Design tailoring of panels for pressure pillowing using tow-placed steered fibers

(1)

Design Tailoring of Panels for Pressure Pillowing Using Tow-Placed

Steered Fibers

(2)

(3)

Design Tailoring of Panels for Pressure Pillowing Using Tow-Placed

Steered Fibers

Proefschrift

ter verkrijging van de graad doctor aan de Technische Universiteit Delft,

op gezag van de rector magnificus, Prof. Dr. Ir. J.T. Fokkema, voorzitter van het College der Promoties,

in het openbaar te verdedigen op dinsdag 9 december 2008 om 12.30 uur door

Ahmad ALHAJ AHMAD

Master of Science in Mechanical Engineering Design, University of Manchester Institute of Science and Technology (UMIST), England

(4)

Dit proefschrift is goedgekeurd door de promotor: Prof. Dr. Z. Gürdal

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof. Dr. Z. Gürdal Technische Universiteit Delft, promotor

Prof. Dr. M. Hyer Virginia Tech

Prof. Dr. V. V. Toropov University of Leeds Prof. Dr. Ir. R. Akkerman Universiteit Twente

Prof. Dr. Ir. R. Benedictus Technische Universiteit Delft

Dr. M. M. Abdalla Technische Universiteit Delft, adviseur Prof. Dr. Ir. A. Rothwell Technische Universiteit Delft, reservelid

Publisher: TUDelft, Faculteit Luchtvaart- en Ruimtevaarttechniek Printed by: PrintPartners Ipskamp, Enschede

ISBN: 978-90-9023708-4

Keywords: Variable-Stiffness/Pressure Pillowing/Tow-Placed Fibers/Structural Optimization

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the publisher: TUDelft, Faculteit Luchtvaart- en Ruimtevaarttechniek

(5)

(6)

(7)

Acknowledgements

On completion of this thesis, I would like to extend my sincere gratitude to everyone who has helped making this work possible. Prof. dr. Zafer Gürdal, for his guidance and generosity. Without his support, I would not have finished or even started this research. Special thanks to Dr. Mostafa M. Abdalla, for his continuous encouragement and smart ideas. My committee members also deserve many thanks for accepting to serve in my comittee, Prof. Hyer, Prof. Toropov, Prof. Akkerman, Prof. benedictus, and Prof. Rothwell. Thanks to all staff members of Aerospace Structures Group. To my friends and colleagues in the group Eelco, Agnes, Claudio, Noh, Farid, Tanvir, Roeland, and Ramzi, and my office mates Attila, Christian, Sam, and Julien.

(8)

(9)

Summary

Design Tailoring of Panels for Pressure Pillowing Using Tow-Placed

Steered Fibers

As the use of modern high-performance composite materials becomes more widespread within the aerospace industry, sophisticated technologies are being developed to build advanced high-quality composite structures. In the traditional design of composite laminates, the fibers in each layer are straight and aligned in a particular direction. The fiber orientation angles of these conventional designs are usually restricted to 0, 90, and ± 45 degrees. The idea of using fibers in any configuration other than a straight-line format has been hampered by the inability to implement such a design. The introduction of advanced tow-placement machines has made it possible to fabricate advanced variable-stiffness (V-S) composite structures where the fiber orientation angle varies continuously within each ply and throughout the structure. This manufacturing capability now allows designers of composites to use the fiber orientation angle as design variable in their analysis, not only for each ply as with conventional composites, but at each point within a ply. Consequently, as opposed to traditional composites with straight fibers, the directional material properties of composites can be fully exploited to improve the structural laminate performance.

Pressure pillowing of the fuselage panels is a common problem in thin-walled stiffened aircraft structures. The cabin pressure in a commercial transport aircraft generates a significant pressure differential across the skin. The frames and stringers that are necessary to carry maneuver loads prevent the fuselage skin from expanding as a membrane, and the skin bulges, or “pillows”, within each panel bay under the action of the internal pressure. When the skin is restrained against radial expansion at the stiffener locations, a bending boundary layer is formed causing bending stress concentrations. Such stresses can lead to failure, and hence special attention must be paid to them when designing such structures.

In this thesis, the pressure pillowing problem of fuselage skin panels is addressed using tow-placed steered fibers. The underlying goal of the research is to adopt the variable-stiffness concept based on the tow-placed steered fibers to come up with innovative tailored composite designs that are able to alleviate the pressure pillowing problem. Different models of fuselage panels with and without cutouts, with different levels of complexity are addressed. Semi-analytical and numerical solutions are developed to obtain the linear and geometrically nonlinear responses of the variable-stiffness panels. The design objective is to determine the optimal distribution of the fiber orientation angles (or the fiber paths) over the panels for different structural performance measures, minimum weight, maximum strength, and maximum buckling performance. Different fiber orientation variations are utilized to construct the variable-stiffness panels. Various design scenarios and approaches for different loading cases and boundary conditions are presented and discussed.

Optimal solutions are sought using different optimization methodologies. Optimal designs are obtained for both constant-stiffness straight fiber and variable-stiffness steered fiber laminates. It is shown that by placing the fibers in their optimal spatial orientations, the

(10)

loads of the structures can be significantly improved compared to straight fiber design baselines. Key findings and possible mechanisms that help improve the structural performance of the variable-stiffness laminates are identified and discussed. Finally, the variable-stiffness designs obtained show smooth fiber paths with small curvatures suggesting the feasibility of manufacturing of the designs using advanced tow-placement machines.

(11)

Samenvatting

Aangepast Ontwerp van Panelen voor Drukplooiing door middel van

Vezelgeplaatste Gestuurde Vezels

Naar mate het gebruik van moderne hoogwaardige composietmaterialen meer wijdverspreid raakt binnen de vliegtuigindustrie, worden er verfijnde technieken ontwikkeld om composieten constructies van hoge kwaliteit te kunnen vervaardigen. In het ontwerp van traditionele composietlaminaten liggen de vezels in elke laag recht en zijn georiënteerd in een gegeven richting. De vezeloriëntatiehoeken voor deze conventionele ontwerpen zijn doorgaans beperkt tot 0, 90, en ±45 graden. Het idee om vezels in welke andere willekeurige configuratie dan de rechte te gebruiken werd belemmerd door het onvermogen dergelijke ontwerpen te verwezenlijken. De introductie van geavanceerde vezelplaatsingsmachines heeft het mogelijk gemaakt om composieten constructies van variabele stijfheid (V-S) te fabriceren waarbij de vezeloriëntatiehoek continu varieert binnen elke laag en door de gehele constructie. Nu geeft deze productiemogelijkheid composietontwerpers de ruimte om verzeloriëntatiehoeken niet alleen per laag te gebruiken als ontwerpvariabele zoals voor conventionele composieten, maar op elk punt in een laag. Dientengevolge, in tegenstelling tot traditionele composieten met rechte vezels, kunnen de richtingsafhankelijke materiaaleigenschappen van composieten ten volle worden benut om de structurele prestaties van een laminaat te verbeteren.

Drukplooiing van de romppanelen is een veelvoorkomend probleem in dunwandige verstijfde vliegtuigconstructies. De cabinedruk in een commercieel transportvliegtuig genereert een significant drukdifferentieel over de huid. De spanten en langsliggers die nodig zijn om de manoeuvrebelastingen te dragen voorkomen dat de romphuid uitzet als een membraan, en de huid puilt, of “plooit”, binnen elke paneelbaai tengevolge van het werk verricht door de interne druk. Als de huid ter plaatse van een verstijver beperkt is om uit te zetten in radiale richting vormt zich een op buiging belast grensgebied, hetgeen leidt tot buigspanningconcentraties. Tengevolge van zulke spanningen kan de constructie bezwijken, en dient er dus speciaal aandacht hieraan te worden geschonken tijdens het ontwerpen van dergelijke constructies.

In dit proefschrift wordt het drukplooiingsprobleem behandeld voor romphuidpanelen die gebruik maken van vezelgeplaatste gestuurde vezels. Het onderliggend doel van het onderzoek is toepassing van het variabele stijfheidsconcept gebaseerd op vezelgeplaatste gestuurde vezels om innovatieve aangepaste ontwerpen te vinden die in staat zijn het drukplooiingsprobleem te verlichten. Verschillende modellen romppanelen met en zonder opening, met verscheidene graden van complexiteit worden behandeld. Semi-analytische en numerieke oplossingen worden ontwikkeld om de lineaire en geometrisch niet-lineaire respons van de variabele stijfheid panelen te verkrijgen. Het ontwerp doel is de optimale verdeling van vezeloriëntatiehoeken (of vezelpaden) voor de panelen vast te stellen voor diverse soorten van structurele prestatie, minimaal gewicht, maximale sterkte, en maximale knikprestatie. Verschillende vezeloriëntatiehoekvariaties worden toegepast om de variabele stijfheidspanelen te construeren. Verscheidene ontwerpscenario’s en –benaderingen worden

(12)

Optimale oplossingen worden gezocht met behulp van verschillende optimalisatiemethodieken. Optimale ontwerpen worden verkregen voor zowel constante stijfheidslaminaten met rechte vezels als variabele stijfheidslaminaten met gestuurde vezels. Het wordt aangetoond dat het plaatsen van de vezels in hun optimale ruimtelijke oriëntatie het drukplooiingsprobleem kan verlichten, en de capaciteit om krachten te dragen evenals de kniklasten significant kan verbeteren in vergelijking tot uitgangsontwerpen met rechte vezels. De belangrijkste bevindingen en mogelijke mechanismen die helpen de structurele presetaties te verbeteren worden geïdentificeerd en besproken. Tenslotte tonen de verkregen variabele stijfheidsontwerpen vloeiende vezelpaden met kleine krommingen, hetgeen de haalbaarheid van productie doormiddel van geavanceerde vezelplaatsingsmachines suggereert.

(13)

Chapter 1 Introduction

1.1 Motivation

As the use of modern high-performance composite materials becomes more widespread within the aerospace industry, sophisticated and innovative technologies are being developed to build composite structures that are more affordable, manufacturable and structurally efficient. Composite structures are designed to make use of multiple layers of fiber-reinforced materials. Traditionally, the fibers in each layer are aligned straight and parallel with each other. Within a particular layer, the fiber orientation is fixed. This is opposed to having a fiber orientation that varies spatially from point to point within a layer or a group of layers. The idea of using fibers in anything other than a straight-line format has been hampered by the inability to implement such a design. However, the introduction of the advanced tow-placement machines has eliminated this barrier, marking a watershed event in the manufacturability of composite structures. Tow-placement machines are high-precision robot computer controlled machines to produce repeatable high quality composite components [1,2]. In the fiber-placement process, individual prepreg tows are fed into a fiber placement head to be placed onto a work surface, along prescribed paths, as a single fiber band. The tow-placement head can accommodate from 16 up to 24 (or more) individual tows. Considering a typical tow width of 3.2 mm, the result is a tow-band ranging from 5.1 cm up to 7.6 cm in width. Machines with larger number of tows and/or tows or slit-tape with larger width can also be used to increase the band of material that can be placed on the surface. Moreover, the machines are capable of cutting and restarting individual tows permitting flexible coverage patterns to build laminated panels. Such flexibility in steering fiber tows provides new design possibilities that can be used to improve the structural performance of composite laminates.

One of the primary advantages of using fiber-reinforced laminated composites in structural design is the ability to change the stiffness and strength of the laminate by designing the laminate stacking sequence in order to improve its performance. This flexibility to design the stacking sequence of the laminate is typically referred to as laminate tailoring. Tailoring is traditionally achieved by keeping the fiber orientation angle within each layer constant throughout a component resulting in constant-stiffness structure. By limiting each layer to a

(18)

Introduction

single orientation over the entire component the designer is unable to fully exploit the directional material properties offered by advanced composites (defined here as ones in which the fiber orientation angle is allowed to vary continuously throughout the structure), because the laminate construction will be dictated by the location that has the most critical stress state. One method of creating variable-stiffness (V-S) composite structure is by changing the fiber orientation angle continuously within the lamina (layer or ply). Allowing the fibers to curve within the lamina constitutes an advanced tailoring option to account for non-uniform stress states, as well as providing other structural advantages such as the alteration of principal load paths. The idea of tailoring composite structures based on variable-stiffness concept is not limited to curvilinear (steered or curved) fibers. A simple method for improved tailoring based on variable-stiffness concept is the introduction of discrete jumps in the laminate construction by adding patches of additional layers with different fiber orientations into the component. For example, Biggers [3,4] achieved tailoring by redistributing the layers with specified orientations across the planform of rectangular plates to create beneficial stiffening patterns against compression and shear buckling. By varying the stiffness properties of composite laminates from one point to another, the design space is expanded as compared to the classical stacking sequence design problem. As a consequence, higher performance and/or lighter structures can be obtained.

Currently advanced composites are being used, on a large scale, in commercial aircrafts. For instance, the commercial aircraft Boeing 787 Dreamliner is the first commercial jet ever to have the majority of its primary structure including the tail, wing and fuselage made of advanced composite materials [5].

Realizing the potential benefits offered by the advanced composites, along with the feasibility of manufacturing of tailored composite designs using tow-placement machines, this research represents a departure from the traditional usage of composite materials and adopts the variable-stiffness concept based on curvilinear fibers to explore improvements that can be achieved in the structural performance of aircraft components, particularly, those in the transport commercial aircrafts.

1.2 Literature Review

For designers, an attractive and interesting subject for fiber orientation angle tailoring is laminates with circular holes. One of the first approaches to improve compressive load carrying capacity of composite laminates with an open hole was the experimental work of Yau and Chou [6], who inserted metal pins into woven fabric prior to curing, pushing the fiber tows apart to create a molded hole. The resulting laminates possessed curved fibers around the hole and showed improved open-hole strength compared to similar laminates with equal-sized drilled holes. Hyer and Charette [7] were among the first to investigate the influence of fiber orientation angle around a cutout for a flat plate with a circular hole subjected to axial tensile loads. They chose the fiber orientations so that the fibers in a particular layer were aligned with the principal stress directions in that layer. Their finite element solution demonstrated significant improvement in the load carrying capacity of the curvilinear fiber design compared to a conventional quasi-isotropic design. In a similar work, Huang and Haftka [8] optimized the fiber orientations near a hole in a single layer of multilayer composite laminate subjected to axial tensile loads. This single layer was divided into two fields, a small near field around the hole and relatively large field away from the

(19)

Chapter 1

hole where the fiber orientations were fixed in the applied load direction. The results showed that the load carrying capacity of composite laminates with hole can be greatly increased through the optimization of continuous fiber orientation distribution within a small area around the hole. Gains in buckling performance of a simply supported plate with a circular center hole was explored by Hyer and Lee [9] using curvilinear fiber format. Gradient-search technique and sensitivity analysis were used to determine the optimal fiber orientation angles in different regions of the plate modelled by a grid of finite elements. It was shown that the curvilinear fibers move the load away from the unsupported hole region of the plate to the supported edges, thus increasing the buckling capacity.

In an attempt to have a manufacturable representation of fiber orientation angles, Nagendra et al. [10] performed a research to develop a method of optimizing tow paths. The tow paths were defined by passing a single curve through a set of control points based on a single cubic Non-Uniform Rational B-Splines (NURBS). Using finite elements, they studied optimal frequency and buckling load design of laminated composite plates with a central hole subject to deformations, ply failure, and interlaminar stress constraints, and demonstrated increased performance.

In an effort to integrate realistic fabrication techniques into the design of laminates with curvilinear fiber layers, research carried out by Gürdal and Olmedo [11,12] introduced a fiber path definition and formulated closed-form and numerical solutions for simple rectangular plates. The proposed fiber orientation variation was generated from a base curve that changes its orientation angle linearly from one end of the panel to the other, while taking into account constraints on the radius of curvature of the fiber paths. Buckling response of the panels was investigated by varying the fiber orientation in the direction of the applied loads and in a direction perpendicular to the loadings. Improvements in the buckling load of up to 80% over straight fiber configurations were found. In a follow-up design study by Waldhart et al. [13], a parametric study of a small set of variables used in the fiber path definitions indicated increased buckling performance due to the stiffness variation, which in turn caused favourable re-distribution of the internal stresses. Also, additional mechanisms that help improve the buckling load of panels were identified. Considerations for the manufacturability of the layers, based on estimates of the limitations of the advanced tow-placement machines, were also included to ensure the manufacturability of the designs. The promising results of the analytical and numerical research mentioned above established the need for experimental validation of the findings. In this framework, Tatting and Gürdal [14] manufactured several variable-stiffness panels that showed improved performance relative to their straight fiber counterparts. Then, Wu et al. [15,16] tested those panels and confirmed the increased load carrying capacity of the variable-stiffness panels, with gains of up to 3-5 times the compressive buckling load of a straight fiber panel. Further design and experiment work performed by Jegly et al. [17] demonstrated, again, that considerable improvements in the buckling load and the failure load can be achieved using the steered fibers compared to traditional composite structures with straight fibers.

Investigation of the optimal designs for both constant- and variable-stiffness rectangular composite plates for minimum compliance was performed by Setoodeh et al. [18]. In these latter studies, the lamination parameters are used as design variables instead of fiber orientation angles, thus reducing the number of design variables. Furthermore, the formulation guaranteed that the solution is optimal, benefiting from the fact that the optimization problem is convex. A disadvantage of the approach is that the optimal distribution of the lamination parameter values does not immediately translate to a unique

(20)

Introduction

fiber orientation angle distribution. Although the actual stacking sequence was unknown, the results showed that substantial improvements in stiffness can be gained by using variable-stiffness designs. In a step further, they proposed a methodology to generate curvilinear fiber paths from the lamination parameter distribution [19]. Minimum weight design of composite structures with curved fibers subject to stress constraints was studied by Parnas et al. [20]. In their study they constructed a bi-cubic Bezier surface for layer thickness representation and cubic Bezier curves for fiber orientation angles and used coordinates of the control points as design variables.

The previous studies indicate that the variable-stiffness concept has great potential for improving the structural performance of composite structures. Despite the fact that these studies established ground work and provided good knowledge to the designers about tailoring the variable-stiffness composite laminates, those studies were limited to investigate simple models mostly with simple loading cases and linear responses, such as panels with or without holes subjected to a single load. Probably some of structures studied may be perceived to be applicable to aircraft structures, however, aircraft structures have, in general, more complicated models and usually subjected to combined in-plane and out-of-plane loadings, where geometric nonlinearities appear to have a significant role in determining the structural behaviour of the components as with wing and fuselage structures.

1.3 Pressure Pillowing Problem

A challenging problem in aircraft structures associated with thin-walled stiffened structures is the so-called pressure pillowing. Pressurized fuselages and fuel tanks are typical examples of structures in which pressure pillowing is observed. In the case of fuselages, the cabin pressure of a commercial aircraft causes a significant pressure differential across the skin. An unstiffened fuselage would carry this internal pressure load as a shell in membrane response, like pressure vessels. However, internal longitudinal and transverse stiffeners (stringers and frames as shown in Figure 1.1) are necessary to carry maneuver loads. The presence of these stiffeners prevents the fuselage skin from expanding as a membrane, and the skin bulges, or “pillows” (see Figure 1.2), within each panel bay under the action of the internal pressure. When the skin is restrained against out-of-plane expansion at the stiffener locations, a bending boundary layer is formed causing bending stress concentrations. Such stresses can lead to failure, and hence special attention must be paid to them when designing such structures.

An experimental and analytical study of the nonlinear response and failure characteristics of internally pressurized constant-stiffness composite fuselage panels with clamped edges was performed by Boitnott et al. [21].The study showed that the graphite-epoxy specimens failed at their edges where the magnitudes of local bending gradients and interlaminar stresses are maximum. Minguet et al. [22,23,24] investigated the effects of axial skin loads and internal pressure loads on a composite materials bonded fuselage panel consisting of a skin, frames and stringers. The focus of the investigations was on the failure mechanisms at the interface between a thin laminated skin and co-bonded frame. Results showed that for the dominant failure mode, fracture initiated within the skin laminate at the tip of the frame flange due to bending in the skin. In a related work [25], the latter author demonstrated the importance of using geometrically nonlinear analysis to capture the skin loads at the edge of the frame. In the latter work, a combination of local analysis model and test data was used to determine

(21)

Chapter 1

interlaminar stresses and strain energy release rates along the length of the frame. Results showed that the frame starts debonding during the application of combined axial and pressure loads and the debonding starts at the flange corner near the stringer and progresses along the frame length towards the middle of the bay.

Figure 1.1: Stiffened structure of a pressurized fuselage (courtesy of Fiber Metal Laminates Center of Competence, Delft University of Technology)

Figure 1.2: Pillowed panel

1.4 Research Objective

The underlying goal of this research is to adopt the variable-stiffness concept based on the tow-placed steered fibers to come up with innovative tailored composite designs that are able to alleviate the pressure pillowing problem. Different models of fuselage panels with and without window openings, with different levels of complexity are addressed. Semi-analytical and numerical solutions are developed to obtain the structural responses of variable-stiffness panels. The design objective is to determine the optimal distribution of the fiber orientation angles (or the fiber paths) over the panels for different structural performance measures (minimum weight, maximum strength, and maximum buckling performance). The optimal variable-stiffness designs are compared to their constant-stiffness counterparts. Possible mechanisms that help improve the variable-stiffness designs are identified and discussed.

1.5 Thesis Layout

Following this chapter, Chapter 2 will introduce briefly the variable-stiffness laminates where different fiber path definitions from the literature will be described. Also, a new proposed definition of the fiber orientation variation based on Lobatto-Legendre polynomials will be demonstrated.

(22)

Introduction

In Chapter 3 description and analysis of different models of fuselage panels under combined loads with and without window openings, with different levels of complexity, will be demonstrated. Semi-analytical and numerical solutions will be developed to obtain the linear and geometrically nonlinear responses of the variable-stiffness panels. The analysis models will initially be simple. One-dimensional ones covering the fundamental aspects of the nonlinear coupling of pressure and compression loads based on one-dimensional plates. Next the model will be expanded to represent a two-dimensional nonlinear plate model based on Rayleigh-Ritz solution. Finally, a finite element model covering more detailed configurations such as a plate with a fuselage window opening will be introduced. Also, the failure analysis of variable-stiffness laminates will be presented. The analysis models developed in Chapter 3 will then be used in the subsequent chapters, namely Chapters 4,5,6, and 7, to perform design studies for various design problem formulations appropriate for the models

As a first step of the research, in Chapter 4 the problem of the pressure pillowing will be treated as a dimensional plate. Linear and geometrically nonlinear analyses of the one-dimensional plate models will be performed. Optimal fiber paths will be determined for minimum weight subject to stress constraints for different loading cases. Optimal variable-stiffness designs will be compared with their optimal constant-variable-stiffness counterparts.

In Chapter 5, a fuselage skin panel bounded by two frames and two stringers will be modeled as a two-dimensional plate. The geometrically nonlinear response will be obtained using the Rayleigh-Ritz approach. The optimal fiber paths will be determined for maximum load carrying capacity, where one layer in the laminate lay-up (angle-ply laminate) will be designed using different fiber orientation definitions. Optimal designs for both straight fibers and steered fibers will be obtained for different load cases and different aspect ratios. The reason behind the improved load carrying capacity of the variable-stiffness laminates over the constant-stiffness ones will be identified and discussed.

In Chapter 6, using the ABAQUS [26] Scripting Interface, a Python script will be developed to perform the linear and geometrically nonlinear finite element analyses of variable-stiffness panels. The optimal fiber paths, within each layer of the laminate stacking sequence for maximum strength and maximum buckling performance, will be determined. Optimal variable-stiffness designs will be compared with their optimal constant-stiffness and quasi-isotropic counterparts. Mechanisms that help improve the buckling performance of the variable-stiffness designs will be identified and discussed.

Similar to Chapter 6, a fuselage skin bounded by two frames and two stringers will be addressed in Chapter 7 this time with a typical centrally located window opening (cutout). Various boundary conditions for both the outer edges of the panel and conditions around the cutout will be modelled. Different stiffness tailoring scenarios using various fiber orientation variations will be utilized to design the optimal fiber paths for maximum strength and buckling performance.

Finally, concluding remarks and significant conclusions drawn from the results presented in the preceding chapters will be made in Chapter 8.

(23)

Chapter 2 Variable-Stiffness Composite Laminates

2.1 Introduction

There are many ways to create a variable-stiffness composite laminate including, but not limited to, dropping or adding plies in the laminate, varying the fiber volume fraction in a layer, and changing the fiber orientation within a lamina. DiNardo and Lagace [27] used the variable-stiffness concept based on dropped plies to investigate the buckling of panels. They showed that the buckling performance is driven by the changes in stiffness along the panel. Martin and Leissa [28] studied the buckling of a rectangular composite sheet composed of variably spaced straight and parallel fibers, and achieved improvement in the buckling performance. Allowing the fiber orientation to vary spatially from point to point in a continuous manner within the lamina represents a more recent concept to construct variable-stiffness laminates. Unlike conventional straight-fiber layers, curvilinear fiber paths can not be described by a single orientation angle. Optimal design techniques for conventional straight-fiber composites typically involve changing the fiber orientation angle of each layer and the total number of layers in the laminate [29], resulting in a large but finite number of design variables. The design problem for advanced composite structures is, however, a much more complex one. In addition to the number of layers in the laminate, the fiber orientation angle at each point of each layer might be a potential design variable. Therefore, more time will be required to perform the analysis, design and optimization of this class of structures. In attempt to cope with such problems, researchers utilized and proposed simple functions to describe the fiber orientation variations, thus reducing the number of design variables. In the next sections some of the fiber orientation variations, from the literature, will be briefly described, and a new proposed definition of the fiber orientation variation based on Lobatto-Legendre polynomials will be introduced.

(24)

Variable-Stiffness Composite Laminates

2.2 Linear Variation of Fiber Orientation Angles

A simple definition that employs a unidirectional variation based on linear function for the fiber orientation angle of the individual layers proposed by Gürdal and Olmedo [11].

Consider a rectangular panel with length a and width b. The fiber orientation angle based on a linear variation along one of the axes, say x, is given by [11]

( )

(

)

(

)

0 1 1 1 0 0 1 2 for 0 2 2 2 for 2 a T T x T x a x a T T x T T x a a θ  − + ≤ ≤  =  ₋ ₊ ₋ _{≤ ≤}  (2.1)

where T0 is the fiber orientation angle at the panel center, x = a/2, and T1 is the fiber

orientation angle at the panel ends, x = 0 and x = a. When the angle varies in the y-direction,

θ = θ(y), x and a should be replaced by y and b, respectively in Equation 2.1. In this case the

angle T0 is the fiber orientation angle (still measured from the longitudinal panel axis x) at the

center section of the panel y = b/2, and T1 is the fiber orientation angle at the transverse edges

of the panel, y = 0 and y =b.

For the form demonstrated above, a representation of a single layer has been suggested to be <T0|T1> [11]. For example, a variable-stiffness laminate with a ±<0o|45o> stacking sequence, where the fiber orientation is a function of x, is shown in Figure 2.1.

a b

x y

Figure 2.1: Variable-stiffness laminate constructed using linear variation (±<0o|45o>).

It should be mentioned that the fiber orientation function presented above is the simplest form of the linear variation. More general definition and details about the linear variation of fiber orientation angle can be found in Refs. [11,12,13].

(25)

Chapter 2

2.3 Circular Arc Variation (Constant Curvature Path)

Similar to the fiber orientation variation presented in the previous section, Gürdal et al. [30] developed a more recent definition based on circular arcs instead of linear variation of the orientation angle. The same parameters T0 and T1 have also been used to define the constant

curvature fiber path. One of the advantages of this definition is producing courses of constant curvatures which better corresponds to the manufacturing constraints of a tow-placement machine.

An example schematic of the circular arc variation is shown in Figure 2.2. Each segment is a constant curvature arc with radius R, which is equivalent to the reciprocal of the curvature. The curvature and related radius can be calculated geometrically from the parameters T0, T1

and d (see figure 2.2) as [30] 1 0 ( 1) (sin sin ) 1 ( ) , floor[ ] k T T x x k R d d κ = = − − = (2.2)

Note that the curvature is a signed quantity to determine which side of the arc that the center of curvature is located. The center of each arc can be similarly calculated as [30]

{

} {

_{

0 0

_}

}

₁ ₀

0 1

1 1

sin , cos , even _cos _cos

, , floor[ ],

sin sin

sin , cos , odd

c c kd R T kS R T k _x _T _T x y k S d d T T kd R T kS R T k − +   ₋   =  = = − − +     (2.3)

where the floor function truncates the real number to the nearest integer that is less or equal to it and S represents the span of one arc in the y-direction.

(26)

The fiber orientation angle is given as [30]

0 1 0

1 0 1

sin (sin sin ) , even

sin ( ) , floor[ ]

sin (sin sin ) , odd

x T T T k k d x x k d x T T T k k d θ     + −  −        =  =    ₊ ₋ ₋     _ _    (2.4)

2.4 Multi-Parameter Fiber Orientation Angle Representation

The linear and circular arc variations presented in the previous sections are predefined curves that represent limited classes of fiber orientation variations. In general, this kind of curves results in a relatively narrow design space. On the other hand, for an expanded design space, the fiber orientation angles can be treated as continuous design variables at every point or element of each layer of the laminate without any restrictions. This results in problems with fiber continuity and manufacturability [7,9,18]. In order to overcome the shortcomings mentioned above, a new definition of the fiber orientation variation based on Lobatto-Legendre (or Lobatto for brevity) polynomials is introduced in this section.

Assume a rectangular domain as shown in Figure 2.3.

a

b

x

y

Figure 2.3: Rectangular domain and coordinate system.

The normalized coordinates ξ and η are defined as follows

2 2 , x a y b a b ξ = − η= − (2.5) such that − ≤ ≤ − ≤ ≤1 ξ 1, 1 η 1

(27)

Chapter 2

( )

1 1

( ) ( )

0 0

,

m n ij i j i j

T L

L

θ ξ η

− −

ξ

η

= =

=

∑ ∑

(2.6)

where m and n are number of basis functions used in

ξ

and

η

directions, respectively, Tij are

unknown coefficients (they will be called Lobatto coefficients throughout the thesis) and Li

are the Lobatto polynomials defined as

( )

₁ 1

( )

, 2

i i

L ξ ξ P₋ µ µd i

−

=

_∫

≥ (2.7)

where L₀

( )

ξ =1 and L₁

( )

ξ =ξ, or in a recursive form

( )

(

1

( )

2

( )

)

1 , 2 i i i L P P i i

ξ

=

ξ

₋

ξ

− ₋

ξ

≥ (2.8)

where Pi are the Legendre polynomials given by

( 1) ( 2)

( ) [(2 1) ( 1) ] / , 2

i i i

P

ξ

= i−

ξ

P ₋ − −i P ₋ i i ≥ _(2.9)

and where P₀( )ξ =1 and P₁( )ξ =ξ .

For example, the first few Lobatto polynomials are

( )

(

)

( )

(

)

2 2 2 3 2 4 2 4 4 5 1 1 , 1 , 2 2 1 1 1 6 5 , 3 10 7 . 8 8 L L L L ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ = − = − = − + = − + (2.10)

In order to visualize the fiber orientation variation given by Equation 2.6, it is better to consider the variation along one of the axes, say x (orξ). In this case the fiber orientation angle may be defined by the following form

1 0 ( ) ( ) m i i i T L

θ ξ

−

ξ

= =

_∑

(2.11)

For example, considering three and four coefficients in the fiber angle expansion, the fiber orientations are given by the following relations, respectively,

2 0 1 2 1 ( ) ( 1) 2 T T T θ ξ = + ξ+ ξ ξ − ( ) ₀ ₁ 1 ₂ ( 2 1) 1 ₃

(

1 6 2 5 4

)

2 8 T T T T θ ξ = + ξ+ ξ ξ − + − ξ + ξ

(28)

To demonstrate the fiber angles defined by these functions, their distributions are plotted in Figures 2.4 and 2.5 for different values of T2 and T3, respectively, assuming that T0 = 15o and T1 = 30o for the first function, and T0 = 15o, T1 = 30o and T2 = 10o for the second function. It

can be noticed from Figure 2.4 that when T2 = 0 the fiber orientation variation becomes linear

resulting in two parameters linear representation same as the one given in Equation 2.1. It is quite clear from Figures 2.4 and 2.5 that by increasing the number of Lobatto coefficients more freedom is achieved to represent the fiber orientation angle variations. Consequently, there is better flexibility to vary the stiffness over the structure and make use of the directional properties of the fibers.

Normalized Length, F ibe r O ri ent at ion A ngl e, D egr ee s -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 -30 -15 0 15 30 45 60 75 _T 2=0 o T₂=30o T₂=60o T₂=-30o T₂=-60o ξ

Figure 2.4: Fiber angle distribution for three coefficients.

Normalized Length, F ibe r O ri ent at ion A ngl e, D egr ee s -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 -15 0 15 30 45 60 T₃=0o T₃=30o T₃=60o T₃=-30o T3=-60 o ξ

(29)

Chapter 3 Problem Description and Analysis of Panel Models

3.1 Introduction

In this thesis, a fuselage skin with and without window opening (cutout) bounded by two frames and two stringers (a panel bay, see Figure 1.1) will be considered for addressing the pressure pillowing problem. For the analysis of the pressure pillowing problem, the fuselage skin can be modelled using different levels of complexity, but in general, requires a two-dimensional plate or shallow shell modelling under combined pressure and in-plane loads. For large-radius fuselages, the fuselage skin of a panel bay has only a slight curvature. In the present study curvature effects will be ignored and the panel will be treated as a flat plate subjected to pressure and in-plane loads.

Composite pressurized aircraft fuselages consist of skin panels with co-cured or adhesively bonded frames and stringers. As shown in Figure 3.1, the internal pressure load causes the skin to bulge or pillow within each panel bay, and this deformation creates local bending moments along the skin-to-frame or skin-to-stringer interface. Similarly, the in-plane pressure-induced tensile loads in the panel also create local bending moments at the same interface. These moments are responsible for creating shear and peel stresses that may cause the frame or stringer to debond from the panel. In addition to the pressure and the in-plane tensile loads, the fuselage skin may also experience in-plane shear or compressive loads that may result from fuselage torsion or bending, respectively. One of the most important and critical load cases, for example, is a pillowed (deflected or buckled) fuselage panel loaded by in-plane compressive loads. Such a situation is, actually, similar to a compressively loaded post-buckled panel, where the out-of-plane deformations result in severe stress state leading to failure.

(30)

Problem Description and Analysis of Panel Models

Figure 3.1: Local deformation along frame flange due to internal pressure and in-plane loads.

It should be mentioned that analysis and modelling of the pressure pillowing problem may not be an easy task. Particularly, the rotation of the skin due to bending in the vicinity of the stringers and frames couple with membrane forces in the skin to cause a geometrically nonlinear response. Such a response can be difficult to be obtained analytically and computationally expensive making it very difficult to put into optimization environment. Therefore, in this thesis, the pressure pillowing problem will be investigated in steps, starting from simple models to complicated ones.

In the subsequent sections different analysis methodologies for different panel models will be utilized to tackle the pressure pillowing problem.

3.2 One-Dimensional Panels

As a first step, in this research, the fuselage skin will be modelled as a one-dimensional problem. The idea behind this simplification is to get an insight into the implementation of the variable-stiffness concept for simple panel response such that results are easier to judge and understand.

A special case of the laminated plates that can be treated as one-dimensional problem is the laminated plates. When the width of the laminated plate is small compared to the length along one of the axes, say x, and the lamination scheme, and the loading is such that the

(31)

Chapter 3

displacements are functions of x only, the laminate treated as one-dimensional problem [31]. In such a case, all the derivatives with respect to y-coordinates are zero.

A fuselage skin section bounded by two frames is shown in Figure 3.2. It is assumed that the skin is flat along the x-direction with a number of periodic frames separated by distance L, while in the transverse direction there are no stringers. A panel section of unit width in the y-direction with two frames represented as supports against out of plane deformation can be treated as one-dimensional plate (see Figure 3.3). Two loading cases will be considered. The first case (Case I) is a one-dimensional plate subjected to pressure and the second one (Case II) is a one-dimensional plate under combined pressure and in-plane compressive loads.

Figure 3.2: Typical stiffened panel section.

(32)

3.2.1 Calculation of Stresses

Using the classical lamination theory, the laminate strains are expressed in the following form [31]

{ }

ε =

{ }

ε0 +_z

{ }

κ _(3.1)

where

{ }

ε0 _and

{ }

κ _{denote the mid-plane strains and curvatures, respectively. For} one-dimensional problems, the strain and the curvature are given by

2 0 2 0 2 1 , 2 xx xx u w w x x x ε =∂_∂ + ∂_∂  κ = −∂   _∂ _(3.2)

where u0 and w are the mid-plane displacements. Since one of the loading cases is a combined compression and out-of-plane loadings a nonlinear form of strain-displacement relations is used.

3.2.2 One-Dimensional Plate Analysis

The two loading cases that are considered for the design require two different analysis models. The first case (Case I) is a one-dimensional plate subjected to pressure and the second one (Case II) is a one-dimensional plate under combined pressure and in-plane compressive loads. For the first case the analysis is assumed to be linear, while geometric nonlinearities are included in the analysis of the second case. Clamped-clamped boundary conditions simulating the panel edges at the frame locations are assumed for both loading cases.

3.2.2.1Case I: One-Dimensional Plate Subjected to Pressure

Consider a laminated beam with length L and width b, and subjected to uniformly distributed load, q, as illustrated in Figure 3.4.

(33)

Chapter 3

For one-dimensional laminated plates it is assumed that Myy and Mxy are zero everywhere in

the plate. Thus, the classical laminated theory constitutive equations for symmetric laminates, in absence of in-plane forces, can be reduced to the following equation [31]

2 * 11 2 xx w D M x ∂ _{= −} ∂ (3.3)

In order to cast Equation (3.3) in a familiar form, the following quantities are introduced

xx bM M = , * 11 * 11 3 12 D I b D h E yy xx = = , 12 3 bh I_yy = _(3.4)

Equations (3.3) and (3.4) give 2 2 xx yy w M E I x ∂ _{= −} ∂ (3.5)

For a linear analysis and due to the absence of the in-plane loads Equations (3.1) give 2 2 xx xx w z z x

ε

=

κ

= − ∂ ∂ (3.6)

thus Equations (3.5) and (3.6) give

yy xx xx I E M z =

ε

(3.7)

In Equation (3.7), computing the strain

ε

_xxrequires bothE and M to be determined. The term _xx

xx

E can be computed straightforward using Equation (3.4). Since the problem at hand is a

statically indeterminate, an expression for the internal bending moment within the panel can be obtained using the superposition method. That is, the clamped-clamped panel can be decomposed into two simply supported ones.

It can be shown that the bending moment expression is

) 1 ( 8 ) ( 2 2

ξ

= qL −c− M (3.8) where L x 2 =

ξ

, and c is given by

(34)

∫

− − − = ₁ 1 1 1 2 1 ) 1 (

ξ

d E d E c xx xx (3.9)

It should be reminded that Exx is a function of x (or more specifically,

ξ

) as long as θ is a

function of x. For Exx = const.,

3 2

=

c .

3.2.2.2Case II: One-Dimensional Plate under Combined Pressure and In-Plane

Compressive Loads

Consider a one-dimensional laminated plate subjected to uniformly distributed load q, and an

axial compressive load Nxx as illustrated in Figure 3.5.

Figure 3.5: Clamped-clamped one-dimensional plate under pressure and axial compressive loads.

In this case the equations of motion for one-dimensional laminated plates based on the classical lamination plate theory are given by [31]

0 = ∂ ∂ x N_xx 2 2 2 2 0 xx xx M w N q x x ∂ ₊ ∂ _{+ =} ∂ ∂ (3.10) Recalling Equation (3.3) 2 * 11 2 xx w D M x ∂ _{= −} ∂ (3.11)

Similar to Equation 3.11, the in-plane constitutive equations can be reduced to the following relation

(35)

Chapter 3 xx xx

A

N

* 11 0

=

ε

(3.12)

and from Equations (3.2) we have

2 0 0 1 2 xx u w x x ε =∂_∂ + ∂_∂    (3.13)

The above set of nonlinear differential equations can be solved using the multiple-shooting method¶.

In order to show a sample result of the nonlinear analysis of the panel under pressure and in-plane compressive loads, a laminate with all 0o layers, where L/b = 40 and the total thickness

h = 0.4 in, is considered. The pressure load is q = 15 psi. The composite material is a typical graphite-epoxy with stiffness and strength properties given in Table 3.1 [32]. The values of the strains along half the panel in the upper layer of the laminate (z = h/2), for different values of the in-plane compressive load, are shown in Figure 3.6.

Table 3.1: Material properties. Material properties Graphite-Epoxy

E1 207 GPa (30×106 psi) E2 5 GPa (0.75×106 psi) ν12 0.25 G12 2.6 GPa (0.375×106 psi) Xt 1035 MPa (150×103 psi) Yt 41 MPa (6×103 psi) S 69 MPa (10×103 psi) Xc 689 MPa (100×103 psi) Yc 117 MPa (17×103 psi)

It can be seen from Figure 3.6 that increasing the in-plane compressive load increases the strain level in the panel, making it more susceptible to failure. This reveals the importance of investigating pressurized fuselage panels loaded by in-plane compressive loads.

¶

From the Microsoft International Mathematics and Statistics Library, using Fortran 90 subroutine DBVPMS, Solving a (parameterized) system of differential equations with boundary conditions at two points, using a multiple-shooting method.

(36)

Problem Description and Analysis of Panel Models Normalized length, S tr ai n, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.003 -0.002 -0.001 0 0.001 0.002 N_xx/(qL) = 0 N_xx/(qL) = 1 N_xx/(qL) = 2 ξ ε xx

Figure 3.6: Strains along half panel in the upper layer of the laminate (z = h/2).

3.3 Two-Dimensional Panels

A pressurized fuselage skin with and without cutout bounded by two stringers and two frames will be modelled as a two-dimensional plate as shown in Figure 3.7. The panel is loaded in two steps. In the first loading step, a uniform pressure P is applied. It is assumed that the pressure load also includes in-plane loads on the panel due to pressurization of a fuselage of radius Rf. The in-plane loads are approximately translated into an axial tensile load Fx, and a hoop tensile load Fy, calculated as

2

f x

PR

F = b F_y=PR a_f (3.14)

where a and b are the length and the width of the plate, respectively. The pressure and hence the tensile loads are incremented by means of a scaling factor λ1 in the first loading step. It should be noted that the relations given by Equations 3.14 are valid only for unstiffened cylindrical shells because the skin loads will be reduced according to the stiffness contributions from the stringers and frames. However, for the sake of simplicity, these contributions are neglected in the present study. For a given fuselage with known ring and longitudinal stiffeners, their affect on the skin loads can easily be incorporated. In principle, the averaged applied loads will also change as some of the skin panels might buckle resulting in load redistribution. Such interactions, however, require a consideration of the complete structure and not a single panel and are beyond the scope of the present work. In the second loading step, an additional axial compressive load B

x

F , which may result from fuselage bending, is applied. This latter load is incremented by means of a scaling factor λ2 while keeping λ1 fixed at the end of the first loading step. Various boundary conditions for both the outer edges of the panels and conditions around the cutout will be modelled.

(37)

Chapter 3

Semi-analytical and numerical solutions will be developed to obtain the linear and geometrically nonlinear responses of the variable-stiffness panels. Depending on the size of the design space of the problem and the computation time needed for performing the nonlinear analyses, semi-analytical solution based on the Rayleigh-Ritz approach and the finite element method will be utilized to analyze the panel without cutout. For the analysis of the panel with cutout, on the hand, the finite element method will be used.

Figure 3.7: Fuselage panel model.

3.3.1 Rayleigh-Ritz Approach

The Rayleigh-Ritz (R-R) approach will be used to obtain the nonlinear response of the panels without cutouts.

Assuming that the plate is thin, such that the Kirchhoff hypothesis is valid, the laminate strains are given by Equations 3.1, where the mid-plane strains and curvatures for two-dimensional panels are given by

{ }

2 0 2 2 0 2 0 2 0 0 2 0 0 0 2 1 2 1 , 2 2 xx _xx yy yy xy xy u w w x x _x v w w y y y u v w w w y x x y x y ε _κ ε κ κ γ  _∂ __∂ _  _ _ ∂  ₊ _ _  _ ₋ _ ∂ ∂    _ _∂ _   _ _   _ _   _ _ _ _ ∂ ∂ ∂         =  = +    =  = −  ∂ ∂  _∂                     _∂ _∂ _{∂ ∂} _∂  ₊ ₊  ₋   _∂ _∂ _{∂ ∂}  _ _{∂ ∂} _   ε _k _(3.15)

where u0, v0 , and w are the mid-plane displacements.

(38)

W

U

−

=

Π

(3.16)

where U is the strain energy, and W is the potential energy of the external loads. For symmetric and balanced laminates, the strain energy in terms of the mid-plane strains and curvatures is given by 0 2 0 2 0 0 0 2 11 22 12 66 2 2 2 0 0 11 22 12 16 26 66 ( ) ( ) 2 ( ) 1 2 ₂ ₂ ₂ a b xx yy xx yy xy xx yy xx yy xx xy yy xy xy A A A A U dxdy D D D D D D ε ε ε ε γ κ κ κ κ κ κ κ κ κ  ₊ ₊ ₊ ₊    = _ _ + + + + +  

∫ ∫

(3.17)

where Aij and Dij are the in-plane and out-of-plane stiffnesses, respectively. For straight fiber panels, the Aij and Dij may be moved out of the integral since they are independent of x and y. However, for variable-stiffness panels the Aij and Dij are functions of the panel coordinates and must remain as part of the integrand.

The potential energy of the external loads is given by

0 0 0 1 ( ) ( ) 2 ( ) 0 0 [ ] a b B x x a y y b x x a W =

λ

F u ₌ +F v ₌ +P

∫ ∫

wdxdy +

λ

F u ₌ (3.18) The displacement functions are assumed as follows,

1 ( , ) t w i i i n w x y a = =

∑

Φ 0 0 1 2 ( , ) _i u_i i t n x u x y b b a ₌ = +

∑

Φ 0 ₀ 1 2 ( , ) _i v_i i t n y v x y c c b = = +

∑

Φ (3.19)

where n_t is the number of terms, ai, bi, and ci are the Ritz coefficients, and 0

0 (x a) b =u ₌ and 0 0 (y b)

c =v ₌ represent the edge displacements. Depending on the choice of the functions w

i

Φ , u i

Φ , and v i

Φ , different boundary conditions can be modelled. Since the panel edges are bounded by different structural elements like stringers, frames and other adjacent panels, the boundary conditions can be very complex. For the Rayleigh-Ritz method, linearly varying deformable straight edges, clamped at (x = 0,

x = a) simulating the frame locations and simply supported at (y = 0, y = b) simulating the stringer locations, are considered. This implies that the out-of-plane displacement of the stringers, due to bending, will be neglected. The boundary conditions can be summarized as follows:

At x = 0,

u0(0,y) = 0 v0(0,y) = c₀ y

b w(0,y) = 0 w,x(0,y) = 0

(39)

Chapter 3 u0(a,y) = b0 v0(a,y) = 0 y c b w(a,y) = 0 w,x(a,y) = 0 At y = 0, v0(x,0) = 0 u0(x,0) = b₀ x a w(x,0) = 0 At y = b, v0(x,b) = c0 u0(x,b) = 0 x b a w(x,b) =0

where a comma (“,”) in subscript indicates derivative with respect to the variable following it. For the assumed boundary conditions the displacement field is given by

( )

1 1

( , ) _ij cos cos sin

i j t t n n _i _x _i _x j y w x y a a a b π π π = =         − + =

∑ ∑

− 0 0 1 1 2 2 ( , ) _ijsin sin i j t t n n x i x j y u x y b b a a b π π = = = +

∑∑

0 0 1 1 2 2 ( , ) sin sin t ij i j t n n y i x j y v x y c c b a b π π = = = +

∑∑

(3.20)

Note that the number of the assumed terms for in-plane displacements u0 and v0 is twice that of out-of-plane displacement in w. This is done to ensure that the in-plane equilibrium is adequately satisfied.

By using the stationary conditions of total potential and minimizing with respect to Ritz coefficients ai, bi, and ci (Note that the matrix aij has been converted to a vector ai containing the elements of aij row by row. For example, for nt = 2, a1= a11, a2= a12, a3= a21, and a4 =

a22. The same conversion applies to bij and cij) the general equilibrium equations for a symmetric and balanced laminated composite plate are obtained (see Appendix A for details), 1 2 (λ F_x λ F_xB) K b_ilub _i K c_iluc _i K_ijluaaa a_i _j 0 − + + + + = 1 0 vb vc vaa y il i il i ijl i j F K b K c K a a λ − + + + =

K a_il _i+K_iklwbab a_i _k +K_iklwcac a_i _k +K_ijklwaaaa a a_i _j _k − Ρ =λ₁ _l 0

(3.21)

The first two equations in Equations 3.21 are linear in the Ritz coefficients bi and ci. Thus, the above three equations can be reduced to a single nonlinear equation by eliminating bi and ci from the third equation using the first two equations. Then the final set of nonlinear equations which define an equilibrium load-deflection path is solved for ai, and by back substitution the coefficients bi and ci are computed. Different techniques are available for the tracing of nonlinear equilibrium paths [34]. In this work, the normal flow algorithm [35] is utilized because of its robustness and efficiency [34].

(40)

The nonlinear analysis performed using the Rayleigh-Ritz method is verified with that of commercial finite element package ABAQUS using S4R element. For this purpose, a sixteen-ply laminate with [±45o]4s lay-up is considered. Each ply has a constant thickness t = 0.01 in.

The composite material is the graphite-epoxy with stiffness and strength properties given in Table 3.1. The initial pressure load is P = 15 psi and the fuselage radius is assumed to be Rf = 100 in. The initial loads Fx and Fy are proportional to the pressure load, as given by Equations 3.14, and the subsequently applied compressive load is FxB = -2Fx. The number of the assumed terms is n_t= 3. The values of the stresses in the principal material directions are

compared for different designs. For the upper ply (z = h/2, on the outside surface of the fuselage) and for the lower one (z = - h/2, on the inside surface of the fuselage) of a square panel (10×10 in), for example, the stresses at the panel center as functions of the load factor are shown in Figure 3.8 and Figure 3.9, respectively. It is clear that the Rayleigh-Ritz analysis model developed agrees well with the finite element analysis. It has been found that the maximum error, in terms of stresses, is within less than 10%, which makes it adequate for preliminary design purposes.

λ₁ σ 1 , p si 00 1 2 5000 10000 15000 20000 25000 R-R FEM λ₂ 0 1 12 λ₁ σ 2 , p si 00 1 2 200 400 600 800 R-R FEM λ₂ 0 121 λ₁

σ

12

,

p

si

00 1 2 2000 4000 6000 8000 R-R FEM 0 λ2 121

Figure 3.8: Stresses in the principal material directions of the [±45o]4s design using R-R

Design tailoring of panels for pressure pillowing using tow-placed steered fibers

Design Tailoring of Panels for Pressure Pillowing Using Tow-Placed

Steered Fibers

Design Tailoring of Panels for Pressure Pillowing Using Tow-Placed

Steered Fibers

Summary

Design Tailoring of Panels for Pressure Pillowing Using Tow-Placed

Steered Fibers

Samenvatting

Aangepast Ontwerp van Panelen voor Drukplooiing door middel van

Vezelgeplaatste Gestuurde Vezels

Contents

Chapter 1

Introduction

1.1

Motivation

1.2

Literature Review

1.3

Pressure Pillowing Problem

1.4

Research Objective

1.5

Thesis Layout

Chapter 2

Variable-Stiffness Composite Laminates

2.1

Introduction

2.2

Linear Variation of Fiber Orientation Angles

( )

(

)

(

)

2.3

Circular Arc Variation (Constant Curvature Path)

{

} {

{

}

}

2.4

Multi-Parameter Fiber Orientation Angle Representation

a

b

x

y

( )

( ) ( )

,

T L

L

θ ξ η

ξ

η

=

∑ ∑

ξ

η

( )

( )

∫

( )

( )

( )

(

( )

( )

)

ξ

ξ

ξ

ξ

ξ

ξ

( )

( )

( )

( )

_{

_}

_∫

_∑