MASTER'S THESIS - Fakultet strojarstva i brodogradnje · PDF fileSection 11 2.2.3.1.3. ......
Transcript of MASTER'S THESIS - Fakultet strojarstva i brodogradnje · PDF fileSection 11 2.2.3.1.3. ......
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
Maja Hećimović Master’s Thesis
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
Maja Hećimović Master’s Thesis
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
torsion load case ....................................................................................................................... 31
Maja Hećimović Master’s Thesis
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
BWBLE and SWBLE models .................................................................................................. 44
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
torsion load case ....................................................................................................................... 46
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
BWBLE and SWBLE models .................................................................................................. 48
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
Maja Hećimović Master’s Thesis
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
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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
Table 2 Data used for comparison of T3 displacements for pure bending load case along the
span (BWB and SWB models) ................................................................................................. 49
Table 3 Data used for comparison of T3 displacements for pure bending load case along the
span (BWB and SWB models) ................................................................................................. 51
Table 4 Data used for comparison of T3 displacements for pure bending load case along the
span (BWB and SWB models) ................................................................................................. 52
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
Maja Hećimović Master’s Thesis
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
Maja Hećimović Master’s Thesis
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
Maja Hećimović Master’s Thesis
Faculty of Mechanical Engineering and Naval Architecture IX
kombiniranog opterećenja. To je vidljivo zbog razlika u pomacima između modela sa i bez
napadnog ruba.
Maja Hećimović Master’s Thesis
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 2
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)
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Faculty of Mechanical Engineering and Naval Architecture 3
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.
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 4
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.
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 5
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)
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 6
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.
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 7
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 8
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.
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 9
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 10
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 11
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.
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 12
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
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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.
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 14
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)
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 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)
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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)
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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)
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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
Maja Hećimović Master’s thesis
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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.
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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
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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.
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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)
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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.
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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.
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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]
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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
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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
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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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 29
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]
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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
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Faculty of Mechanical Engineering and Naval Architecture 31
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
Maja Hećimović Master’s thesis
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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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 33
4.2. Mesh
Figure 33 SADE shell wing box with leading edge model (SWBLE)
Figure 34 SADE shell wing box model (SWB)
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 34
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)
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 35
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).
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 36
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 37
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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 38
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 39
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 40
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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 41
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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 42
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 43
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)
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 44
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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 45
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 46
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 47
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
Maja Hećimović Master’s thesis
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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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 49
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
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Faculty of Mechanical Engineering and Naval Architecture 50
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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 51
Figure 64 Normalized comparison of R2 displacements between BWBLE and SWBLE models in a case of pure torsion
Table 3 Data used for comparison of T3 displacements for pure bending load case along the span (BWB and SWB models)
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
Maja Hećimović Master’s thesis
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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
Table 4 Data used for comparison of T3 displacements for pure bending load case along the span (BWB and SWB models)
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
Maja Hećimović Master’s thesis
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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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 54
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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 55
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 56
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 57
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]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 58
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.
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T3_wb_b_2
T3_wb_b_1
T3_wb_app_b
y[mm]
T3_w
b[m
m]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 59
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
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delta_b_T3
y[mm]
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delta_s_T3
y[mm]
y[mm]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 60
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.
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T3_BWB_ap
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y[mm]
T3[m
m]
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delta_b_R2
y[mm]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 61
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.
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R2_BWB_ap
R2_SWBLE
R2_SWB_ap R2
[rad]
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 62
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.
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 63
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
Maja Hećimović Master’s thesis
Faculty of Mechanical Engineering and Naval Architecture 64
9 ATTACHMENTS
I CD-R Disc