aerospace and ocean engineering department a new scheme for the optimum design of stiffened...

Aerospace and Ocean Engineering Department

A New Scheme for The Optimum Design of Stiffened Composite Panels with Geometric Imperfections

By

M. A. Elseifi

ZGürdalE.Nikolaidis

Sponsored in part by NASA Langley Research Center


Outline

Introduction

Effect of imperfections on the nonlinear response of stiffened panels.

Probabilistic and Convex Models

Current and suggested models for the uncertainty in the imperfections.

Manufacturing Model

One-dimensional curing model and process induced imperfections.

Design Optimization Problem

Results and Conclusions


Non-Linear Elastic Behavior of Stiffened Panels

Local Postbuckling

Panel buckles into half-wavelengthsequal to the width between stiffeners.

Global (Euler) Postbuckling

Panel buckles into one half-wavelengthalong its length.

Modal Interaction

Local and global modes have equal critical loads.

Introduction

GlobalPostbucklingLocal Postbuckling

1

P/Pcr

/ cr1

ElasticLimit

ModalInteraction


Current Scheme for Design of Panels with Imperfections(Perry, Gürdal, and Starnes, Eng. Opt. 1997)

DesiredResponse

Estimate fornominal imperfection

profile

Nonlinear analysisand design

optimization OutputDesign

+


Geometrically Nonlinear Analysis of Stiffened Composite Panels(Stoll, Gürdal, and Starnes, 1991)

NLPAN: Non-Linear Panel Analysis,

- Finite strip method

- Linked plates of any cross-section

Displacements are assumed to have the following general form :

N

1i

N

1jijji

N

1iiiL uqququu }{}{}{}{

The imperfection shape is expressed as :

{ } { }u q uoio

ii

N

1


Addition of a Model for Uncertainties in Imperfections(Elseifi, Gürdal, and Nikolaidis, AIAA J., 1999)

ImperfectionModel

Estimate forImperfectionParameters

WeakestPanel Profile

+

DesiredResponse




Non-probabilistic Convex Model (Ben-Haim and Elishakoff, J. of Applied Mech., 1989)

The objective is to determine the minimum elastic limit load.

Let be a vector whose components are the amplitudes of the N dominant mode shapes that represent the initial imperfection profile of the panel.

Let E ( ) represent the elastic limit load of the panel whose initial imperfection profile is given by .

Let be a nominal imperfection profile, which depends on the manufacturing process.

q

q

q

oq

N

iii

o uqu1

}{}{


Assume that varies on an ellipsoidal set of initial imperfection spectra :

where the size parameter and the semiaxes n are based on experimental data. Thus Z () can be chosen to represent a realistic ensemble of panels.

The elastic limit for an initial imperfection spectrum to first order in is:

nq

N

n n

ooo

q

Eqq

1

)()(

N

n n

nZ1

22

2

:),(

oq


Explicit relationship between the minimum elastic limit and the parameters defining the initial imperfections spectrum

is the elastic limit of the “weakest” panel. Z is an ensemble, which has been constructed to represent a realistic range of panels.

Evaluate the minimum elastic limit as varies on the previous convex set.

])([min),(1

),(nq

N

n n

o

Zo

q

Eq

N

nq

nn

oo

q

Eq

1

2)()(),(

q1

q2


Probability Distribution Function of Elastic Limit for = 0.02405

0

0.2

0.4

0.6

0.8

1

1.2

Elastic Limit

Probability o

f F

ailure

Convex Model Prediction = 0.877492

oq1


ConvexModel

ImperfectionParameters

Convex Model in the Design Loop

oq

*Weakest

Panel Profile

+

DesiredResponse




ConvexModel

ImperfectionParameters

Closed Loop Design Scheme with Manufacturing Model

oq

*Weakest

Panel Profile

+

DesiredResponse



ManufacturingModel

DesignParameters


One-Dimensional Curing Model for Epoxy Matrix Composites (Loos and Springer, 1983)

No resin flow (top or edge).

No energy transfer by convection.

No chemical diffusion.

No void formation.

Bleeder

Composite

Tool plate

To , Po

Lb

Li

To , Po

xz

Process Induced Imperfections in Laminated Composites(Elseifi, Gürdal, and Nikolaidis, 1998)


Energy Equation

Hz

TK

zt

cT .)(

)(

Density of composite

c Specific heat of composite

K Thermal conductivity perpendicular to plane of composite

T Temperature of composite

H.

Rate of heat generation by chemical reaction. Function of the degreeof cure

Initial Conditions

0

)(

zTT i

0

0

t

Lz

Boundary Conditions

LzattTT

zattTT

u

l

)(

0)(

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Mom

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men

ts

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Panel Profile Generation

x

yz

Mesh Point (i,j)

Only imperfections in skin.

The skin surface is discretized into a number of mesh points.

The 1-Dimensional curing simulationis applied at every mesh point.

8

1ii b

y

l

xiqw

sinsin

Curvatures-deflection relationships

8

1

2 sinsin)(i

ix b

y

l

xi

l

iq

8

1

2 sinsin)(i

iy b

y

l

xi

bq

8

1

coscos))((2i

ixy b

y

l

xi

bl

iq

Assumed imperfection profile shape


Experimental Validation


00.2

0.40.6

0.80

0.2

0.4

0.6

0.8-1

0

18X, Y<0

0.20.4

0.60.8

Model Prediction

Experimental Scanning


Simulation of Different Sources of Imperfection

Only uncertainties incurred in the constituent (primitive) material properties.

A random number is generated at each mesh point.

The material properties variations are assumed Gaussian.

The random numbers generated are independent from one point to the other.

Material Property

Prob. Dist.


Results

X Y

Z

1168 mm

37 mm

178 mm

Stiffened Panel Geometry and Dimensions

Compressive Design Load : 56,000 N

Inplane Shear Design Load : 14,000 N

Material: Hercules AS4/3502 Graphite/Epoxy

Design Problem

Minimize Panel Weight : W = t ( ns As + nb Ab)

Such that : No failures for a balanced symmetric laminate


First Proposed Closed Loop Design Scheme

Manufacturing Model

ConvexModel

Initial Population

NonlinearAnalysis

Fitness Processor

Genetic Processors

NO

NO

DesiredFailure

Load


Second Proposed Closed Loop Design Scheme

GeneticProcessors

Initial Population

ManufacturingModel

ConvexModel

NonlinearAnalysis

Fitness Processor

NO


Results of the First Design Scheme

Starting Imperfection Profile (Random)

First Optimization Result

Panel Mass(Kg)

failureP

(Newton) PliesLaminate

S: Skin B: Blade

0.523743 59500 (S)-[6]

(B)-[26]

s]0[ 3

s]45/0/90/0/90/0/45[ 62


First Optimum Cured Profile

Failure Load : 40740 N

Second Optimization Result

Panel Mass(Kg)

failureP


S: Skin B: Blade

0.539223 60000 (S)-[8]

(B)-[18]

s]90/45/45/0[

s]90/45/90/45/0/90/90/45/45[


Second Optimum Cured Profile

Failure Load : 53200 N

Third Optimization Result

Panel Mass(Kg)

failureP


S: Skin B: Blade

0.611980 60000 (S)-[10]

(B)-[16]s]90/0[ 23

s]45/45/90/0/45/0[ 2

Actual Failure Load : 56200 N


Results of the Second Design Scheme

Panel Mass(Kg)

failureP


S: Skin B: Blade

0.497427 57000(56320)

(S)-[10]

(B)-[4]

s]45/0/45[ 3

s]0/90[

0.535611 60000(58200)

(S)-[10]

(B)-[8]s]45/0[ 3

s]0/90/0[ 2

0.535611 57000(56520)

(S)-[10]

(B)-[8]s]45/0/45/0[ 2

s]0/90/0[ 2

Actual failure load of the optimum : 56320 N


Comparison with existing optimization results

Design ToolPanel Mass

(Kg)failureP


S: Skin B: Blade

NLPANOPT

withImperfection

0.583725 31000

(S)-[8]

(B)-[16]s]0/90/45/45[

s]0/90/45/45[ 6

Scheme 1 0.611980 56200

(S)-[10]

(B)-[16]

s]90/0[ 23

s]45/45/90/0/45/0[ 2

Scheme 2 0.554704 56320

(S)-[10]

(B)-[10]

s]45/0/45/0[ 2

s]0/90/0[ 22


Concluding Remarks

A convex model has been introduced for the analysis of uncertainties in geometric imperfections.

A one-dimensional curing model has been extended to calculate the process-induced curvatures in epoxy matrix composite. A procedurewas suggested for the incorporation of uncertainties in the primitive material parameters as a source of imperfections.

It was shown that panels designed with empirically assumed imperfectionswere not able to carry their design load when applied along with acorresponding realistic imperfection profile.

It was demonstrated that incorporating the panel’s manufacturing information early in the design process results in panels capable of carryingrequiredloading without much increase in weight.

aerospace and ocean engineering department a new scheme for the optimum design of stiffened...

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