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State Transition Approach to Reliability

Based Design of Composite Structures

Varun Sakalkar* and Prabhat Hajela

†

Rensselaer Polytechnic Institute, Troy, NY, 12180

The paper presents a new approach to the optimal design of composite structural systems in

the presence of uncertainties. A progressive damage propagation model for composites,

based on transformation field analysis (TFA), is used to develop a design for reliability that

includes multiple failure modes typical in laminated composite structural systems. This

analysis involves multiscale computations, with complex interactions between response

parameters at different length scales. Not all failure modes are equally catastrophic and may

only result in degraded structural performance. A design problem formulation based on a

state transition approach is introduced, and allows for the handling of multiple failure

modes in a rational manner. This methodology termed as a system effectiveness approach,

models designer preference as to acceptability of degraded performance, and is used to

develop optimal designs. A comparison of these designs against those obtained from a more

widely used competing risk methodology provides insight into the advantages of the new

approach.

I. Introduction

Analytical and computational modeling is at the very core of designing high performance composite structural

systems for aerospace applications. The behavioral response available from such models is used to provide the ‘what

if’ input to the design process. There is concern that inherent modeling errors, if not quantified appropriately, can

introduce significant deviations in the behavior of the physical system from predicted values. Composite structures,

with distinctly different failure modes and patterns from homologous material systems, are particularly vulnerable to

this deficiency. Failure in composite structures is typically obtained from mechanistic models of damage that span

length scales ranging from the material constituent level to component or macro structural level. Each level of

modeling has uncertainties which influence the behavioral response used to guide design decisions. The proposed

paper will focus on a novel state transition [1] based design tool that is especially applicable to developing damage

tolerant composite structures, with an explicit focus on risk related to operating degraded (by progressive damage)

structural systems. Model development and analysis for the composite systems is based on a transformation field

analysis (TFA) framework.

Methods to characterize progressive damage in a composite have been studied extensively in literature. A number of

models for fiber debonding, matrix cracking, delaminations at crack intersections and free edges, fiber breaks, and

final ply failure have been developed. Models for interfacial debonding and sliding were developed initially by

solving a solitary inclusion problem with appropriate interfacial discontinuities [2-7]. A shear lag model for

debonding with sliding resistance was developed by Hutchinson and Jensen [8]. More recently, Dvorak and

co-workers [9-11] have used the transformation field analysis with equivalent eigenstrain to model the local fields

due to interfacial de-cohesion. For both micromechanical and intermediate scale models, however, it is imperative to

include a non-deterministic treatment so as to model the uncertainty associated with their predictive capability. At a

fundamental level, the relationship between damage/failure modes and the microstructure of composite should

define the manner in which these structures should be designed to resist the long-term influence of mechanical,

hygrothermal and chemical loading. This can be facilitated by adoption of suitable yield/failure criteria based on

measurable mechanical and strength properties of the constituent phases and their interfaces [12-20]. Given

significant variations in these properties, it would be necessary to include stochastic aspects of specific damage

mechanisms, such as Weibull distributions for interface and ply strengths. Some preliminary efforts in this regard

*Corresponding author, PhD Candidate, Department of Mechanical, Aerospace, and Nuclear Engineering, AIAA

Student Member. †Professor, Department of Mechanical, Aerospace, and Nuclear Engineering, AIAA Fellow.

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th4 - 7 May 2009, Palm Springs, California

AIAA 2009-2179

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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have been implemented in modeling of composites with progressive reinforcement debonding [9,11,21]. However,

most widely used analytical or numerical mechanistic models do not explicitly model uncertainty in the prediction

of damage onset or progression. Several recent publications have focused on using probabilistic design approaches

for the design of composite structures. Chamis and Abumeri [22] present a Monte Carlo approach along with FORM

for the dynamic buckling analysis of composite shell structures. Liu and Mahadevan [23] use a Monte Carlo

approach for the progressive fatigue damage analysis of composite laminates. Abumeri and Chamis [24] use a

double-loop strategy (probabilistic constraints within an optimization problem) for the design of composite shell

structures, while Soliman and Kapania [25] use FORM for reliability assessment of first ply failure analysis of

composite cylinders. Arbocz and Hilburger [26] present a probabilistic analysis for the buckling of composite shells

using the Monte Carlo and the first-order second-moment methods. Current work attempts to provide system level

reliability estimation by utilizing the progressive failure characteristics of the micro TFA model.

A critical step towards estimating system reliability is the identification of all possible modes of failure that

contribute significantly to the overall failure probability. For simple systems, or systems that are statically

determinate, it may be possible to identify these limit states by inspection. However, for a complex structure with

high redundancy, this step may prove to be the more difficult [27]. A failure mode approach can be used for this

purpose [28] which identifies all possible failure sequences – this process that may be impractical for large-scale

indeterminate structures [29,30]. However, it is unlikely that all failure modes contribute equally to the overall

system failure. Therefore, much of the recent work focuses on efficient methods to identify only the significant

failure modes of a complex structure. Mahadevan and Raghothamachar [31] present an approach based on a

combination of Monte Carlo sampling and analytical methods to identify significant failure modes of a complex

structure. Mehr and Tumer [32] present a detailed approach to handling reliability issues for large-scale aerospace

systems at NASA. They assert that most generic system reliability techniques, such as the failure mode approach,

may not be adequate for large-scale aerospace systems, where multiple engineering teams work concurrently on

different aspects of the design. Instead, a dedicated approach that caters to their specific requirements may need to

be developed. Vittal [1] describes a state transition approach to evaluate system reliability, where probabilities of a

system transitioning from one failure to another are considered. This approach has been adapted for the design of

laminated composite structures in the current paper.

Computation of the system level reliability involves characterization of uncertainty in damage prediction models as

the first step towards in developing effective tools for designing damage tolerant composite structures. In this

context, probabilistic approaches have shown significant promise as design tools. These approaches are based on

combining nondeterministic analysis methods with design optimization algorithms. Central to this methodology are

methods by which to compute the reliability indices that are used as performance metrics in the design optimization.

Probabilistic design algorithms such as FORM/SORM have been proposed and are widely used in design practice

[28]. The numerical efficiency of these approaches is directly dependent upon the methods used in the deterministic

analysis of composites for damage initiation and progression. The following section summarizes the transformation

field analysis framework that has been shown to be effective in this context, and can be adapted to include

uncertainties in geometric and material properties of the composite structure.

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Applied Laminate

Stress

Stiffness matrix +

Coordinate

transformation matrix

Ply Stresses

Mori-Tanaka Averaging

+

Concentration Factors

Phase Stresses

Failed

Phases

Eigen Stresses

Corresponding to

Failure

Transformation Matrix

+

Influence Functions

UPDATED Phase

stresses

Laminate Strains

Zero Eigen

Stresses

Levin’s Equation

+

Influence Functions

Ply Eigen Stresses

Compliance Matrix +

Coordinate

Transformation Matrix

Eigen Strains

+

UPDATED

Laminate Strains

-Phase material properties

- Laminate geometry

- Volume fraction

- Phase material properties

- Volume fraction

- Failure Modes

-Ultimate Strength Values

- Phase material properties

- Volume fraction

- Phase material properties

- Laminate Geometry

- Volume fraction

1

2

3

4

5

NO

YES

Figure 1. Micro-TFA model

II. Progressive Failure of Laminated Composites

This section summarizes the main features of the micro-TFA model. Fig. 1 shows the flow of computations in

the transformation field analysis based analysis of composite structures. In this figure, the entities in the oval are

intermediate response quantities. The entities in the rectangular boxes are coupling variables and are numbered 1

through 5. Each coupling variable transforms one input response to the corresponding output response. For example,

the stiffness matrix and co-ordinate transformation matrix listed in box 1 transform applied laminate stress into ply

stress. The curly braces indicate the quantities needed to derive each coupling variable. The figure clearly

demonstrates the coupled nature of the analysis.

The fibrous laminate under consideration consists

of n fully bonded thin elastic plies, Fig. 2. The ply

thickness is denoted as ih , i = 1, n, such that ih h=∑ ,

the total thickness of the laminate. Hence, /i ic h h= is

the volume fraction of ply i, such that 1ic =∑ . Two

coordinate systems are defined as shown in Fig. 2, one

is overall ( jx , 1,2,3j = ), and one is local

( kx , 1,2,3k = ). Fiber orientation of the i-th ply is

given by angle iϕ between the local 1x and the overall

1x axes. In the present work, only laminates consisting

of identical plies and symmetric layups are considered.

Membrane mechanical loads confined to the mid-plane

1 2x x of the laminate are considered. For symmetric

laminates, they cause uniform stresses 11σ , 22σ and

12σ as shown in Fig. 2.

Properties of the unidirectionally reinforced plies are given in terms of elastic moduli of the fiber (f) and matrix (m)

materials, and their volume fractions, ,f mv v where 1f mv v+ = . The matrix is assumed to be isotropic with Young’s

modulus mE and Poisson’s ratio mν . The fiber is transversely isotropic with f

LE and f

TE as the longitudinal and

transverse Young’s moduli, fLν and

fTν the associated Poisson’s ratios, and

fLG the longitudinal shear modulus.

Consequently, the unidirectional plies are transversely isotropic with overall elastic properties analogous to those of

the fiber ( LE , TE , Lν , Tν , LG ). The overall properties of each ply can be either measured or computed from the

local properties using a micromechanical model [33].

Figure 2. Geometry and loading of a fibrous laminate.

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4

Assuming that loading includes both membrane forces, [ , , ]1 2 12=NNNN N N N , and bending moments,

[ , , ]1 2 12=MMMM M M M , the classical laminate analysis (CLA) approach allows for the following load-response relations

for an undamaged laminate.

oε ′ ′ ′= + +A B fN MN MN MN M κ ′ ′ ′= + +C D gN MN MN MN M (1)

Where,

2 2 3 3

1 1 1

1 1 1

1 1( ) , ( ) , ( )

2 3

n n n

i i i i i i i i i

i i i

h h h h h h− − −

= = =

= − = − = −∑ ∑ ∑A L B L D L (2a)

2 2

1 1

1 1

1( ) , ( )

2

n n

i i i i i i

i i

h h h hλ λ− −

= =

= − = −∑ ∑f g (2b)

1( ) −′ ′= −A I B B A , 1−′ ′= −B A BD , 1−′ ′= −C D BA , 1 1[ ]− −′ = −D D BA B (3a)

′ ′ ′= − −f B g A f , ′ ′ ′= − −g C f D g (3b)

and the arrays ′f and ′g are overall eigenstrains and curvature. The eigenstrains may be caused by a local event in

any of the constituent materials, including damage or material degradation. The transformation field analysis

approach centers on finding an auxiliary transformation stress field in the plies and/or their fiber and matrix phases

such that the magnitude of the corresponding net stress components, which violate the underlying failure criteria, are

zero. The fiber and matrix stresses for an undamaged laminate subjected to a known overall load are first

determined. Next, the relevant failure criteria are examined for all plies, and the violating stress components are

identified. These stress components are then set to zero and the eigenstresses necessary to satisfy equilibrium are

then computed. These newly computed eigenstresses alter the stress fields in the undamaged constituents and these

have to be re-examined for additional failure. The process is repeated until no further damage is found under the

applied overall loading before incrementing the load to the next level. This coupled multiscale analysis is depicted in

Figure 3.

Constituent

level (micro)

- Volume fraction

- Microstructure assumptions

- Materia l strength values

Lamina level

(macro)

- Ply thickness

- Ply orientation ply

stress

Laminate

load

constituent

stress

Damage criterion +

TFA system of

equationsphase

eigenstress

Laminate level

analysis

Eigenstress calculations

ply

eigenstress

laminate

stress

laminate

eigenstrain

Σ

Total laminate

response Repeat until no further damage at

current load-level

Figure 3. Multiscale TFA data flow

In summary, there are computations at two distinct length scales. At the macro ply and laminate level, the relevant

equations are as follows.

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1

1

1

1

1

, ,

,

{ ) { )

N

i i

i

N

i i i i i i i i

i

N

i i i ij j

j

L L h

L L

P N Q M U

σ ε λ ε σ µ λ λ

σ ε λ ε σ µ

σ λ

−

=

−

=

=

= + = + =

= + = +

= + +

∑

∑

∑

(4)

Here σ and ε are the total laminate stress and strain, respectively; L is the overall stiffness matrix; λ and µ are

the laminate eigenstresses and eigenstrains and subscripted quantities refer to these responses in the ith

ply. At the

material or phase level, a corresponding set of constitutive relations can be written as follows.

1

.

,

, r r r r r r r r

T Ti f f f m m m

r r i r s s

s f m

L L

c A c A

B F

σ ε λ ε σ µ

λ λ λ

σ σ λ

−

=

= + = +

= +

= + ∑ (5)

Here rσ and rε are the local stress and strain fields respectively; rL is the local stiffness matrix; rλ and rµ are the

local eigenstresses and eigenstrains, all for phase “r”; rA and rB represent strain and stress concentration factors for

the fiber and matrix; rsF are transformation influence functions which depend on mutual constraints of the phases

and their elastic moduli. It is clear from the two sets of equations that to complete the ply and laminate stress

computations, the eigenstresses are required. These eigenstresses in turn must be completed by the microscale

computation that takes as an input the ply stresses computed at the macroscale. This coupled analysis is both

complex and computationally cumbersome, requiring the computation of stress distribution functions, stress

concentration factors, and stress and strain transformation influence functions; these quantities are obtained from

geometrical considerations and from the phase material properties. A more complete description of the TFA

approach is available in [34].

A representative stress strain response obtained from a multiscale TFA analysis is shown Fig. 4. It shows the

comparison of the results predicted by the micro-TFA model against available experimental data for a carbon-epoxy

[0, +/-45, 90]s laminate. Fig. 4 shows stress-strain characteristics ε22 vs. σ22, when the laminate is subjected to a

biaxial loading σ22:σ11 = 2:1, the solid blue line represents the results obtained from the micro-TFA model, the black

stars represent the experimental data from Ref. [35,36], and the dotted red line represents the results obtained from

the ply discount model, which has been widely used in the literature [34]. Note that the ply discount model considers

a single ply as the basic element instead of material phases. Additionally, when a failure occurs in a ply, its stiffness

is not taken into account for further analysis.

A key feature of the Micro-TFA damage model developed as part of this research is its ability to reveal

various stages of laminate damage as the load is increased (stress control method). In Fig. 4, the letters m and f

denote matrix and fiber failure, for a particular ply, respectively. Subscripts 11, 22, and 12 denote the mode of

failure in the normal, transverse, and shear directions, respectively. From Fig. 4, we can also observe that the micro-

TFA model predicts results that closely match the experimental data. The first and second failure strains are denoted

by 1 2,

f fε ε respectively.

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Ply discount

1fε 2fε

Figure 4. Stress strain response for a [0, +/-45, 90]s laminate with progressive damage

2.1 Response Approximation using Radial Basis Neural Network

The computational approach described in the previous section is integrated into a multi-parameter reliability

analysis of a composite structural system. This approach is computationally demanding, and must be repeated

several times to evaluate the performance functions required for reliability analysis. Further integration of the

approach into an iterative design environment would be computationally prohibitive, and surrogate models of

analysis provide an alternative approach. The present work adopts a surrogate model based on a Radial Basis

Function (RBF) neural network, and this approximate model is used in lieu of the actual micro TFA evaluation.

Details of the RBF model functions are not included in this paper but are available in Ref. [37-39].

Building a RBF model requires that training sets or samples be first created. Several exact computations of the

function ( )g X were performed so as to cover the range of values of x that span the design space. The number of

sampling points required to accurately model the performance function is dependent on the number of random

variables, the nonlinearity of the problem considered and the assigned computational accuracy. Guidelines on the

“design” of sampling points can be found in various statistical textbooks and neural network monographs [40-42].

These sampling data points were then used as the training and testing data in the RBF training so as to

approximately represent the performance function ( )g X . A sample set of results showing the training and validation

mean squared error for the approximation of the first and second failure strain against exact TFA computations is

shown in Fig. 5. The micro TFA model was setup for a [0, +/-φ , 90] s laminate with φ and fiber volume fraction

( rc ) as the independent variables. The radial basis network modeled the TFA analysis and yielded a surrogate

approximation between input variable φ and rc , and the output quantities1 2,

f fε ε . The fiber and matrix material

properties assumed for the TFA calculations are shown in Tables 1a and 1b. A total number of 225 training points

were selected and 40 validation points were used in testing the approximate model. A small training and validation

error at convergence (given in Fig. 5) shows that the simulated RBF network output compares well with the actual

TFA solution.

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Table 1a Mechanical properties of fiber [34]

Material LE (GPa) TE (GPa) LG (GPa) TG (GPa) Lν uσ (GPa) fρ (kg/m

3)

AS4 Carbon 26.2 262.6 15 7 0.2 3.35 24

Table 1b Mechanical properties of matrix [34]

Material E (GPa) ν uTσ (MPa) uCσ

(MPa) uτ (MPa) Lη mρ (kg/m

3)

3501-6 epoxy 4.2 0.34 69

250 50 0.3

12

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200 250

No

rma

lize

d M

ean

sq

ua

re e

rro

r (%

)

Epoch

Epsilon_f1 (Strain at first failure)

Epsilon_f2 (Strain at second failure)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25

Norm

alize

d M

ean

Sq

ua

re E

rro

r

(%)

Epoch

Epsilon_f1 (Strain at First Failure)

Epsilon_f2 (Strain at Second Failure)

Figure 5a. Training Error of the Radial Basis network Figure 5b. Validation Error of the Radial Basis

network

III. Reliability based design optimization

The design optimization problem involving nondeterministic design variables and constraints requires a

modified problem formulation. A system objective J is to be minimized as shown in Eq. (6), and is a function of the

design variables x that may be nondeterministic in nature. In such cases, the statistical characteristics of the function

J would be considered as the design objectives. The constraint vector g is also a function of these design variables,

and in contrast to a deterministic constraint, the optimization problem formulation considers bounds on probability

of constraint violation. In the following problem statement, fP is the probability of failure and is bounded by an

upper limitallowfP . Other measures of reliability can also be used in lieu of the probability of failure.

min ( )

. . ( ( ) 0)

x

allowf f

J x

s t P g x P≤ ≤ (6)

The computation of the probability of failure and other estimates of reliability has been researched extensively.

A widely used approach is the first-order reliability method that is both efficient and effective. A first-order second-

moment approximation of nonlinear limit states using normal variables (or equivalent normal variates of non-normal

variables) is typically required in these approaches. The failure probability is estimated by formulating and solving

an optimization problem. In this approach, all the variables are transformed to a standard normal space as

' ( ) /x xx x µ σ= − (7)

where x are the original normal variables and 'x are the standard normal variates. In this reduced space, the

reliability index is defined as

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( '*) ( '*)Tx xβ = (8)

Where '*x is the minimum distance point (most probable point of failure or MPP) from the origin of the reduced

space to the limit state boundary. An optimization problem [28] can be formulated to obtain this reliability index as

'min ( ') ( ')

. . ( , ') 0

T

xx x

s t g x x

β =

=

(9)

Finally, the failure probability is approximately calculated using the cumulative distribution function (CDF) of the

standard normal distribution as

( )fp β= Φ − (10)

It is not sufficient, however, to only compute probabilities of failure. The subject of estimating system reliability is

more involved, and probability of failure is one step in that estimation.

3.1 System Reliability Assessment

The component reliability estimation described in the previous section must be extended to include situations where

a system of multiple interacting components is to be considered. A structural system composed of more than one

structural element may also include multiple modes of failure, such as those resulting from large deflections, or from

buckling, shear, and corrosion, among others. Evaluation of the reliability of such a system involves multiple limit

state functions, and obtaining independent failure probability estimates for each constraint. Component-level

reliabilities can be estimated using the techniques described above. The final system reliability is likely to be a

function of the probabilities for the individual modes of failure, and any correlation between the different modes.

To obtain estimates of system failure probabilities, the traditional approach is to model the system as either a series

or a parallel system [43,28]. Series systems are those where the failure of the system occurs when any of the

individual limit states is violated. Such systems are also called weakest link systems: a statically determinate

structure is a typical example. In this case, the probability of failure is the union of the individual failure

probabilities. If iEF is the failure event corresponding to the i-th failure mode, the system failure probability is

given as [28,29]

( )series iP P EF= ∪ , (11)

The probability of the union is not trivial to calculate because of the presence of joint probabilities of one or more

events. Alternatively, system failure might occur only if some or all the elements fail (parallel system), and for

which case the system failure probability is the probability of intersection of the individual failure events as follows.

( )parallel iP P EF= ∩ , (12)

Statically indeterminate or redundant systems can be termed parallel systems. In practice, a structural system might

need to be decomposed into a set of series or parallel systems, where the final system reliability (or failure

probability) evaluation requires repeated applications of the above relations. Several approximations to the above

series and parallel reliabilities are in use. Conservative first-order series bounds are given as [27,28]

[ ] [ ]max ( ) 1 1 ( )i series iP EF P P EF≤ ≤ − ∏ − , (13)

Narrower second-order bounds that are widely used can be found in Haldar and Mahadevan [28]. For a parallel

system, first-order bounds are given as

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[ ] [ ]1

max 1 1 ( ) , 0 min ( )

n

i parallel i

i

P EF P P EF

=

− − ≤ ≤

∑ , (14)

where the upper bound is exact if the events are perfectly correlated. It is important to note that if the individual

failure probabilities in the above equations are obtained using a first or second order approximation to the limit state

(FORM/SORM), then the accuracy of the above bounds may be affected further.

In some applications it is important to distinguish between the different failures modes. For purposes of improving

reliability, it is essential to identify the cause of failure down to the component level and, in many applications,

down to the actual physical cause of failure.

The failure time of a system with two or more failure modes can be modeled with a series-system or competing

risk model. Each risk is like a component in a series system. When one component fails, the system (i.e., product)

fails. Each unit has a potential failure time associated with each failure mode. The observed failure time is the

minimum of these individual potential failure times. Most generic system reliability techniques, such as the failure

mode approach, may not be adequate for large-scale aerospace systems, where multiple engineering teams work

concurrently on different aspects of the design. Instead, a dedicated approach that caters to their specific needs may

need to be developed. Vittal [1] describes a state transition approach to evaluate system reliability or effectiveness,

and where probabilities of a system transitioning from one failure to another are taken into consideration.

3.2 State Transition Method for System Effectiveness

The state transition method is an alternative approach for estimating risk in a system with multiple failure modes

(limit states). The concept is loosely patterned on ideas developed in the military operations research community in

the field of system effectiveness analysis. The approach is particularly significant in that it can compute the

probability of a system existing in different states (failure modes) as a function of system operation history and

failure mode relationship. This makes it possible to develop accurate risk models and “manage” the life of the

system in a manner that maximizes usage and minimizes risk. In the context of composite design using TFA, each

failure mode corresponds to progressive damage in the various constituent plies.

One of the early attempts to quantify system effectiveness is available as a report published by the Weapon

Effectiveness Industry Advisory Committee (WSEIAC) in 1965 [44]. Effectiveness is defined as a product of

availability (A), the dependability (D), and the capability (C) of the system under consideration, and expressed

mathematically as the following matrix product.

E A D C= × × (15)

In expanded form, this product is obtained as follows.

{ }

11 12 1 1

21 22 2 2

1 2 3

1 2

j

j

i

ji i ij

D D D C

D D D CE A A A A

CD D D

=

�

�…

�� � � �

�

(16)

In Eq. 16, ‘E’ is the System effectiveness and is a measure of the extent to which a system will perform its mission;

it is function of the probability to survive various states of damage, as well as the consequence of arriving in a

particular state (condition). ‘A’ is the availability, a measure of the system condition at the start of a mission. For the

purpose of the system degradation, it is the probability that a system is put into service in various conditions ranging

from new (full life), to a fully degraded state with no life. Ai is the probability of a system starting its mission state.

In context of composites it can be the probability of a particular ply to be available at any point of time during its

service life. ‘D’ is the dependability indicated by a transition matrix, where elements of the matrix dij represent the

probability of a system starting in state ‘i’ and ending in state ‘j’. ‘C’ is the capability, a measure of the system’s

ability to achieve a mission objective. For system-lifing applications, a system with full life will complete its desired

objective that may involve supporting given loads or surviving applied temperature gradients.

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This model is extremely useful in that it provides a snapshot of the system or part in terms of various operational

states or part conditions. In many problems of probabilistic design, the probability of interest is the reliability of a

system, and the algorithm (as discussed in section II) usually produces a static number. By formulating the limit

states for each failure mode, a given design can be evaluated in terms of its probability of surviving these failure

modes. This clearly recognizes the fact that all failure modes are not as severe and risk to the system depends on the

type of failure as well as its combination with other failure modes.

In the context of a composite design problem, each of the failure modes like the ones described in Fig. 4 can be

included in the design problem. The feasibility of this approach can be illustrated using a composite design problem

involving two failure modes as follows.

Mode I: The failure strain at the point of first failure ( 1fε in Fig. 1) does not exceed an allowable value (0.5%).

Mode II: The failure strain at the point of second failure ( 2fε in Fig. 1) does not exceed an allowable value

(0.75%).

The failure mode II is assumed to be more critical. Let PI, PII be the probability of failure due to failure mode I, II

respectively. The following state transition tree can be developed to indicate the nature of the progressive damage.

Composite

Laminate

Fail in

mode II

Fail in

mode I

State 3

State 2

State 1

Yes

PII

No

1-PII

No

1- PI

Yes

PI

Figure 5. State Transition Tree

The various states are as follows.

State 1 – The undamaged laminate.

State 2 – A benign failure where the laminate has excessive preliminary failure strain.

State 3 – A complete failure where the laminate has excessive secondary failure strain.

The transition rules diagram can be expressed as shown in Fig. 6. It shows how the system can either stay in its

present state or transition to one of the others with a certain probability.

State

1

2

3

Figure 6. State Transition Diagram

The state transition matrix then takes the following form, representing the probabilities of transitioning to one of the

other states.

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11

11 12 13

22 23

33

0

0 0

d d d

D d d

d

=

(17)

In the above, the components of D can be expressed as:

11 12 13

22 23 33

(1 )(1 ); (1 ) ;

(1 ); ; 1.0

I II II I II

II II

d P P d P P d P

d P d P d

= − − = − =

= − = = (18)

Note that the probabilities of modes I and II defined here as PI and PII are computed using a typical FORM

approach. The starting vector A can be assumed to be [ ]1.0 0 0 , i.e., indicating that the laminate starts with no

damage at all. The capability vector (C) can be assumed of the form [ ]1.0 0.5 0 which means that if the laminate

is in state 1 than it is 100% capable of accomplishing the mission; in state 2 it has 50% capability, and in state 3, it is

incapable of performing the desired mission. Here, the second entry of the capability vector ( 2c ) can take any value

from 0 to 1 and the effect of this choice on the overall results will also be shown in a successive section.

Using classical system reliability methods, reliability of a system with two or more failure modes can be modeled

with a series-system or competing risk model. For the simple case of two failure modes as in this problem, the

system reliability would be written as follows.

(1 )(1 )system I IIR P P= − − (19)

The proposed system effectiveness technique can be extended to handle multiple failure modes; it provides a

convenient way to handle the interaction between the modes that is not as obvious in probabilistic reliability based

approach. For the design of composite systems where damage in various modes may occur progressively, correlating

the individual failure modes needs to be given relative importance. Not all damage modes will be equally

catastrophic for system performance, and more meaningful composite structural designs are possible by constraining

the system effectiveness in contrast to designs based strictly on system reliability.

The calculation of individual failure mode probabilities is done by considering the limit state corresponding to each

failure mode and solving them using a FORM based approach explained in the previous section. For example, the

probability of failure corresponding to the first failure mode limit state is shown below in Eq. 20.

1 1

( )f allow

I fP P ε ε= ≤� � (20)

Here, the ~ overscript denotes a random variable and 1fε� and 1allowε� are the random variables corresponding to first

failure strain and the allowable strain at first failure in the laminate. Pf is the probability of failure of this limit state

obtained using FORM.

3.3 Composite System Reliability and Effectiveness Design Formulations

In context of composite design optimization, a wide array of design problem formulations can be deployed. The

optimization can be single or multi-objective, and can be a discrete or continuous programming problem. In the

presence of uncertainty in design variables and problem parameters, the system reliability or system effectiveness

constraints can be incorporated as part of the design statement.

The suitability of system effectiveness approach in probabilistic optimization is illustrated through a comparative

study of deterministic and nondeterministic design optimization solutions as follows.

Case A: Deterministic optimization formulation - the composite design problem is first formulated as a

deterministic multiobjective optimization problem that seeks to minimize the weight of the laminate and maximize

American Institute of Aeronautics and Astronautics

12

the modulus of elasticity ( 11E ) in the direction of unidirectional in-plane loading (11-direction from Fig. 1). The

design variables are chosen as the volume fraction of the fiber material ( rc ) and the ply angle φ in a [0, +/-φ , 90]s

laminate. The design problem statement is explicitly stated as follows.

11

1 1

2 2

:

:

. .

0.2 0.9

30 60,

f allow

f allow

r

Min Weight

Max E

s t

c

Integer

ε ε

ε ε

φ

≤

≤

≤ ≤

≤ ≤

(21)

The ply angles φ are deterministic integer design variables bounded between 30 deg and 60 deg. This is a design

problem with both continuous and integer design variables. A genetic algorithm (GA) based optimization solution

was implemented in this problem given the mixed nature of the design variables and expected multimodalities in the

design space. The GA based approach required a very large number of function evaluations and the use of the RBF

approximation was critical to this implementation. The RBF model describing the relationship between rc andφ to

the outputs 1fε and 2fε was used in the optimization. A flowchart showing the process of creating the RBF based

surrogate model and its subsequent integration into the GA based search is shown in Fig. 7. The multiobjective

problem is solved using a weighted objective function approach described in [46]. This formulation gives the Pareto

optimal set of solutions shown in Fig. 8, and which will be used for comparison with nondeterministic cases.

Design variables (x)

E.g. ,rc φK samples

using Space

filling Latin

Hypercube

design

ith iteration

Micro-TFA

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 130

35

40

45

50

55

60

cr

φ

(i+1)th iteration

ith Response (y)

E.g. First and

second failure

strains

Generating Training and validation data set for Neural Network

Generate

approximation

to y=

Using a Radial

Basis Function

Network

( )f x

GA based

optimizer for

mixed integer

programming

Figure 7. High fidelity Global optimization using a Radial basis function

American Institute of Aeronautics and Astronautics

13

3 4 5 6 7 8 9 10 110

1

2

3

4

5x 10

10

Weight (kg)

E11 (

lam

ina

te)

(N/m

2)

Pareto Frontier for a deterministic optimization formulation

Figure 8. Pareto optimal solutions for the optimization formulation in Eq. 21

Case B: Nondeterministic optimization formulation (reliability constraint) - the composite optimization problem is

formulated as a non deterministic multiobjective optimization problem seeking to minimize the mean weight

(Weightµ ) of the laminate while also maximizing the mean axial modulus of elasticity ( 11Eµ ) of the laminate. The

design variables are the volume fraction of the fiber material and the ply angle φ in a [0, +/-φ , 90]s laminate. The

design problem statement is explicitly stated as follows.

11

1 1 2 2min

:

:

. .

; (1 )(1 ) (1 ( ))(1 ( ))

0.2 0.9

30 60,

f allow f allowsystem system I II f f

r

Min Weight

Max E

s t

R R where R P P P P

c

Integer

µ

µ

µ

ε ε ε ε

φ

≥ = − − = − ≤ − ≤

≤ ≤

≤ ≤

� � � �

�

(22)

Here, rc� is a normal random variable ( ,0.01)r rc N cµ=� � with its mean value ( rcµ� ) as the design variable in the

optimization problem. The allowable failure strains follow a normal distribution, with values

as 1 2(0.5,0.05); (0.75,0.075)allow allowN Nε ε= =� � . The system reliability (using Eq. 19) and system effectiveness

(using Eq. 15) are computed for each of the solutions on the Pareto front. The results obtained are presented in Table

2 and also shown on Fig. 9.

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Table 2 Pareto solution for Case B formulation

Volume Fraction Ply angle Weight Laminate Axial Stiffness Rsystem Esystem

0.898 57 10.780 4.979E+10 1.000 1.000

0.898 56 10.775 4.949E+10 1.000 1.000

0.870 57 10.445 4.114E+10 1.000 1.000

0.789 57 9.465 2.766E+10 1.000 1.000

0.757 56 9.081 2.456E+10 0.997 0.999

0.734 56 8.811 2.278E+10 0.982 0.991

0.730 55 8.763 2.248E+10 0.975 0.986

0.727 57 8.720 2.224E+10 0.969 0.983

0.717 57 8.604 2.158E+10 0.943 0.970

0.714 55 8.563 2.135E+10 0.924 0.958

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.01

7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0

E_sy

stem

, R

_sy

stem

Weight (Kg)

R_system

E_system

Figure 9. Pareto solution for Case B formulation

Case C: Nondeterministic optimization formulation (effectiveness constraint) – this case involves the solution of a

multiobjective optimization problem as before, with the addition of system effectiveness based constraints. The

problem formulation is as follows.

11

min

:

:

. .

0.2 0.9

30 60,

system

r

Min Weight

Max E

s t

E E

c

Integer

µ

µ

µ

φ

≥

≤ ≤

≤ ≤

�

(23)

American Institute of Aeronautics and Astronautics

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This problem formulation is the same as in case (B) except that reliability constraint is replaced by effectiveness

constraint. The results for this problem formulation are summarized in Table 3 and depicted graphically in Fig. 10.

Table 3 Pareto solution for Case C formulation

Volume Fraction Ply angle Weight Laminate Axial Stiffness Rsystem Esystem

0.899 56 10.784 4.977E+10 1.000 1.000

0.896 56 10.755 4.886E+10 1.000 1.000

0.880 55 10.563 4.358E+10 1.000 1.000

0.863 56 10.354 3.925E+10 1.000 1.000

0.777 57 9.330 2.648E+10 0.999 0.999

0.730 57 8.763 2.250E+10 0.975 0.987

0.728 57 8.741 2.237E+10 0.972 0.985

0.715 57 8.584 2.147E+10 0.938 0.967

0.710 56 8.517 2.111E+10 0.915 0.956

0.695 57 8.337 2.021E+10 0.825 0.909

0.80

0.85

0.90

0.95

1.00

1.05

7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0

Esy

stem

, R

syst

em

Weight (Kg)

R_system

E_system

Figure 10. Pareto solution for Case C formulation

3.4 Discussion

In Tables 2 and 3, it is evident that the reliability is bounded between 0.825 and 1.0, and effectiveness is bounded

between 0.909 and 1.0. For the same design on the Pareto front, the reliability is always lower than effectiveness.

This is due to the fact that competing risk model emphasizes both failure modes equally whereas the state transition

approach places less emphasis on the first failure mode. Thus, the effectiveness metric provides a more realistic

estimate of the true system capability and is based on the acceptance by the designer that the residual load carrying

capacity of the structure continues to be significant. This is also demonstrated in the second formulation with the

effectiveness constraint. The designs generated in Tables 2 and 3 are in general lighter than the designs generated by

the reliability based approach. In this case study, first failure strain was considered a benign failure, i.e., from

operational perspective a laminate with initial strain failure has still completed a part of its mission. This “risk

reduction” is captured by the effectiveness based formulation and is finally reflected in the optimal designs

produced.

American Institute of Aeronautics and Astronautics

16

Another aspect of effectiveness approach is to understand the impact of the terms in the capability vector “C” (only

2c can be chosen independently in the case of 2 failure modes). A detailed sensitivity study was done by varying 2c

from 0.2 to 0.9 for both Case B and C formulations. The resulting values for systemE and systemR was plotted against

each other as shown in Fig. 11 and Fig. 12, for Case B and C formulations, respectively. This clearly shows that as

2 0c → the systemE values approach systemR . In other words, for a Pareto design point the corresponding systemR is the

lower bound to systemE for all values of 2 0c > . Thus, the value of 2c can be a designer preference to penalize or relax

a particular failure mode based on prior knowledge of the system, expert input or system configuration. In general,

for N failure modes there would be 2N-1 states. So, the column vector (2 1,1)NC − will have 2N-3 values which can be

independently chosen by the designer as per their preference.

0.9 0.92 0.94 0.96 0.98 10.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Esystem

Rsyste

m

Comparison of Esystem

and Rsystem

for formulation "B"

c2=0.2

c2=0.3

c2=0.4

c2=0.5

c2=0.6

c2=0.7

c2=0.8

c2=0.9

c2=0.2

c2=0.5

c2=0.9

Figure 11. Sensitivity of solutions to designer definitions of system capability (Case B)

American Institute of Aeronautics and Astronautics

17

0.9 0.92 0.94 0.96 0.98 1

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Esystem

Rsyste

m

Comparison of Esystem

and Rsystem

for formulation "C"

c2=0.2

c2=0.3

c2=0.4

c2=0.5

c2=0.6

c2=0.7

c2=0.8

c2=0.2

c2=0.5

c2=0.8

Figure 12. Sensitivity of solutions to designer definitions of system capability (Case C)

IV. Closing Remarks

The paper describes a new optimization problem formulation for laminated composite structures that are subject to

progressive failure. The TFA approach for analysis is shown to be effective in predicting the failure loads, and also

accounting for residual load carrying capacity of damaged composites. The method can be extended to provide a

probabilistic analysis tool that takes into consideration uncertainty in material and geometric parameters of the

laminate. However, as shown in this paper, the approach is computationally cumbersome due to coupling across the

length scales. In the presence of uncertainties, the issue of computational costs is further exacerbated, and defies

integration into an optimization framework without recourse to the use of function approximations. In the present

paper, a radial basis neural network is used to construct a surrogate model of the TFA approach that was coupled to

a genetic algorithm based optimization methodology. The genetic search was preferred over traditional

mathematical programming due to the mixed design variable space involving both continuous and discrete variables,

and also because of the expected multimodal behavior of the design space.

The optimization problem formulation is based on a concept of system effectiveness that models the degraded

behavior of a structure undergoing progressive damage during service. System effectiveness reflects the ability of

the structure to perform a required mission, and is a convenient tool to model designer preference in the optimization

problem formulation. The method uses routine uncertainty modeling tools to build an effectiveness measure that can

be used as an objective criterion in the optimization formulation. The paper compares numerical results obtained in

this new approach against those derived from more traditional system reliability formulations. The latter are known

to yield more conservative designs, and not as amenable to including designer preferences. Additional work on

implementing the proposed numerical scheme to larger scale design problems is currently in progress, and will be

reported in another publication.

V. Acknowledgements

The support received for this work under a grant from the AFOSR (Award no: FA 9550-05-1-0140) is gratefully

acknowledged.

American Institute of Aeronautics and Astronautics

18

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