1

Multidisciplinary Optimization Standardization Approach for Integration and Configurability

MOSAIC Project

Task 6

WING–BOX STRUCTURAL DESIGN OPTIMIZATION

Report 6

3D Finite Element Model of DLR-F6 aircraft Wing-Box Structure, Created in PATRAN and Analyzed in

NASTRAN

By

Mostafa S.A. Elsayed, M.Sc. Ph.D. Candidate

Amandeep Sing M.Sc. Student

Ramin Sedaghati, Ph.D, P.Eng.

Associate Professor Principle Investigator for Task 6

Department of Mechanical and Industrial Engineering Concordia University

Sponsor’s Ref. No: CRIAQ 4.1-TASK 6

September 2006

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3D Finite Element Model of DLR-F6 aircraft Wing-

Box Structure, Created in PATRAN and Analyzed in

NASTRAN

Abstract

The current report is a summary of the work done in stages three and four

of TASK 6 of the MOSAIC project. The third stage is completed where a

methodology for the design optimization of the spars and spar caps of the wing

box is presented. The methodology is based on the incomplete diagonal tension

theory where the spars are considered in the diagonal tension state of stress. A set

of constraints are applied with an objective function of mass minimization which

generated an acceptable results. The data obtained from stage three along with

the data of the stiffened panels and the ribs are all employed to generate the 3D

finite element model of the wing box structure. MSC.PATRAN is used as a

modeler while MSC.NASTRAN is used as analyzer. The generated 3D FEM is

validated by testing the performance of the model due to the application of a set

of aerodynamic loads representing normal cruising conditions. On the other hand

a stick model of the 3D finite element model is generated where the flexibility

method is used to evaluate the model stiffness properties. Also, a set of empirical

formulas generated by the Bombardier aerospace are used to generate the

stiffness properties of the ideal stick model of similar aircraft wing-box. The

empirical stick model performance is compared with the performance of the 3D

FEM and its stick model which showed a great agreement. The comparison

showed that the design methodology followed in this project stage is a

conservative design where the model generated is stiff model compared with the

ideal one. On the other hand the design methodology succeeded in achieving a

weight reduction in the wing-box structure as previously explained in previous

reports.

Key Words: Wing-Box, Spars and spar caps, stick models, Diagonal

Tension, Multi Disciplinary Design Optimization (MOD).

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Section I Overview

I-1 Description of the Project:

The objective of task 6 in the MOSAIC project is to improve the available

structural analysis modules in the Bombardier Aerospace and perform a

structural design optimization of the wing box by adding an optimization loop

around the analysis code. The objective is to design a wing-box more rapidly and

automatically. Task 6 is divided into four stages.

Stage I: Optimization of one skin stringer panel: (finished)

Stage I explained in details the procedure to optimize one skin-stringer

panel consists of one stringer with one stringer spacing (or pitch) of skin in the

chord wise direction and the distance between two ribs in the span wise direction.

Skin-stringer panels on the upper and lower wing covers are considered. The load

acting on the panels is taken to be constant (i.e. same load acting on all panels)

which resulted in identical dimensions for all panels. Stage-I provides a

methodology to obtain the optimum dimensions for a skin-stringer compression

panel with a minimum mass under six constraints namely crippling stress,

column buckling, up-bending at center span (compression in skin), down-

bending at supports (compression in stringer outstanding flange), inter-rivet

buckling and beam column eccentricity. It also provides optimum design

variables for panels under tensile loading with fatigue life as a design constraint

with same objective function (Minimum mass for panel). A panel on the lower

wing cover is designed for Damage Tolerance. (For more details refer to report II

and III)

Stage II: Load Redistribution: (Finished)

Stage II presented the methodology for calculating the actual load

experienced by each skin-stringer panel when arranged on the airfoil profile at

any span wise section of the wing. The number of stringers required on the upper

and lower wing covers is obtained by dividing the width of the wing-box by their

corresponding stringer pitch obtained from stage I. These panels are then re-

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arranged on the actual airfoil profile at certain span wise section. Each panel now

experiences different magnitude of compressive or tensile load depending on its

relative location with respect to the centriodal axes of the section.

The optimum dimensions for panels on upper and lower wing covers are thus

obtained using stage-I optimization program with the new calculated design load

which resulted in a different optimum dimensions for each panel according to its

location. (For more details refer to report IV).

Stage III: Optimization of the Spars and Spar Caps: (Finished)

This stage is an extension to stage II. In this stage the development of the

optimization tools to include the spars thickness and web cap dimensions will be

considered.

Stage IV: 3D FE Model of the wing box: (Finished)

This stage is the subject of the current report. Please read below for

details.

The DLR-F6 aircraft has been chosen as a practical example to apply the

optimization methodology under investigation.

I-2 DLR-F6 Aircraft Geometry and Wing Details:

The geometry and load details are taken from DLR-F6 aircraft [5]. The

actual wind tunnel model geometry is shown in Figure (1).

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y y*

Fig. (1) DLR-F6 wind tunnel model [5]

Fig. (2) DLR-F6 wind tunnel model geometry [5]

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In Figure (2), Axes x, y and z denote the coordinate system for the aircraft body

and axes x*, y* and z* refer to the wing coordinate system. The wing with nacelle

is defined in wing coordinate system and is placed in the body system with x and

z translations of 13.661 in. and -1.335 in respectively with a dihedral of 4.787

degrees.

The nacelle is located at 8.189 in. from the wing origin. The projected wing semi-

span is 23.0571 in. The wing is defined by a number of airfoil sections at different

stations along the wing span as shown in Figure (3). The shape of the airfoil at

each station is selected based on the aerodynamics and holds the shape of the

wing.

Fig. (3) DLR-F6 wing showing different airfoil sections [5]

Figure (3) shows a number of airfoil sections that are defined at different η along

the wing span, where η is the normalized coordinate defined as *

*

sy

=η .

The front spar is usually positioned at 15% of chord and the rear spar at 65% of

chord measured from the leading edge. The enclosed area between the spars as

shown is called the wing-box.

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In order to test the optimization, the wind tunnel geometry is scaled by a factor

λ=20 to build an approximately realistic aircraft model. The scaled model

dimensions of the wing are given below:

The wing reference area for the scaled model is S=90148 in.2 and the semi-span

in wing coordinate system is s*= 463.3 in. The average chord length of the wing is

746.97=avC in. and the mean aerodynamic chord length is 18.111=macC in.

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Section II For initial sizing of the wing-box structure, it is necessary to define the

different loads experienced by an aircraft during its maneuvering.

II-1 Loads Acting on an aircraft wing-box:

An aircraft wing structure is mainly subjected to three kinds of loading

a) Aerodynamic loads in the form of lift and drag forces and

pitching moments.

b) Concentrated forces due to landing gear connections, power

plant’s nacelle connections, connections to the fuselage, connection

with the controlling surfaces structures like ailerons…etc.

c) Body forces in the form of gravitational forces and inertia forces

due to wing structural mass.

The stress analysis of the wing-box requires a complete identification of all the

loads acting on its structure.

II-1-a Aerodynamic Loads:

Generally, an aircraft flying in air is subjected to aerodynamic loads [2, 3].

The lift produced by the aircraft balances its weight and the drag force balances

the thrust produced by the aircraft as shown in Figure (4).

Fig. (4) Lift & Weight and Drag & Thrust balancing the Aircraft [5]

Initial Sizing of Wing-Box Structure

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Figure (5) shows different rotational motions exhibited by an aircraft. Pitching

moment is expressed about the center of gravity of the aircraft.

Fig. (5) Pitch, Yaw and Roll motions of an Aircraft [5]

The loads experienced by an aircraft wing are usually expressed in terms of

aerodynamic coefficients [2], namely, the lift coefficient ( LC ), the drag

coefficient ( DC ), the pitching moment coefficient ( MC ), the normal force

coefficient ( NC ) and the tangential force coefficient ( TC ). All these coefficients

are usually calculated using CFD solutions, as shown in figure (6), and are

verified by wind tunnel tests since testing an actual aircraft is quite cumbersome

and expensive.

Fig. (6) DLR-F6 CFD model, used to calculate aerodynamic loads [5]

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The above mentioned aerodynamic coefficients are all defined as below:

SqLCL∞

= (1)

SqDCD∞

= (2)

cSqMCM∞

= (3)

SqNCN∞

= (4)

SqTCT∞

= (5)

Where S is the wing reference area; for airfoils a reference length is required

rather than an area; thus the chord or length of the airfoil section is used for this

purpose. ∞q is the free stream dynamic pressure calculated as:

221 Vq ρ=∞ (6)

Where ρ and V are the density of air and speed of the aircraft (calculated from

Mach number, M) respectively. Since the speed of sound varies with the density

of air, it is required to determine the density of the air through which the aircraft

is flying. To compute this, the chart shown in Table (1), called the International

Civil Aviation Organization Table (ICAO) is always used. It can be noticed that as

the altitude increases, the density of air decreases and so does the speed of sound.

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Table 1: Variation of density of air and speed of sound with altitude

When a wind tunnel is used to collect aerodynamic data, first the actual lift force

L is measured then it is converted to a non-dimensional coefficient LC using

equation (1). All the complex aerodynamics has been hidden away in the lift

coefficient. It is noticed that LC depends on the angle of attack (α ), Mach

number (M) and Reynolds’s number (Re). To summarize, the lift coefficient it

becomes a function of three variables,

LC = f (α , M, Re) (7)

The CFD solution for wing-body-pylon-engine (wing-mounted engine) case

giving the lift coefficient and pitching moment coefficient for DLR-F6 aircraft

wing at test conditions of Mach = 0.75; CL=0.5 (CL is the overall lift coefficient);

o-0.0111=α and Re = 0.300E7 is given in Tables (2) and (3).

Altitude (ft)

Density of Air

( 3/ mkg )

Speed of Sound

(m/s) 0 1.2249 340.4076

1000 1.1894 339.2758 2000 1.1548 338.0926 3000 1.1208 336.9094 4000 1.0878 335.7262 5000 1.0554 334.5429 6000 1.0239 333.3083 7000 0.9930 332.1250 8000 0.9626 330.9418 9000 0.9332 329.7072 10000 0.9044 328.5239 15000 0.7709 322.4021 20000 0.6524 316.1773 25000 0.5488 309.7982 30000 0.4581 303.2647 35000 0.3798 296.6284 40000 0.3015 295.1880 45000 0.2370 295.1880 50000 0.1865 295.1880 55000 0.1469 295.1880

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Table 2 Variation of Lift Coefficient (vs) η

*

*

sy

=η LC c

0.1274 0.4328 158.8756 0.1651 0.4580 149.9319 0.2029 0.4784 140.9653 0.2409 0.4908 131.9525 0.2793 0.4926 122.8697 0.3180 0.4864 113.6916 0.3572 0.5141 104.3917 0.3971 0.5483 94.9413 0.4377 0.5698 91.3885 0.4792 0.5899 88.1641 0.5219 0.6068 84.8560 0.5657 0.6212 81.4495 0.6111 0.6340 77.9280 0.6582 0.6439 74.2717 0.7074 0.6502 70.4573 0.7589 0.6553 66.4566 0.8133 0.6504 62.2348 0.8711 0.6353 57.7487 0.9330 0.5861 52.9427 1.0000 0.4832 47.7443

Tables (2) shows the variation of the local lift coefficient at different stations

along the wing span where “ LC ” is the local lift coefficient at a specific span

coordinate and “c” is the local chord length at that span coordinate.

Figure (9) shows the variation of the lift coefficient along the wing span. From

table (2) and by using equation (1), the lift force per unit length along the wing

span can be calculated, as shown in figures (7) and (8).

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Fig. (7) lift coefficient LC (vs) normalized wing span coordinate η

Fig. (8) Lift Force L per unit length (vs) η

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Table (3) Pitching Moment Coefficient about Local Quarter Chord (vs) η

*

*

sy

=η MqcC

0.1274 -0.0958 0.1651 -0.0929 0.2029 -0.0939 0.2409 -0.0987 0.2793 -0.1071 0.3180 -0.1201 0.3572 -0.1374 0.3971 -0.1461 0.4377 -0.1381 0.4792 -0.1325 0.5219 -0.1280 0.5657 -0.1249 0.6111 -0.1231 0.6582 -0.1221 0.7074 -0.1228 0.7589 -0.1222 0.8133 -0.1203 0.8711 -0.1165 0.9330 -0.1128 1.0000 -0.1093

Table (3) shows the values of pitching moment coefficient about quarter chord

length along the wing span. These data are represented graphically in Figure (9).

From table (3) and by using equation (3), the pitching moment about quarter

chord length can be calculated, as shown in Figure (10).

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Fig. (9) Pitching moment coefficient about quarter chord MqcC (vs) η

Fig.(10) Total Pitching Moment (about Quarter Chord) (vs) η

Integration of the curve in Figure (8) along the spanwise direction gives the shear

force distribution on the wing as shown in Figure (11). The bending moment

distribution along the wing span can also be obtained by integrating the shear

force distribution as shown in Figure (12).

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Fig. (11) Shear Force (vs) η

Fig. (12) Bending moment (vs) η

The loads calculated in Figures (10), (11) and (12) are still not the actual DESIGN

loads. They need to be scaled up by applying suitable scaling factors as these

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loads are too small to use for sizing the wing box. The conditions of Mach = 0.75;

CL=0.5 and Re = 0.3E7 is a cruise condition. Hence, a design condition of 2.5g

maneuver is considered here and the obtained loads are multiplied by a factor 2.5

to make them the actual DESIGN loads. Also an additional safety factor of 1.5 is

applied over these loads.

All these external aerodynamic loads will be resisted by internal reactions in the

wing structure. The design of the stiffened panels is based on the assumption that

the stringers are the members which are responsible about the bending

resistance, while the skin is designed to just carry in plane stresses in the form of

in plane shear stresses and tensile stresses, but its resistance to compressive

stresses is very limited due to its instability under slightly compressive loads. The

variation of the bending stress along the stiffened panels will generate a flexural

shear flow in the plane of the airfoil.

II-1-b Concentrated Loads:

Concentrated forces acting on the wing-box structure are acted primarily on

the wing ribs, which by its turn redistribute these forces in to the wing section in

the form of shear flow.

The following is a summary for different cases of concentrated loads and the

corresponding rib stiffeners arrangements:

1) If the concentrated force is applied in the plane of the rib, then the

stiffener should be aligned with the line of action of the force.

2) If placing the stiffener to be aligned with the load is impossible due to

some openings in the rib, cutouts…etc, then placing two inclined stiffeners

is also acceptable, since each stiffener will carry a component of the load in

its direction.

3) If the load is out of plane of the rib, then placing three stiffeners

perpendicular to each other is also acceptable since each stiffener will

carry a component of the force in its direction, as shown in figure (13)

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Fig. (13) the stiffeners arrangement in a shear web subjected to out-of-plane

concentrated force [6]

4) If the load is normal to the web, then design of stronger flanges to carry

the load in bending then transfer it to the web.

II-1-c General definition of wing station external

loads:

A wing station “j” is subjected to two type of loading

1- Shear forces in the form of:

a. Vertical shear force jZV

This vertical shear force includes:

(i) The total lift summation from the wing tip till the ‘jth’ wing

station which can be obtained from figure (11) for the DLF-6

aircraft.

(ii) Wing structural weight (body forces) included in the wing

portion extending from the wing tip till the ‘jth’ station. It is

important to note that in the conceptual design stage the size of

the wing parts is not yet determined. Accordingly, the weight of

the wing portions will not be available. Alternatively, an

approximate value for the distribution of the wing weight along

the wing span can be obtained from previously designed

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airplanes, or the weight may not be included in the initial sizing

process, then it can be included later then an iteration design

process can be conducted for a suitable convergence for the wing

weight.

(iii) Inertia forces (body forces), where the mass of the wing portion

structure must be multiplied by the acceleration of flight in the

vertical direction.

(iv) Non-structural mass forces due to the fuel tank weight…etc. in

the form of weight and inertia forces.

b. Horizontal shear force jXV

This horizontal shear force includes:

(i) The total drag summation from the wing tip till the ‘jth’ wing

station.

(ii) Inertia forces (body forces), where the mass of the wing portion

structure must be multiplied by the acceleration of flight in the

horizontal direction.

(iii) Non-structural mass forces due to the fuel tank mass…etc. in the

form of inertia forces.

2. Twisting moment

a wing station “j” is subjected to twisting moment ‘ jM ’ the sources of this

twisting moment are

(i) The pitching moment jqcM . The pitching moment about quarter

chord location for DLR-F6 can be obtained from figure (10).

(ii) Twisting effect of lift forces.

The lift force is always calculated with respect to the aerodynamic

center of the wing cross-section which with an acceptable

approximation considered as the airfoil quarter chord location. This lift

force at the quarter chord has a twisting effect with the value of jXj

Z eV

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where jXe is the horizontal distance between jth station quarter chord

and shear center.

(iii) Flange forces twisting moment ‘ jfM ’:

Since the aircraft has a tapered wing, which implies that the stiffeners

are not perpendicular to the airfoil cross-section but they have

inclination angle in the Y-Z plane as well as in the Y-X planes. These

inclinations generate a flange force components in the three space

directions jiXf

F,

, jiYf

F,

and jiZf

F,

.

In the calculation of the shear flow around the airfoil cross-section the

in-plane forces are of quite importance to the calculations. jiXf

F,

and

jiZf

F,

are the ‘ith’ stringer in the ‘jth’ wing station flange forces in the X

and Z directions respectively. These forces are generating a flexural

shear effect as well as a twisting effect on the airfoil cross-section.

(iv) Twisting effect of the drag forces:

The general shape of the airfoil is shown in figure (14) and the drag

forces are always considered as acting horizontally through the airfoil

chord line, as shown in the following figure

Fig. (14) Airfoil main lines

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If the line of action of the drag forces is not passing with the airfoil

shear center, then a twisting effect takes place with magnitude jZj

X eV

where jZe is the vertical distance between wing station shear center

and its chord line.

(vi) Twisting effect due to wing portion weight:

Once the wing weight included in the design process, a twisting effect

of the wing portion weight must be introduced, the magnitude of this

twisting moment is ‘ 0eW j ’ where jW is the weight of the wing portion

extending from the wing tip till wing station ‘j’ and 0e is the horizontal

distance between the airfoil centroid (center of gravity) and its shear

center at that wing station.

II-2 Initial Sizing of Stiffened Panels

Refer to reports one, two, three and four for details of stiffened panels sizing.

II-3 Initial Sizing of Wing-Box Ribs

Refer to report five for details of wing rib stress analysis and design.

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II-4 Initial Sizing of Wing-Box Spars and Spar’s

Caps

II-4-1 Literature reviews about spars and wing box

Generally, there are two categories of approaches to deal with the design and

analysis of spar and wing box. One is the traditional method, which was

introduced in detail by Kuhn and al.[1], Bruhn [2], and Niu [3]. By using column

and plate buckling and crippling analysis theory combined with empirical

equations or curves, the dimensions of spars and wing box can be decided and

optimized. The advantages of this method are simple and quick calculation with

coarse accuracy. The main obstacle is that the empirical equations and curves can

only be used to the specific materials and the configuration of structures.

Therefore, the applications are limited. Another category is finite element

analysis, which has been used more and more, especially for composite wing box

structure. With the developments of integrated CAD/CAE software and reducing

cost of computer hardware, a considerable amount of research has been

conducted in this field, especially for Multi-Disciplinary Optimization (MDO).

The main obstacle of this method is the increased computational cost and the

unacceptable solution time because of the nonlinear analysis and the so many

iteration procedures. Therefore, the objective of stage 3 is the section sizing

optimization of spars and wing box by using traditional method and stage 4 is

whole wing box optimization by using FEA method based on the results of stage 1

to stage 3.

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II-4-2 Design of Spars Web and Caps

Each wing box structure has affront spar assembly, a rear spar assembly

and the ribs. The air load act directly on the wing covers which transmits the

loads to the rib. The rib transmits the load in shear to the spar web and

distributes the load between them in proportion to the web stiffness. The spar

web and caps are mainly subjected to bending and shear loading. The depth of

the web is usually large as compared to depth of the cap, therefore bending stress

in the web are neglected. It is assumed that caps develop the entire bending

resistance and shear flow is constant over the web.

Fig. (14) Spar Cap Assembly

The spars are approximately located early in the design phase during the

selection and layout of the wing box size. A natural tendency is to locate the front

spar at a constant chord location, between 5% to 20% chord. The front spar

location should be selected to ascertain the space provisions in the leading edge

device and to maximize the box volume for fuel containment and structural

rigidity. The rear spar is usually located between 60% to 80% chord. The rear

spar location is subject to as many or more influences as the front spar. Spar caps

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are used to connect spar web to the skin of the wing box as shown in Figure 1.

Generally, T-type and L-type caps are used for the aircraft structures. The cross

sectional area and other design parameters for these sections are:

For a T-section as shown in Figure 2

Fig. (15) T-Type Spar Cap

2211 TBTBAcap += (8)

( )

))(2()2(

2211

21222

11

BTBTBTBTTBY

+++

= (9)

21221

323

121 )()(

3)(

3YTBATBBTBBI capx −+−−−+= (10)

where capA , Y and xI are cap cross sectional area, second moment of area about

centroidal axis and centroidal distance from the top of cap.

For a L-section as shown in Figure 3

2211 TBTBAcap += (11)

)(2

)(

121

12122

TBBTTBBY

−+−+

= (12)

25

3

)))(()(( 3121

31

321 TYTBYBYBTI x

−−−+−= (13)

Fig. (16) L-Type Spar Cap

The eh , uh and eh are calculated as:

Yhhe 2−= , )( 2 YBhh eu −−= , 2

)( 2 YBhh uc−

−= (14)

The applied shear flow in the web can be written as:

eh

Vq = (15)

where V is the applied shear force on the beam. The applied shear stress in the

web can be calculated as:

tqfs = (16)

where t is the thickness of web.

II-4-2-a Objective Function

The objective of the optimization problem is to minimize the mass of the spar

web and caps assembly while preventing against any type of failure. The design

26

variables are the thickness of the spar and dimensions of the caps. The objective

function can be stated as:

Objective Function= LAht cap )2( + (17)

where h is the height of the spar web and L is the length of the bay.

II-4-2-b Constraints

In order to optimize the wing box structure, the design must satisfy a set of

constraints, e.g. material failure and buckling must not occur anywhere within

the configuration. The present work is mainly concentrated on following

constraints

II-4-2-b -1) Spar Web Failure

II-4-2-b-2) Spar Cap Failure

II-4-2-b-2-1) Crippling failure

II-4-2-b-2-2) Bending failure

II-4-2-b -1 Spar Web Failure

The two basic types of web design are shear resistant type and diagonal tension

field type. A shear resistant web is one that carries its design load without

buckling of the web. The design shear stress is not greater than the buckling

shear stress for the individual web panels and the web have sufficient stiffness to

keep the web from buckling as a whole. It is realized that the buckling web stress

is not a failing stress as the web will take more before collapse of the web take

place, thus in general web is not loaded to its full capacity for taking load.

27

Therefore, diagonal tension type web are generally used for the design of spars of

an aircraft. In the diagonal tension webs, buckling of the web is permitted with

shear loads being carried by diagonal tension stresses in the web. At the buckling

load panel buckles into the diagonal folds and additional loading is taken by

diagonal tension produced in these folds. The equations required for the analysis

are presented here and the detail description of theory of incomplete diagonal

tension can be referred from [1].

The stresses in the web subjected to incomplete diagonal tension depend on the

diagonal tension factor which is measure of degree of loading of structure above

its buckling strength. The diagonal tensional factor can be calculated as:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛=

s_cr

s10 F

flog5.0tanhK (18)

where s_crF is the shear buckling stress of the web. The shear buckling stress of the

web can obtain by following steps:

(i) Calculate the flat plate buckling coefficient sK for inplane shear

loading using the following polynomial:

16.497r13.668r2.0808r0.3401r0.0293r0.001K 3456s +−−+−= (19)

where

dhr c=

(ii) Calculate the following ratio:

2

SS_R dtEKF ⎟⎠⎞

⎜⎝⎛= (20)

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(iii) The shear buckling stress of the web is obtained from the

following polynomial approximations:

( ) ( ) ( )( ) ( )

0F 0F

12.8190.8472FF0.0141F0.0001

85.197F&0FF108F102F103F

85.971F42.5F

S_Rs_cr

s_R2

s_R3

s_R

S_RS_R4

s_R75

s_R96

s_R12

s_cr

S_Rs_cr

==

++−+

<>×−×+×−=

>=−−−

(21)

The maximum shear stress in the web corresponding to the above calculated

diagonal tension factor can be calculated as:

( )212

ss_max KC1)CK(1ff ++= (22)

where 1C and 2C are the stress correction factors. The factor 1C is to allow for the

fact that the angle α of the diagonal tension differs from 45 degrees and can be

obtained as:

1)(2Sin

1C1 −=α

(23)

The factor 2C is the stress concentration factor arising from flexibility of the cap

and can be obtained as:

4wd1C4wd0.0713(wd)0.2267(wd)

0.2434(wd)0.1156(wd)0.0211(wd)0.0013(wd)C

2

2

34562

>=<−+

−+−=

(24)

where wd is the cap flexibility factor and can be obtained as:

41

cTe )I(Ih4tdwd ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

= (25)

29

where TI and CI represent the second moment of area of the tension and the

compression flanges respectively (with respect to their centroidal axis).

The allowable maximum shear stress in the web can be obtained from the

following relation:

For o40α =

34.8527.272K4.2935K124.62K243.12K186.3K52.612KF 23456alls, +−++−+−= (26)

For o45α =

34.92428.05K19.277K53.707K92.897K41.756K1.1689KF 23456alls, +−++−+−= (27)

The allowable maximum shear stress at o45α = is good approximation for the

most of the problems. The optimization problem is constrained such that

maximum shear stress is less than the allowable maximum shear stress. The

optimization constraint can be written as:

0Ff alls,maxs, ≤− or (28)

01Ff

alls,

maxs, ≤− (29)

II-4-2-b-2 Spar Cap Failure

The crippling and bending failures are two main modes of failures in the spar cap.

The cap is designed such that its resist both types of failures

30

II-4-2-b-2-1 Crippling Failure

The cap resists two types of axial compressive stresses, compressive stress caused

by bending moment and compressive stresses caused by diagonal tension. The

compressive stress in the cap caused by beam bending moment can be written as:

cape

b AhMf = (30)

The compressive stress caused by the diagonal tension in the web can be written

as:

tanα2A

KVfcap

F = (31)

The crippling failure in the cap is caused by combination of bf and Ff . To compute

the allowable crippling stresses of the cap, the section is broken down into

individual segments and each segment n has width a width b and a thickness t

and will have either one or no edge free. The allowable crippling stress for each

segment n is found from the applicable material test curve or from the following

empirical formulas:

If segment n has free edge:

0.788

n

cyn

n

nccn E

FtbFcyn6424.0F

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡= (32)

If segment n has no edge free:

0.7882

n

cyn

n

nccn E

FtbFcyn1819.1F

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡= (33)

31

where cynF and nE are the allowable compression yield stress and the modulus of

elasticity of segment n. The allowable crippling stress for the entire section is

computed by taking a weighted average of the allowable for each segment:

∑∑=

nn

ccnnncc tb

FtbF (34)

II-4-2-b-2-2 Bending Failure

In addition to the compressive stress, the cap is also subjected to bending

moment. The bending moment is known as secondary bending moment and

produce by incomplete diagonal tension in the web. The secondary bending

moment can be obtained as:

12

tanαdtfCKM2

S3max = (35)

where 3C is the stress concentration factor and can be obtained from the

following equation:

4wd0.6C4 wd 10.0092(wd)0.0341(wd)

0.0332(wd)0.0084(wd)0.001(wd)(wd)105C

3

2

345653

≥=<+−+

−+−×= −

(36)

The secondary bending stress in the cap can be obtained as:

capx

sb IYTBMf

,

12max )( −+= (37)

The cap is subjected to both compressive and bending stress simultaneously,

therefore the margin of safety of the cap is combination of both compressive and

bending failure. The margin of safety for the cap can be calculated as:

32

1

Ff

Fff1MS

tu

sb

cc

Fb−

++

= (38)

The constraint on the optimization problem is imposed such that margin of safety

of cap is always greater than zero. The nonlinear optimization constraint can be

written as:

0

Ff

Fff11

tu

sb

cc

Fb≤

++

− (39)

II-4-3 Numerical Validation

The optimization problem formulated above is validated by comparison with

design example solved by Niu [3] and Abdo [4]. The design parameters for the

problem are:

n560000lb.iMand8.0ind30000lb,V14in.,h ====

The material properties of the web and caps are:

Web-7075-T6 bare sheet Caps-7075-T6 Extrusion

psiFpsiFpsiF

sucy

tu

4800071000800001010.5E 6

===×=

psiFpsiFpsiF

sucy

tu

4400074000820001010.7E 6

===×=

In the present case, the optimization problem is solved by considering three

different cases:

Case 1: All design variables are independent

Case 2: Lengths of both flanges are constrained to be equal.

33

Case 3: Lengths and thickness of both flanges are constrained to be equal

The results obtained from the present optimization algorithm along with those

obtained by Niu [3] and Abdo [4] are presented in Table 4. The optimum

dimensions of the spar cap are given in Table 5.

Table 4: Results obtained by Niu and M.Abdo

Acap T

Total Cross. Area K he hu hc Fscr Fs,all fs,max

MS

(web)

MS Cap

Niu 0.918 0.085 3.026 0.25 12.08 10.4 9.7 8772 29700 29387 0.01 0.03 M.Abdo 1.01 0.085 3.21 0.26 12.12 10.51 9.71 8145 29383 29242 0.0048 0.067 Case 1 0.7946 0.0855 2.626 0.2804 12.122 10.85 10.21 7680.1 29255.8 29273 0.0006 0.0002 Case 2 0.8177 0.0822 2.6869 0.3047 12.784 11.52 10.889 6702.2 28993.9 29009 0.0005 0.0004 Case 3 0.8181 0.0821 2.6873 0.3047 12.798 11.48 10.828 6706.3 28995 29994 0 0.0003

Table 5: Dimensions of cap

B1 B2 T1 T2 Case 1 1.0986 2.0491 0.1617 0.3011 Case 2 1.6315 1.6315 0.2396 0.2616 Case 3 1.669 1.669 0.2451 0.2451

It can be seen that total cross sectional area of the web-caps assembly is reduced

significantly by using present method. It can also be observed that the diagonal

tension factor obtained at the optimum design using present method is more

than that obtained by both Niu [3] and Abdo[4], and web is subjected to large

diagonal tension. The minimum cross sectional area is obtained for case 1 where

all design variables are independent. For case 2, an additional constraint is

imposed on the optimization algorithm such that lengths of both flanges are

equal. The additional constraint is imposed to obtain more symmetrical design

and results in decrease in the number of design variables. The additional

constraint results in a little heavier design than the previous case, but still much

34

lighter than that obtained by Niu [3] and Abdo[4]. The thickness of the web is

decreased and web is subjected to higher diagonal tension field. To obtain more

symmetrical design, case 3 is considered. It is assumed that both flanges have

equal length and thickness. It can be that very small increase in the mass of the

structure is observed by imposing this additional constraint, and insignificant

change has been observed in the diagonal tension factor and thickness of the web.

II-4-3.4 Conclusion

The optimization problem formulated above generate very accurate results, and

even better than other formulations. The optimization algorithm will be used to

size web spar and spar caps at each section of wing box. Furthermore, the

comparison between T type and L type section will be also be made, and effect of

numbers of uprights on the optimum design will also be investigated. The

optimum dimensions of spar web and caps obtained from optimization process

will be used to build conceptual wing box model.

35

Section III

Since the thickness of the skin as well as the width of the skin-stringer

panels are the two design variables of the optimization process of stages one and

two, then the output of these two stages are the dimensions of the skin thickness

along each panel pitch at each wing station. These dimensions are determined

through an optimization process for mass minimization as an objective functions.

Considering station 21 as an example, the output of stages one and two is

tu=[0.08 0.08 0.08 0.08 0.08 0.08]

bu=[3.64 3.64 3.64 3.64 3.64 3.64]

tL=[0.07 0.07 0.07 0.07 0.07]

bL=[4.22 4.22 4.22 4.22 4.22]

Where “tu” and “tL” are the wing skin thicknesses along the upper and the lower

skin panels, respectively. While “bu” and “bL” are the width of the skin-stringer

panels along the upper and the lower skin, respectively.

Using the stringer’s pitch along each skin-stringer panel, the number of stringers

as well as the coordinates of the stringers along the upper and the lower wing

skin are determined in wing coordinate system. Taking station 21 as an example,

the coordinates of the stringers along the upper and the lower skin are presented

as

x=[245.02 246.84 250.48 254.12 257.76 261.40 265.04 268.98 268.98 264.01 259.79 255.57

251.35 247.13 245.02]

z=[43.96 44.16 44.48 44.68 44.79 44.80 44.71 44.52 40.41 39.70 39.29 39.06 39.05 39.25 39.44]

These two vectors represent the x and z coordinates of each stringer at station 21.

It is important to mention that, the fist and last component in the x and z vectors

represent the location of the front spar upper and lower caps respectively. While

the 8th and the 9th component represent the rear spar upper and lower cap

respectively. The rest represent the coordinates of the stringers locations along

the upper and the lower skin.

To insure moment of inertia maximization, a set of relations are adopted to relate

skin thickness and panel width with the rest dimensions of the skin-stringer

panel. Recalling from the first report the details concerning the ‘Z’ stringer

3D Finite Element Model of DLR-F6 Aircraft Wing-Box Structure, Created

in PATRAN and Analyzed in NASTRAN

36

Fig. (17) Panel geometry definition using ‘Z’ stringer [7]

Where

( )aasts

ew

aFaf

sa

sa

ssa

tbAtbb

ttandbbstringersflangeequalfor

tttifbtiftb

4.1

7.03.0312.13.006808.2

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

==

=>=<+=

(40)

‘ stA ’ is the stringer cross-section area, and it can be represented as

( ) fw

fwfwst ttbttbA ⎟⎠⎞

⎜⎝⎛ ++−=

22 (41)

And ‘ eb ’ is the effective width of the skin [10], where,

sk

skse

EKtb

ση

= (42)

where ‘K’ is the skin buckling coefficient and it takes the values

37

s

stb

11032.64062.3 >=<=s

s

s

stb

forKortb

forK

Between the above values there is a gradual transition, as plotted in this figure

Fig. (18) Variation of compression panel skin buckling constant with skin cross-

section aspect ratio [6]

‘η ’ in equation (18) is the plasticity reduction factor which is determined using

the following equation

sk

skt

EE

=η (43)

Where ‘ skE ’ and ‘ sktE ’ are the elastic and tangent modulii of the skin,

respectively While ‘ skσ ’ is the skin axial stress.

For practical use, the design curves for the skin stringer panels can be used,

where

38

LNindexloadofvalueslowfor

bb

LNindexloadofvaluehighfor

bb

e

s

e

s

3.1:1.1

1

=

=

(44)

Where ‘N’ is the axial load intensity, and it can be calculated using the equation

shscbMN = (45)

And ‘L’ is the effective column length, or the distance between two successive

ribs.

The output of station three is the dimensions of the spars. A wing spar is

composed of a spar web, upper and lower spar cap and a group of uprights to

reinforce the spar against collapse when it is subjected to incomplete diagonal

tension.

Also, the output of the optimization process of the wing rib is the thickness of the

rib web, number and center location of lightening holes and the diameters of the

lightening holes. Refer for the report five for more details.

All there dimensions are employed to build the 3D finite element model of the

wing.

III-1 Organization of the Finite Element Model

The wing-box is divided into 20 bays extending between 21 stations as

shown in Figure (20). The stations are named from station 1 at the wing root to

station 21 at the wing tip. While the bays are named from bay 2001, extending

from station one to station two, to bay 2120, extending from station 20 to station

21.

All the elements in a bay are numbered so that the first four digits in its index

represent the name of the bay (i.e. an element in bay 2120 has the index

2120xxx).

39

y

Fig. (19) DLR-F6 wing lay-out

MSC.PATRAN is used as the modeler to build the finite element model. Each bay

is grouped into five groups namely the skin, the stringers, two groups

representing each rib bounding the bay in the span wise direction and a group for

the load card and its rigid elements which are used to load distribution.

III-2 Building the Finite element model

MSC.PATRAN is used to build the finite element model. Following are the steps

used to create bay 2120, extending between stations “21” and “20” in the current

wing-box

1- in MSC.PATRAN a new data base is created and named DLR-F6-wing-box-

3D-FEM

2- The model tolerance is set to the default. Analysis type is set to “structure”

while the analysis code is chosen to be MSC/NASTRAN.

3- The set of coordinates representing the locations of the stringers and the spar

caps at stations “20” and “21” are used to generate these points in

MSC.PATRAN as shown in Figure (20).

40

Fig. (20) Points representing the locations of the spar caps and the stringers at

stations “20” and “21”

4- A new group is created and named “2120_stringers”, then post this group as

the current group. Using the points created in the previous step a group of

lines is generated between pairs of points extending from station “21” to

station “20”. The order of creating these lines must start by the lines

representing the spar caps, then the lines representing the stringers are

created in the order starting from the points near the front spar then proceed

towards the rear spar. The reason of this order is that the stringers run out

always takes place at the rear spar, i.e. a difference in the number of stringers

between two stations means that there is a stringer run out equal to the

difference between the number of stringers between the two successive

stations and these run outs take place at the rear spars, as shown in the next

figure.

41

Fig. (21) Group of lines representing the spar caps and stringers in bay 2120

From figure (21) it can be noticed that there is a point on the top skin and another

one on the lower skin that are not employed in generating the stringer lines, this

indicates that these two points are a run out of two stringers in bay 2019.

5- A group is created and named as “2120_skin” and posted as the current

group. Use the lines generated in the previous step to generate surfaces

extending between adjacent lines in the chord wise direction as shown in the

following figure.

42

Fig. (22) Group of surfaces representing the bay skin and the spars webs

6- Once the geometry of the bay is created, finite elements can be generated.

It is well known that increasing the number of elements in the model enhances

the accuracy of the results but it increases the model cost. Accordingly, it is

required to keep the minimum number of elements necessary to obtain

acceptable accurate results. To do so, the number of elements along the bay is

selected to be two elements in the span wise direction, and after finishing the

whole bay model, the bay is tested for an arbitrary load and the results are

obtained. Then the number of elements is increased to three in the span wise

direction and the model is resubmitted to NASTRAN for analysis. The results

obtained are compared with the results obtained from the pervious step. If a

significant change is obtained in the results then, it is required to re-increase the

number of elements and test again. A change in the result with in 0.05% doesn’t

43

require additional refining of the model. It has been found that three elements in

the span wise direction results in acceptable results.

9- Elements Properties:

After creating the finite elements, the elements properties should be applied.

The stringers are modeled by beam elements with a Z shape cross-section. The

details of the Z-shape cross-section are shown in the next figure.

Fig. (23) The Z-shape cross-section of the beam element used in the PBEAML

Card for stringer modeling [8]

A comparison between the dimensions of this Z-shape cross-section and the

dimensions obtained from the optimization process in stages one and two, shows

that these DIM1, DIM2, DIM3 and DIM4 dimensions can be calculated by simple

transformations as follows

21 w

at

bDIM −= (46)

wtDIM =2 (47)

aw tbDIM −=3 (48)

aw tbDIM +=4 (49)

Since the dimensions of the stringers vary from one station to the other, an

interpolation process is used to obtain the dimensions of all elements between

stations.

44

This is done by defining the dimensions in PATRAN as fields, where a local

coordinate is created at each station with its Z-coordinate directed in the span

wise direction. Set of fields are created in PATRAN defined in the station local

coordinate, representing the variation of the dimension in the span wise

direction. As an example, consider the dimension DIM1 of the Z-shape stringer

extending between stations 21 and 20, this dimension is defined in PATRAN as a

field on the form

ZDIMDIMDIMDIM )20_121_1(20_11 −+= (50)

Similarly for all the other dimensions.

The spar caps are also modeled by beam elements but with L-shape cross-section

as shown in the following figure

Fig. (24) The L-shape cross-section of the beam element used in the PBEAML

Card for spar caps modeling [8]

45

Fig. (25) the model stringers after applying the properties in PATRAN

The skin is modeled by SHELL elements, where the thickness of the shells is also

defined by fields representing the variation of the skin thickness in the span wise

direction.

7- modeling of ribs:

a group of points is generated to represent the perimeter of the rib, these points

have the same y-coordinate of the station, while its x-coordinate has the same x-

coordinate of the corresponding stringer, while its z-c00rdinate can be defined by

the following equation

ssr DIMzz 4−= (51)

Where rz is the z-coordinate of the rib perimeter point, sz is the z-coordinate of

the corresponding stringer while sDIM 4 is a dimension belongs to the stinger

corresponding to this rib point.

46

The rib web is modeled by QUAD4 elements with PSHELL card for its properties.

While the perimeter of the rib and the lightening holes are reinforced by beam

elements.

The following figure shows a complete bay modeled in PATRAN.

Fig. (26) Complete bay modeled in PATRAN

III-3 Model Verification

Early model verification is very important before proceeding for the whole

finite element model. Early detection of errors is very important, since detection

of errors in the advanced stages is very costly and time consuming.

The model can be verified by either of the following two methods

a) Model verification.

b) Modeling methodology verification.

47

a) Model verification:

The finite element model of the DLR-F6 wing-box bay, created in

PATRAN in the previous section, can be tested in NASTRAN for an

arbitrary value of loading. Then the result obtained from NASTRAN

is compared with the analytical solution of such model with the

same loading. The complementary internal virtual work theory of

idealized beams is used to calculate the deflection of this wing bay

under the effect of a flexural bending force.

NASTRAN Analysis

The model created in the previous section is loaded by an arbitrary load of

1000 lbs acting at the bay centeroid. To apply a loading to the bay at its centroid,

a grid point is created at the section centroid, then this grid point is connected to

the skin-stringers connectivity grid points by a group of RBE2 elements, with its

independent degrees of freedom are at the centroidal grid point, and its

dependent degrees of freedom are at the skin-stringers connectivity grids. A load

of 1000 lbs is applied in the negative z-direction at the centroidal grid point. The

model is fixed at all the skin-stringers connectivity grid points of the opposite bay

station. Then, the model is submitted to NASTRAN for linear static analysis,

which resulted in a deflection of 0.0126 in., as shown in the following figure.

48

Fig. (27) Deflection of a wing-box bay due to an arbitrary force loading

it is clear from figure (31) that the deflection is the result of superposition of a

combined loading. Since the bay section centroid does not coincide with the its

shear center, then the force applied at the centroid has a bending, shear as well as

a torsion effect.

The deflection obtained from the NASTRAN analysis is verified by the analytical

results.

b) Modeling methodology verification.

Another method to verify the finite element model is to verify the modeling

methodology it self. By solving the deflection of simple box beam structure

subjected to an arbitrary loading, then modeling the same structure and analyzes

it in NASTRAN. If the results coincide, then the modeling procedure is correct.

49

III-4 Complete 3D Finite Element Model of the DLR-

F6 Wing-Box

Once the model is verified, the work can proceed towards creating the full finite

element model of the wing-box, as shown in the following figure

Fig. (28) DLR-F6 wing-box finite element model

The model is submitted to NASTRAN for linear static analysis. The wing-box is

subjected to static loading represents normal cruising conditions, applied along

the wing-box elastic axis which produced a maximum deflection at the wing tip of

magnitude 17.7 in. as shown in the following figure.

50

Fig. (29) DLR-F6 wing-box deflection due to cruising conditions loads

Completing the entry of the material card in the bulk data file, to include the

ultimate stresses of the material in tension and in compression, generates the

margins of safety of the finite elements. The analysis showed that the margins of

safety are in the zero one interval which indicates proper sizing of the model.

51

III-5 Post-processing of the Wing-Box Finite

Element Model:

Deformation of an aircraft wing during flight has significant consequences

on the aerodynamic performance. Predicting an accurate value of the bending

and twisting of the wing in flight depends on the fidelity of the finite element

model of the wing-box. Validation of the finite element model means making sure

that the structural response of the model reproduces the structural response of

the real wing within an acceptable accuracy.

To find the deflection and the twisting experienced by the current wing-box due

to the applied aerodynamic loads, the deflection and the twisting experienced by

the wing-box elastic axis are plotted against the normalized span wise coordinate

“η ” as shown in the following figure

Fig. (30) Deflections in z-direction (vertical) experienced by the DLR-F6 wing-

box elastic axis under the effect of cruising conditions aerodynamic loads

η

Def

lect

ion

in Z

-dire

ctio

n

52

Fig. (31) Deflections in x-direction (in plane bending) experienced by the DLR-F6

wing-box elastic axis under the effect of cruising conditions aerodynamic loads

η

Def

lect

ion

in X

-dire

ctio

n

53

Fig. (32) Twisting angle around the y-direction (torsional) experienced by the

DLR-F6 wing-box elastic axis under the effect of cruising conditions aerodynamic

loads

Since these deformations experienced by the wing-box are just an interpretation

of the structural stiffness properties, then it is more convenient to calculate the

model stiffness properties while the deformations can vary based on the loading

conditions.

To evaluate the equivalent moment of inertia and torsional rigidity of the model,

two shear center nodes are created at the extremities of each wing bay, those two

nodes are attached to the structure, as previously explained, by rigid bodies

whose its dependent degrees of freedom are specified at an arbitrary number of

grid points of the skin-stringers connectivity points, while its independent

degrees of freedom are specified at a single grid point of the shear center. The

next step is to load the node, which is towards the wing tip, by three sets of unit

load moments. The first set moment is along the x-axis to calculate the vertical

η

Twis

ting

arou

nd y

-axi

s

54

bending moment of inertia. The second set is the moment along the y-axis to

calculate the torsional stiffness rigidity and the third set is along the z-axis to

predict the horizontal bending stiffness. The wing-box rotations in the x, y and z

directions due to the corresponding applied moments are computed using

NASTRAN and then the corresponding values of the stiffness are calculated using

the equations

( )( )

ij xx

ijjix

SEI

θθηη−

−=→

*

)( (52)

( )( )

ij yy

ijjiy

SGJ

θθηη−

−=→

*

)( (53)

( )( )

ij zz

ijjiz

SEI

θθηη−

−=→

*

)( (54)

Where "* 2.463=S is the semi-span of the DLR-F6 wing. iη and jη are the

normalized coordinates of the two stations i and j respectively.

The stiffness properties of the 20 wing bays are calculated and plotted against the

wing normalized coordinateη , as shown in the following figures.

55

η

xEI

η

yGJ

Fig. (33) Distribution of the vertical bending stiffness of the DLR-F6 wing-box

along its span

Fig. (34) Distribution of the torsional stiffness of the DLR-F6 wing-box along its

span

56

η

zEI

Fig. (35) Distribution of the horizontal bending stiffness of the DLR-F6 wing-box

along its span

III-6 Model Stiffness Validation

A methodology to estimate the stiffness distribution of a new wing using the

stiffness distributions of Bombardier’s existing wings was developed by M. Abdo

et.al. [9]. This methodology is based on a set of empirical relations that are

generated for predicting the ideal stiffness of an arbitrary aircraft wing-box. To

obtain those empirical relations which are applicable to all these wings, the data

of the stiffness properties of a group of existing Bombardier’s aircrafts were

normalized. One of the normalization techniques used is that, the stiffness of the

existing wing structure is divided by the stiffness of a solid block material

bounded by the leading and trailing edge of the wing, which referred to

as CATIAEI )( , this is because CATIA was used for the calculation of these solid

wing stiffness, as shown in the following figure.

57

Fig. (36) DLR-F6 wing airfoil sections

Figure (39) shows the airfoils sections of the DLR-F6 wing at 21 wing stations. At

each wing station, the airfoil section is padded, and then the measure tool bar is

used to calculate the stiffness of each wing section.

The data obtained from CATIA are used along with the empirical relations to

predict the ideal stiffness properties of such aircraft wing-box.

For xEI the behavior of the normalized stiffness appeared to be different

outboard and inboard of the break in the plan form, consequently different

relations were used to fit the data of the empirical relations, as follows

( )( ) ( )

( )( ) 1≤≤=

≤≤+−−−

=

ηη

ηηηηηηη

BreakBreakCATIAx

FEMx

BreakRootRootRootRootBreak

RootBreak

CATIAx

FEMx

forREIEI

forRRR

EIEI

(55)

*SyBreak

Break =η (56)

58

*SyRoot

Root =η (57)

Where RootR is the ( ) ( )CATIAxFEMx EIEI ratio at Rootηη =

And BreakR is the ( ) ( )CATIAxFEMx EIEI ratio at Breakηη =

It was determined that 03.0=RootR and 1.0=BreakR provides an acceptable fit for

the airplanes.

For yGJ the following empirical relations was developed

( )( ) 002.0=

CATIAy

FEMy

GJ

GJ (58)

For zEI the following empirical relations was developed

( )( ) 007.00103.0 += η

CATIAz

FEMz

EIEI

(59)

The previous empirical relations are used to predict the stiffness properties of the

ideal wing-box of the DLR-F6 aircraft, and it is compared with the current FEM

stiffness properties, as shown in the following figures.

59

o o o o

+ + + Ideal FEM Model

Current FEM Model

η

yGJ

Fig. (37) Comparison between distributions of the vertical bending stiffness of the

DLR-F6 wing-box along its span in the ideal FEM and current FEM

Fig. (38) Comparison between distributions of the torsional rigidity of the DLR-

F6 wing-box along its span in the ideal FEM and current FEM

o o o o

+ + + Ideal FEM Model

Current FEM Model

η

xEI

60

o o o o

+ + + Ideal FEM Model

Current FEM Model

η

zEI

Fig. (39) Comparison between distributions of the horizontal bending stiffness of

the DLR-F6 wing-box along its span in the ideal FEM and current FEM

It is clear that there is a great agreement between the current NASTRAN FEM

and the ideal FEM of such a wing, predicted by the empirical relations.

III-7 Wing-Stick Model of the DLR-F6 Wing-Box Full

Finite Element Model

Different levels of wing structural models have been used for wing static

aeroelastic analysis and optimization, ranging from simple models based on

analytical or empirical expressions to complex finite element structural models.

The difficulty is to find or develop aeroelastic models that are sufficiently simple

to be called thousands of times during optimization, but are sophisticated enough

to accurately predict wing deformations in bending as well as twisting. Simplified

beam finite element model of aircraft wing-box structure, also known as stick

model, are often used for aerodynamics-structure interaction, such models can be

used for either static or dynamic aeroelsatic analysis [9].

61

To build the stick model of the DLR-F6 wing-box, each wing bay is modeled by

only one bar element representing the bay elastic axis, where the entry of that bar

element are extracted as equivalent parameters from the full finite element model

of that bay.

Following is the entry necessary for the CBAR card used to construct the stick

model in NASTRAN along with its PBAR card.

Where EID…element unique ID

PID…element property unique ID

GA, GB…grid point identification number, the two shear centers

identification number of the wing bay two stations.

X1, x2, x3…components of the element orientation vector

MID…material unique identification number

A…area of the element, equivalent area of the wing-box bay

I1, I2, J…equivalent bending stiffness in two planes, and torsional stiffness

of the wing-box bay

K1, K2…equivalent area factors for shear of the wing-box bay

All these equivalent parameters must be extracted from the wing-box full FEM.

the equivalent stiffnesses of the wing-box are already obtained in the previous

section where their values can be calculated from the following relations, based

on a unit load deformations, where the loads are applied in the directions of the

principle inertia of the wing-box cross-section

62

( )( )

ij xx

ijjix

SE

Iθθ

ηη−

−=→

*1)( (58)

( )( )

ij yy

ijjiy

SG

Jθθ

ηη−

−=→

*1)( (59)

( )( )

ij zz

ijjiz

SE

Iθθ

ηη−

−=→

*1)( (60)

While the equivalent area and equivalent area factors for shear needs further

processing in NASTRAN and PATRAN to be calculated.

III-7-1 Evaluation of the Equivalent EA’s and GK’s of

the Wing-Box

The process of evaluating the equivalent area and shear factors span wise

distribution is different from that of the bending and torsional stiffness

evaluation. In this process NASTRAN is executed for three load subcases for each

wing-box bay. In this process, the three degrees of freedom related to rotation in

all skin-stringers connectivity grid points at all wing station are frozen. While the

translation degrees of freedom are kept free. The elastic axis nodes are connected

to the skin-stringer connectivity grid points by RBE2 elements with its dependent

degrees of freedom specified at the skin and its independent degrees of freedom

specified at the shear center grid points.

The first subcase, a unit force in the y-direction, the axial direction, is applied at

the bay shear center grid point, in order to calculate the equivalent area, as

follows

( )( )( )

ij yy

ijji DD

SE

A−

−=→

*1 ηη (61)

In the second subcase, the structure is loaded by a unit load at the shear center in

the z-direction, to calculate the shear factor in the z-direction as follows

63

( )( )( )

ij zz

ijji DD

SGA

K−

−=→

*

11 ηη

(62)

Similarly, the structure is loaded by a unit force in the x-direction to calculate the

shear factor in the x-direction, as follows

( )( )( )

ij xx

ijji DD

SGA

K−

−=→

*

21 ηη

(63)

Where “D” denotes the displacement deformation in certain direction.

This process is applied to the wing box 20 bays, where all the elastic properties of

the 3D model are extracted and applied to construct the stick model, the results

of the calculations are shown in the following table

Table (6) the PBAR card entries necessary to generate wing stick model

I1 I2 J A K1 K2 1.0e+003 * 4.2220 1.0e+003 * 3.3290 1.0e+003 * 2.5298 1.0e+003 * 1.9032 1.0e+003 *1.3495 1.0e+003 *1.0322 1.0e+003 *0.7285 1.0e+003 *0.5558 1.0e+003 * 0.4592 1.0e+003 *0.3814 1.0e+003 *0.3071 1.0e+003 *0.2367 1.0e+003 *0.1850 1.0e+003 *0.1440 1.0e+003 *0.1156 1.0e+003 *0.0812 1.0e+003 *0.0610 1.0e+003 *0.0497 1.0e+003 *0.0406 1.0e+003 *0.0329

1.0e+004 *2.58941.0e+004 *2.16791.0e+004 *1.68811.0e+004 *1.30301.0e+004 *0.81511.0e+004 *0.73901.0e+004 *0.53051.0e+004 *0.44321.0e+004 *0.36761.0e+004 *0.30131.0e+004 *0.23791.0e+004 *0.17411.0e+004 *0.13561.0e+004 *0.10591.0e+004 *0.08411.0e+004 *0.05761.0e+004 *0.04341.0e+004 *0.03471.0e+004 *0.02801.0e+004 *0.0224

1.0e+004 *1.0549 1.0e+004 *0.8692 1.0e+004 *0.6786 1.0e+004 *0.5353 1.0e+004 *0.3961 1.0e+004 *0.3099 1.0e+004 *0.2335 1.0e+004 *0.1774 1.0e+004 *0.1477 1.0e+004 *0.1213 1.0e+004 *0.0958 1.0e+004 *0.0716 1.0e+004 *0.0553 1.0e+004 *0.0431 1.0e+004 *0.0348 1.0e+004 *0.0233 1.0e+004 *0.0177 1.0e+004 *0.0145 1.0e+004 *0.0119 1.0e+004 *0.0098

31.0505 31.1960 29.5022 28.3718 26.8366 26.9245 26.2704 23.3872 20.9122 18.8147 16.3768 13.7504 11.6117 9.9506 8.9070 6.8692 5.7850 5.3652 5.0415 4.9649

0.2303 0.2423 0.2374 0.2356 0.2311 0.221 0.2190 0.2048 0.2075 0.2056 0.2014 0.1935 0.1879 0.1890 0.1827 0.1792 0.1788 0.1790 0.1814 0.1728

1.0244 1.0150 0.9807 0.8337 0.8602 0.8203 0.8243 0.8379 0.8386 0.8344 0.8366 0.8411 0.8454 0.8389 0.8357 0.8333 0.8280 0.7729 0.7679 0.7155

These data are used to generate the wing stick model in NASTRAN, as shown in

the following figure

64

Fig. (40) Stick model of the DLR-F6 wing-box structure

Another stick model is created to represent the ideal stick model of this aircraft,

the two stick models are loaded with the same loading that was previously

applied to the 3D FEM to test the behavior of the models, as shown in the

following figure.

65

Fig. (41) the stick model deformed under the effect of the cruising conditions

aerodynamic loads

A comparison diagrams are generated to compare between the performances of

the three models, namely the 3D FEM, the stick model representing the 3D FEM

and the ideal stick model generated using the Bombardier aerospace empirical

formulas, as shown in the following figure.

66

o o o o

+ + + Ideal Stick Model

3D FE Stick Model

> > > > 3D FE Model

η

Axi

al D

efor

mat

ions

(y d

irect

ion)

[inc

h]

Fig. (43) Comparison between the 3D FEM, 3D FE stick Model and the Ideal stick model axial deformations (y direction) [inch]

η

o o o o

+ + + Ideal Stick Model

3D FE Stick Model

> > > > 3D FE Model V

ertic

al D

efor

mat

ions

(y d

irect

ion)

[inc

h]

Fig. (42) Comparison between the 3D FEM, 3D FE stick Model and the Ideal stick model vertical deformation [inch]

67

The comparison of the three models reveals that the design of the 3D FEM is a

conservative design where its deformations are always less than the deformations

experienced by the ideal stick model, generated using the Bombardier Aerospace

empirical formulas, when the two models are loading with the same loading.

Also the comparison curves validate the stick model generated using the stiffness

properties extracted from the 3D FEM which is very useful in optimization and

aeroelastic analyses processes.

o o o o

+ + + Ideal Stick Model

3D FE Stick Model

> > > > 3D FE Model

η

Twis

ting

defo

rmat

ions

(aro

und

y-ax

is) [

degr

ee]

Fig. (44) Comparison between the 3D FEM, 3D FE stick Model and the Ideal stick model twisting deformation (y direction) [degree]

68

References

1. Kuhn P., Peterson J. and Levin L. “A summary of diagonal tension, Part1-

methods of analysis,” NACA Technical Note 2661

2. Bruhn, E.F. “Analysis and Design of Flight Vehicle Structures”, Jacobs &

Associates Inc., June 1973

3. Niu M. “Airframe Stress Analysis and Sizing,” Hong Kong, Conmilit Press

Ltd., 1997.

4. Abdo M., Piperni P.,Isikveren A.T., Kafyeke, F. “Optimization of a

Business Jet,” Canadian Aeronautics and Space Institute Annual General

Meeting, 2005.

5. http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpw, 3rd AIAA CFD Drag

Prediction Workshop, San Francisco, 2006.

6. Bruhn E.F, Analysis and Design of Flight Vehicle Structures, Jacobs &

Associates Inc., June 1973

7. M. Abdo, P. Piperni, F. Kafyeke “Conceptual Design of Stinger Stiffened

Compression Panels”.

8. http://www.mscsoftware.com/support/online_ex/Library.cfm

9. M. Abdo, R. L’Heureux, F. Pepin and F. Kafyeke “Equivalent Finite

Element Wing Structural Models Used for Aerodynamics-structures

Intraction”, Canadian Aeronautic and Space Institute 50th AGM and

Conference, 16th Aerospace Structures and Materials Symposium 28-30

April 2003.

10. M. Abdo, P. Piperni, F. Kafyeke “Conceptual Design of Stinger Stiffened

Compression Panels”.