Semester Thesis

Dimensioning of an access panel for the

xed leading edge of a commercial aircraft

Jesús Ignacio Maldonado Covarrubias

Centre of Structure Technologies D-MAVT

ETH Zürich

Supervision

Michael Winkler

Prof. Dr. Paolo Ermanni

June 2011

IMES-ST 11-019

Department of Mechanical and Process Engineering Centre of Structure Technologies ETH Zurich Leonhardstrasse 27 CH-8092 Zurich

IMES-ST/SA Maldonado.docx Page 1

Eidgenössische Technische Hochschule Zürich Swiss Federal Institute of Technology Zurich

Student:

Maldonado, Jesús

ETH-Nr: 10-937-241 Departement: MAVT

Hochschule (if external student):

Thesis:

Title: Dimensioning of an access panel for the fixed leading edge of an commercial aircraft

Kind of Thesis: SA Semester: FS 2011

Supervisor: Prof. Dr P. Ermanni

Advisor: Winkler, Michael

Start of the work: 21/02/2011

Intermediate presentation (Zwischenpräsentation): 24/03/2011

Final presentation (Endpräsentation): 19/05/2011

Deadline delivery final report: 03/06/2011

Introduction Within the framework of the EU-project COALESCE2 new concepts for fixed leading edges of commercial aircrafts on the basis of the A320 are under investigation. The project focuses on new technologies and materials which shall reduce the manufacturing costs for the wing leading edge. Among other things the access panels are analyzed. Composites and aluminum is considered as potential material for the panels. Due to high material cost for composite solutions a high speed machined aluminum design with blade stiffeners will be investigated as a low cost option within this student thesis. The dimensioning of this panel for defined loads and to fulfill certain criteria has not been made so far. The thesis will be done in cooperation with the according project partners.

Figure 1: Fixed leading edge

IMES-ST/SA Maldonado.docx Page 2

Objectives The first goal of the thesis is to investigate a given initial design of an access panel. One panel will be investigated in detail with the help of finite element analyses. The finite element model and a suitable definition of the boundary conditions have to be developed. The exact number and direction of stiffeners is thereby not defined so far. Only the shape of the stiffeners is defined due to manufacturing restrictions. Wing bending and differential pressure will be considered as loads. Regarding the criteria air tolerances, strength, stability of the stiffeners and strength of the connections will be investigated. The criteria will be evaluated with the help of the simulation results and with the help of analytical considerations. First of all, it has to be investigated whether chordwise, spanwise or combined chordwise/spanwise stiffeners are preferable. After the decision of the alignment of the stiffeners, the second goal is to find a solution with a reduced mass. This will be done by parameter studies. Within these studies, the number, thickness, height, length and position of the stiffeners can be changed in order reduce the mass. At the end a lightweight solution has to be available which is fulfils the requirements.

Work breakdown The work will be mainly subdivided in the following tasks:

1. Surface model: A surface model as a preparation for the FE simulations has to be built.

2. FE model: A FE model has to be built which considers the load cases of wing bending and pressure and all relevant boundary conditions. Afterwards, several criteria have to be fulfilled by the panel. In detail air tolerances, strength, stability of the stiffeners and strength of the connections will be considered. The model includes the different possibilities of stiffeners alignment.

3. Design Improvement: The decision concerning the stiffeners alignment will be made. Afterwards, the design has to be improved in order to reduce the mass. This will be made by parameter studies.

The organization of the work is depicted in following table. The estimated deadlines are also shown.

Date 21.

Feb

28.

Feb

07.

Mrz

14.

Mrz

21.

Mrz

28.

Mrz

04.

Apr

11.

Apr

18.

Apr

25.

Apr

02.

Mai

09.

Mai

16.

Mai

23.

Mai

30.

Mai

Task Week 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22Literature ResearchSurface modelFE modelMid-Term Presentation XDesign ImprovementFinal Presentation XWriting of the report

Abstract

The dimensioning of an aluminum access panel is the main goal of this semester thesis.The problem is solved by means of three dierent approaches. An adaptive analytical platemodel is built to describe the orthotropic stiened access panel deections. A nite elementanalysis is used to prove in detail that the structure fullls all design criteria. The structuraloptimization algorithm ensures a reduction of mass to a minimum. Design improvementsfrom every phase of the study are included in the nal model. The main programs used arePatran, CATIA, Nastran, ANSYS and MATLAB.The nite element analysis reveals local eects that increase deection, which the analytical

model is unable to consider with its current formulation. An increased number of stienerswith optimized dimensions increase the local skin stiening and provide compliance of allrequirements. Design observed criteria are: strength, air tolerances, fastener strength andbuckling load. The balance is found between isotropic and orthotropic designs, neither a thickpanel skin nor a few large stieners will provide the best solution.

i

Zusammenfassung

Die Dimensionierung eines Access Panels ist das Ziel dieser Semesterarbeit. Das Problemwird mit drei verschiedenen Ansätzen gelöst. Eine adaptives orthotropes Plattenmodell wirdverwendet, um die Verformung der versteiften Platte zu beschreiben. Eine Finite-Elemente-Analyse wird anschliessend durchgeführt, um im Detail zu zeigen, dass die Struktur alleAnforderungen erfüllt. Zudem wird ein Strukturoptimierungsalgorithmus angewendet, umdie Masse der Platte zu reduzieren. Design Verbesserungen jeder Phase der Studie werdenschliesslich in das endgültige Modell aufgenommen. Die wichtigsten verwendeten Programmesind CATIA, Nastran, ANSYS und MATLAB.Die Finite Elemente Analyse zeigt, dass lokale Eekte die Verformung erhöhen, die das

analytische Modell mit seiner derzeitigen Formulierung nicht berücksichtigt. Eine grössereAnzahl von Versteifungen mit optimierten Abmessungen erhöht die lokale Versteifung derPlatte, so dass alle Anforderungen erfüllt werden. Das Gleichgewicht wird zwischen isotropenund orthotropen Konstruktionen gefunden, das heisst, dass weder eine dicke Platte noch einePlatte mit wenigen grossen Versteifungen die beste Lösung ergibt.

ii

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Structure of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 State-of-the-art 42.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Analytical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Isotropic plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Orthotropic plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Plate stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 General notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 Global objective function formulation . . . . . . . . . . . . . . . . . . 10

3 Problem denition 113.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Project requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Operating conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 Design factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.3 Repair Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Access panel requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.1 Air tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.2 Panel strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.3 Stieners' stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.4 Connections strength . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.1 Aerodynamic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.2 Wing bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Analytical modeling 164.1 Initial approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Beam model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1.2 Isolated orthotropic model . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Adaptive orthotropic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

iii

4.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Finite element modeling 235.1 CAD surface model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Modeling considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2.1 Fasteners modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2.2 Prying-eect considerations . . . . . . . . . . . . . . . . . . . . . . . 245.2.3 Rotational constraints on rib zones . . . . . . . . . . . . . . . . . . . 265.2.4 Element selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2.5 Support landings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2.6 Wing bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 Comparison to analytical model . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Optimization 306.1 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.2 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3 Optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.3.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.3.2 Algorithm execution . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Results 35

8 Conclusions 40

9 Outlook 41

Appendix A 42

Appendix B 45

iv

Nomenclature

AISI American Iron and Steel Institute

ANSYS Analysis System

CATIA Computer Aided Three-Dimensional Interactive Application

CC Cruise Conditions

COALESCE2 Cost Ecient Advanced Leading Edge Structure 2

FE Finite Element

FLE Fixed Leading Edge

GOF Global Objective Function

LL Limit Load

MATLAB Matrix Laboratory

UL Ultimate Load

v

List of Figures

1.1 Structure of thesis work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 Air tolerances denition from [13] . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Conventions for pressure from [12] . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Cross-sections considered in the iterative beam model. . . . . . . . . . . . . . 164.2 Prole of deection, beam model with variable cross-section. . . . . . . . . . 174.3 Adaptive orthotropic plate model regions. . . . . . . . . . . . . . . . . . . . 194.4 Prole of deection, adaptive orthotropic plate model. . . . . . . . . . . . . . 214.5 Sensitivity analysis graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1 Surface model extracted from solid geometry . . . . . . . . . . . . . . . . . . 235.2 Beam and bush elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Illustration of prying eect, as taken from [14]. . . . . . . . . . . . . . . . . . 255.4 Position of middle ribs above access panel, halfway along the spanwise length. 265.5 Tria6 mid-node elements shown in the triangular extremity of the stieners . . 275.6 Local deection eects from inner skin of access panel. . . . . . . . . . . . . 295.7 Stress distribution on LC-0-1. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.1 Real constants denition for ANSYS automatic model. . . . . . . . . . . . . 316.2 Iteration graph from GOF in run 27. . . . . . . . . . . . . . . . . . . . . . . 34

7.1 Optimal mass search results . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.2 Final optimized CATIA model . . . . . . . . . . . . . . . . . . . . . . . . . . 367.3 Region where one node causes a reserve factor Rstress = 0.99 for LC-1-1. . . 387.4 Extraction of maximum stiener stresses for buckling considerations. . . . . . 387.5 LC-1-0, the largest deection contribution to all load cases. . . . . . . . . . . 39

vi

List of Tables

3.1 List of load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1 Wing bending information extracted from [1]. . . . . . . . . . . . . . . . . . 285.2 Comparison of error magnitudes of analytical model vs. FE model. . . . . . . 29

6.1 Design parameters in optimization vector. . . . . . . . . . . . . . . . . . . . 316.2 Constraints for the Global objective function . . . . . . . . . . . . . . . . . . 33

7.1 Comparison of mass between models . . . . . . . . . . . . . . . . . . . . . . 377.2 Reserve factors for the nal access panel dimensions . . . . . . . . . . . . . 38

vii

1 Introduction

1.1 Motivation

The following semester thesis is product of working under the framework of Cost EcientAdvanced Leading Edge Structure 2 (COALESCE2) project, a collaboration of ETH Zürichand other academic and industrial partners. Within the framework of the EU-project CO-ALESCE2, new concepts for xed leading edges of commercial aircraft on the basis of theA320 are under investigation. The project focuses on new technologies and materials whichshall reduce the manufacturing costs for the wing xed leading edge (FLE).The objectives ofthe project are:

Development of a xed leading edge structure which is over 30% more cost ecientthan state-of-the-art structures

Examination of conventional and alternative leading edge ight control mechanisms

Simulation of assembly process and structural performance

During the accomplishment of these objectives, a more specic need is found, setting thegoals for the following study. Composites and aluminum is considered as potential materialfor the panels. Nevertheless, due to high material cost for composite solutions a high speedmachined aluminum design with blade stieners will be investigated as a low cost option.Thus, this work is focused on the dimensioning and denition of design criteria for the accesspanels.

1.2 Objectives

The thesis is divided into chapters which show the work structure mainly based on chrono-logical order. Nevertheless, a general structure will be useful to have a clear idea of the wholedesign process.Initial objectives of the semester thesis, as derived from the semester thesis agreement, are:

1. Literature research and software familiarization

a) Obtain sources of information and collect all relevant data, equations and consid-erations regarding the semester thesis.

b) Learn to use the software required by the industry partners, and evaluate all resultson it. Relevant software packages are:

i. Dassault Systèmes' CATIA

1

ii. MSC's Nastran and Patran

2. Analytical model of access panel

a) Use the available literature to characterize the panel deection.

b) Should the literature be insucient, adapt the existing models for the best ap-proximation.

3. Solid and surface model

a) Generate a modied CATIA model from the original one provided by existingproject les.

b) Extract an adapted surface model to simplify mesh construction on Patran.

4. Finite element model

a) Take advantage of surface model to create corresponding mesh of the access panel.

b) Set all boundary conditions and load cases for analysis.

5. Design improvement

a) Perform parametric studies on access panel to determine relevant design variables.

b) Improve design to nd a low mass solution.

1.3 Structure of work

During the course of the semester thesis, the possibility of applying optimization algorithmswas obtained, therefore the nal initial objective of design improvement changed from para-metric studies into structural optimization. Figure 1.1 depicts the approach at objectives.This scheme was used throughout course of the work, and all chapters on this thesis arebased on it.Chapter 2 provides an overview of the state-of-the-art in the relevant elds. This back-

ground is divided in the three dierent elds from gure 1.1. The design problem is denedin chapter 3, where all preliminary conditions are described. The rst approach is containedin chapter 4, with a full description of the analytical model. All information regarding thenite element modeling is included in chapter 5. The nal stage, explaining an optimizationalgorithm construction, is located in chapter 6. Important sections of the code from chapters4 and 6 are included as appendix.The results are presented in chapter 7, where the nal design and details are listed. Chapter

8 contains the conclusions derived from this thesis. An outlook is also included in chapter 9.

2

Figure 1.1: Structure of thesis work

3

2 State-of-the-art

This chapter provides necessary knowledge and latest methods to address the design problemof this thesis. All topics are grouped according to the previously mentioned structure of thethesis work in gure 1.1.

2.1 Literature review

As preparation step, dierent sources were investigated to obtain an insight in the relevantelds.Buckling and strength of stieners have been a major consideration ever since before the

days of ight. Since the 18th century, engineers have been tasked with creating structuresto withstand enormous loads, such as stiened bridges shown in [11]. An increasing numberof orthotropic plate theories have arisen, beginning with Huber (1914), later perfected byMassonet (1950) (all examined in [18]), and nally reaching the models by Cusens and Pama[11], which are used in this thesis.Several sources provide an approach to deection modeling of stiened plates and beams.

While some authors prefer to skip all development of the equations and fall straight intogeneralized (rather empirical) models for stiened plate design, such as Michael C. Niu in[10], others prefer a more analytical approach. Troitsky's analysis in [18] begins with a suitedchronological development of the equations, explaining important events that trigger theiradvance. In the last chapters about orthotropic integral stieners plate theory, he recognizesthe most used and practical set of equations from Cusens and Pama [11]. They provide acomprehensive collection of the most precise equations, with comments on any addition orcorrection to the models, assessing their applicability as well.The FEM Guideline in [3] from the COALESCE2 project provides specic approaches for

the design, from the element selection, to modeling specics, and software usage recommen-dations. Another useful source is the documentation les from MSC software, specically thepart 3 of the Reference manual: Finite element modeling [9].Finite Element (FE) analysis is a powerful engineering tool widely used in the modern world.

Clearly stated by G. Kress in [6], they provide a notorious advantage through the capabilityof nding approximations to true solutions that are not accessible to exact analytical solutionmethods.The foundation strength of all review models is considered as a selection criteria for including

them in the numerical programs.

4

2.2 Analytical models

2.2.1 Isotropic plate

For the considerations of this thesis, a plate is a thin-layered isotropic linear element, whosethickness is relatively small compared to its width and length. This element is loaded with lin-early distributed loads (Qx, Qy), moments (Mx,My,Mxy) and a surface load in the directionof z. The isotropic plate follows the theory as introduced by E. Mazza in [8].The displacements are considered with the following approach, similar to beam theory:

uz = w(x, y) + f (x, y, z)

ux = −z · w′x (2.1)

uy = −z · w′y

Where w is the vertical displacement of the plate middle-plane. Strains are derived fromthe kinematic relations.

εxx = −z · w′xx εzz = f′z

εyy = −z · w′yy εxz = 12 · f′x (2.2)

εxy = −z · w′xy εyz = 12 · f′y

The function f is selected to comply with:

σzz = (2µ+ λ) · f′z + λ(−zw′xx − zw′yy) = 0 (2.3)

Where µ and λ are Lamé constants. Plate stiness is dened as:

D =Et3

12(1− ν2)(2.4)

After deriving the moment equations and making a balance of forces, the nal plate equationcomes to light:

D · (w′xxxx + 2w′xxyy + w′yyyy) = p(x, y) (2.5)

44w =p(x, y)

D(2.6)

In the case of a simply supported plate, the deection can be describe with a mathematicalapproach as a double sum of trigonometric functions, as presented in [17] and [8], like:

w(x, y) =

∞∑m

∞∑n

wmn · sinmπ

ax · sin

nπ

by (2.7)

5

The load is also a sum of terms in the following manner:

p(x, y) =∞∑m

∞∑n

pmn · sinmπ

ax · sin

nπ

by (2.8)

Where Pmn denes the force which loads the plate. An even pressure sets this coecientas:

pmn=16p0

π2mnfor m, n = 1, 3, 5... (2.9)

Equations (2.7), (2.8) and (2.9) are set into equation (2.6) and solved to obtain wmn.Now w(x, y) can be obtained, and if x and y are set to 1

2 , the maximum deection can befound. Further denitions for the load, and a solution example for a specic width/lengthratio can be found in [8].

2.2.2 Orthotropic plate

The orthotropic approach denes plate stiness to be dependent upon dierent variables in xor y direction. Dierent plate stiness coecients mainly aect the general plate equations(2.5) and (2.6). The bending rigidities are dened in [11] as Dxx and Dyy, the couplingrigidities by D1 and D2, and the torsional rigidities as Dxy and Dyx. The result leads to:

Dx∂4w

∂x4+ 2H

∂4w

∂x2∂y2+Dy

∂4w

∂y4= p(x, y) (2.10)

Where:

2H = (Dxy +Dyx +D1 +D2) (2.11)

Following Cusens' and Pama's model for slab stiened with solid ribs, the plate is consideredwith dierent heights and thicknesses on the stieners, and to some extent considers also theeect of the Poisson's ratio in the intersection areas of them. Since the value of 2H can onlybe noticeably high when the stiener's eccentricities ex and ey are large, a recommended andsimplied equation is also provided. The resultant collection of equations is:

Dxx = D +E∗tx

6bx

[hx −

(ex −

t

2

)]2

· (2hx + ex + t)−

(ex −

t

2

)2

(ex + t)

(2.12)

Dyy = D +E∗ty

6by

[hy −

(ey −

t

2

)]2

· (2hy + ey + t)−

(ey −

t

2

)2

(ey + t)

(2.13)

2H = Bxy +Byx + 2D (2.14)

6

Where Bxy, Byx and E∗ are dened as:

Bxy =

(Gkt3xh

bx

)(2.15)

Byx =

(Gkt3yh

by

)(2.16)

E∗ =E(

1− ν2txty

bxby

) (2.17)

2.2.3 Plate stability

For considerations on stiener stability, a relevant approach used by Mazza in [8], whichconsiders a plate simply supported in three sides, with one free side. The approach fordeection is:

w = y · sin

(mπx

a

)(2.18)

After denition of the potential energy and the displacement of the load introduction point,the critical distribute force is:

Q = N critx =

π2D

b2

(6(1− ν)

π2+m2λ−2

)λ =

a

b(2.19)

Where D is the plate stiness as expressed in (2.4), a is the length of the free side, and bthe remaining perpendicular sides. The variable m indicates the buckling forms, and shouldbe considered as m = 1 for conservative analysis.

2.3 Finite Element Analysis

The main steps of a structural FE analysis, from [6], can be summarized as follows:

1. Preprocessing

a) Importing of data

b) Construction of geometry

c) Binding of geometry

d) Element equations

e) System of equations

f) Boundary conditions

7

2. Main processing

a) Equation system solution

3. Postprocessing

a) Element displacements

b) Stress outputs

c) Strength criteria

d) Visualization

While the mentioned steps are fully valid for a structural analysis, the FE specic outputsmay dier from uid FE analyses, thermal analyses, and many others.The 2008r1 version of Patran software is mainly divided into the following modules:

Geometry Provides commands to work with geometry, considering this as a broad group ofall model data which are not nite elements.

Elements A preliminary step is required for any surface or solid: a mesh seed is createdto guide the construction of a mesh, which can also be governed by a mesh controlcommand. The user must create isolated mesh seeds for every surface. Once the meshis ready to be created, selection of element type, mesher, topology, target geometry listand properties are the next step. The equivalence command (targeted only to nodes)is a powerful and easy way to bind elements into a single FE model.

Loads/Boundary Here the user is able to create, modify, tabulate, plot or delete boundaryconditions in the model.

Materials In this module the user can input material properties. Materials may be isotropic,orthotropic, uids, among others. It is recommended to input only elastic modulusand poisson ratio on isotropic materials, as the software will calculate shear modulusautomatically.

Properties This module provides specic property input options for all created elements,and creates a list for each one indicating the application region. 2D element's thicknessand material can be changed here. 1D element's cross-section may be also denedhere. Orientation of the cross-section can be shown with the menu command: Display,Load/BC/Elem props; and setting Beam Display options to 3D. Further details areprovided in the Patran software documentation.

Load Cases All relevant Load cases are built here, considering dierent combinations of setspreviously established in the Loads/Boundary module.

Analysis In this module the main processing is executed. When the action option Analyze isselected, the desired load cases can be specied and turned into sub cases, which willbe used for the analysis. The Nastran solver will not run if the Conguration Utility isnot correctly modied to locate the executable le. The action option Access Results is

8

used to select the output le from the solver. For a new analysis, the user must deletethe existing job solution le with the corresponding command.

Results Here the results can be displayed by means of a deformed model, or by using afringe which maps the selected criteria using a colored scale. Fringe, deformed plot,and animation options are also available.

The specics regarding element formulation, systems of equations, shape functions and othersFE concepts are very problem-dependent. Further details are found on [6]. Modern FEprograms like MSC Patran provide a useful graphic user interface, skipping all detailed problemformulations.

2.4 Optimization

2.4.1 General notions

Optimization is a relatively modern discipline, nevertheless the nature of it dates way backfrom the beginning of engineering. Once a satisfactory solution for a determined problem isfound, the designer is driven by logical sense to minimize the consumption of resources to thelowest minimum possible, achieving the same solution at a higher eciency. This translatesinto lower manufacturing costs, increased production volumes, and ultimately larger protmargins for the involved parties.Optimization can be expressed as algorithms which examine a search space of variables

and analyze a given problem with them, nding a minimum of an established function. Thealgorithm formulation can be mathematical or evolutionary, as described by G. Kress in [5]:

Mathematical programming: These ...assume the optimization objective to be formulatedin terms of continuous and, often, at least twice dierentiable functions that shouldalso be convex.[5]. Mathematical programs usually take less numerical eort that theircounterparts, the evolutionary.

Evolutionary algorithms: Such types of programs are assisted by stochastic or heuristicmethodologies, succeeding on discrete search spaces where discontinuities cause math-ematical analyses to fail. This is the situation for many practical applications, wheresolution spectrum can only consider predetermined shapes or proles.

A useful approach at programming optimization problems from Eschenauer is also presentedand described in [5], and consists of the three following columns:

Structural model: The problem is translated into a structural formulation, which must besolved to obtain the desired output variables. These can then be evaluated to assessthe status of the current iteration. Usually an FE solution software will handle thesecalculations.

Optimization Algorithm: This column comprises the iteration control system, which im-ports an initial variable vector x0 and computes the required results. The algorithm

9

keeps analyzing the resulting improved vector xi in each iteration, creating an objectivefunction whose minimum is adjusted to reect the optimized solution.

Optimization model: The bridge between the results of the structural model in every iter-ation and the optimized solution is created by this column. Respecting all constraintswill be complicated without a systematic approach. At this point, constructing a GlobalObjective Function (GOF) will require transforming all constraints and structural resultsinto one continuous function.

2.4.2 Global objective function formulation

There are several aspects to be considered while formulating a GOF.

Local optimality: For structural models, failure is a local concern, and minimizing the peaksof the governing failure criteria is always the main task. This is expressed as:

minmaxσeqv(spatial position) (2.20)

The statement (2.20) may also be aected by constraints on the input design variablesof the structural model, in which case a function to continuously describe the behavioris required, namely the Global Objective Function, or GOF.

Constraints: These are classied into equality constraining functions hj(x) or inequalityconstraining functions gi(x), whose behavior must be:

gi(x) ≤ 0 i = 1, 2, ...,m

hj(x) = 0 j = 1, 2, ..., n

Where m and n are the amount of corresponding constraints. They dene a feasi-ble region where the function evaluates to the structural results, but when violated,a penalty term is included in the GOF, providing a higher function evaluation, andtherefore causing the optimization algorithm to avoid current x input values.

Pseudo objective function: Since the GOF is now actively linked with the constraints, it isalso named Pseudo function Ω. Interior or exterior point penalty methods are used tointroduce these constraints into the function. Using exterior point penalty methods,the Pseudo function converts to (2.21).

Ω(x,R) = Rm∑i=1

max[0, gi(x)]2 +Rn∑j=1

hj(x)2 (2.21)

Further useful denitions are found in [5].

10

3 Problem denition

3.1 Conventions

For the following study, these conventions are considered:

The radix symbol, namely the decimal mark, will be represented by a point. Example:93.45 Pa.

Three-digit grouping symbol will be represented by a comma. Example: 1,240.05 m.

The units along the document will be expressed in the International system of units(SI).

The Coordinate system for FE modeling is similar to the front spar system: X =facingport side, Y =facing tail, Z =facing upwards.

3.2 Project requirements

Even if the objectives for this thesis are focused on compliance of design criteria and massreduction, the overall cost reduction objective of the project is always kept in mind. COA-LESCE2 project documents provide a detailed denition of all requirements and considerationsfor this dimensioning study, further details can be found on [13]. Main aspects are describedin the following sections.

3.2.1 Operating conditions

Three main operating conditions are considered:

Cruise Conditions (CC): These are the basic operating conditions.

Limit Load (LL): These load conditions are dened as: LL = 2.5× CC

Ultimate Load (UL): Equivalent to UL = 1.5× LL = 3.75× CC

All design criteria are scaled and considered accordingly throughout the thesis.

3.2.2 Design factors

Factor of safety This factor corresponds to the category of the FLE component. Any Cate-

gory C structure is dened in [13] as a structure whose failure does not aect the ULcapability of the aircraft, i.e. structure may detach. Access panels are considered intothis category. These factors refer to CC, LL and UL, as previously dened.

11

Defects and damage Consideration is given to defects and damage should they occur duringmanufacturing and assembly processes. For stress analysis, a guideline of Rdd = 1.1 isused in the design to allow for likely damage and defects.

Damage tolerance and fatigue As a Category C structure, a reserve factor of 1 will be usedto account for in-service fatigue and damage tolerance, as specied in [13].

Variability The design considers variability of the material. For unqualied materials thedesign incorporates some conservatism, and for stress analysis in metallic structures thereserve factor is set as Rvar = 1.06.

3.2.3 Repair Philosophy

Easiness of repair is also a concern in all stages of the project. Integral stieners examinedin this thesis provide a low cost solution, but also have the advantage of fully visible surfaceswhich can be inspected without further complications. Further nal considerations of theproject will determine if the structure is repairable by means of a permanent repair; or if thedamage is outside allowable damage limits, then the damaged component is either replacedbefore next ight or a simple temporary repair is applied and the component is replaced atthe next scheduled maintenance check.

3.3 Access panel requirements

For the access panel, a set of requirements was established, all listed in [13].

3.3.1 Air tolerances

Deection limits on the FLE skin are also an important consideration. The aircraft's aero-dynamic variability on performance and reliability will be proportional to the variations onthe outside skin prole. Therefore, a ±δ tolerance on this variation is determined in therequirements document [13], as shown in the gure 3.1.These tolerances apply to cruise conditions.

3.3.2 Panel strength

Failure criteria is dependent of material. The considered aluminum alloy Al 7475T7351 hasthe following properties at room temperature:

E = 71,000 MPa

Poisson ratio ν = 0.33

Ultimate tensile strength: Ftu = 450 MPa

Ultimate bearing strength: Fbru =710 MPa

Ultimate shear strength: Fsu = 290 MPa

12

Figure 3.1: Air tolerances denition from [13]

Tensile yield strength: Fty = 365 MPa

Bearing yield strength: Fbry = 565 MPa

Compressive yield strength: Fcy = 415 MPa

3.3.3 Stieners' stability

Stability is considered individually for spanwise and chordwise stieners. The operating loadsare compared to the buckling loads: the stieners are not allowed to have a reserve factorlower than 1 for LL, as dened in [13]. The formula (2.19) is adjusted to obtain the criticalstress in the stiener:

σcrit =π2Et2

12b2(1− ν2)

(6(1− ν)

π2+b2

a2

)(3.1)

The free side dimension is considered as the top side of the stieners. All remaining sidesare analyzed as simply supported, allowing for some conservativity, since they are integralstieners completely xed to the panel. All terms of σcrit are dened in chapter 2.

3.3.4 Connections strength

Fasteners are considered as annealed AISI 302 stainless steel screws, with Young's modulusE = 190 GPa, ultimate strength Ftu = 655 MPa and tensile yield strength: Fty = 260 MPa.

13

Figure 3.2: Conventions for pressure from [12]

3.4 Loads

The panel is subjected to two main operating loads, described in the following subsections.

3.4.1 Aerodynamic pressure

Loads come from the ight envelope, as shown in the COALESCE2 baseline loads document[12]. Pressures are expressed in Pascals and dened as positive if they tend to explode theD-nose outwards and negative if the D-nose tends to collapse inwards, as shown in gure 3.2.The access panels are located in the η position between 0.667 and 0.811, which dene

in a unitary scale the distance from aircraft center coordinate frame to the wing tip. Moredetails on wing sections can be found on [12]. There are two access panels installed in thesepositions, however for the study of this semester thesis, only the access panel with the largestdierential pressure is selected:

Maximum average dierential pressure between η positions [0.667,0.714] is: 4Pmax =9, 490 Pa

Minimum average dierential pressure between η positions [0.667,0.714] is: 4Pmin =−22, 360 Pa .

4Pmax and 4Pmin are given as UL.

In any analysis, these loads are multiplied by Rdd and Rvar factors.

3.4.2 Wing bending

Wing bending must also be considered. The information regarding the displacements ofthe landings is obtained from the excel spreadsheet [1] for the COALESCE2 Project whichcontains details about the front spar deection. The lower area from this spar is geometricallylocated in the same zone as the access panels, making the deformation prole valid for itsuse.

14

Load Case Name Description

LC-1-0 Minimum PressureLC-2-0 Maximum PressureLC-0-1 Upward BendingLC-0-2 Downward BendingLC-1-1 Minimum Pressure and Upward BendingLC-1-2 Minimum Pressure and Downward BendingLC-2-1 Maximum Pressure and Upward BendingLC-2-2 Maximum Pressure and Downward Bending

Table 3.1: List of load cases

3.5 Load cases

Load Cases are constructed and identied with: LC-X-Y, where

X letter: Pressure denition, which can take values of 1 for minimum pressure or 2 formaximum pressure.

Y letter: Bending denition, which can be 1 for upward bending, or 2 for downward bending.

A value of 0 indicates no load in the corresponding letter. The table (3.1) shows a shortdescription for each load case. This information is placed into the load cases module onPatran.

15

4 Analytical modeling

For describing the deection of the access panel, and following the work structure of this thesispresented in 1.1, the rst approach was an analytical model. There were two initial approachesuntil a nal adaptive model was obtained, which considers isotropic and orthotropic platebehavior. The reason for this analytical phase was to nd a good estimate for all dimensionsby means of an initial parametric study. In a simplied and conservative approach for theanalytical modeling, the panel is considered as simply supported.

4.1 Initial approaches

4.1.1 Beam model

The problem was initially addressed as a beam bending simplication, where only a sectionwith one stiener was considered. The model considers a variability of the second moment ofinertia, namely the cross section, along the length of the equivalent simply supported beambeing deected. The dierent cross-sections and their position in the access panel are shownin gure 4.1. The height of the cross-section, where there is an angle in the extremity of thestiener, also changes with every step along the analysis. The iteration begins from the leftof gure 4.1 and ends on its right side. An equivalent beam is constructed considering onlyone stiener, with stiener height and stiener pitch as relevant dimensions. As the iterationposition shifts through the beam, cross-section dimensions are changed.This model obeys the equation of elastic curve for a simply supported beam, as explained

by P. Beer in [4] and shown in equation (4.1).

d2y

dx2=M(x)

EI(4.1)

Figure 4.1: Cross-sections considered in the iterative beam model.

16

Figure 4.2: Prole of deection, beam model with variable cross-section.

As the cross-section changes from the thickened outer plate, to the skin plate and thestiener dimensions, the moment of inertia is adjusted accordingly. This is done using theParallel-axis theorem, shown in (4.2).

Ix =

ˆ

A

y2dA = Ix′ +Ad2 (4.2)

The equations were programmed with iterations on MATLAB. The resultant prole fromthe beam deection can be seen in gure 4.2.Deections obtained with this model obtained a realistic accuracy. Nevertheless, such a

simplied model is useful mainly as a reference for later examinations, as it carried severalassumptions that diered from reality, such as:

Simply supported spanwise edges

Unsupported chordwise edges

Ability to consider only chordwise stieners

4.1.2 Isolated orthotropic model

An orthotropic plate model was sought as an approach to obtain more realistic solutions.Models and derivation process from [8] proved themselves quite useful at this stage. Afterliterature research, consideration of stieners in a plate was found to be wide subject, treatedby many authors. A useful understanding on the matter was provided by [18], and deeperdetails were presented in [11].The equations for orthotropic balance (2.10) and torsional coupling (2.11) are set into

the general plate equation (2.6) to generate a new equation that describes an orthotropicbehavior.Moment denition gives:

17

Mxx = D11

(− ∂w0

∂x2

)+D11ν21

(− ∂w0

∂y2

)

Myy = D22

(− ∂w0

∂y2

)+D22ν21

(− ∂w0

∂x2

)

Mxy = D66

(− ∂w0

∂x∂y

)After the balance, the pressure is found to be dened as:

P = D11∂4w0

∂x4+D22

∂4w

∂y4+ ν21(D11 +D22)

∂4w

∂x2∂y2+ 4D66

∂4w

∂x2∂y2(4.3)

At this point, equations from [8] have an identical plate stiness D coecient, leading toEq. (2.5). Deection follows the mathematical approach of fourier series from Eq. (2.7), butadjusted to the orthotropic terms. The nal transformation of the general isotropic plate Eq.(2.6) to orthotropic yields:

0 =∞∑m

∞∑n

wmn

[D11

(mπ

a

)4

+ (ν21(D11 +D22) + 4D66) (4.4)

·

(mπ

a

)2(nπ

b

)2

+D22

(nπ

a

)4 ]− Pmn

· sin

mπ

ax · sin

nπ

by

An abbreviation is introduced:

2H = ν21(D11 +D22) + 4D66 (4.5)

And the unknown wmn, part of the double sum approach (2.7), is now equal to:

wmn =PmnD11

(mπ

a

)4

+ 2H ·

(mπ

a

)2(nπ

b

)2

+D22

(nπ

a

)4 (4.6)

Pmn is selected depending on the type of pressure. For the case of this study, a uniformpressure over the plate is dened as:

Pmn =16p0

π2mnfor m, n = 1, 3, 5, 7...

To nd the maximum deection at the middle point, x and y positions are set, and a newvariable is introduced:

18

Figure 4.3: Adaptive orthotropic plate model regions.

x =a

2, y =

b

2, ψ =

b

a

The maximum deection is then found as:

wmax =16p0

π6a4 ·

k∑m

k∑n

ψ4

mn · (D11m4ψ4 + 2H ·m2n2ψ2 +D22n4)·sin

(mπ

2

)·sin

(nπ

2

)(4.7)

This equation is programmed on MATLAB to obtain values. In this program, for compu-tational time reasons, ∞ from the sum in Eq. (4.4) is only iterated up to a k value. Bestapproximations are achieved with value ranges of k = [150, 200]. Large values only result inexcessive computation time without substantial gain on accuracy.

4.2 Adaptive orthotropic model

A full orthotropic model was constructed which considered specic areas of the panel, applyingplate equations to isotropic and orthotropic regions correspondingly. The access panel wassplit into these regions for the model, as depicted in gure 4.3.The iterative numerical models built before had worked suciently good, therefore the

same approach was considered. The new model is able to:

Distinguish from isotropic and orthotropic areas automatically. (Feasible and unfeasibleareas for each set of equations)

Build separate stiness constants for each section.

Scale the equation boundaries to each area size.

19

Be always actively analyzing the Fourier series, but neglecting plate deformations in theunfeasible areas.

Create a nal deformation plot for examination and sensitivity analysis.

The model was adjusted according to the requirements, and programmed in MATLAB. Foraccurately describing the deection of the access panel of the FLE, the analytical model wasdened as:

w(x, y) =

Ω∑i

wi

xi, yi ∈ Ω : wi = wmn

xi, yi /∈ Ω : wi = wxp(4.8)

Where Ω refers to the dened number of access panel regions, namely the isotropic outerand inner skin, and orthotropic stiened area. Deection wmn comes from the Fourier serieswhen the xi, yi coordinates correspond to the desired area. Otherwise, wxp is calculated asan extrapolation from previous deections, neglecting the plate formula from feasible areasand describing a prole that describes no contribution to deection on the unfeasible areas.The deection alternatives are described as:

wxp = w(xi−1, yi) + [w(xi, yi−1)− w(xi−1, yi−1)] (4.9)

wmn =

∞∑m

∞∑n

Pmn · a4i

ψ4

mn · (Dxxim4ψ4 + 2Hi ·m2n2ψ2 +Dyyin4)(4.10)

·sin

(mπ

ai(xi − uxi)

)· sin

(nπ

ψiai(yi − uyi)

)for m, n = 1, 3, 5...

Chordwise plate length is ai, and Pmn is taken as previously dened. All orthotropic plateconstants are obtained from [11], and are adjusted dierently in every iteration when the arearegions require them to be so. They are dened as:

Dxxi =

D +E ∗ tx

6bx

[hx −

(ex −

t

2

)]2

· (2hx + ex + t)−

(ex −

t

2

)2

(ex + t)

i

(4.11)

Dyyi =

D +E ∗ ty

6by

[hy −

(ey −

t

2

)]2

· (2hy + ey + t)−

(ey −

t

2

)2

(ey + t)

i

(4.12)

2Hi = Bxyi +Byxi + 2Di (4.13)

20

Figure 4.4: Prole of deection, adaptive orthotropic plate model.

Di =Et3i

12(1− ν2)(4.14)

Other required variables like Bxy, Byx, E∗are also adjusted accordingly to each iteration.Chosen variables for the model are:

ψi =bi

ai(4.15)

uxi = x1(Ωi)− x1(Ω1) (4.16)

uyi = y1(Ωi)− y1(Ω1) (4.17)

The introduced ψ factor helps simplify the equation, and the uyi and uxi factors shift theinitial boundaries for the plate equations. Figure 4.4 shows a deformation prole created by themodel. Notice the dierent change of deformations from coordinates to others, demonstratingisotropic and orthotropic (stiened) deection.

4.3 Sensitivity Analysis

Once the model was complete, it was tested with dierent congurations to investigate theinuence of several variables. Some of them proved to be relevant, as shown in 4.5. Whilethe stieners have indeed an inuence on deection, the most important role is taken by theskin thickness. Nevertheless, it cannot be changed easily, because its eect on mass is high.All variables have a linear impact on mass. The height of the chordwise stieners has more

21

Figure 4.5: Sensitivity analysis graphs

inuence than the amount of stieners as well. Spanwise stieners seem to be unnecessary,but later approaches will reveal their importance. At this stage of the work, there is a clearimpression of trade-os between variables, where parametric studies or optimizations are thebest option.

22

5 Finite element modeling

For the results regarding this project, MSC's Patran was the software of choice, in which agood estimation obtained from the sensitivity analysis was modeled. This is the followingstage in the semester thesis according to the work structure in gure 1.1.

5.1 CAD surface model

The CAD geometry required by COALESCE2 has a complex curvature along the plate skin,and even if it can barely be seen at rst sight, the model should take it into account. Thereforethe nal results must be based in the skin plate as extracted from the aerodynamic prolesdetermined by industry partners.An initial CATIA solid model was provided from start, out of which this skin curvature

features could be extracted. The model required adjustments, as the integral stieners weremodeled from scratch.As import le for Patran, a surface geometry was required. Surfaces were then extracted

from the solid model for this purpose, including panel skin and stieners. The nal model isshown in gure 5.1.Later attempts to generate a mesh revealed that element distortion was proportional to

individual surface complexity. The surface model had to be adjusted a few times, until allshapes were mostly reduced to single rectangular splits. The mesh seed could then be easilyplaced on the sides of the squares, and the mesh is created uniformly.

5.2 Modeling considerations

The access panel is split into dierent areas:

Edge skin thickness

Figure 5.1: Surface model extracted from solid geometry

23

Inner skin thickness

Chordwise stieners

Spanwise stieners

Equivalent middle rib contact zones

Each area was congured with its corresponding skin thickness.

5.2.1 Fasteners modeling

Fasteners were modeled according to a suggestion from M. Brethouwer in [2]. The goal of hisapproach was to accurately obtain the deection of the panel during the analyzed conditions.In previous works [7], the fasteners have been modeled as MPC, which indeed served totransfer displacements and rotations, however it doesn't provide any information about thefasteners. Another eect had to be considered, namely the prying eect, which causes stresson the edges of the panel. The latter is addressed in the next subsection.To model fasteners, bush and beam elements were set into the model. Bush elements

represent the fasteners, and they can only be loaded in compression. They work with a springconstant, which was determined from:

K =F

uz(5.1)

uz =

ˆ

L

εz =

ˆ

L

F

AE(5.2)

The fasteners are considered as annealed AISI 302 stainless steel, with E = 190 GPa,length of 1.8mm (half of skin thickness in fastener area), and diameter Ø = 6.35mm. Theresult was K = 3, 342, 861.84N/mm. The beams were set with a square cross-section of sidelength l = 3mm, merely to provide enough support to consider the prying eect.Figure 5.2a depicts the location of fasteners in the model, while gure 5.2b shows the

beam and bush elements in vertical position, connecting upper and lower elements, whichcorrespond to the landings and access panel.

5.2.2 Prying-eect considerations

As previously said, boundary conditions on the access panel are fully modeled with fastenersand prying-contact zones, as suggested in [2]. When the bush elements are subject to tension,the beam elements absorb the stress, transferring it to the panel. Nonetheless, partners ofthis project have regarded these criteria as non-critical, due to the redistribution of loads allover the contact zones between the panel and the landings.Figure 5.3 from [14] shows an example of prying eect.

24

(a) Location of fasteners (b) Zoomed view tobush elements

Figure 5.2: Beam and bush elements

Figure 5.3: Illustration of prying eect, as taken from [14].

25

Figure 5.4: Position of middle ribs above access panel, halfway along the spanwise length.

5.2.3 Rotational constraints on rib zones

The area corresponds to contact zones with equivalent properties set to simulate the con-straints imposed by the middle ribs. These limit the rotations of the contacting surfaces, asthe ribs are considered to be notoriously stable compared to the bare panel skin. The middleribs can be seen as reference in gure 5.4. Because of their stiened construction, they areconsidered to constrain rotations of Y and X axis on the contacting elements.

5.2.4 Element selection

To describe the behavior of the access panel, Quad4 shell elements were set all over the model.Prior to the beginning of this study, a conclusion was made between academic and industrialpartners to employ shell elements instead of oset beams for modeling the stieners. Thisprocedure was decided due to the benets of higher accuracy from the results that could beobtained with this element, and also because stieners were no longer a secondary prioritybut the main focus of the work. A better mapping of stress on the stiener was also anadvantage.On the extremities of the stieners, a triangle section was cut at a determinate angle to

eliminate excessive material, which in turn resulted in triangular surfaces. As mentioned insection 4.6 of [6], standard 3-node triangular shell elements are unable to map bending stressand deformations accurately by nearly half. Since the introduction of bending from the plateinto the stieners takes place in these elements, this issue could not be left out. Therefore theselection on these areas was changed to Tria6 elements, or middle-node triangular elements,which have increased precision for bending applications. Figure 5.5 shows one side of stienerswith tria6 elements in dierent color.

5.2.5 Support landings

An area above the outer skin was modeled to simulate the landings, or surfaces where theaccess panel is connected. In this areas, there are connections to the fasteners and beam

26

Figure 5.5: Tria6 mid-node elements shown in the triangular extremity of the stieners

elements for prying-eect. This area is given a thickness of 8mm, and its edges are constrainedin rotations and displacements. This way the landings hold their place when the loadcasedoes not include wing bending. Corresponding displacements can be introduced if the loadcase also includes wing bending.

5.2.6 Wing bending

Introducing the wing bending into the access panel was another important condition. Thiscould be done in a simple manner by setting displacement on the edges of the access panel,but a more realistic approach was suggested and chosen, using important input from a wing-bending calculation le provided by the project partners [1].The purpose of this le was characterizing the deection of the front spar regarding the

wing bending, and the results could also be extracted since the access panel rests on the loweredge of the front spar. The le also accounts for osets in Z direction.Displacements are dependent upon the mentioned oset. A measure from the spar coor-

dinate system to the panel surface equals to 41 mm. Category C structures like the accesspanel are not UL critical, therefore a LL should be sustained. Oset is then selected as:

Zoffset = −41mm · LL = −102.5mm

The oset is taken as −100 mm. This data is typed into the excel spreadsheet, whichcalculates the 4x and 4z for the selected node. A row of nodes was selected every 100mm,as shown in table 5.1a. The resulting displacements per row are shown in 5.1b.

5.3 Comparison to analytical model

The load cases previously dened were programmed and introduced in the solver. For thisstudy, MSC Software's Nastran was used during all runs. There were no complications usingthe software, other than correctly setting the conguration utility from Patran.

27

X position Y position Z position0.0 0.00 -100.00100.00 0.00 -100.00200.00 0.00 -100.00300.00 0.00 -100.00400.00 0.00 -100.00500.00 0.00 -100.00600.00 0.00 -100.00700.00 0.00 -100.00800.00 0.00 -100.00900.00 0.00 -100.001000.00 0.00 -100.001100.00 0.00 -100.001200.00 0.00 -100.001300.00 0.00 -100.001400.00 0.00 -100.00

(a) Position of nodes for bending calculation

4X Step 4Z0.0 0 0.0-0.15567 1 -0.0178-0.31135 2 -0.07122-0.46705 3 -0.16024-0.62279 4 -0.28487-0.77859 5 -0.44512-0.93445 6 -0.64097-1.09038 7 -0.87243-1.2464 8 -1.1395-1.40253 9 -1.44218-1.55876 10 -1.78046-1.71513 11 -2.15436-1.87164 12 -2.56387-2.02829 13 -3.00898-2.18512 14 -3.4897

(b) displacement deltas on specicnodes due to wing bending

Table 5.1: Wing bending information extracted from [1].

Results were obtained for tested load cases, examined on both stress and displacement re-quirements. An initial result indicated compliance with requirements, however an optimizationwas still part of the approach, and from those results a nal adjustment is made.The local eects resulting from excessively thick stieners and large areas between them

is depicted in gure (5.6). The upper cut-section shows how the deection forms a waveand the areas aected will no longer fully obey the analytical model. The areas close to thestieners will follow the predicted deection prole by the orthotropic model, as shown inthe lower cut-section. This demonstrates the utility of the FE analysis to validate analyticalresults.Figure 5.7 illustrates a deformation plot obtained when analyzing loadcase LC-0-1, already

with an improved conguration. A comparison of accuracy of the analytical model with theFE model is shown in table 5.2. The columns analytical and Nastran show the totaldeection corresponding to their test congurations. The Variance 1 shows the increaseof each conguration with respect to conguration a. The Variance 2 shows the variationof deection values between analytical model and Nastran, with regard to each single testcongurations, from a to g. These error values exist due to the local eects in the plate.

28

Figure 5.6: Local deection eects from inner skin of access panel.

Figure 5.7: Stress distribution on LC-0-1.

Test congurations Max deection [m] Variance 1 Variance 2

Chord-S Span-S Inner Skin Edge skin Analytical Nastran Analyt FE Analyt:FE

a 2 2 2 4 5.88E-04 4.89E-04 16%b 4 4 2 4 5.88E-04 4.18E-04 0.00% -0.01% 28%c 6 6 2 4 5.88E-04 3.91E-04 0.00% -6.42% 33%d 2 2 4 4 5.88E-04 1.80E-04 0.00% -63% 69%e 2 2 6 4 5.88E-04 1.19E-04 0.00% -34% 79%f 2 2 2 6 1.74E-04 4.21E-04 -70% -14% -141%g 2 2 2 8 7.35E-05 3.55E-04 -58% -16% -383%

Table 5.2: Comparison of error magnitudes of analytical model vs. FE model.

29

6 Optimization

Once the rst stage of FE model and analysis was ready, it was time for design improvement.As stated in the agreement, parametric studies were initially intended for this purpose. How-ever, the ability to proceed with optimization algorithms was obtained and a program waselaborated for this purpose.An approach with mathematical programming was chosen for simplicity and time-eciency.

Constraints were provided by manufacturing, safety and the sensitivity analysis. Threecolumns method was used, as described before in chapter 2. The model only considersthe most critical conditions for stiener placements, namely the maximum and minimumpressure. It is also simply supported on the edges, and therefore only considers load caseswithout wing bending (LC-1-0 and LC-2-0). More options for complexity can be added if theproject requires them, and a sucient timespan is determined to work on this.

6.1 Structural model

The problem was translated into a structural formulation using ANSYS software. The programwas written on the Mechanical APDL interface, where the whole structure is automaticallybuilt using the inputs from the MATLAB algorithm. The program is written with all loadsand design criteria scaled to UL, also including Rdd and Rvar factors. When the ANSYS codeis executed, the following operations take place:

1. Parameter and variables denition.

2. Material, element properties and real constants denition (Isotropic, aluminum alloy,thicknesses).

3. Cyclic keypoint construction according to number of chordwise and spanwise stieners(*DO cycles).

4. Cyclic areas construction according to amount and position of keypoints.

5. Binding of areas with dierent thicknesses according to region.

6. Meshing with shell181 elements, with controlled numeration for all stieners and meshrenement on triangular stiener extremities (Bending introduction).

7. Denition of boundary conditions, loads and constraints, as dened for LC-1-0.

8. Static solution, with storing of stress and deection data.

30

Figure 6.1: Real constants denition for ANSYS automatic model.

Vector variables Design parameters

stx_hi Chordwise stiener heightstx_th Chordwise stiener thicknesssty_hi Spanwise stiener heightsty_th Spanwise stiener thicknessSkt Skin thicknessSa Stiener extremity angle

Table 6.1: Design parameters in optimization vector.

9. Localized comparison of spanwise and chordwise stieners to analyze buckling, withdata storage.

10. Global instability solution, with buckload analysis and data storage.

11. Cycle repeat from steps to 7 to 10, changing the loadcase to LC-2-0.

12. Writing of results le.

Figure 6.1 depicts step 5 of the operations, showing a specic property for each area accordingto the region. The complete code can build automatically any panel geometry as required,but the number of chordwise stieners is restricted to Nsx = [1, 20] , and chordwise stienersare also restricted to Nsy = [0, 9].

6.2 Optimization Algorithm

The algorithm is controlled by MATLAB by a preset function called fminsearch, which ndsthe minimum of an unconstrained multivariable function using the derivative free Simplexmethod. The function starts with an initial estimate x0 and continues the search with xivalues as obtained from the simplex construction. The selected variables are shown in table6.1.Properties of chordwise and spanwise stieners (height and thickness), skin thickness, and

extremity angle of every stiener. The resulting variable vector was:

design parameters : x = ′stx_hi′,′ stx_th′,′ sty_hi′,′ sty_th′,′ Skt′,′ Sa′ (6.1)

31

The clear advantage of this MATLAB algorithm was the time eciency due to the otherwiserequired time for programming the simplex. The code was structured taking B. Schläpfer's[16] as a general guide. All relevant commands were adjusted to the thesis requirements,variables and constraints were added or deleted as necessary.

6.3 Optimization model

In order to create a continuous function including all constraints, the Pseudo-function ap-proach for a GOF was used, as described in [5]. This method integrates penalty terms forviolation of constraints. The GOF analyzes terms with regard to:

Geometry There were manufacturing limitations to the geometry that could be created dueto the stability of the cutting tools, therefore a minimum thickness of 1.8 mm wasset for the stieners. Inner skin is also subjected to lightning safety standards, and aminimum thickness of 2.0mm is necessary. The sensitivity analysis from the analyticalmodel also provided useful information: additional constraints helped to reduce thesearch space of the variables for the GOF. A good initial vector was also obtained fromthe analytical approach. Any violation of these constraints activates a penalty term.

Design The access panel must withstand any buckling in either direction of the stieners,as well as global instability. Utilization factors were established, which must not exceeda unitary value. Also the displacement was subject to a tolerance, namely the ±δ airtolerance described before. The stress distribution in the panel should also remain underthe UL admissible stress.

Mass The term regarding mass of the panel is always actively analyzed and minimized.

6.3.1 Constraints

A detailed table of constraints for the GOF is shown in table 6.2, where chordwise stienersare referred as (x), and spanwise stieners as (y). If none of the constraints are violated, thesearch for minimum of mass is the only active term. Mass and design terms were normalized,and geometry terms scaled individually. ANSYS code transforms the scaled geometric termsinto International System units. Since the original air tolerance for the FLE is considered fromCC, they have to be adjusted in the optimization, which considers UL for both load cases.

±δnew = (±δ@CC)(LL)(UL) = (0.8mm)(2.5)(1.5) = 3.0mm

The nal GOF is introduced as:

Ω(M,di, gj , R,Rj) = R(M/Mexpected) +i∑1

Rmax[0, di]2 +

j∑1

Rjmax[0, gj ]2 (6.2)

Where di are the design constraints, gj are the geometrical constraints, and Rj the penaltyfactors. The expected mass was set to 3kg, value that has no relevant inuence on the result,

32

Constraint GOF term formulation Term type

minmass R(M/Mexpected) Massdisplacement ≤ 3mm Rmax[0, (D −Dadmissible)]2 Designyield utilization ≤ 1.0 Rmax[0, ((kY/kYadmissible)− 1)]2 DesignInstability utilization ≤ 1.0 Rmax[0, ((kI/kIadmissible)− 1)]2 DesignBuckling(x) utilization ≤ 1.0 Rmax[0, ((kBx/kBxadmissible)− 1)]2 DesignBuckling(y) utilization ≤ 1.0 Rmax[0, ((kBy/kByadmissible)− 1)]2 Designstiffener angle ≥ π/4 g1=0.785-Sa; R1max[0, g1]2 Geometrystiffener angle ≤ π/2 g2=Sa-1.57; R2max[0, g2]2 GeometryHeight(x) ≥ height(y) g3=sty_hi-stx_hi; R3max[0, g3]2 GeometryHeight(x) ≥ 10mm g4=1-stx_hi; R4max[0, g4]2 GeometryHeight(x) ≤ 60mm g5=stx_hi-6; R5max[0, g5]2 GeometryHeight(y) ≥ 0mm g6=-sty_hi; R6max[0, g6]2 GeometryHeight(y) ≤ 30mm g7=sty_hi-3; R7max[0, g7]2 GeometryThickness(x) ≥ 1.8mm g8=1.8-stx_th; R8max[0, g8]2 GeometryThickness(y) ≥ 1.8mm g9=1.8-sty_th; R9max[0, g9]2 GeometrySkin Thickness ≥ 2.0mm g10=2-Skt; R10max[0, g10]2 Geometry

Table 6.2: Constraints for the Global objective function

but had to correspond on magnitude to a real estimation. More details can be found amongthe MATLAB code created for this thesis.

6.3.2 Algorithm execution

Figure 6.2 illustrates the behavior of the last execution of the optimization program runningwith pseudo-function (6.2). The top plot from shows the GOF diminishing as the simplexalgorithm from fminsearch evaluates and nds the unconstrained minimum; the lower plotshows the evolution of the design constraints: a value above 1.0 equals a violation and there-fore a penalty in the GOF. The understanding of behavior and tendencies of the optimizationalgorithm is improved in every run.After the rst dozen of runs, it was evident that the algorithm was very sensitive to local

minima. The simplex then rarely changes the stiener quantities: Nsx (chordwise stieners)and Nsy(spanwise stieners), which were originally part of the vector variables in table 6.1.If it did change, since the current thickness and height setups were optimal for the startingNsx,Nsy conguration, a higher GOF value is found just across the border of the newNsx,Nsy conguration.A selection of congurations was made, and several local minima were investigated. Since

the initial estimate was already compliant to requirements, a factor was include to cause anaggressive behavior of the algorithm. This factor causes the optimization model to marginallymeet already violated design constraints in order to nd the best conguration. It alsoaccounts for a small safety margin for initial boundary slopes that come from accuratelymodeling the fasteners, which is important because the ANSYS model considers a simply

33

Figure 6.2: Iteration graph from GOF in run 27.

supported plate. Since all runs have the same conditions, the chosen optimum is taken fromthese results collection, and a nal adjustment of the Patran FE model is done afterward withthe selected conguration. The design terms variate freely over their feasible areas in orderto accommodate an optimal solution, nding the minimum mass for many possible stienercongurations, namely the amounts of chordwise and spanwise stieners. Execution timesuntil a minimum is found last between 6 and 8 hours.

34

7 Results

The work was initially addressed as a bending problem: directly following beam theory. Thepossibility of adding stringers was considered due to the benecial beam bending properties,even though it added complexity to the panel. The mass also increases due to the addedmaterial on the stringers, thus the number of stieners had to be limited.The obtained data from the sensitivity analysis clearly indicated that large stieners had a

clear eect on total mass. The trade-o between performance and mass had to be addressed,which was the later purpose of optimization.Figure 7.1 illustrates the results obtained from running the algorithm, investigating several

relevant local minima. The center of the pentagon represents a low mass solution, radiallyincreasing its value. Chordwise stiener congurations are represented with dierent lines,which coincide with certain spanwise stiener congurations on the pentagon's vertexes.The real problem is the local phenomena, where thin skin deformation is proportional to

area size between stieners. Increasing the close-to-skin stiness is the main concern, ratherthan looking simplistically into moment of areas. Ultimately, skin deection will be mostlyunaected by stringers or relatively tall stieners.The height and thickness ratios should be optimized for this purpose, as one variable is

strongly dependent from the other in terms of design constraints evaluation. They shouldnot be addressed individually, but rather with an optimization approach, or at the very least,with a complete multidimensional parametric study.From gure 7.1, the global minimum for the investigated dimensioning congurations is

selected:

Nsx = 8, Nsy = 1

The algorithms and models construct the panel with this conguration in each side of themiddle ribs, therefore the CATIA solid model has a total of 16 chordwise ribs and one singlespanwise rib, which is split in the middle due to the ribs. Figure 7.2 presents the nal CATIAsolid model.The FE model was adjusted according to the latest optimization run. Table 7.1 shows

a comparison of mass reduction. The initial model was provided at the beginning of thissemester thesis. The improved model is obtained from the sensitivity analysis. The optimizedmodel is the result of the optimization algorithm. The initial model is taken from the rstsuggestion for this study (which did not fulll design criteria), and it's then adapted tomarginally comply with requirements. The cost reduction is obtained from COALESCE2metallic versus composite le [15], where it can be seen that the integral stiener accesspanel takes around 63% less time to be manufactured, and since adding or removing stienersfrom the design only alters the path of the CNC machining, the inuence on the cost can be

35

Figure 7.1: Optimal mass search results

Figure 7.2: Final optimized CATIA model

36

Design type Initial design Improved design Optimized design Composite

Origin Airbus UK Sensitivity analysis Algorithm COALESCE2

Total mass 2.4518 2.3237 2.1137 1.475

Decrease of mass

from Initial design0% 5.2% 13.7% 39.8%

Increase of mass

from composite66.2% 57.5% 43.3% 0%

Cost reduction approx. 70% approx. 70% approx. 70% 0%

Table 7.1: Comparison of mass between models

neglected. It may even reduce it a little, since smaller stieners mean a thinner initial blockof aluminium for machining.Table 7.2 depicts the obtained reserve factors for the required design criteria. Using linear

static analysis, the results obtained from the rst four load are summed to be valid for thecombinations of the remaining load cases. Requirements are given with regard to dierentoperating conditions in the documents, and even if they were adjusted accordingly duringthe analysis, they are presented in the same operating conditions as they were given inthe documentation, namely CC, LL or UL. This way they can be directly compared to thedocument requirements without further factor transformations. Factors from defects, damageand variability are always considered. Buckling and Instability considerations are extractedfrom ANSYS. Reserve factors for stress are given in LL and UL because the ratio of governingstress criteria on them is not the same as the factor, therefore they cannot be scaled. Therelevant reserve factors are related to:

Rdisp: Maximum air tolerance

Rslope: Maximum mean slope (waviness)

Rstress: Maximum stress criteria

Rfast: Maximum fasteners strength

Rbuck−x: Buckling load of chordwise stieners

Rbuck−y: Buckling load of spanwise stieners

Factors indicated as N/R are not relevant to buckling considerations because on the givenload case, the stiener is under tension. The most critical reseve factor shown is Rstress =0.99 at UL in LC-1-1, but is not considered critical due to the fact that only one node reachesthe stress level that contributes to that factor, as shown in gure 7.3. Moreover, this nodeis located exactly at the base of a spanwise stiener, where the manufacturing radius cannotbe considered in the FE model. Therefore, in reality it would have lower stress values in thezone.Figure 7.4 illustrates how the stress in the stieners is isolated and extracted for buckling

analysis. Figure 7.5 shows a plot with the largest deformation from all four basic load cases,namely LC-1-0.

37

Load case Rdisp Rslope Rstress Rfast Rbuck−x Rbuck−y

Operatingconditions

CC CC LL UL LL UL LL LL

LC-1-0 3.36 1.86 4.38 3.60 603.72 1013.93 N/R N/RLC-2-0 8.80 4.88 11.45 9.41 1585.37 2662.60 29.94 549.35LC-0-1 3.33 1.85 1.65 1.36 398.77 669.73 6.24 25.59LC-0-2 7.69 4.27 3.34 2.74 955.88 1605.39 15.09 61.86LC-1-1 2.15 1.19 1.20 0.99 240.15 403.33 13.80 29.14LC-1-2 2.72 1.51 1.89 1.56 370.02 621.44 N/R 87.68LC-2-1 3.57 1.98 1.45 1.19 318.63 535.13 5.17 24.45LC-2-2 5.44 3.02 2.58 2.12 596.33 1001.53 10.03 55.60

Table 7.2: Reserve factors for the nal access panel dimensions

Figure 7.3: Region where one node causes a reserve factor Rstress = 0.99 for LC-1-1.

Figure 7.4: Extraction of maximum stiener stresses for buckling considerations.

38

Figure 7.5: LC-1-0, the largest deection contribution to all load cases.

39

8 Conclusions

After analyzing the results, several concluding statements arisen:

Increased amount of Chordwise stieners Chordwise stieners do not act as beam ele-ments in the panel, but rather as plate stiening components, providing additionalorthotropic properties to the access panel.

Reduce height/thickness ratio The material function is mostly used by limiting thin innerskin deformations, therefore an increased height is unnecessary. Thickness of the sti-eners shall also be kept to a minimum, but allowing stable operation risk-free frombuckling.

Inclusion of spanwise stieners If the orthotropic properties are improved, relevant defor-mations will still arise from local phenomena. The small rectangular sections betweenthe stieners will act as single thin plate regions, increasing total deection. By includ-ing spanwise stieners, these area sizes are easily reduced, and so is their inuence.

Find a balance: Orthotropic & Isotropic The best design will usually lay between completeisotropy and orthotropy. A thick unstiened plate, as isotropic solution, will still havelarge deections. On the contrary, large and thick stieners will provide good orthotropy,but their mass will also be large. If the number of stieners are reduced for lowermass, large local phenomena will arise, yet again contributing to large deections. Theoptimum often rests on a balance between both.

Meeting of requirements The achieved solution fullls all design requirement in the mostecient way.

Optimummass solution The mass is optimized within an acceptable range. The reservefactors indicate ecient use of the material for limiting deformations and stress.

Cost ecient The resources saved from the manufacturing costs could be relocated to re-duce weight in more critical parts of the aircraft. Nonetheless, integral stiened panelsare a cost and mass ecient solution for this dimensioning study.

40

9 Outlook

The work done with this semester thesis opens possibilities for future engagements:

Advantage of optimization algorithm the optimization algorithm is ready to be used withany geometrical model as desired. It is matter of modifying the FE program code andchanging variables as required. Other components may be also analyzed, taking fulladvantage from the work done.

Bird strike requirement While the consideration of these requirements was not inside thescope of the thesis, safety requirements are important for the FLE. The approachwould be dierent as well, with possibilities for kinematic and non- impact analysis.Optimization opportunities may also be found on this topic.

Adapt to nal FLE design The access panel was dimensioned according to the latest CADspecications. Should the FLE models be modied, the must be take into considera-tions. Drive shafts and stiening nerves in ribs may require minor adjustments to thenal access panel dimensions.

Advantage of analytical model The generated model is applicable to any stiened or un-stiened plate. It could be applied to similar deection problems, using the latestapproach models available.

Expansion of analytical model Local eects and wing bending displacements are not eval-uated in the analytical model. The applicability of the model would benet frommodifying it to consider these aspects.

41

Appendix A

MATLAB relevant code lines for the adaptive orthotropic plate model.

1 % BEGIN OF PROGRAM − SEMESTER THESIS 11−0192 % PRELIMINARY DATA & VARIABLES3

4 % Material Input Properties:5

6 v= 0.33;7 E= 71e9; % Pa8 G= 26.9e9; % Pa9 rho= 2650; % density, [kg/m^3], KO824210 Rp02= 300e6; % Pa, KO824211 Rm= 390e6; % Pa, KO824212

13 % Stiffener Plate Configuration (raw variables, input data here)14

15 sh= 0.020; % Stiffener height (x, Chordwise stiffener)16 st= 0.0018; % Stiffener thickness (x, Chordwise stiffener)17 shy= 0.010; % Stiffener height (y, Spanwise stiffener)18 sty= 0.0018; % Stiffener thickness (y, Spanwise stiffener)19 Ots=0.10403; % Offset to Starboard20 Otp=0.12319; % Offset to Port21 Otr=0.11889; % Offset to ribs22 Ote= 0.000; % Offset to thickened edges23 Dtm= 0.69; % Distance to middle plane of ribs [m]24 Drs= 0.142; % Distance of Rib minimum spacing25 Gh =sym('Gh'); % Girder height26 Gl =sym('Gl'); % Girder lenght27 Skt= 0.002; % Skin thickness [m]28 Ste= 0.008; % Skin thickness on edges29 Etw= 0.0274; % Expanded thickness width30 Fo =0.016875; % Fasteners −centerline to perimeter− offset [m]31 Sa =(pi()/4); % Stiffener borders angle [rad]32 CL =0.204886; % Chordwise Stiffened Panel Lenght33 SL =1.384; % Spanwise Stiffened Panel Lenght34 L =0.1776; % Effective pin−ended plate lenght35 Nsp= 7; % Number of stiffeners in Port side (x) Direction36 Nss= 7; % Number of stiffeners in Starboard side (x) Direction37 Nsy= 1; % Number of stiffeners in Spanwise (y) Direction38

39 % Cusens, Pama 1975; for solid ribs stiffened slab, all following lines40 Bxy= G*0.3*st^3*sh/sp;41 Byx= G*0.3*sty^3*shy/spy;42 H2= Bxy + Byx + E*(Skt^3/(6*(1−v^2))); % Cusens, Pama43 Estar= E/(1− (v^2*st^2*sty^2/sp/spy));44 Eprime= E/(1−v^2);45 ex= (st*sh*(sh+Skt))/(2*Eprime/Estar*Skt*sp + st*sh); % excentricity from plate neutral plane to stiffened ...

plate neutral plane46 Dxx= E*(Skt^3/(12*(1−v^2))) + Estar*st/6/sp*((sh−(ex−Skt/2))^2 *(2*sh+ex+Skt)−(ex−Skt/2)^2*(ex+Skt)); ...

%Cusens, Pama47 ey= (sty*shy*(shy+Skt))/(2*Eprime/Estar*Skt*spy + sty*shy); % excentricity from plate neutral plane to ...

stiffened plate neutral plane48 Dyy= E*(Skt^3/(12*(1−v^2))) + Estar*sty/6/spy*((shy−(ey−Skt/2))^2 *(2*shy+ey+Skt)−(ey−Skt/2)^2*(ey+Skt));49 % End of Cusens, Pama50

51 % GENERAL PLATE DEFORMATION, WHOLE STIFFENED PANEL (X,Y)52

53 psi=(SL−2*Ote−2*Etw)/(CL−2*Ote−2*Etw);54 jx=50; % segmentation in X direction, even number55 jy=338; % segmentation in Y direction, even number56 j3=39; % Set number of summations, odd number for even pressure distribution on Lagrange plate.57 a=(CL−2*Ote−2*Etw);58 y=zeros(jy+1,1);59 x=zeros(jx+1,1);60 whole=zeros(jx+1,jy+1);61 plate1=zeros(jx+1,jy+1);62 wxy=0;63 last=zeros(1,jy+1);64 maxlast=0;

42

65 nx=0;66 ny=0;67 wstep_x=zeros(jx+1,jy+1);68 wstep_y=zeros(jx+1,jy+1);69 whole_fixa=zeros(jx+1,jy+1);70 whole_fixb=zeros(jx+1,jy+1);71 whole_fixc=zeros(jx+1,jy+1);72 for nx=0:jx73 x(nx+1)=nx*CL/jx;74 for ny=0:jy75 y(ny+1)=ny*SL/jy;76

77 if ((0<=x(nx+1) && x(nx+1)<=Etw) || ((CL−Etw)<=x(nx+1) && x(nx+1)<=CL)) || ((0<=y(ny+1) && ...y(ny+1)<=Etw) || ((SL−Etw)<=y(ny+1) && y(ny+1)<=SL))

78 plate1(nx+1,ny+1)=1;79

80 % Fourier cycles for Etw plate section (outer thickened plate)81

82 H2_a= E*(Ste^3/(6*(1−v^2))); % Cusens, Pama83 D_a= E*(Ste^3/(12*(1−v^2)));84 psi_a=SL/CL;85 a_a=CL;86

87 fourier_a=0;88 for m=1:j389 if mod(m,2)~=0;90 for n=1:j391 if mod(n,2)~=0;92 % Sample from Mazza's S−Analysis93 % f=1/(m*n*(m^2+ (n^2)/4)^2)*sin(m*pi()/2)*sin(n*pi()/2);94 % fourier_a= fourier_a+f;95 f_a=(psi_a^4)/(m*n*(D_a*m^4*(psi_a^4)+ H2_a*(psi_a^2)*m^2*n^2+ D_a*n^4)) ...

*sin(m*pi()*(x(nx+1))/a_a) *sin(n*pi()*(y(ny+1))/(psi_a*a_a));96 fourier_a= fourier_a+f_a;97 end98 end99 end100 end101

102 % End of Fourier cycles for Etw plate section (outer thickened plate)103 whole_fixa(nx+1,ny+1)= −16*a_a^4*p0/(pi()^6)*fourier_a;104

105 if 0<x(nx+1) && 0<y(ny+1)106 ystep_0=whole_fixa(nx+1,ny);107 ystep_1=whole_fixa(nx+1,ny+1);108 xstep_0=whole_fixa(nx,ny+1);109 xstep_1=whole_fixa(nx+1,ny+1);110 end111 else112 whole_fixa(nx+1,ny+1)=whole_fixa(nx,ny+1) + (whole_fixa(nx+1,ny) − whole_fixa(nx,ny)); % The magic ...

is here113

114 if ((Etw<x(nx+1) && x(nx+1)<=(Etw+Ote)) || ((CL−Etw−Ote)<=x(nx+1) && x(nx+1)<(CL−Etw))) || ...((Etw<y(ny+1) && y(ny+1)<=(Etw+Ote)) || ((SL−Etw−Ote)<=y(ny+1) && y(ny+1)<(SL−Etw)))

115 plate1(nx+1,ny+1)=2;116

117 H2_b= E*(Skt^3/(6*(1−v^2))); % Cusens, Pama118 D_b= E*(Skt^3/(12*(1−v^2)));119 psi_b=(SL−2*Etw)/(CL−2*Etw);120 a_b=(CL−2*Etw);121

122 % Fourier cycles for Ote plate section (inner skin−only plate)123

124 fourier_b=0;125 for m=1:j3126 if mod(m,2)~=0;127 for n=1:j3128 if mod(n,2)~=0;129 % Sample from Mazza's S−Analysis130 % f=1/(m*n*(m^2+ (n^2)/4)^2)*sin(m*pi()/2)*sin(n*pi()/2);131 % fourier_b= fourier_b+f;132 f_b=(psi_b^4)/(m*n*(D_b*m^4*(psi_b^4)+ H2_b*(psi_b^2)*m^2*n^2+ D_b*n^4)) ...

*sin(m*pi()*(x(nx+1)−Etw)/(a_b)) *sin(n*pi()*(y(ny+1)−Etw)/(psi_b*a_b));133 fourier_b= fourier_b+f_b;134 end135 end136 end137 end138

139 % End of Fourier cycles for Ote plate section (inner skin−only plate)140 whole_fixb(nx+1,ny+1)= −16*a_b^4*p0/(pi()^6)*fourier_b;141

142 if 0<x(nx+1) && 0<y(ny+1)143 ystep_0=whole_fixb(nx+1,ny);

43

144 ystep_1=whole_fixb(nx+1,ny+1);145 xstep_0=whole_fixb(nx,ny+1);146 xstep_1=whole_fixb(nx+1,ny+1);147 end148 else149 whole_fixb(nx+1,ny+1)=whole_fixb(nx,ny+1) + (whole_fixb(nx+1,ny) − whole_fixb(nx,ny));150

151 if ((Etw+Ote)<x(nx+1) && x(nx+1)<(CL−Etw−Ote)) || ((Etw+Ote)<y(ny+1) && y(ny+1)<(SL−Etw+Ote))152 plate1(nx+1,ny+1)=3;153

154 fourier_c=0;155 for m=1:j3156 if mod(m,2)~=0;157 for n=1:j3158 if mod(n,2)~=0;159 % Sample from Mazza's S−Analysis160 % f_c=1/(m*n*(m^2+ (n^2)/4)^2)*sin(m*pi()/2)*sin(n*pi()/2);161 % fourier_c= fourier_c+f_c;162 % f_c=(psi^4)/(m*n*(Dxx*m^4*(psi^4)+ H2*(psi^2)*m^2*n^2+ Dyy*n^4)) ...

*sin(m*pi()*(x(nx+1)−Etw−Ote)/(CL−2*Etw−2*Ote)) *sin(n*pi()*(y(ny+1)−Etw−Ote)/(SL−2*Etw−2*Ote));163 f_c=(psi^4)/(m*n*(Dxx*m^4*(psi^4)+ H2*(psi^2)*m^2*n^2+ Dyy*n^4)) ...

*sin(m*pi()*(x(nx+1)−Etw−Ote)/a) *sin(n*pi()*(y(ny+1)−Etw−Ote)/(psi*a));164 fourier_c= fourier_c+f_c;165 end166 end167 end168 end169

170 whole_fixc(nx+1,ny+1)= −16*a^4*p0/(pi()^6)*fourier_c;171

172 if 0<x(nx+1) && 0<y(ny+1)173 ystep_0=whole_fixc(nx+1,ny);174 ystep_1=whole_fixc(nx+1,ny+1);175 xstep_0=whole_fixc(nx,ny+1);176 xstep_1=whole_fixc(nx+1,ny+1);177 end178 end179 end180 end181

182 whole(nx+1,ny+1)= whole_fixa(nx+1,ny+1)+whole_fixb(nx+1,ny+1)+whole_fixc(nx+1,ny+1);183

184 end185 end186

187 figure(8);188 subplot(1,1,1); surf(y,x,whole);189 axis([0 CL 0 SL])190 axis tight191 title('Deformation of Access panel as orthotropic and isotropic plate','fontsize',13,'fontweight','b')192 ylabel('Length of Panel in ';'Chordwise direction [m]','fontsize',13,'fontweight','b')193 xlabel('Length of Panel in ';'Spanwise direction [m]','fontsize',13,'fontweight','b')194 zlabel('Deformation profile [m]','fontsize',13,'fontweight','b')195

196 % Made by: Jesus I. Maldonado C. Diverse source citations for equations197 % can be found among on code.198 % March of 2011, ETHZ.

44

Appendix B

ANSYS relevant code lines used in the structural model of the optimization algorithm.

1 !!!!!!!!!!!!!!!!!!!!2 ! Keypoints3 !!!!!!!!!!!!!!!!!!!!4

5 k,1,0,0,06 k,2,CL,0,07 k,3,CL,SL,08 k,4,0,SL,09 k,5,Etw,Etw,010 k,6,CL−Etw,Etw,011 k,7,CL−Etw,SL−Etw,012 k,8,Etw,SL−Etw,013 k,9,0,SL/2−0.5*Drs,014 k,10,0,SL/2+0.5*Drs,015 k,11,CL,SL/2−0.5*Drs,016 k,12,CL,SL/2+0.5*Drs,017 k,13,Etw,SL/2−0.5*Drs,018 k,14,Etw,SL/2+0.5*Drs,019 k,15,CL−Etw,SL/2−0.5*Drs,020 k,16,CL−Etw,SL/2+0.5*Drs,021 !Ribs section22 k,17,0,SL/2−0.5*Drs+0.025,023 k,18,Etw,SL/2−0.5*Drs+0.025,024 k,19,CL−Etw,SL/2−0.5*Drs+0.025,025 k,20,CL,SL/2−0.5*Drs+0.025,026 k,21,0,SL/2+0.5*Drs−0.025,027 k,22,Etw,SL/2+0.5*Drs−0.025,028 k,23,CL−Etw,SL/2+0.5*Drs−0.025,029 k,24,CL,SL/2+0.5*Drs−0.025,030

31 !*If,Nsx,GT,15,THEN32 !Nsx=1533 !*ENDIF34

35

36 ! DO cycles for stiffeners construction keypoints37 *DO, k, 1, Nsx, 138

39 ! x stiffener positions40 k,k+100,Etw,Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),041 k,k+300,Etw,SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),042 k,k+200,CL−Etw,Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),043 k,k+400,CL−Etw,SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),044 k,k+500,0,Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),045 k,k+700,0,SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),046 k,k+600,CL,Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),047 k,k+800,CL,SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),048

49 ! Stiffener upper keypoints50 k,k+150,Etw+abl,Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),stx_hi51 k,k+350,Etw+abl,SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),stx_hi52 k,k+250,CL−Etw−abl,Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),stx_hi53 k,k+450,CL−Etw−abl,SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),stx_hi54

55 *IF,Nsy,GE,1,THEN56 *DO, j, 1, Nsy, 157

58 ! y stiffener positions59 k,j+900,Etw+j*(CL−2*Etw)/(Nsy+1),0,060 k,j+910,Etw+j*(CL−2*Etw)/(Nsy+1),Etw,061 k,j+920,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2−Drs/2,062 k,j+930,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2−Drs/2−ably,sty_hi63 k,j+940,Etw+j*(CL−2*Etw)/(Nsy+1),Etw+ably,sty_hi64 k,j+950,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2−Drs/2+0.025,065

66 k,j+1000,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2+Drs/2−0.025,067 k,j+1010,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2+Drs/2,0

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68 k,j+1020,Etw+j*(CL−2*Etw)/(Nsy+1),SL−Etw,069 k,j+1030,Etw+j*(CL−2*Etw)/(Nsy+1),SL−Etw−ably,sty_hi70 k,j+1040,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2+Drs/2+ably,sty_hi71 k,j+1050,Etw+j*(CL−2*Etw)/(Nsy+1),SL,072

73 ! stiffeners intersections74 k,2000+k*50+j,Etw+j*(CL−2*Etw)/(Nsy+1),Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),075 k,3000+k*50+j,Etw+j*(CL−2*Etw)/(Nsy+1),Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),sty_hi76 k,4000+k*50+j,Etw+j*(CL−2*Etw)/(Nsy+1),Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1),stx_hi77 k,5000+k*50+j,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),078 k,6000+k*50+j,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),sty_hi79 k,7000+k*50+j,Etw+j*(CL−2*Etw)/(Nsy+1),SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1),stx_hi80

81 *ENDDO82

83 *END IF84

85 *ENDDO86

87 !!!!!!!!!!!!!!!!!!!!88 ! Areas89 !!!!!!!!!!!!!!!!!!!!90

91 !Constant areas92 a,11,20,19,1593 a,24,12,16,2394 a,13,18,17,995 a,22,14,10,2196 a,20,24,23,1997 a,18,22,21,1798 a,6,2,601,20199 a,1,5,101,501100 a,16,12,801,401101 a,10,14,301,701102

103 ! Controlling IF cycle for areas construction segmentation104

105 *IF,Nsy,EQ,0,THEN106

107 ! DO cycles for areas without spanwise stiffeners108

109 *DO, k, 1, Nsx, 1110 ! a,900+j,920+j,940+j,960+j111 ! a,1000+j,1020+j,1040+j,1060+j112 *IF,k,EQ,1,THEN113

114 a,5,6,201,101115 a,14,16,401,301116 a,1,2,6,5117 a,3,4,8,7118

119 a,15,19,18,13120 a,23,16,14,22121 a,19,23,22,18122

123 *ELSEIF,k,GT,1,AND,k,LE,Nsx124 a,99+k,199+k,200+k,100+k125 a,299+k,399+k,400+k,300+k126 a,99+k,100+k,500+k,499+k127 a,599+k,600+k,200+k,199+k128 a,299+k,300+k,700+k,699+k129 a,799+k,800+k,400+k,399+k130

131 *IF,k,EQ,Nsx,THEN132

133 a,15,13,100+k,200+k134 a,7,8,300+k,400+k135 a,13,9,500+k,100+k136 a,11,15,200+k,600+k137 a,8,4,700+k,300+k138 a,3,7,400+k,800+k139 *ENDIF140 *ENDIF141 a,18,19,23,22142 a,100+k,200+k,250+k,150+k143 a,300+k,400+k,450+k,350+k144

145 *ENDDO146

147 *ELSE148

149 ! DO cycles for areas with spanwise stiffeners150

151 *DO, k, 1, Nsx, 1

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152 *DO, j, 1, Nsy, 1153

154 ! Areas definition155

156 ! extremities157 *IF,k,EQ,1,THEN158 *IF,j,EQ,1,THEN159 a,5,1,900+j,910+j160 a,5,910+j,2000+k*50+j,100+k161 a,14,22,1000+j,1010+j162 a,14,1010+j,5000+k*50+j,300+k163 !rib mid section164 a,22,18,950+j,1000+j165 !spanwise stiffeners166 !a,910+j,2000+k*50+j,3000+k*50+j,940+j167 !a,1010+j,5000+k*50+j,6000+k*50+j,1040+j168 *IF,Nsy,EQ,1,THEN169 a,2,6,910+j,900+j170 a,6,200+k,2000+k*50+j,910+j171 a,23,16,1010+j,1000+j172 a,16,400+k,5000+k*50+j,1010+j173 !rib mid section174 a,19,23,1000+j,950+j175 *ENDIF176

177 *ELSEIF,j,GT,1,AND,j,LE,Nsy178 a,900+j,910+j,910+(j−1),900+(j−1)179 a,910+(j−1),910+j,2000+k*50+j,2000+k*50+(j−1)180 a,1000+j,1010+j,1010+(j−1),1000+(j−1)181 a,1010+(j−1),1010+j,5000+k*50+j,5000+k*50+(j−1)182 a,949+j,950+j,1000+j,999+j183

184 *IF,j,EQ,Nsy,THEN185 a,2,6,910+j,900+j186 a,6,201,2000+k*50+j,910+j187 a,23,16,1010+j,1000+j188 a,16,401,5000+k*50+j,1010+j189 !rib mid section190 a,19,23,1000+j,950+j191 *ENDIF192

193 *ENDIF194 !spanwise stiffeners195 a,910+j,2000+k*50+j,3000+k*50+j,940+j196 a,1010+j,5000+k*50+j,6000+k*50+j,1040+j197 *ELSEIF,k,EQ,Nsx198

199 *IF,j,EQ,1,THEN200 a,18,13,920+j,950+j201 a,13,100+k,2000+k*50+j,920+j202 a,4,8,1020+j,1050+j203 a,8,300+k,5000+k*50+j,1020+j204 *IF,Nsy,EQ,1,THEN205 a,15,19,950+j,920+j206 a,15,920+j,2000+k*50+j,200+k207 a,7,3,1050+j,1020+j208 a,7,1020+j,5000+k*50+j,400+k209 *ENDIF210

211 *ELSEIF,j,GT,1,AND,j,LT,Nsy212 a,920+j,950+j,950+(j−1),920+(j−1)213 a,920+j,920+(j−1),2000+k*50+(j−1),2000+k*50+j214 a,1020+j,1050+j,1050+(j−1),1020+(j−1)215 a,1020+j,1020+(j−1),5000+k*50+(j−1),5000+k*50+j216 *ELSEIF,j,EQ,Nsy217 a,920+j,950+j,950+(j−1),920+(j−1)218 a,920+j,920+(j−1),2000+k*50+(j−1),2000+k*50+j219 a,1020+j,1050+j,1050+(j−1),1020+(j−1)220 a,1020+j,1020+(j−1),5000+k*50+(j−1),5000+k*50+j221

222 a,15,19,950+j,920+j223 a,15,920+j,2000+k*50+j,200+k224 a,7,3,1050+j,1020+j225 a,7,1020+j,5000+k*50+j,400+k226 *ENDIF227

228 a,11,15,200+k,600+k229 a,13,9,500+k,100+k230 a,3,7,400+k,800+k231 a,8,4,700+k,300+k232 !spanwise stiffeners233 a,920+j,930+j,3000+k*50+j,2000+k*50+j234 a,1020+j,1030+j,6000+k*50+j,5000+k*50+j235 *ENDIF

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236

237 ! Middle areas definition238

239 *IF,k,GE,2,THEN240

241 a,99+k,100+k,500+k,499+k242 a,599+k,600+k,200+k,199+k243 a,299+k,300+k,700+k,699+k244 a,799+k,800+k,400+k,399+k245 !spanwise stiffeners246 a,3000+(k−1)*50+j,2000+(k−1)*50+j,2000+(k)*50+j,3000+(k)*50+j247 a,6000+(k−1)*50+j,5000+(k−1)*50+j,5000+(k)*50+j,6000+(k)*50+j248 *IF,j,EQ,1,THEN249 a,99+k,2000+(k−1)*50+j,2000+(k)*50+j,100+k250 a,299+k,5000+(k−1)*50+j,5000+(k)*50+j,300+k251 *IF,Nsy,EQ,1,THEN252 a,199+k,200+k,2000+k*50+j,2000+(k−1)*50+j253 a,399+k,400+k,5000+k*50+j,5000+(k−1)*50+j254 *ENDIF255 *ELSEIF,j,GT,1,AND,j,LE,Nsy256 a,2000+(k−1)*50+(j−1),2000+(k−1)*50+j,2000+k*50+j,2000+k*50+(j−1)257 a,5000+(k−1)*50+(j−1),5000+(k−1)*50+j,5000+k*50+j,5000+k*50+(j−1)258 *IF,j,EQ,Nsy,THEN259 !a,2000+(k−1)*50+(j−1),2000+(k−1)*50+j,2000+k*50+j,2000+k*50+(j−1)260 !a,5000+(k−1)*50+(j−1),5000+(k−1)*50+j,5000+k*50+j,5000+k*50+(j−1)261 a,199+k,200+k,2000+k*50+j,2000+(k−1)*50+j262 a,399+k,400+k,5000+k*50+j,5000+(k−1)*50+j263 *ENDIF264 *ENDIF265

266 *ENDIF267

268 !stiffener areas definition269 *IF,j,EQ,1,THEN270 a,100+k,2000+k*50+j,3000+k*50+j,4000+k*50+j,150+k271 a,300+k,5000+k*50+j,6000+k*50+j,7000+k*50+j,350+k272 *IF,Nsy,EQ,1,THEN273 a,250+k,4000+k*50+j,3000+k*50+j,2000+k*50+j,200+k274 a,450+k,7000+k*50+j,6000+k*50+j,5000+k*50+j,400+k275 *ENDIF276 *ELSEIF,j,GT,1,AND,j,LE,Nsy277 a,2000+k*50+(j−1),2000+k*50+j,3000+k*50+j,3000+k*50+(j−1)278 a,3000+k*50+(j−1),3000+k*50+j,4000+k*50+j,4000+k*50+(j−1)279 a,5000+k*50+(j−1),5000+k*50+j,6000+k*50+j,6000+k*50+(j−1)280 a,6000+k*50+(j−1),6000+k*50+j,7000+k*50+j,7000+k*50+(j−1)281 *IF,j,EQ,Nsy,THEN282 a,250+k,4000+k*50+j,3000+k*50+j,2000+k*50+j,200+k283 a,450+k,7000+k*50+j,6000+k*50+j,5000+k*50+j,400+k284 *ENDIF285 *ENDIF286

287 *ENDDO288 *ENDDO289

290 *ENDIF291

292 ! Area Attributes293

294 ! Skin thickness295 asel,all296 asel,s,loc,z,0297 asel,r,loc,x,Etw,CL−Etw298 asel,r,loc,y,Etw,(SL/2)−(Drs/2)299 aatt,1,1,1300

301 asel,all302 asel,s,loc,z,0303 asel,r,loc,x,Etw,CL−Etw304 asel,r,loc,y,(SL/2)+(Drs/2),SL−Etw305 aatt,1,1,1306

307 asel,all308 asel,s,loc,z,0309 asel,r,loc,x,Etw,CL−Etw310 asel,r,loc,y,(SL/2)−(Drs/2)+Mrw,(SL/2)+(Drs/2)−Mrw311 aatt,1,1,1312

313 ! Skin thickness on edges314 asel,all315 asel,r,loc,x,Etw,CL−Etw316 asel,r,loc,y,Etw,SL−Etw317 asel,inve318 aatt,1,2,1319

48

320 ! Skin + Ribs equivalent thickness321 asel,all322 asel,s,loc,z,0323 asel,r,loc,y,(SL/2)−(Drs/2),(SL/2)−(Drs/2)+Mrw324 aatt,1,5,1325

326 asel,all327 asel,s,loc,z,0328 asel,r,loc,y,(SL/2)+(Drs/2),(SL/2)+(Drs/2)−Mrw329 aatt,1,5,1330

331 ! Spanwise stiffeners skin thickness332 *DO, j, 1, Nsy, 1333 asel,all334 asel,s,loc,z,0.0001,sty_hi335 asel,r,loc,x,Etw+(CL−2*Etw)/(Nsy+1)*j336 aatt,1,4,1337 *ENDDO338 ! Chordwise stiffeners skin thickness339 *DO, k, 1, Nsx, 1340 asel,all341 asel,s,loc,z,0.0001,stx_hi342 asel,r,loc,y,Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1)343 aatt,1,3,1344 asel,all345 asel,s,loc,z,0.0001,stx_hi346 asel,r,loc,y,SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1)347 aatt,1,3,1348 *ENDDO349 ! Meshing350 asel,all351 asel,s,loc,z,0352 AESIZE, all, 0.01,353 mshkey,2354 amesh,all355

356 ! preparation for buckling calculations357

358 ! Spanwise stiffeners skin thickness359 *DO, j, 1, Nsy, 1360 asel,all361 asel,s,loc,z,0.0001,sty_hi362 asel,r,loc,x,Etw+(CL−2*Etw)/(Nsy+1)*j363 AESIZE, all, 0.01364 mshkey,2365 amesh,all366 esel,all367 esel,s,real,,4368 *get,numstyelemmin,elem,0,num,min369 esel,all370 esel,s,real,,4371 *get,numstyelemmax,elem,0,num,max372 *ENDIF373 *ENDDO374

375 ! Chordwise stiffeners skin thickness376 *DO, k, 1, Nsx, 1377 asel,all378 asel,s,loc,z,0.0001,stx_hi379 asel,r,loc,y,Etw+k*((SL−2*Etw−Drs)/2)/(Nsx+1)380 AESIZE, all, 0.01381 DESIZE, , , , , , , 0.01, ,382 mshkey,2383 MSHAPE,0384 amesh,all385 asel,all386 asel,s,loc,z,0.0001,stx_hi387 asel,r,loc,y,SL/2+Drs/2+k*((SL−2*Etw−Drs)/2)/(Nsx+1)388 AESIZE, all, 0.01389 DESIZE, , , , , , , 0.01, ,390 mshkey,2391 MSHAPE,0392 amesh,all393

394 esel,all395 esel,s,real,,3396 *get,numstxelemmin,elem,0,num,min397 esel,all398 esel,s,real,,3399 *get,numstxelemmax,elem,0,num,max400

401 *ENDDO

49

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