............................................................................................................. 47

5.1.1. BWBLE and SWBLE ............................................................................................ 47

5.1.2. BWB and SWB ...................................................................................................... 49

5.2. Pure torsion ................................................................................................................ 50

5.2.1.1. BWBLE and SWBLE ......................................................................................... 50

5.2.1.2. BWB and SWB .................................................................................................. 52

5.3. Combined bending and torsion .................................................................................. 53

5.3.1.1. BWBLE and SWBLE ......................................................................................... 54

5.3.1.1.2. R2 displacements ............................................................................................ 55

5.3.1.2. BWB and SWB .................................................................................................. 56

5.3.1.2.1. T3 displacements ............................................................................................ 56

5.3.1.2.2. R2 displacements ............................................................................................ 57

6 COMPARISON OF THE STRUCTURAL DISPLACEMENTS BETWEEN MODELS

WITH LE (BWBLE and SWBLE) and WITHOUT LE(BWB and SWB) .............................. 58

6.1 T3 displacements ....................................................................................................... 58

6.2 R2 displacements ....................................................................................................... 60

6. CONCLUSION ................................................................................................................. 62

8 REFERENCES ................................................................................................................. 63

9 ATTACHMENTS ............................................................................................................. 64


Faculty of Mechanical Engineering and Naval Architecture III

LIST OF FIGURES

Figure 1 EBAM(Eccentric beam actuation mechanism) in its stowed and deflected positions . 1

Figure 2 Comparison of real shell and beam wing planform ..................................................... 4

Figure 3 Neutral beam axes and perpendicular elements ........................................................... 5

Figure 4 SWBLE model with neutral beam axes and section lengths denoted with points ....... 5

Figure 5 SWB model with its neutral beam axes and neutral beam axes of SWBLE model for

comaparison ............................................................................................................................... 6

Figure 6 Listing coordinates of relevant point of each cross-section ......................................... 7

Figure 7 Cross-section visualization a ....................................................................................... 9

Figure 8 Beam tapered model visualization a ............................................................................ 9

Figure 9 Beam tapered model visualization b .......................................................................... 10

Figure 10 First cross-section and second cross-section areas for each element (BWBLE

model) along wing span ........................................................................................................... 18

Figure 11 First cross-section and second cross-section areas for each element (BWB model)

along wing span ........................................................................................................................ 18

Figure 12 Example of a FEMAP neutral file provided by TORO program for the definition of

tapered beam element property stiffness characteristics .......................................................... 20

Figure 13 Example of a FEMAP interactive form for the definition of tapered beam element

property stiffness characteristics .............................................................................................. 20

Figure 14 Geometry of BWBLE model ................................................................................... 21

Figure 15 Meshed Beam model(BWBLE) ............................................................................... 22

Figure 16 Meshed Beam model (BWB) ................................................................................... 22

Figure 17 Constrained beam model (BWB) ............................................................................. 22

Figure 18 Plane characteristics ................................................................................................. 24

Figure 19 Calculation of an approximated shear center ........................................................... 25

Figure 20 Beam model(BWBLE) loaded with pure bending load case ................................... 25

Figure 21 Beam model(BWB) loaded with pure bending load case ........................................ 25

Figure 22 Beam model (BWBLE) displacements (T3) with pure bending load case .............. 26

Figure 23 Beam model (BWB) displacements (T3) with pure bending load case ................... 26

Figure 24 Beam model (BWBLE) loaded with pure torsion load case .................................... 27

Figure 25 Beam model (BWB) loaded with pure torsion load case ......................................... 27

Figure 26 Displacements (R2) of elements (BWBLE) in case of pure torsion load case ........ 28

Figure 27 Displacements (R2) of elements (BWB) in case of pure torsion load case ............. 28

Figure 28 Example of aerodynamic center, neutral axis and chord origin positions for fifth

element ..................................................................................................................................... 29

Figure 29 Beam mode(BWBLE) loaded with combined torsion and bending load case ........ 30

Figure 30 Beam model (BWB) loaded with combined torsion and bending load case ........... 30

Figure 31 Displacements (T3) of elements (BWBLE model) in case of combined bending and

torsion load case ....................................................................................................................... 31

Figure 32 Displacements (T3) of elements (BWB model) in case of combined bending and



Faculty of Mechanical Engineering and Naval Architecture IV

Figure 33 SADE shell wing box with leading edge model (SWBLE) ..................................... 33

Figure 34 SADE shell wing box model (SWB) ....................................................................... 33

Figure 35 Constrained shell model (SWBLE) ......................................................................... 34

Figure 36 Constrained shell model (SWB) ............................................................................. 34

Figure 37 Example of geometry cross-section with four points for needed for definition of

shell nodes ................................................................................................................................ 35

Figure 38 Calculation of a_na and b_na distances ................................................................... 36

Figure 39 Load distribution along the span in the case of pure bending for BWBLE and

SWBLE models ........................................................................................................................ 37

Figure 40 Load distribution along the span in the case of pure bending for BWB and SWB

models ...................................................................................................................................... 37

Figure 41 Loaded shell model (SWBLE) in case of pure bending load case ........................... 38

Figure 42 Loaded shell model (SWB) in case of pure bending load case................................ 38

Figure 43 Displacements (T3) of elements (SWBLE) in case of pure bending load case ....... 39

Figure 44 Displacements (T3) of elements(SWB) in case of pure bending load case ............. 39

Figure 45 Load distribution along the span in a case of pure torsion for BWBLE and SWBLE

models ...................................................................................................................................... 40

Figure 46 Load distribution along the span in a case of pure torsion for BWB and SWB

models ...................................................................................................................................... 41

Figure 47 Loaded SWBLE model in case of pure torsion load case ........................................ 41

Figure 48 Loaded SWB model in case of pure torsion load case ............................................ 41

Figure 49 Displacements (R2) of elements (SWBLE) in case of pure torsion load case......... 42

Figure 50 Displacements (R2) of elements (SWB) in case of pure torsion load case ............. 42

Figure 51 Load distribution along the span in a combined bending and torsion load case for

BWBLE and SWBLE models .................................................................................................. 44

Figure 52 Load distribution along the span in a combined bending and torsion load case for


Figure 53 Loaded SWBLE model in case of combined bending and torsion load case .......... 45

Figure 54 Loaded SWB model in case of combined bending and torsion load case ............... 45

Figure 55 Displacements (T3) of elements (SWBLE) in case of combined bending and torsion

load case ................................................................................................................................... 45

Figure 56 Displacements (R2) of elements (SWBLE) in case of combined bending and


Figure 57 Displacements (T3) of elements (SWB) in case of combined bending and torsion

load case ................................................................................................................................... 46

Figure 58 Displacements (R2) of elements (SWB) in case of combined bending and torsion

load case ................................................................................................................................... 46

Figure 59 Comparison of T3 displacements for pure bending load case along the span for


Figure 60 Normalized comparison of T3 displacements between BWBLE and SWBLE

models in a case of pure bending ............................................................................................. 48

Figure 61 Comparison of T3 displacements for pure bending load case along the span for

BWB and SWB mode .............................................................................................................. 49


Faculty of Mechanical Engineering and Naval Architecture V

Figure 62 Normalized comparison of T3 displacements between BWB and SWB models in a

case of pure bending ................................................................................................................. 49

Figure 63 Comparison of R2 displacements for the pure torsion load case (BWBLE and

SWBLE models ........................................................................................................................ 50

Figure 64 Normalized comparison of R2 displacements between BWBLE and SWBLE

models in a case of pure torsion ............................................................................................... 51

Figure 65 Comparison of T3 displacements for torsion load case along the span for BWBLE

and SWBLE models ................................................................................................................. 51

Figure 66 Comparison of R2 displacement for the pure torsion load cas e(BWB and SWB

models) ..................................................................................................................................... 52

Figure 67 Normalized comparison of R2 displacements between BWBLE and SWBLE

models in a case of pure torsion ............................................................................................... 52

Figure 68 Comparison of T3 displacements for torsion load case along the span for

Beam(BWB) and Shell (SWB)model ...................................................................................... 53

Figure 69 Comparison of T3 displacements for combined bending and torsion load case along

the span for BWBLE and SWBLE models .............................................................................. 54

Figure 70 Normalized comparison of T3 displacements between BWBLE and SWBLE

models in a combined bending and torsion load case .............................................................. 54

Figure 71 Comparison of R2 displacements for combined bending and torsion load case

along the span for BBWBLE and SWBLE models .................................................................. 55

Figure 72 Normalized comparison oR2 displacements between BWBLE and SWBLE models

in a combined bending and torsion load case ........................................................................... 55

Figure 73 Comparison of T3 displacements for combined bending and torsion load case along

the span for BWB and SWB models ........................................................................................ 56

Figure 74 Normalized comparison of T3 displacements between BWB and SWB models in a

combined bending and torsion load case .................................................................................. 56

Figure 75 Comparison of R2 displacements for combined bending and torsion load case

along the span for BWB and SWB ........................................................................................... 57

Figure 76 Normalized comparison of R2 displacements between BWB and SWB models in a

combined bending and torsion load case .................................................................................. 57

Figure 77 Approximation used to determine T3_wb displacement on the place of node of

BWBLE model ......................................................................................................................... 58

Figure 78Comparison of T3 displacements between BWBLE and BWB models along span 59

Figure 79 Comparison of T3 displacements between SWBLE and SWB models along span 59

Figure 80 Comparison of T3 displacements for four models (BWBLE, BWB, SWBLE, SWB)

on positions of beam nodes of BWBLE model ........................................................................ 60

Figure 81 Comparison of R2 displacements between BWBLE and BWB models along span 60

Figure 82 Comparison of R2 displacements between SWBLE and SWB models along span 61

Figure 83 Comparison of R2 displacements for four models (BWBLE, BWB, SWBLE, SWB)

on positions of beam nodes of BWBLE model ........................................................................ 61


Faculty of Mechanical Engineering and Naval Architecture VI

LIST OF TABLES

Table 1 Data used for comparison of T3 displacements for pure bending load case along the

span (BWBLE and SWBLE models) ....................................................................................... 48


span (BWB and SWB models) ................................................................................................. 49





Table 5 Data used for comparison of T3 and R2 displacements for combined bending and

torsion load case along the span (BWB and SWB models) ..................................................... 55

Table 6 Data used for comparison of T3 and R2 displacements for combined bending and

torsion load case along the span (BWB and SWB models) ..................................................... 56


Faculty of Mechanical Engineering and Naval Architecture VII

SAŽETAK

Analiza utjecaja oblikovanog napadnog ruba na krutost cijelog krila. Dvije vrste modela krila

su uspoređene, bazirane na grednim i plošnim (ljuskastim) konačnim elementima. Jedna grupa

modela ima oblikovani napadni rub a druga grupa modela ima samo torzijsku kutiju. Gredni

modeli su pripremljeni prema plošnim (ljuskastim) konačnim elementima koristeći FEMAP,

Solidworks i TORO programe, tako da imaju približno jednake karakteristike krutosti.Tri

vrste opterećenja su primjenjene na pripremljene modele, čisto savijanje, čista torzija i

kombinirano savijanje i torzija. Relevantni pomaci za svako opterećenje su uspoređeni duž

raspona krila.

Ključne riječi:oblikovani napadni rub, metoda konačnih elemenata,karakteristike krutosti


Faculty of Mechanical Engineering and Naval Architecture VIII

PROŠIRENI SAŽETAK

Cilj ovog rada bio je istražiti utjecaj oblikovanog napadnog ruba na krutost cijelog krila. Da bi

se to moglo postići, dva modela krila su uspoređena, jedan sa mehanizmom za oblikovanje

napadnog ruba i jedan samo sa torzijskom kutijom. Oba modela su za proizvoljni putnički

zrakoplov sa 150-sjedala prethodno korišten u EU FP7 SADE projektu. Dodatni zadatak bio

je usporediti rezultate između dva modela krila diskretizirana sa različitim konačnim

elementima (plošnim(ljuskastim) i grednim konačnim elementima).Dva plošna konačna

elementa (sa i bez oblikovanog napadnog ruba) su napravljena tijekom EU FP7 SADE

projekta. Gredni modeli (sa i bez oblikovanog napadnog ruba) su pripremljeni koristeći

konstrukcijske elemente plošnih modela. Plošni modeli su importirani u Solidworks i

presječeni u 11 dijelova duž raspona krila.Relevantne točke sa svakog kraja odsječenog dijela

su korištene za definiranje poprečnih presjeka. Karakteristike krutosti poprečnog presjeka su

izračunate koristeći TORO program. Nakon toga bilo je moguće definirati krutost suženog

konačnog grednog elementa.To je ostvareno unošenjem karakteristika presjeka svakog

suženog konačnog elementa.u neutralni format programa FEMAP.

Nakon što su gredni modeli (sa i bez oblikovanog napadnog ruba) napravljeni, bilo je moguće

opteretiti sve modele (dva gredna i dva plošna) i usporediti rezultate. Primjenjena opterećenja

su bila čisto savijanje, čista torzija i kombinirano savijanje i torzija. Zbog razlika u

modeliranju sa različitim vrstama konačnih elemenata nije bilo moguće primjeniti opterećenja

na istim mjestima u oba modela (grednom i plošnom) i zato su opterećenja primjenjena na

gredne elemente prilagođena da simuliraju iste pomake i na plošnim modelima. Raspodjele

opterećenja duž raspona krila za svaki slučaj opterećenja su prikazane.Za svaki slučaj

opterećenja odgovarajući pomaci su izlistani i rezultati su uspoređeni za plošni modele sa

napadnim rubom i gredni model sa napadnim rubom, odnosno plošni model bez napadnog

ruba i gredni model bez napadnog ruba. Rezultati su također prikazani duž raspona krila. Bilo

je moguće usporediti samo T3 pomake (u smjeru z-osi) za sva četiri modela u slučaju

kombiniranog savijanja i torzije. To je zbog toga što samo za tu vrstu opterećenja, opterećenje

djeluje na istom mjestu (aerodinamički centar) za sva četiri modela. Rezultati pokazuju da su

razlike između plošnih i grednih modela manje nego razlike između grednog elementa bez

napadnog ruba i grednog elementa sa napadnim rubom. Iz ostvarenig rezultata može se reći da

su gredni modeli ispravno zadani i da imaju slične karakteristike krutosti kao plošni elementi.

Također, vidljivo je da oblikovani napadni rub ima utjecaj na karakteristike krutosi za slučaj


Faculty of Mechanical Engineering and Naval Architecture IX

kombiniranog opterećenja. To je vidljivo zbog razlika u pomacima između modela sa i bez

napadnog ruba.


Faculty of Mechanical Engineering and Naval Architecture X

ABSTRACT

The influence of the morphing LE structure on the stiffness of a whole wing was analysed.

Two types of wing models were compared based on finite element method (FEM) shell and

beam models, one set with morphing leading edge mechanism and one set with only wing box

(WB) structure. Beam models were prepared based on shell models using FEMAP,

Solidworks and TORO software. Three load cases were applied to the models prepared, pure

bending, pure torsion and combined bending and torsion. Structural displacements were

obtained and results presented across a wing span for each load case.

Key words: morphing leading edge, FEM, stiffness characteristics, tapered beam element.

Maja Hećimović Master’s thesis

Faculty of Mechanical Engineering and Naval Architecture 1

2 INTRODUCTION

Morphing in the aeronautical field is adopted to define „a set of technologies that increase a

vehicle's performance by manipulating certain characteristics to better match the vehicle state to

the environment and task at hand.“ There is neither an exact definition nor an agreement between

the researchers about the type or the extent of the geometrical changes necessary to qualify an

aircraft for the title „shape morphing“[13]. The objective of morphing activities is to develop high

performance aircraft with wings designed to change shape and performance substantially during

flight to create multiple-regime, aerodynamically efficient, shape changing aircraft.[13]

The SADE(SmArt high lift Devices) project [2] task is to develop a viable morphing wing to

replace the conventional wing. The key goals were to develop a natural laminar wing and reduce

the high lift noise footprint. To meet these objectives, the slotted slat was replaced by a seamless

droop nose mechanism. Secondly, the trailing edge of the flap was identified as a potential

morphing zone to directly increase the lifting performance.

The morphing mechanism used in this project is the eccentric beam actuation mechanism

(EBAM)[1]. The aim of the mechanism is to define wing airfoil shape and provide support against

aero-loads. The device can change wing airfoil depending on flight conditions, while the target

shape is achieved by rotation of the beam so the disks deflect the skin. In the cruise condition,

EBAM is in its stowed position.

Figure 1 EBAM(Eccentric beam actuation mechanism) in its stowed and deflected positions



2.1 Aims and objectives

Unlike the traditional slats which are structurally discontinued from the wing structure, a

morphing leading edge (LE) is part of the wing structure. The geometric change of the morphing

LE which normally takes more than 15% of the wing chord in dimension may have significant

influence on the structural behaviour of the whole wing. For wings with morphing LE, the change

of structural geometry will influence the bending and especially the torsion stiffness and

aeroelastic effect. This will lead to the impact on the aerodynamic performance of the whole wing.

The aim of this study is to analyse the influence of the morphing LE structure on the stiffness in

comparison with a wing modeled with only wing box (WB) without LE structure, for a generic

150-seat passenger aircraft previously used in the EU FP7 SADE project. An additional goal of

this study was to compare the results between two types of wing finite element method (FEM)

models that are usually called shell model and beam model. For the shell based wing model, FEM

shell elements were used to model the skins, spars and ribs. In this model, stringers, rib and spar

caps were usually modeled with beam finite elements. Both shell models are prepared during the

EU FP7 SADE project [2] by the Cranfield University, and were provided for this thesis courtesy

of Dr. Shijun Guo. Beam models were prepared based on the structural characteristic of those two

shell models.

For the beam model, a beam element was used to model the entire stiffness of the wing structure

between two cross sections. The beam model is usually used in an early stage design when the

first information on wing structural behavior or mass distribution is needed. The shell model is

typically used in the later phases of a design, when more accurate results are needed.

The study in this thesis includes comparison of the structural displacements between four models:

shell wing box (SWB),

shell wing box with leading edge (SWBLE),

beam wing box (BWB),

beam wing box with leading edge (BWBLE)



Load cases applied on the models that are used to compare the structural displacements are:

1) pure bending,

2) pure torsion,

3) combined bending and torsion.

Term constraint was used to FEMAP for boundary conditions. Constraints were based on [1].

The total force applied to the wing for load cases 1 and 3 was equal to one half of the airplane

weight [1] :

(1)

After that linear static analysis was performed using FEMAP NASTRAN [3] and obtained

displacement results are shown and compared below.



3 CALCULATION OF STIFFNESS CHARACTERISTICS

In order to prepare the beam model it was necessary to calculate stiffness characteristics at the

relevant cross-sections.

For a general wing with sweep angle and taper, the beam model does not represent an exact wing

planform. In the beam model, all cross section are placed perpendicular to the beam axis. This is

the axis in the neutral axis of the wing structure, therefore the planform at root and tip cannot be

represented correctly as illustrated in Figure 2.[4]

Figure 2 Comparison of real shell and beam wing planform

2.1. Relevant cross-sections

In order to obtain correct geometric characteristic of the beam, surfaces generated from SWBLE

FEM model (using FEMAP export FEM to surface command)were imported into Solidworks and

three referent cross sections were determined at the position of wing root, wing kink and wing tip

Figure 3. The Solidworks model contained surfaces of front spar, rear spar, leading edge skin,

wing box upper skin and wing box lower skin. Since with the FEMAP export command it was not

possible to obtain the position of stringers from the shell model, their position on those three

sections was added manually, as sketch points on the intersection of the exported wing surfaces

and three reference planes. Airfoil coordinates of those three cross-sections at the position of the

stringers were measured in Solidworks and imported into TORO, which was used to calculate

centers of gravity for each cross-section. The calculated centers of gravity were then used to

determine neutral beam axes along the span. One axis goes from the wing root to the wing kink

and the other axis from wing kink to wing tip, as visible on Figure 3.For a BWB beam model three

new cross-sections (at wing root, wing kink and wing tip) were defined and the same procedure as

for determination of neutral axes of BWBLE model was applied.



Figure 3 Neutral beam axes and perpendicular elements

In order to obtain correct coordinates for calculation of beam elements cross sections, two new

coordinate systems were created in which they-axes were defined in the direction of two beam

axes.

Figure 4 SWBLE model with neutral beam axes and section lengths denoted with points

Ro

ot

stre

awis

e se

ctio

n (

RSS

)

Kin

k st

reaw

ise

sect

ion

(K

SS)

Tip

str

eaw

ise

sect

ion

(TS

S)

CG RSS

CG KSS

CG TSS

Beam axis 1

Beam axis 2

CG RSS

Beam element cross section (between root and kink)

Beam element cross section (between kink and tip)



Figure 5 SWB model with its neutral beam axes and neutral beam axes of SWBLE model for comaparison

The next step was to cut the wing into 11 geometric sections (using Solidworks Trim surface

option)[5] at the positions where property changes exist in the respective shell models. Sections

were defined as perpendicular to the neutral axes (first 4 sections perpendicular to the first beam

axes and the remaining seven perpendicular to the second beam axes).

Section 4 from the respective shell models was split into two sections, since Section 4was placed

at the position of the wing kink and belonged to two different beam axes. Section 1 at the wing

root from the shell model was not the same as section 1 needed for the correct definition of beam

modeling the wing root. In order to obtain a correct first cross section (wing root) of beam section

1, the first cross-section(wing root) of section 1 of the shell model was changed using Trim

surface and Extend surface options in Solidworks. Using this, first cross-section of beam model

section 1 was defined as perpendicular to the first neutral axis and therefore relevant for

calculation of stiffness characteristic of beam model. There was no need to change section 11(at

wing tip) of the shell model because that cross-section was approximately perpendicular to second

neutral axis.



2.1.1. Determination of relevant point coordinates

In order to obtain the coordinates for the 11 geometric sections, they were imported into FEMAP

as Parasolid files. The procedure was different then for determination of coordinates for three

reference sections completed previously, because that procedure was time consuming and error

prone. Since for the beam elements it was necessary to obtain coordinates on the 22 cross sections,

it was therefore also necessary to find a solution which would enable execution of this task in a

realistic time frame. Sections which were placed on the first neutral beam axis and sections placed

on the second axis, were given coordinates in the first and second coordinate system respectively.

After the geometric sections were imported into FEMAP, each cross-section was meshed by

meshing two lines(upper and lower) connecting front and rear spar(for BWBLE and BWB model)

and two lines(upper and lower) which define leading edge(just for BWBLE model). The

geometric sections described did not contain stringers, therefore meshing the lines was needed in

order to obtain the same number of nodes as number of stringers for the relevant cross-section.

Additionally, each cross-section was labeled with two points which define front spar and two

points which define rear spar. Leading edge lines always contained the same number of nodes for

every cross-section, since the number of stringers did not change along wing span. Beside those

nodes, four additional nodes were added in order to better describe curvature of the leading edge.

Node coordinates of relevant points in the appropriate coordinate system were listed by the

FEMAP’s List Geometry Nodes command. The relevant point coordinates were inserted into MS

Excel and input files for TORO [6] were prepared.

Figure 6 Listing coordinates of relevant point of each cross-section



2.2. TORO program

TORO was used to calculate cross-section data such as area, moments of inertia, center of gravity,

shear center and warping and torsional stiffness.

Since tapered beam formulation was used for generation of beam models, 11 elements needed 22

cross section stiffness characteristics calculations, therefore22 TORO input files for BWB model

and 22 TORO input files for BWBLE model were prepared. As a result, TORO provided 22

output files with the information about cross-section stiffness characteristics for each tapered

beam model.

2.2.1. TORO input files

TORO’s principle of work was to divide cross-section into elements by defining each element

with two nodes. After node coordinates were provided it was necessary to define element order by

adding two nodes to each element.

TORO input file per each cross-section was prepared in MS Excel, consisting of relevant point

coordinates (which are the same as node coordinates) and cross-section element order (defined by

nodes). In relevant points (nodes) where stringers were placed it was necessary to add their

dimensions. For each element it was possible to define thickness and material. The material used

in both (BWBLE and BWB) beam models was aluminum. After the files were prepared in MS

Excel, they were saved as input files for TORO in the .dat format.



2.2.2. Cross-section visualization

TORO data files were visualized using USCS ShipExplore application in order to check correct

preparation of the data (Figure 6). From the visualization below, it was visible that calculation of

the stiffness characteristics was made with realistic geometry of the wing structure, including

realistic geometry of the stringers.

Figure 7 Cross-section visualization a

Figure 8 Beam tapered model visualization a



Figure 9 Beam tapered model visualization b

2.2.3. TORO Calculation

2.2.3.1.Method for generation of the primary response fields in bending and restrained

torsion of thin-walled structures [7]

2.2.3.1.1. Modeling philosophy for Primary Response in Concept Design

For the concept design structural evaluation of the primary response the beam idealization of a

ship/bridge/wing is often used. A primary strength calculation provides the dominant response

field (Demand) for design feasibility assessment. The evaluation is based on extended beam

theory, which needs cross-sectional characteristics. These are obtained using analytical methods,

which can be very complicated for real combinations of open and closed cross-sections.

x

px+x

Transverse srip (S1-S2) with external loading p, warping fields u and 1D / 2D FEM idealization



Application of energy based numerical methods gives an opportunity for an alternative approach

to the given problems. The method is based on decomposing a cross-section into the line finite

elements between nodes i and j with coordinates (yi, zi), (yj, zj); element thickness te; material

characteristics (Young’s modulus E / shear modulus G); material efficiency RN and RS (due to

cutouts, lightening holes, etc.) with respect to normal/shear stresses.

Using the FEM approach, a procedure is developed for calculating the set of cross-sectional

geometric and stiffness characteristics at position x denoted Gx with the following elements:

Cross-section area A

Centre of gravity YCG, ZCG,

Shear/torsion center YCT, ZCT

Moments of inertia with respect to the centroid: IY , IZ, Iyz, Ip ; principal: I1, I2, 0-angle of

axis-1 w.r.t. Z-axis

Horizontal and vertical bending:

Flexural stiffness EIZ , EIY

Shear stiffness GAV , GAH ,

Cross-section axial stiffness EA

Torsion stiffness GIT

Warping stiffness EIW

2.2.3.1.2. Calculation of Response for a Transverse Strip with a Complex Cross

Section

The shear flow and stiffness characteristics of the cross section in bending and torsion are usually

calculated using analytical methods. Such calculations become rather complicated for multiple-

connected cross section graphs with a combination of open and closed (cell) contours. Application

of numerical methods based on the energy approach offers an elegant alternative. The procedure is

based on section decomposition into finite elements, as first introduced by Herman and Kawai. In

the sequel, the method of calculation as described in [8, 9] is presented. It has been successfully

used in practical calculations since its development for [10]. The simplest decomposition of thin-

walled cross-section (symmetric or not) into line finite elements (segments) is shown in Fig. 1.

These segments form boundary of the stiffened panel macro-elements for the feasibility

evaluation.



The methodology is based on applying the principle of minimum total potential energy (П) with

respect to parameters which define the displacement fields of the structure. The primary

displacement field (following classical beam theory) is defined via displacements and rotations of

the cross section as a whole. Secondary displacement field u2(x,y,z)≡u(x,y,z) represents warping

(deplanation) of the cross section. For piecewise-linear FEM idealization of the cross-section,

divided into n elements, with shape functions N in the element coordinate system (x, s), the

warping field reads:

j

i

ee

exx

e

u

u

l

s

l

ssu 1)( T

0uN (2)

Element strain and stress fields ε and σ are obtained from the strain-displacement and stress-strain

relations:

j

i

ee

e

xs u

u

lls

u 11TuB

and ee

xs

e

xs GG uBT (3)

j

i

ee

exx

e

u

u

l

s

l

ssu 1)( T

0uN (4)

The total potential energy of the x -long transverse strip of the beam, with the cross-section

divided into n elements, reads:

e

eeeee

e eV eSn eSeV

T xSsusFVdSsusxpdV FuuKuTTT

2

1d)()(d

2

1)(),(

(5)

where ),( sxp is the external loading on two cross sections (S1 and S2) of the strip. Minimization of

П leads to the classical FEM matrix relation K2D u2D = F2D (shortened to K u = F). The element

stiffness matrix for the proposed linear displacement distribution along the line element (the same

for bending and torsion) reads:

11

11e

eee

e

l

RStGK ,. (6)

where RS is the prescribed shear efficiency



2.2.3.1.3. Cross-Sectional Shear Stress Distribution Due To Bending

In the case of bending, the net external load (due to bending moments M(x+Δx) and M(x)) is the

normal stresses:

xxI

sxQxsx

xsxsxxsxp C

SS

)(

)()(),(),(),(),(

12

, where ξc(s) is distance from the point to

N.A.The load vector for a nonsymmetrical cross-section in, e.g., bending about the z axis reads:

3

sin

2

6

sin

2

3

sin

2

6

sin

2)()()()( 2

2

2

2

2 eee

ic

eee

ic

YZeee

ic

eee

ic

Y

YZZY

ee

y

e

e

zy

e

zllz

llz

Illy

lly

IIIIE

RNtxQExxQx

FF

(7)

For bending around the Y and Z axes, the matrix relations K u = F with uu )(xQ can be

converted into expressions FuK for the warping due to unit load F . For node warping ui(x),

unit warping )(xu must be multiplied by Q(x). This enables the assessment of shear stresses e

Y or

e

Z from the expressione = G (u2j-u2i) / l

e in each element e between nodes i and j. If necessary, it

is possible to calculate shear stress distribution )(se

xs more accurately, from the mean stress ke

xs

obtained from FEM, and the contribution to each element calculated analytically

)()( 11 ss u

ke

uu

ke

xsu

e

xs u = y or z from expression (for symmetrical section):

eicjcicycic

e

Y

ee

e

y

e

zz

e

xsl

szzszzz

l

EI

RNEGQs

22

1

3 )(

2

TuB

(8)

In this case, the sectional characteristics and shear centre are easily obtained. The shear/torsion

centre position reads:

nYQ

el

e

C

ee

YSC

nZQ

el

e

C

ee

ZSCdsdtZdsdtY

1010

; (9)

where e

Cd is the normal distance from the centroid to e.



The shear stiffness for bending about the Y and Z axes, GAV , GAH reads:

1

0

2

1

H

1

0

2

1

V

))((;

))((

e

e

e

el

e

zQz

e

xse

e

e

el

e

yQy

e

xs

RSdstG

GARSdstG

GA

(10)

2.2.3.1.4. Corrected Normal Stresses due to the Influence of Shear (Shear Lag)

The normal stress must be corrected for stress arising from a longitudinal change of the warping

field and normal stress due to correcting bending moment (Mc), compensating for the loss of cross

section equilibrium:

ec

y

i

yz

e

i

y

ei

y

c

xRNuupE

x

uE

Eε

e

el

e

cy

c

x

c

ydstszsM

0

)()( (11)

The total normal stress correction in node i reads

c

y

i

yz

eeY

c

YZZY

icYZicZei

y

cT

xuupEEM

IIIE

yIzIRN

2

(12)

The approximate value of normal stress for simultaneous bending about axes y and z for node i

reads:

i

y

cT

xic

e

Y

Yei

z

cT

xic

e

Z

Zei

xzE

EI

MRNyE

EI

MRN

(13)

2.2.3.1.5. Calculation of Warping and Primary Shear Stresses due to Pure Torsion

A transverse strip of a thin-walled beam of length x is subjected to torsion loading. The

displacement field of the middle line of thin walled elements can be expressed using the warping

function 0

)(t

su , rotation 0

)(ts

xv around the center of twist, twist rate (xx ,

) and angle(x

)of the

twist reads:

)()(),(

0,0xsusxu

xxt

, )()(),,(

00T0xdxvtsxv

xtts

(14)

Where dT is the normal distance from the element to the center of torsion. The strain (with 0s )

and stress fields read:

xx

xxx

xs

x

ds

u

u

,T

,

ε ,

xx

xxx

ds

uG

uE

,T

,

Εεσ (15)



The total potential energy of a section is given by the standard expression:

V

T WVWUΠ d 2

1εσ ,

(16)

After summation of all elements and transformation of local element displacements eeuNu T and

loads Fe into global displacements u and loads F we get:

)

2

1

2

1(

2

T

TT2

,

eeee

e

e

xxGtldRSxU FuuKu

(17)

Where

el

eeee dstGRS0

TBBK and

el

eeeee dsdtGRS0

TBF (18)

Minimization of total potential energy leads to two sets of equations:

(1) 0'

(1D beam torsion) and (2) 0

u (2D cross-section warping).

A second set of equations, FuKuu

0UΠ , enables determination of the unit

warping field.

The primary shear stresses on the elements which are parts of closed contours (cc) and open

sections (os) can now be calculated as functions of 1D twist rates θ,x(x) (to be obtained from the

first relation for 1D beam torsion):

e

j

i

eexx

eccke

xsd

u

u

llG

T,

)( 11 and

2,

)(

max

e

xx

eoske

xs

tG

(19)

2.2.3.1.6. Calculation of Torsional and Warping Stiffness of Thin-Walled Structures

To solve the equation for 1D beam free torsion, the torsion stiffness of elements which are parts

of the open eo and closed cells ec can now be calculated using the known unit warping field u :

e

ee

oe

e RStl

GGI

3

3

To; ee

e

ijee

ce

e RSdl

uutlGGI

2

TTc and

GIT=GITo+GITc

(20)

Warping stiffness is calculated using the expression:

e

jjii

e

ee

e

WRNuuuu

tlEEI

)(

3

22 (21)



Using GIT and EIW , the matrix K1D for the 1D beam problem can be formed and relevant

parameter distributions θ(x), θ,x(x), θ,xx(x), θ,xxx(x) can be determined for use in shear stress

calculations.

2.2.3.1.7. Normal and Secondary Shear Stresses due to Restrained Warping

Restrained warping of a thin-walled beam will induce (a) normal stresses in a cross-section and (b)

secondary shear stresses which will balance the longitudinally non-uniform distribution of normal

stresses. This additional mechanism will influence the strain energy and work, so an iterative

solution may be needed for greater accuracy.

Let u(x,s) = )()(,

xsuxx

be the warping field in the cross-section calculated from the case of free

torsion. Normal stresses are caused by restraining the warping, and vary along the x axis. They are

given by:

)()(

,xsu

xEE

xxxxw

or e

ixxx

ei

xwRNuE

, , (22)

Let u2(x,s) be the secondary displacement field containing a displacement correction due to

restrained warping. The total potential energy of a transverse strip consists of the internal energy

generated from the fields ε2 and σ2 (based on u2) and the additional work done by the strip axial

load px on the secondary displacements u2. If the change of u2 along strip length x is neglected, the

total potential energy reads:

e e eV eS

xee Ssxux

pxV

s

uGΠΠ

2

2

2

2 d),(d2

1

(23)

The net external load Δ px due to restrained warping reads:

xRNxsuEsxsxp e

xxxx

e

xwx )()(),(),(

, (24)

and the total potential energy of the element, using the same shape functions as before, reads:

el

eeee

xxxx

e

el

eeeeee sRNEtxsRSGtxΠ0

T

,

T

2

0

2

TT

2)d()d(

2

1uNNuuBBu (25)

Minimization of the total potential energy with respect to the unknown displacement field u2 leads

to:

FuKFKu

2220)(0 xΠ

u (26)



where: K is the global stiffness matrix as before, u2 is the global vector of unknown displacements

2,2uu

xxxx , F is the global load vector FF

xxxx, . The element load and the secondary shear

stresses (constant on element) read:

e

xxxx

j

ieeee

xxxx

e

u

ultERN FF

,,3161

6131

;

(27)

xxxxe

ije

e

ijeeke

xsl

uuG

l

uuG

s

uG

,

22222)2(

(28)

The shear stress distribution can be calculated more accurately along the element (similar to the

bending case) from the known element average stress )2(ke

xs , the direction of shear stress flow, local

element contribution )(2

s and its average ke

2 using expression )()(

22

)2()2(

ss keke

xs

e

xs . After

rearranging, it reads:

xxxxe

ij

i

e

jie

ee

e

ijee

xss

l

uusu

luu

RS

RNE

l

uuGs

,

222)2(

23)

2

1()(

(29)



2.2.4. TORO output files

After TORO output files were obtained, stiffness characteristics of 44 cross-sections (22 per each

beam model) were known. In order to check results of the output files, Figure 10 (for BWBLE

model) and Figure 11 (for BWB model) below were created. It can be seen that areas of first

cross-section (blue dots on diagram) of each element were almost continuously dropping along the

wing span. The same followed for the second area of each element (red dots). It can be seen that

area 2 of the previous element (red dot) is slightly bigger than area 1 of following element (blue

dot) for each element. That is because after each element, the number of stringers reduces. The

only exception was element 4, which was split in two because of wing kink at that position,

therefore the number of stringers remained the same through element.

Figure 10 First cross-section and second cross-section areas for each element (BWBLE model) along wing span

Figure 11 First cross-section and second cross-section areas for each element (BWB model) along wing span



It can be seen from Figure 11 relevant for the BWB model, that first and second cross-sections of

each element along the wing span have similar distribution as for the BWBLE model.

The same comparison was made for moments of inertia for both models. After accuracy of the

output files was checked and stiffness characteristics for all cross-sections of both models were

known, it was possible to start preparing beam models in FEMAP.



3. TAPERED BEAM FEM MODELS OF SADE WING

3.1. Definition of cross-section properties for tapered element

For definition of a tapered element in FEMAP, data for the cross sections of each element were

needed. In order to reduce time constraints and the possibility of error while entering a large

amount of stiffness characteristics data directly using interactive form, it was decided to prepare

the beam stiffness properties characteristics in a FEMAP neutral file. An application in C# was

developed to print the data calculated by TORO in a FEMAP neutral file for all 44 sections. This

operation was repeated for each element, creating property data for the Beam model as a FEMAP

neutral file. It is important to mention that the material used in all modeling was aluminum.

Figure 12 Example of a FEMAP neutral file provided by TORO program for the definition of tapered beam element

property stiffness characteristics

Figure 13 Example of a FEMAP interactive form for the definition of tapered beam element property stiffness

characteristics



BWB and BWBLE FEM node coordinates were defined at the intersection of the respective beam

axes and cross section where the stiffness characteristics were calculated (in the same reference

coordinate system as SWB and SWBLE models).

3.2. Geometry

Positions of beam element nodes were obtained by entering three points which represented the

center of gravity of the first three referent cross-sections (at wing root, wing kink and wing tip).

Those three points were then connected with a line, as shown below for BWBLE model. The same

procedure was applied to the BWB model except that coordinates of points were different

(because BWB model is defined on the different beam axis).

Figure 14 Geometry of BWBLE model

3.3. Mesh

Both, BWB and BWBLE models were discretized by 11 elements (same as number of sections).

Node coordinates were defined by splitting beam axes into 11 sections and entering those

coordinates (in the same reference coordinate system as SWB and SWBLE models) into FEMAP.

A picture of the meshed beam model can be seen below. Since it was not possible to show the real

beam model cross section geometry, the cross section was visualized by the available idealized

trapezoidal cross-section (drawn in Solidworks and imported into FEMAP using General section

option). The purpose of this was simply to reduce error while entering the load data and to have

some perception of the size of the cross sections.



Figure 15 Meshed Beam model (BWBLE)

Figure 16 Meshed Beam model (BWB)

3.4. Constraints

Degrees of freedom were fixed at at the wing root, as shown below.

Figure 17 Constrained beam model (BWB)



3.5. Loads

There were 12 nodes along the beam model but load is not applied to the first node (node in which

is fixed constraint), therefore the overall force (in case of combined load and bending) or moment

(in case of torsion and bending) was divided by 11.

Since neutral beam axes are not in the same direction as the y-axis of reference coordinate system,

in order to correctly apply moments around y-axis, two coordinate systems (one for first beam axis

and one for second beam axis) were defined in FEMAP. By this method it was possible to apply

momentum around the beam axis and create almost independent bending and torsion loading.

Load cases with pure bending and torsion were under investigation in order to determine accuracy

of beam element model for each of those dominant types of a loading on a wing.



3.5.1. Pure bending load case

The force (1) acting at each node was defined as:

(30)

This force is taken to equal half of the plane weight as shown in Figure 18

Figure 18 Plane characteristics

If a state of pure bending is to be achieved, the force which causes that bending must be

accompanied by the torsion moment. Since the neutral axis and shear center do not coincide for

this type of cross-section, an additional torsion moment was applied to cancel the moment

produced by the force applied along the neutral axis.

(31)

where is x-coordinate of neutral axis point (beam node) and is approximated shear

center. Neutral beam axis point was placed between the second cross-section of the previous

element and first cross-section of the following element. Those two cross-sections have different

shear centers due to the fact that number of stringers reduces per each element. Therefore,

approximated shear center is calculated. Note, position of shear center is one of the characteristics

previously calculated by TORO.

Since each element had different position of center of gravity and different position of shear

center, this momentum was different for each element.

The picture below shows an example of the calculation of approximated shear center between

elements 5 and 6.



Figure 19 Calculation of an approximated shear center

That moment must be applied to the beam model in the opposite direction to cancel the moment of

the load of force , therefore force F causes pure bending. Since each element had

different position of center of gravity and different position of shear center, this momentum was

different for each element.

Figure 20 Beam model (BWBLE) loaded with pure bending load case

Figure 21 Beam model (BWB) loaded with pure bending load case

21520

21530

21540

21550

21560

21570

21580

21590

21600

21610

21620

0 5000 10000

EL5_2_sc

EL6_1_sc

x_na

app_sc_position

x[mm] y[mm]



3.5.1.1. Results

Figure 22 Beam model (BWBLE) displacements (T3) with pure bending load case

Figure 23 Beam model (BWB) displacements (T3) with pure bending load case



3.5.2. Pure torsion load case

The amplitude of torsional moment applied to the Beam model was chosen to cause rotation

around the y-axis of approximately 2 degrees. It was taken that that momentum is:

(32)

The amount of torsional moment applied to each node of the beam element was:

(33)

Figure 24 Beam model (BWBLE) loaded with pure torsion load case

Figure 25 Beam model (BWB) loaded with pure torsion load case



3.5.2.1. Results

Figure 26 Displacements (R2) of elements (BWBLE) in case of pure torsion load case

Figure 27 Displacements (R2) of elements (BWB) in case of pure torsion load case



3.5.3. Combined torsion and bending load case

The force (1) acting at each node was defined as:

(34)

An additional moment must be added due to the fact that force acts at the center of pressure and it

can be added to the model only in neutral axis. That moment is:

(35)

Where is the x-coordinate of the aerodynamic center for each element and is x-

coordinate of the neutral beam axis point.The procedure for calculation of is shown in

combined shell load section. That load was applied at the airfoil center of pressure, which is the

same as the aerodynamic center for a symmetric airfoil and it is located at one quarter of the

chord.

The center of pressure of an aircraft is the point where all of the aerodynamic pressure field may

be represented by a single force vector with no moment. A similar idea is the aerodynamic center

which is the point on an airfoil where the pitching moment produced by the aerodynamic forces is

constant with angle of attack [12].

Figure 28 Example of aerodynamic center, neutral axis and chord origin positions for fifth element

19600

19800

20000

20200

20400

20600

20800

21000

21200

21400

21600

0 2000 4000 6000 8000 10000

Tc_1/4_EL5

x_na_EL5

O_EL_5

x[mm]

y[m]



Figure 29 Beam mode (BWBLE) loaded with combined torsion and bending load case

Figure 30 Beam model (BWB) loaded with combined torsion and bending load case



3.5.3.1. Results

Figure 31 Displacements (T3) of elements (BWBLE model) in case of combined bending and torsion load case

Figure 32 Displacements (T3) of elements (BWB model) in case of combined bending and torsion load case



4 SHELL MODELS

Both shell models were prepared during the EU FP7 SADE project [2] by the Cranfield

University. Models were provided as NASTRAN bdf models, imported into FEMAP and saved

into FEMAP’s modfem format.

4.1. Geometry

The Shell model geometry includes ribs, stringers, front and rear spars, skins and leading edge

actuation mechanism. The number and dimensions of stiffeners per section and thicknesses of the

plane elements varied along the span. [2]



4.2. Mesh

Figure 33 SADE shell wing box with leading edge model (SWBLE)

Figure 34 SADE shell wing box model (SWB)



4.3. Constraints

As for the beam model shown previously, degrees of freedom at one end of the model were fixed. [2].

Figure 35 Constrained shell model (SWBLE)

Figure 36 Constrained shell model (SWB)



4.4.Loads

4.4.1. Definition of nodes

In order to apply equivalent loads to beam and shell models it was necessary to define the most

appropriate nodes in the shell model at which to apply loads, so that the final shell models load

distribution would be equivalent to the beam load distribution.

It was decided to use four points; two points at the front spar (one at the top and one at the bottom)

and two points at the rear spar(one at the top and one at the bottom) per equivalent beam node

position element. Each shell model section was imported into FEMAP and points were listed

using FEMAP List Geometry Point command. Points were listed for each section’s second cross-

section.

Figure 37 Example of geometry cross-section with four points for needed for definition of shell nodes

After points were listed for all sections, their coordinates were used to find closest shell nodes. It

was needed to find four nodes per second cross-section of each element using FEMAP List Model

Nodes command.

Those nodes are grouped (one group for front spar and one group for rear spar) and used for

application of loads on shell model. In the end there were 22 groups along the span (11 groups for

front spar and 11 groups for rear spar).



4.4.2. Pure bending load case

It was necessary to calculate forces acting on the front and rear spar, that produces the pure

bending load case in which torsion could be neglected.

The sum of forces acting at front and rear spar must be the same as the force acting at the neutral

axis in the Beam model (1).

(36)

(37)

(38)

where and are distances of front and rear spar from the neutral axis for each section.These

distances were obtained by finding fictional nodes between nodes 1 and 2(front spar) and nodes 3

and 4(rear spar). After fictional nodes were found (with approximately the same z-coordinate as

neutral beam axis node) it was possible to calculate (distance between front spar and neutral

axis) and (distance between neutral axis and rear spar). It was assumed that both distances

( ) are in the direction of x-axis. Neutral axis offset ( ) is distance between neutral

point and shear center

Figure 38 Calculation of a_na and b_na distances



The force on each spar was divided into two nodes, so the forces applied in each group in FEMAP were:

-front spar

(39)

-rear spar

(40)

The distribution of loads applied to each spar along the span are shown on diagram below.

Figure 39 Load distribution along the span in the case of pure bending for BWBLE and SWBLE models

Figure 40 Load distribution along the span in the case of pure bending for BWB and SWB models

-6000000

-4000000

-2000000

0

2000000

4000000

6000000

8000000

10000000

-50000

-45000

-40000

-35000

-30000

-25000

-20000

-15000

-10000

-5000

0

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

F_pb_beam_wb F1_pb_shell_wb F2_pb_shell_wb M_pb_beam_wb

y[mm]

F[N] M[Nmm]



Figure 41 Loaded shell model (SWBLE) in case of pure bending load case

Figure 42 Loaded shell model (SWB) in case of pure bending load case



4.4.2.1. Results

Figure 43 Displacements (T3) of elements (SWBLE) in case of pure bending load case

Figure 44 Displacements (T3) of elements(SWB) in case of pure bending load case



4.4.3. Pure torsion load case

The torsional moment applied at each node of the beam model has to be split into two equivalent

forces which act on the front and rear spar, to obtain similar load application for the beam and

shell models in the case of pure torsion. Forces are found in the way that equals

(41)

Where and are distances of the front and rear spar from the shear center. Calculation of

shear center position is shown in section

Torsional moment and distances and were different at each section and therefore force

differed along the span.

In order to reduce the local concentration effect, the force on each spar was divided in two nodes,

so forces applied in each group in FEMAP are:

(42)

Figure 45 Load distribution along the span in a case of pure torsion for BWBLE and SWBLE models

0

5000000

10000000

15000000

20000000

25000000

0

5000

10000

15000

20000

25000

30000

35000

40000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

F1_pt_shell

M_pt_beam

y[mm]

F[N]



Figure 46 Load distribution along the span in a case of pure torsion for BWB and SWB models

Figure 47 Loaded SWBLE model in case of pure torsion load case

Figure 48 Loaded SWB model in case of pure torsion load case

0

5000000

10000000

15000000

20000000

25000000

0

5000

10000

15000

20000

25000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

F1_pt_shell_wb

M_pt_beam_wb

y[mm]

F[N] M[Nmm]



4.4.3.1. Results

Figure 49 Displacements (R2) of elements (SWBLE) in case of pure torsion load case

Figure 50 Displacements (R2) of elements (SWB) in case of pure torsion load case



4.4.4. Combined bending and torsion load case

For the combined load case (acting at the aerodynamic center), forces were calculated by

assuming that:

(43)

(44)

Where and are distances of front and rear spar from aerodynamic center.

(45)

(46)

Where is distance between front and rear spar for each relevant cross-section and is the

position of the front spar (in percentage of chord). The position of aerodynamic center (before or

after front spar position, in percentage of chord) alters the values of and therefore

application of loads.

Positions of the front and rear spar (in chord percentage) are know from [1] for wing root and

wing tip and therefore it was possible to calculate positions of the front and rear spar at each beam

node position.

Also, on the same position along the y-axis, the distance between front and rear spar is known.

(47)

With previous information it was possible to calculate wing chord at required position.

(48)

(49)



The sum of forces acting at front and rear spar must be the same as the force acting at neutral axis

in the Beam model. In order to reduce the local concentration effect, the force on each spar was

divided onto two nodes, so forces applied in each group in FEMAP were:

-front spar

(50)

-rear spar

(51)

Figure 51 Load distribution along the span in a combined bending and torsion load case for BWBLE and SWBLE models

Figure 52 Load distribution along the span in a combined bending and torsion load case for BWBLE and SWBLE models

-35000000

-30000000

-25000000

-20000000

-15000000

-10000000

-5000000

0

-50000

-45000

-40000

-35000

-30000

-25000

-20000

-15000

-10000

-5000

0

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

F_c_beam F1_c_shell F2_c_shell M_c_beamF[N] M[Nmm]

-45000000

-40000000

-35000000

-30000000

-25000000

-20000000

-15000000

-10000000

-5000000

0

-50000

-45000

-40000

-35000

-30000

-25000

-20000

-15000

-10000

-5000

0

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

F_c_beam_wb F1_c_shell_wb F2_c_shell_wb M_c_beam_wb M[Nmm]F[N]

y[mm]



Figure 53 Loaded SWBLE model in case of combined bending and torsion load case

Figure 54 Loaded SWB model in case of combined bending and torsion load case

4.4.4.1. Results

Figure 55 Displacements (T3) of elements (SWBLE) in case of combined bending and torsion load case



Figure 56 Displacements (R2) of elements (SWBLE) in case of combined bending and torsion load case

Figure 57 Displacements (T3) of elements (SWB) in case of combined bending and torsion load case

Figure 58 Displacements (R2) of elements (SWB) in case of combined bending and torsion load case



5 COMPARISON OF THE STRUCTURAL DISPLACEMENTS BETWEEN

BEAM (BWBLE,BWB) AND SHELL (SWBLE,SWB) MODELS

5.1.Pure Bending

The loads and where applied to each node of a BWBLE model and forces

and where applied to each group of nodes placed on front and rear spar along the span of a

SWBLE model respectively. Since forces are acting in the negative direction of z-axis, relevant

displacement is translation in z-direction (T3).

Same procedure as for BWBLE and SWBLE models was used to find T3 displacements in case of

pure bending for BWB and SWB models. Loads applied to beam model were and

and loads applied to SWB model were and .

5.1.1. BWBLE and SWBLE

After T3 translations for each model(BWBLE and SWBLE) in case of pure bending were obtained

displacements were listed and shown in diagram Table 1 .

Since on Figure 59difference in T3 displacments along the span between BWBLE and SWBLE

models cannot be seen precisely. normalized diagram Figure 60was made.

Two different normalizations where done and shown on diagram. First normalization is

(52)

Where are displacements of BWBLE model in case of pure bending along the span and

are displacements of SWBLE model in case of pure bending along the span. Label e in

stands for displacements normalized with displacement of an “each” relevant

node.

Second normalization is

(53)

Where “max” labels that difference between displacements is normalized with displacement of a

node which has biggest displacement



Figure 59 Comparison of T3 displacements for pure bending load case along the span for BWBLE and SWBLE models

Figure 60 Normalized comparison of T3 displacements between BWBLE and SWBLE models in a case of pure bending

Table 1 Data used for comparison of T3 displacements for pure bending load case along the span (BWBLE and SWBLE

models)

It can be seen that BWBLE model has higher T3 displacements than SWBLE model along the

whole span except element 11(wing tip).

-1400

-1200

-1000

-800

-600

-400

-200

0

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

BWBLE_pb_T3

SWBLE_pb_T3

T3[m

m]

y[mm]

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

-0.2

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_bs_pb_e

delta_bs_pb_max

y[mm]

y[mm] BWBLE_pb_T3[mm] SWBLE_pb_T3[mm]

3610 -5 0

5264 -21 -14

6682 -46 -36

7360 -63 -50

8284 -92 -78

9443 -139 -122

11099 -234 -214

12755 -363 -341

14411 -530 -501

16067 -733 -731

19110 -1183 -1200



5.1.2. BWB and SWB

Same procedure as for comparison of displacements between BWBLE and SWBLE models is

used. Normalizations , were used to compare T3 displacements between

BWB ( ) and SWB ( ) models.

Figure 61 Comparison of T3 displacements for pure bending load case along the span for BWB and SWB mode

Figure 62 Normalized comparison of T3 displacements between BWB and SWB models in a case of pure bending

Table 2 Data used for comparison of T3 displacements for pure bending load case along the span (BWB and SWB models)

It can be seen that distribution of differences between T3 displacements for case of pure bending

is not the same as for BWBLE and SWBLE models.

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 5000 10000 15000 20000

delta_bs_wb_pb_e

delta_bs_wb_pb_max

y[mm]

y_wb[mm] BWB_pb_T3[mm] SWB_pb_T3[mm]

3606 -1 -2

5256 -12 -14

6670 -34 -37

7360 -50 -51

8347 -81 -82

9499 -131 -128

11145 -240 -228

12792 -397 -370

14438 -593 -554

16084 -830 -831

19110 -1391 -1434



5.2.Pure torsion

Torsional moment was applied to each node of a BWBLE model and the same force

was applied to each group of nodes placed on front and rear spar along the span of a

SWBLE model.

Load applied to BWB model was and loads applied to SWB model were

and .

Moment is applied along y-axis and relevant displacements for this load case

is rotation around y-axis (R2). Displacements R2 per each beam node of and BWBLE and BWB

models are listed directly from FEMAP as already described in previous sections and used for

comparisons in diagrams.

5.2.1.1.BWBLE and SWBLE

In case of pure torsion load case first normalization is :

(54)

Where are displacements of BWBLE model in case of pure torsion along the span and

are displacements of SWBLE model.

Second normalization is

(55)

Figure 63 Comparison of R2 displacements for the pure torsion load case (BWBLE and SWBLE models

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

BWBLE_pt_R2

SWBLE_pt_R2

R2[rad]

y[mm]



Figure 64 Normalized comparison of R2 displacements between BWBLE and SWBLE models in a case of pure torsion


From diagrams above it can be seen that in case of pure torsion SWBLE model has bigger

displacements almost along entire wing span (except first and second element).

Although T3 displacement is not representative for pure torsion load case diagram below is

obtained to prove that pure torsion load case is applied to models .From Figure 65 it can be said

that T3 displacements are neglectable for beam model and are small for shell model.

Figure 65 Comparison of T3 displacements for torsion load case along the span for BWBLE and SWBLE models

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_bs_pt_e

delta_bs_pt_max

y[mm]

y[mm] BWBLE_R2[rad] SWBLE_pt_R2[rad] BWBLE_pt_T3[mm] SWBLE_pt_T3[mm]

3610 0.0004 0.0003 0.0 0.0

5264 0.0010 0.0010 0.0 0.0

6682 0.0019 0.0022 0.0 0.1

7360 0.0024 0.0026 0.0 0.0

8284 0.0033 0.0038 0.0 -0.1

9443 0.0046 0.0055 0.0 -0.4

11099 0.0071 0.0086 0.0 -1.5

12755 0.0104 0.0126 -0.1 -3.3

14411 0.0147 0.0179 -0.1 -6.3

16067 0.0200 0.0213 -0.1 -12.4

19110 0.0316 0.0374 -0.2 -32.1



5.2.1.2.BWB and SWB

Figure 66 Comparison of R2 displacement for the pure torsion load cas e(BWB and SWB models)

Figure 67 Normalized comparison of R2 displacements between BWBLE and SWBLE models in a case of pure torsion


Same as for models with leading edge, it can be seen that shell model (SWB) has higher R2

displacements along the wing span (except first element where displacements are the same).

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

BWB_pt_R2

SWB_pt_R2

R2

[rad]

y[mm]

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_bs_wb_pt_e

delta_bs_wb_pt_max

y[mm]

y_wb[mm] BWB_pt_R2[rad] SWB_pt_R2[rad] BWB_pt_T3[mm] SWB_pt_T3[mm]

3606 0.0004 0.0004 0.0 0.1

5256 0.0010 0.0011 0.0 0.1

6670 0.0019 0.0022 0.0 0.2

7360 0.0024 0.0028 0.0 0.1

8347 0.0034 0.0040 0.0 0.1

9499 0.0048 0.0058 0.0 -0.2

11145 0.0085 0.0091 -0.1 -1.2

12792 0.0123 0.0137 -0.1 -3.1

14438 0.0158 0.0198 -0.1 -6.4

16084 0.0221 0.0245 -0.2 -14.3

19110 0.0389 0.0455 -0.3 -43.5



Also, T3 displacements are shown as for comparison of models with leading edge.

Figure 68 Comparison of T3 displacements for torsion load case along the span for Beam(BWB) and Shell (SWB)model

5.3.Combined bending and torsion

For combined bending and torsion load case loads and where applied to each node of a

BWBLE model and forces and where applied to each group of nodes placed on front and

rear spar along the span of a SWBLE model respectively. Forces are acting in the negative

direction of z-axis so relevant displacement is translation in z-direction (T3).Also, because there is

a influence of a moment, R2 (around y-axis) displacements are relevant.

Displacements T3 and R2 are relevant and for combined bending and torsion load case for BWB

and SWB models. Loads applied to beam model were and and loads applied to

SWB model were and .

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

BWB_pt_T3

SWB_pt_T3

T3[m

m]

y[mm]



5.3.1.1.BWBLE and SWBLE

Differences between displacements (T2 and R2) and their normalizations are shown between

BWBLE and SWBLE models. Principle of calculation of

is the same as for T3(R2) displacements shown in previous load cases.

5.3.1.1.1. T3 displacements

Figure 69 Comparison of T3 displacements for combined bending and torsion load case along the span for BWBLE and

SWBLE models

Figure 70 Normalized comparison of T3 displacements between BWBLE and SWBLE models in a combined bending and

torsion load case

It can be seen that BWBLE is having higher T3 displacements along span except for last two

elements.

-1,400

-1,200

-1,000

-800

-600

-400

-200

0

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

BWBLE_c_T3

SWBLE_c_T3

T3[m

m]

y[mm]

0

0.2

0.4

0.6

0.8

1

1.2

0 5000 10000 15000 20000

delta_bs_c_T3_e

delta_bs_c_T3_max

y[mm]



5.3.1.1.2. R2 displacements

Figure 71 Comparison of R2 displacements for combined bending and torsion load case along the span for BBWBLE and

SWBLE models

Figure 72 Normalized comparison oR2 displacements between BWBLE and SWBLE models in a combined bending and

torsion load case

Table 5 Data used for comparison of T3 and R2 displacements for combined bending and torsion load case along the span

(BWB and SWB models)

Model SWBLE is having higher displacement R2 along the span.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.00 2000.00 4000.00 6000.00 8000.00 10000.00 12000.00 14000.00 16000.00 18000.00 20000.00

BWBLE_c_R2

SWBLE_c_R2

R2

[rad

]

y[mm]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_bs_c_R2_e

delta_bs_c_R2_max

y[mm]

y[mm] BWLE_c_T3[mm] SWLE_c_T3[mm] BWLE_c_R2[rad] SWLE_c_R2[rad]

3610 -5 0 0.0012 0.0004

5264 -21 -13 0.0032 0.0045

6682 -46 -36 0.0056 0.0093

7360 -63 -50 0.0072 0.0114

8284 -92 -78 0.0093 0.0155

9443 -139 -122 0.0127 0.0206

11099 -234 -214 0.0182 0.0309

12755 -363 -342 0.0242 0.0423

14411 -530 -503 0.0302 0.0537

16067 -733 -735 0.0352 0.0650

19110 -1183 -1208 0.0423 0.0759



5.3.1.2.BWB and SWB

5.3.1.2.1. T3 displacements

Figure 73 Comparison of T3 displacements for combined bending and torsion load case along the span for BWB and SWB

models

Figure 74 Normalized comparison of T3 displacements between BWB and SWB models in a combined bending and torsion

load case

Table 6 Data used for comparison of T3 and R2 displacements for combined bending and torsion load case along the span

(BWB and SWB models)

-1600

-1400

-1200

-1000

-800

-600

-400

-200

0

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

BWB_c_T3

SWB_c_T3

T3[m

m]

y[mm]

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_bs_wb_c_T3_e

delta_bs_wb_c_T3_max

y[mm]

y_wb[mm] BWB_c_T3[mm] SWB_c_T3[mm] BWB_c_R2[rad] SWB_c_R2[rad]

3606 -1 -2 -0.0002 0.0005

5256 -12 -14 0.0016 0.0029

6670 -34 -38 0.0038 0.0065

7360 -50 -52 0.0054 0.0080

8347 -82 -82 0.0075 0.0108

9499 -131 -128 0.0106 0.0144

11145 -240 -228 0.0180 0.0225

12792 -397 -370 0.0235 0.0317

14438 -592 -553 0.0272 0.0415

16084 -830 -827 0.0313 0.0524

19110 -1391 -1422 0.0338 0.0687



It can be seen that difference between T3 displacements between BWB and SWB models along

the span varies and it has not similar distribution as comparison between BWBLE and SWBLE

models.

5.3.1.2.2. R2 displacements

Figure 75 Comparison of R2 displacements for combined bending and torsion load case along the span for BWB and SWB

Figure 76 Normalized comparison of R2 displacements between BWB and SWB models in a combined bending and torsion

load case

It can be seen that for all elements shell model has higher R2 displacement than beam model.

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

BWB_R2

SWB_R2

y[mm]

R2

[rad

]

-2

-1

0

1

2

3

4

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_bs_wb_c_R2_e

delta_bs_wb_c_R2_max

y[mm]



6 COMPARISON OF THE STRUCTURAL DISPLACEMENTS BETWEEN

MODELS WITH LE (BWBLE and SWBLE) and WITHOUT LE(BWB

and SWB)

Combined load case was the only load case where displacements between models with leading

edge (BWBLE and SWBLE) and without leading edge (BWB and SWB) could be done due to the

fact that only in combined load case forces are assumed to act in the same point (aerodynamic

center) for all four beam and shell models. In bending load case position of neutral axis point for

each cross-section is different for models with leading edge and without leading edge and

therefore their displacements are not relevant for the comparison. The same situation is with pure

torsion load case where loads are applied around shear center.

6.1 T3 displacements

Since BWBLE model elements do not have the same length as BWB models it was needed to

approximate either T3 BWBLE displacement or T3 BWB displacement so both displacements are

compared on the same place along y-axis .It was decided to approximate T3 BWB displacements

and example approximation is shown on figure below

Figure 77 Approximation used to determine T3_wb displacement on the place of node of BWBLE model

After approximation was obtained it was possible to compare T3 displacements between BWBLE

and BWB models. Normalization used IS:

(56)

Where is approximated value of BWB model T3 displacement relevant node and is

the T3 displacement of the BWBLE model at that node.

-60

-50

-40

-30

-20

-10

0

0 2000 4000 6000 8000

T3_wb_b_2

T3_wb_b_1

T3_wb_app_b

y[mm]

T3_w

b[m

m]



Figure 78Comparison of T3 displacements between BWBLE and BWB models along span

For shell models, enable comparison them with appropriate beam models, fictional node placed on

the same place as beam node was being found .

After T3 displacements in fictional node were being found for each model (SWBLE and SWB) ,

the same approximation as on Figure 77 applied to beam models was used.

Normalization used for shell models is:

(57)

Where is approximated value of SWB model T3 displacement of a fictional node and

is the T3 displacement of of the SWBLE model at that node.

Figure 79 Comparison of T3 displacements between SWBLE and SWB models along span

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_b_T3

y[mm]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_s_T3

y[mm]

y[mm]



Figure 80 Comparison of T3 displacements for four models (BWBLE, BWB, SWBLE, SWB) on positions of beam nodes of

BWBLE model

6.2 R2 displacements

The same approximation as in T3 displacements is used to compare results between BWBLE and

BWB R2 displacements.

Used normalization is the same as the one for T3 displacements

(58)

Where is approximated value of BWB model R2 displacement in relevant node and

is the R2 displacement of the BWBLE model in that node.

Figure 81 Comparison of R2 displacements between BWBLE and BWB models along span

Approximation used to compare R2 displacements between SWBLE and SWB models is the same

as in previous cases.

-1600

-1400

-1200

-1000

-800

-600

-400

-200

0

0 5000 10000 15000 20000

T3_BWBLE

T3_BWB_ap

T3_SWBLE

T3_SWB_ap

y[mm]

T3[m

m]

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

delta_b_R2

y[mm]



Normalization used is:

(59)

Where is approximated value of SWB model R2 displacement in frictional node and

is the R2 displacement of the SWBLE model in that node.

Figure 82 Comparison of R2 displacements between SWBLE and SWB models along span

Figure 83 Comparison of R2 displacements for four models (BWBLE, BWB, SWBLE, SWB) on positions of beam nodes of

BWBLE model

After results for R2 displacements were seen it was realized that pure torsion load was not applied

to be relevant for comparison between models.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5000 10000 15000 20000 25000

delta_s_R2

y[mm]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 5000 10000 15000 20000

R2_BWBLE

R2_BWB_ap

R2_SWBLE

R2_SWB_ap R2

[rad]



6. CONCLUSION

In this study the aim was to analyse the influence of the morphing LE structure on the stiffness of

a whole wing. In order to achieve that, two wing models were compared, one with morphing

leading edge mechanism; and one modeled with only wing box (WB) structure. Both are models

for the wing of a generic 150-seat passenger aircraft previously used in the EU FP7 SADE project.

An additional goal of this study was to compare the results between two types of wing finite

element method (FEM) models, typically called shell model and beam model. Two shell models

(with and without leading edge) were previously prepared during the EU FP7 SADE project [2] by

Cranfield University and were provided for this thesis courtesy of Dr. Shijun Guo.

Beam models (with and without leading edge) were prepared using structural characteristics of

two provided shell models. Shell models were imported in Solidworks and cut into 11 sections.

Relevant points from each end of a section were used to define cross-sections. Cross-section

stiffness characteristics were calculated using TORO program. Having those characteristics it was

possible to define stiffness of a tapered beam element. That was accomplished by importing

necessary property characteristics of each tapered element cross-section in FEMAP neutral file.

Beam and shell models were fixed at the wing root.

After beam models (with and without leading edge) were obtained, it was possible to apply test

load cases to all models (two beam and two shell) and compare results. The applied load cases

were pure bending, pure torsion and combined bending and torsion. Load distributions in

dependence of wing span for each load case and structural model were shown.

For each load case the most relevant displacements were listed and results were compared for shell

and beam models with and without leading edge. The distribution of the resulting displacements

were presented across a wing span.

Based on the conducted comparison, the next conclusions can be drawn:

From the results obtained, it could be stated that beam models were correctly defined and they

have similar stiffness characteristics as shell models. Also, it was determined that morphing

leading edge has an influence on wing stiffness characteristics, due to differences in T3

displacements between models with and without leading edge.



8 REFERENCES

[1] Guo,S.,Ahmed,S.,”Optimal Design and Analysis of a Wing with morphing High Lift

Devices”,Cranfield University

[2] http://www.sade-project.eu/

[3] FEMAP help

[4] Dorbath,F.,Nagel,B.,Gollnick,V,”Comparison of Beam and Shell Theory for Mass Estimation

in Preliminary Wing Design”,DLR

[5]Solidworks help

[6] TORO help

[7] Zanic, V., P. Prebeg, and S. Kitarovic. "Method for Generation of the Primary Response

Fields in Bending and Restrained Torsion of Thin-Walled Structures." Harbin Gongcheng Daxue

Xuebao/Journal of Harbin Engineering University 27.SUPPL. 2, 2006.

[8] Hughes, O.F.: Ship Structural Design, Wiley, 1983, SNAME 1992.

[9] Zanic, V.: Calculation of shear flow in cross-section of ship in bending.(in Croatian). Proc.

of

[10] CREST Documentation, Croatian Register of Shipping, Split, Croatia, 2004.

[11]Q Fu, S Guo, D. Li, “Optimization of a Composite Wing with a Morphing Leading Edge

Subject to Aeroelastic Effect”, ICMNMMCS-2012, Politecnico di Torino, 18-20 June 2012.

[12] http://en.wikipedia.org/wiki/Aerodynamic_center

[13] Barbarino,S.,Bilgen,S.,Ajaj,R.M.,Friswell,M.,Inman,D.,”Review of Morphing

Aircraft”,Swanesa University,Virginia Tech

http://www.sade-project.eu/

http://en.wikipedia.org/wiki/Aerodynamic_center



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MASTER'S THESIS - Fakultet strojarstva i brodogradnje · PDF fileSection 11 2.2.3.1.3. ... Figure 13 Example of a FEMAP interactive form for the definition of tapered beam ... Maja