Reliability of fiber-reinforced composite laminate plates

Reliability of fiber-reinforced composite laminate plates

Dan M. Frangopola,*, Sebastien Recekb

aDepartment of Civil, Environmental and Architectural Engineering, Campus Box 428, University of Colorado, Boulder, CO 80309-0428, USAbFrench Institute for Advanced Mechanics, IFMA, Campus des Cezeaux, Aubiere 63170, France

Abstract

Many engineering structures ranging from aircrafts, spacecrafts and submarines to civil structures, automobiles, trucks and rail vehicles,

require less weight and more stiff and strong materials. As a result of these requirements, the use of composite materials has increased during

the past decades. In fact during the past five years, we have witnessed exponential growth in research and field demonstrations of fiber-

reinforced composites in civil engineering. Manufacturers and designers have now access to a wide range of composite materials. However,

they face great problems with forecasting the reliability of composites materials. Due to the differences among the properties of materials

used for composites, manufacturing processes, load combinations, and types of environment, the prediction of reliability of composites is a

very complex task. In this study, the reliability of fiber-reinforced composite laminate plates under random loads is investigated. The

background of the problem is defined, the failure criterion chosen is presented, and the probability of failure is computed by Monte Carlo

simulation.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Fiber-reinforced; Composite materials; Laminate plate; Reliability; Monte Carlo simulation

1. Introduction

A composite is made of two types of materials: matrix

and reinforcement. Often, composites use fibers as

reinforcement, such as graphite, glass or polymer fibers.

Fiber-reinforced composites (FRCs) are the most commonly

used composite materials. Usually, fibers have different

mechanical properties than the matrix. FRCs are exhibiting

directional characteristics under loads. This is mainly due to

their naturally anisotropic material behavior.

There is a wide variety of FRCs. The three types of fibers

quoted previously define the three largest families of

composite materials. The type of matrix and the shape of

the fibers provide different materials. For example, fibers

can be put in only one direction in the material or can be

woven or even cut and randomly distributed in the matrix.

Mechanical properties of FRCs are dependent on the

shape of fibers used within the material. For example, if the

fibers are cut in small parts (1–30 mm usually) and put

randomly in the matrix, the resultant composite will have an

isotropic behavior. This type of fiber-reinforced material is

used mainly on low performance parts produced in great

series. If FRCs use fibers that run all the length of the

composite in only one direction, the mechanical properties

will vary with the direction of load. The composite will have

stronger mechanical properties in the direction of fibers and

weaker in any other direction.

There have been a large number of studies on stress and

strength analysis of FRCs and various failure criteria are

used [2–4,6,8,10–14]. Most of these studies are determi-

nistic. However, some of them use statistical models for

describing the failure process but neglect the randomness in

loading. Since it is generally recognized that stress, strength

and failure of FRCs are non-deterministic, concepts and

methods of probability have to be used for the evaluation of

reliability and development of acceptance and design

criteria for FRCs [1,9].

This study focuses on fiber-reinforced composite

materials with unidirectional fibers, especially on laminate

plates under random loads. Since a macro-mechanical level

of analysis is used, only the properties along and

perpendicular to the fiber direction are considered (i.e.

microstructure of the lamina is ignored). At this level, only

the average properties of the lamina are considered

important [14]. The background of the study is defined,

the failure criterion chosen is presented, and the probability

of failure is computed by Monte Carlo simulation.

0266-8920/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10 . 1 01 6 /S02 6 6- 89 2 0( 02 )0 00 5 4- 1

Probabilistic Engineering Mechanics 18 (2003) 119–137

www.elsevier.com/locate/probengmech

* Corresponding author. Tel.: þ1-303-492-7165; fax: þ1-303-492-7317.

E-mail address: [email protected] (D.M. Frangopol).

http://www.elsevier.com/locate/probengmech

2. Background

2.1. Laminate plate

Consider a composite laminate plate with n layers as

shown in Fig. 1. Each layer is a composite material made of

fibers in a matrix. All fibers are oriented in the same

direction within each layer (i.e. the principal direction 1 in

Fig. 2). This orientation produces considerably higher

mechanical properties in principal direction 1 than in the

perpendicular direction (i.e. principal direction 2 in Fig. 2).

The fibers of various layers can be oriented in different

directions in the horizontal (i.e. (x,y)) plane shown in Fig. 1.

Due to these different orientations of fibers in layers, the

laminate plate can be designed specifically for prescribed

load directions and combinations.

2.2. Kirchhoff hypothesis

Due to the small thickness of a layer compared to the

other two dimensions, the layer is considered as a plate.

Therefore, the classical plate theory proposed by Kirchhoff

in the mid 1800s can be used.

A global Cartesian coordinate system is defined as

follows: x and y are two perpendicular axes in the plane of

the plate, and the origin of coordinate z is located in the

geometric midplane of the plate [5].

According to Kirchhoff, a line normal to initial

geometric midplane of the plate will remain straight and

normal to the deformed geometric midplane, despite

deformations caused by loads. A displacement of a point

belonging to a normal line to the plate is composed of a

translation and a rotation. Therefore, the displacements of

all points belonging to a normal line are linked. Following

Ref. [5], the displacements along x, y and z are denoted as

uðx; y; zÞ; vðx; y; zÞ and wðx; y; zÞ; respectively.

The intersection of the normal to the plate with the

midplane is denoted P 0. This point can only translate in the

ðx; yÞ plane when the plate is stressed. The displacements of

P 0 in the directions x, y, and z are denoted u 0(x,y), v 0(x,y),

and w 0(x,y), respectively. The displacements of an

unspecified point of the normal line to the plate can be

obtained by the following relations [5]:

uðx; y; zÞ ¼ u0ðx; yÞ2 z›w0ðx; yÞ

›xð1Þ

vðx; y; zÞ ¼ v0ðx; yÞ2 z›w0ðx; yÞ

›yð2Þ

wðx; y; zÞ ¼ w0ðx; yÞ ð3Þ

The displacement along the z-coordinate is explicit. There-

fore, the behavior of the entire plate may be extrapolated

from the behavior of its geometric midplane. The following

development is summarized from Ref. [5].

The strains in the three directions x, y and z are,

respectively, as follows

1xðx; y; zÞ ¼ 10xðx; yÞ þ zx0

xðx; yÞ ð4Þ

1yðx; y; zÞ ¼ 10yðx; yÞ þ zx0

yðx; yÞ ð5Þ

gxyðx; y; zÞ ¼ g0xyðx; yÞ þ zx0

xyðx; yÞ ð6Þ

where

10xðx; yÞ ¼

›u0ðx; yÞ

›xand x0

xðx; yÞ ¼ 2›2w0ðx; yÞ

›x2ð7Þ

10yðx; yÞ ¼

›v0ðx; yÞ

›yand x0

yðx; yÞ ¼ 2›2w0ðx; yÞ

›y2ð8Þ

g0xy ¼

›v0ðx; yÞ

›xþ

›u0ðx; yÞ

›yand x0

xy ¼ 22›2w0ðx; yÞ

›x›yð9Þ

The midplane strain vector is

{1} ¼

10x

10y

g0xy

8>>><>>>:

9>>>=>>>;

ð10Þ

and the midplane change of curvature and twist vector is:

{x} ¼

x0x

x0y

x0xy

8>>><>>>:

9>>>=>>>;

ð11Þ

Fig. 2. Composite layer.

Fig. 1. Composite n-layer laminate plate.

D.M. Frangopol, S. Recek / Probabilistic Engineering Mechanics 18 (2003) 119–137120

2.3. Layer stresses

Each layer k is a plate having a plate-like stiffness matrix

in principal axes

CðkÞ ¼

CðkÞ11 CðkÞ

12 0

CðkÞ12 CðkÞ

22 0

0 0 CðkÞ44

26664

37775 ð12Þ

with

CðkÞ11 ¼

EðkÞ1

1 2 nðkÞ12n

ðkÞ21

ð13Þ

CðkÞ12 ¼

EðkÞ1 vðkÞ21

1 2 nðkÞ12n

ðkÞ21

¼EðkÞ

2 nðkÞ12

1 2 nðkÞ12n

ðkÞ21

ð14Þ

CðkÞ22 ¼

EðkÞ2

1 2 nðkÞ12n

ðkÞ21

ð15Þ

CðkÞ44 ¼ GðkÞ

12 ð16Þ

where EðkÞ1 and EðkÞ

2 are elastic moduli in directions 1 and

2 of layer k, nðkÞ12 and nðkÞ21 are Poisson ratios in directions

1 and 2 of layer k, GðkÞ12 is the shearing modulus

associated with direction 1 and 2 of layer k, and E1=E2 ¼

n12=n21:

The stiffness matrix links stress, strain, curvature and

twist of the layer k in principal axes as follows:

sðkÞ1

sðkÞ2

tðkÞ12

8>>><>>>:

9>>>=>>>;¼ CðkÞ

1ðkÞ1

1ðkÞ2

gðkÞ12

8>>><>>>:

9>>>=>>>;

ð17Þ

Using Eqs. (4)–(6), Eq. (17) becomes:

sðkÞ1

sðkÞ2

tðkÞ12

8>>><>>>:

9>>>=>>>;¼ CðkÞ

10ðkÞ1

10ðkÞ2

g0ðkÞ12

8>>><>>>:

9>>>=>>>;þ z

x0ðkÞ1

x0ðkÞ2

x0ðkÞ12

8>>><>>>:

9>>>=>>>;

0BBB@

1CCCA ð18Þ

To obtain this matrix in geometrical axes ðx; y; zÞ; a rotation

matrix is introduced as follows

TðkÞ ¼

cos2uðkÞ sin2uðkÞ 2sinuðkÞcosuðkÞ

sin2uðkÞ cos2uðkÞ 22sinuðkÞcosuðkÞ

2sinuðkÞcosuðkÞ sinuðkÞcosuðkÞ cos2uðkÞ2 sin2uðkÞ

266664

377775

ð19Þ

where u (k) is the orientation of fibers in layer k, measured

counterclockwise from the þx axis to the þy axis [5].

Using the rotation matrix (19) and the stiffness matrix in

principal axes, a stiffness matrix �C is defined in geometrical

axes as follows:

�CðkÞ ¼ TðkÞ21CðkÞTðkÞ ð20Þ

Therefore, the stress–strain relations associated with the

layer k are:

sðkÞx

sðkÞy

tðkÞxy

8>>><>>>:

9>>>=>>>;

¼ �CðkÞ

10ðkÞx

10ðkÞy

g0ðkÞxy

8>>><>>>:

9>>>=>>>;2 z

x0ðkÞx

x0ðkÞy

x0ðkÞxy

8>>><>>>:

9>>>=>>>;

0BBB@

1CCCA ð21Þ

2.4. Stress in each layer of a laminate plate

Given the loading and the dimensions of the laminate

plate, the stresses in each direction of the plate are easy to

calculate. Each layer has a different stiffness due to fiber

orientation and/or different material. Therefore, in each

layer a relation is needed between applied loads and

stresses.

An important hypothesis is the assumption of perfect

bond between layers. This implies that the laminate plate

will have the displacement and strain behavior of a

homogenous plate. The following development is summar-

ized from Ref. [2].

Consider a laminate plate subjected to distributed forces

Nx, Ny, Nz, distributed moments Mx, My, Mz and concen-

trated p and distributed q loads normal to the plate.

The contribution of each layer k to the stress-resultants

{N} and stress-couples {M} of the plate is given by:

NðkÞx ¼

ðzk

zk21

sðkÞx dz ð22Þ

NðkÞy ¼

ðzk

zk21

sðkÞy dz ð23Þ

NðkÞxy ¼

ðzk

zk21

tðkÞxy dz ð24Þ

MðkÞx ¼

ðzk

zk21

sðkÞx z dz ð25Þ

MðkÞy ¼

ðzk

zk21

sðkÞy z dz ð26Þ

MðkÞxy ¼

ðzk

zk21

tðkÞxy z dz ð27Þ

For a plate with n layers, the stress-resultant and stress-

couple are obtained by the summation of Eqs. (22)–(27)

over n layers. Substituting Eqs. (22)–(27) into Eq. (21) and

performing this summation results in

{N} ¼ A{1} 2 B{x} ð28Þ

{M} ¼ B{1} 2 D{x} ð29Þ

where the plate midplane stress-resultant vector is {N} ¼

{Nx;Ny;Nxy}T and the plate midplane stress-couple vector is

{M} ¼ {Mx;My;Mxy}T; and

Aij ¼Xn

k¼1

�CðkÞij ðhk 2 hk21Þ ð30Þ

D.M. Frangopol, S. Recek / Probabilistic Engineering Mechanics 18 (2003) 119–137 121

Bij ¼1

2

Xn

k¼1

�CðkÞij ðh

2k 2 h2

k21Þ ð31Þ

Dij ¼1

3

Xn

k¼1

�CðkÞij ðh

3k 2 h3

k21Þ ð32Þ

where hk is the maximum distance (always positive) from

the most distant surface of the layer to the midplane surface.

Solving Eq. (28) leads to the following result:

{1} ¼ A21{N} þ A21B{x} ð33Þ

Substituting Eq. (33) into Eq. (29) gives

{M} ¼ BA21{N} þ ðBA21B 2 DÞ{x} ð34Þ

and

{x} ¼ ðBA21B 2 DÞ21ð{M} 2 BA21{N}Þ ð35Þ

{1}¼A21{N}þA21BðBA21B2DÞ21ð{M}2BA21{N}Þ

ð36Þ

In Eqs. (35) and (36) the total strain of the laminate plate is

defined in function of the applied load and stiffness matrix

of each layer. Now stresses in layer k are obtained using this

layer stiffness matrix and z-coordinate, and the midplane

strain vectors {1} and {x} as follows:

sðkÞx

sðkÞy

tðkÞxy

8>>><>>>:

9>>>=>>>;¼ �CðkÞð{1}2 z{x}Þ ð37Þ

Fig. 3. Tsai–Wu failure ellipsoid; 3D representation.

Fig. 4. Tsai–Wu failure ellipsoid for different values of t12.


sðkÞ1

sðkÞ2

tðkÞ12

8>>><>>>:

9>>>=>>>;¼TðkÞ �CðkÞð{1}2 z{x}Þ ð38Þ

Or, based on Eq. (20)

sðkÞ1

sðkÞ2

tðkÞ12

8>>><>>>:

9>>>=>>>;¼CðkÞTðkÞð{1}2 z{x}Þ ð39Þ

where {x} and {1} are given by Eqs. (35) and (36),

respectively.

3. Failure criterion

As indicated in Ref. [5], there are many issues surrounding

the subject of failure of composite materials and many

mechanisms must be considered when studying failure of

these materials. Knowing the applied loads on the laminate

plate, it is possible to determine the principal stresses in each

layer. A criterion is needed in order to predict if the principal

stresses lead to layer failure. The most commonly used

criteria for failure of polymer–matrix composites are the

maximum stress and the Tsai–Wu criteria [5].

In this study, the failure criterion chosen for fiber-

reinforced composite laminate plates is the Tsai–Wu

criterion [5]. This criterion is used to determine the failure

of an orthotropic material. An orthotropic material has

different mechanical properties in three mutually perpen-

dicular directions denoted as 1, 2, and 3, respectively.

Fiber-reinforced composite materials are considered ortho-

tropic in the principal material coordinate system.

The Tsai–Wu criterion for a composite material plane

layer element subject to stresses in its principal directions is

expressed as [5]:

(a) Survival criterion

F1s1 þ F2s2 þ F11s21 þ F22s

22 þ F66t12

2ffiffiffiffiffiffiffiffiffiF11F22

ps1s2 , 1 ð40Þ

(b) Failure criterion

F1s1 þ F2s2 þ F11s21 þ F22s

22 þ F66t12

2ffiffiffiffiffiffiffiffiffiF11F22

ps1s2 ¼ 1 ð41Þ

Fig. 5. Tsai–Wu failure ellipsoid for different values of s2.

Fig. 6. Tsai–Wu failure ellipsoid for different values of s1.


where

F1 ¼1

sT1

þ1

sC1

ð42Þ

F2 ¼1

sT2

þ1

sC2

ð43Þ

F11 ¼ 21

sT1s

C1

ð44Þ

F22 ¼ 21

sT2s

C2

ð45Þ

Fig. 7. Effect of correlation, r(s1,s2), on the probability of failure in tension–tension. (a) Arithmetic scale and (b) logarithmic scale for failure probability.


F66 ¼1

t F12

!2

ð46Þ

where sCi is the compression strength in the direction

i (negative value), sTi is the tension strength in

the direction i (positive value), and tF12 is the shear

strength in the plane 1–2.

The inequality F1s1 þ F2s2 þ F11s21 þ F22s

22 þ

F66t12 2ffiffiffiffiffiffiffiffiffiF11F22

ps1s2 . 1 is physically impossible

Fig. 8. Effect of t12 on the probability of failure in tension–tension. (a) Arithmetic scale and (b) logarithmic scale for failure probability.


since, for given applied loads, the stress will grow until

Eq. (41) is satisfied and there will be degradation of the

effective properties that will change sCi ; sT

i ; tF12:

However, we may assume that, for given s1, s2, s12,

the event

F1s1þF2s2þF11s21þF22s

22þF66t122

ffiffiffiffiffiffiffiffiffiF11F22

ps1s2.1 ð47Þ

represents failure.

Fig. 9. Probability of failure in tension–tension and in compression–compression for different correlations r(s1,s2). (a) Arithmetic scale and (b) logarithmic

scale for failure probability.


The Tsai–Wu criterion leads to the failure ellipsoid,

whose equation in s1, s2, t12 space is Eq. (41). If a point

ðs1;s2; t12Þ is inside the ellipsoid there is no failure, if it is

on or outside the ellipsoid failure occurs.

Figs. 3–6 are different graphical representations of the

Tsai–Wu ellipsoid. The material is graphite-reinforced

epoxy composite with the following properties (see Ref. [5],

page 396): sT1 ¼ 1500 MPa; sC

1 ¼ 21250 MPa; sT2 ¼ 50

MPa; sC2 ¼ 2200 MPa and tF

12 ¼ 100 MPa:

Fig. 3 shows the 3D representation of the ellipsoid. It is

noted that using the above values, one stress is dominant.

This is the stress in the direction of fibers s1. In this

direction, the layer can support more stress.

Figs. 4–6 show cross-sections of the ellipsoid for

different values of the stresses t12, s2 and s1, respectively.

4. Probability of failure

The purpose herein is to calculate the probability of

failure of a FRC laminate plate under random loads. As

indicated previously (a) the principal stresses associated

with each layer of a laminate plate can be computed, and (b)

there is a failure criterion based on these principal stresses.

The next step is to use this information in a reliability

model. Unfortunately, principal stresses calculation

methods and the Tsai–Wu criterion lead to a very complex

expression to compute the probability of failure analytically.

A direct way to compute this probability of failure is by

Monte Carlo simulation. For this particular study Monte

Carlo simulation is preferable to first and second order

reliability methods since non-linear complex behavior does

not complicate the basic procedure.

4.1. Single-layer laminate plate

This section concentrates on a single layer of graphite-

reinforced epoxy composite material with properties

indicated in Ref. [5], considering principal stresses as

lognormal distributed random variables. The lognormal

distribution was chosen since no information on the type of

distribution for principal stresses was available for this

study. The mean values of s1 and s2 are assumed to satisfy

the relation Eðs1Þ ¼ 25Eðs2Þ: The coefficient of variation of

each random variable is assumed the same Vðs1Þ ¼

Vðs2Þ ¼ Vðt12Þ: Monte Carlo simulation was performed

by using the software MONTE [7]. The results of these

simulations are shown in Figs. 7–9. In these figures, the

deterministic material properties sC1 and sC

2 are indicated in

absolute values.

Fig. 7 shows the effects of the mean value of the principal

stress s1, E(s1), and the coefficient of correlation between

principal stresses, r(s1,s2), on the probability of failure Pf

for the tension–tension case (i.e. s1 . 0, s2 . 0 and

t12 ¼ 0). As indicated, an increase in correlation increases

the probability of failure in the range of interest (say,

Pf , 1023).

Fig. 8 shows the results of simulation for the tension–

tension case without correlation (i.e. s1 . 0, s2 . 0 and

t12 – 0) considering deterministic and random shear stress,

t12. It is noted that the values of t12 are too small to change

significantly the results shown in Fig. 7.

The probabilities of failure in tension– tension and

compression–compression are compared in Fig. 9. As

indicated the reliability in tension–tension is much lower

than in compression–compression. This is due to the

asymmetry of the Tsai–Wu failure ellipsoid shown in Fig. 3.

4.2. Two-layer laminate plate

4.2.1. Uniaxial tension

Consider the glass–epoxy composite laminate plate

constituted of two layers with perpendicular fiber directions

shown in Fig. 10. For the representation of simulation

results the Cartesian coordinate system shown in Fig. 10 is

used, due to its convenience, for the remaining part of this

study. It is noted that this system is different from that used

for the derivation of mechanics aspects of FRCs in Fig. 1.

The only load is Nx. The mechanical properties of the glass-

epoxy composite used for simulations are as follows [5]:

E1 ¼ 55 GPa, E2 ¼ 18 GPa, n12 ¼ 0.25, n21 ¼ 0.08 and

G12 ¼ 8 GPa.

The probability of failure is computed by Monte Carlo

simulation using the software MONTE [7]. The user file of

MONTE has been modified to enable resultant forces input

instead of stress input. The results are shown in Fig. 11. The

layer whose fibers are perpendicular to the applied force (i.e.

layer 1) has a much greater chance to fail before the layer

whose fibers are in the direction of the force (i.e. layer 2). If

layer 1 does not fail before layer 2 (e.g. layer 1 made of a

more resistive material than layer 2, but with same stiffness

matrix), the probability of failure of layer 2 given that layer

1 survives, PðF2lS1Þ; is larger than the probability of failure

of layer 2 alone PðF2lS1Þ . PðF2Þ:

The presence of layer 1 creates twists and changes

of curvature that do not exist when layer 2 is alone.

Fig. 10. Two-layer laminate plate under uniaxial tension.


This produces extra stresses causing failure of layer 2

sooner. For this reason the presence of layer 1 increases

the probability of failure of layer 2. On the other hand,

the presence of layer 2 decreases the probability of

failure of layer 1 alone, PðF1lS2Þ , PðF1Þ:

The probability of failure of the plate, PðFPLATEÞ;

assumed as a weakest-link system, was calculated

by Monte Carlo simulation [1]. The results indicate

that PðFPLATEÞ is almost equal to the probability of

failure of layer 1. This is due to the fact that

Fig. 11. Probability of failure of the layers of the plate shown in Fig. 10. (a) Arithmetic scale and (b) logarithmic scale for failure probability.


the probability of failure of layer 1 is much greater

than that of layer 2.

4.2.2. Biaxial tension

If an additional force Ny is applied on the same plate,

Ny ¼ Nx; in the perpendicular direction to Nx, the

probability of failure of each layer will be the same

(Fig. 12). This is due to the symmetry of the system and

the assumptions that only loads are random variables.

The probability of failure of the plate is greater than

the probability of failure of each layer, due to the

assumption of a series system.

As expected, the probability of failure of the plate under

biaxial tension Nx and Ny is greater than the probability of

the same plate under uniaxial tension Nx (Fig. 13). However,

the increase of the probability of failure in the biaxial case is

not dramatic. This is because the plate may support either Nx

alone, or Ny alone, or both Nx and Ny without significant

change in the probability of failure. However, if the plate is

intended to support only a load in a single direction, it is

Fig. 12. Two-layer laminate plate under biaxial tension: probability of failure of layers and plate. (a) Arithmetic scale and (b) logarithmic scale for failure

probability.


obviously better to use a plate with all fibers in the direction

of the force.

4.3. Thickness of layers effect

Consider a two-layer laminate plate under biaxial tension

and denote t1 the thickness of layer 1, whose fibers are

oriented along y-axis, and t2, the thickness of the layer,

whose fibers are oriented along x-axis (i.e. a [90,0] laminate

plate). The uniform distributed loads Nx and Ny are both

lognormal with mean values of 200 and 100 kN/m,

respectively, and same coefficient of variation VðNxÞ ¼

VðNyÞ ¼ 0:20: The material properties are those specified in

Fig. 10, where the values chosen for the Tsai–Wu failure

criterion are taken from Ref. [5].

What should the thickness ratio of the two layers t1/t2be in order to minimize the probability of failure? We

may wrongly answer that the layer whose fibers’

orientation is 08 (i.e. layer 2) should be twice as thick

as the one whose fibers’ orientation is 908 (i.e. layer 1),

Fig. 13. Comparison of probabilities of failure of a two-layer laminate plate under uniaxial and biaxial tension. (a) Arithmetic scale and (b) logarithmic scale for

failure probability.


Fig. 14. Probability of failure of a two-layer laminate under biaxial tension: effect of layer thickness ratio.

Fig. 15. Two-layer laminate plate under uniaxial tension with the fibers in layer 1 normal to the direction of tension. Three different orientations of fibers in

layer 2.


as it carries a load two times greater. In order to provide

the correct answer, the thickness of layer 1, t1, is varied

keeping thickness of layer 2 constant, t2 ¼ 3 mm: In this

study, the thickness of 3 mm for layers was chosen for

simulation demonstration purposes. The probability of

failure of the plate is indicated in Fig. 14. The best

choice is to eliminate layer 1. In this case, layer 2 under

both loads, Nx and Ny, has maximum reliability.

This proves that adding a layer in order to support a force

perpendicular to the fibers may actually increase the

probability of failure of the plate if the thickness of this

layer is not chosen rationally.

Fig. 16. Two-layer laminate plate under uniaxial tension with the fibers in layer 1 normal to the direction of tension. Effect of orientation of fibers in layer 2 on:

(a) probability of failure of layer 2, (b) probability of failure of layer 1, and (c) probability of failure of the plate.


4.4. Fibers’ orientation effect

Consider the glass–epoxy laminate plate in Fig. 10

constituted of two layers whose thicknesses are 3 mm under

a uniform uniaxial load Nx (see Fig. 15). The first layer’s

fibers are perpendicular to the force Nx (i.e. 908). However,

the orientation of the second layer’s fibers, u2, varies from

90 to 08. The probability of failure of the plate with respect

to the orientation of the fibers of the second layer is plotted

in Fig. 16.

Fig. 16 (continued )

Fig. 17. Two-layer laminate plate under uniaxial tension with the fibers in layer 1 along the direction of tension. Three different orientations of fibers in layer 2.


When u2 increases the probability of failure of layer 2

under the random lognormal distributed load Nx increases

(Fig. 16(a)), but the probability of failure of layer 1 may

increase or decrease (Fig. 16(b)). This is due to the presence

of a layer carrying a load perpendicular to its fibers. The

probability of failure of the plate shows a minimum for

u2 ¼ 308 (Fig. 16(c)).

The inverse problem is also addressed herein. The

angle of the first layer’s fibers is set to 08 and the second

layer’s fiber orientation angle varies from 0 to 908

(Fig. 17). In this case, as indicated in Fig. 18(a), the

probability of failure of the second layer increases when

u2 increases, but the probability of failure of layer 1 does

not necessarily increase when u2 increases (see

Fig. 18(b)). However, the lowest probability of failure

is when u2 ¼ 08: The probability of failure of the plate is

minimal for fibers’ orientation of 08 in both layers

(Fig. 18(c)).

Fig. 18. Two-layer laminate plate under uniaxial tension with the fibers in layer 1 along the direction of tension. Effect of orientation of fibers in layer 2 on: (a)

probability of failure of layer 2, (b) probability of failure of layer 1, and (c) probability of failure of the plate.


4.5. Four-layer laminate plate

Finally, the four-layer laminate plate [0,45, 2 45,0] in

Fig. 19 is considered. The orientation of fibers of the two

external layers (i.e. 1 and 4) is 08. The orientations of fibers

of the internal layers 2 and 3 are 45 and 2458, respectively.

The thickness of all layers is the same, ti ¼ 3 mm; and the

uniaxial uniform distributed tension Nx is lognormal with

20% coefficient of variation.

Since the internal layers have the fibers oriented in

perpendicular directions, the plate is not symmetric but

its behavior under the uniaxial uniform distributed load

has symmetry. This implies that, for a given load Nx,

stresses are the same in layers 1 and 4 or in layers 2 and

3. For this reason, the external layers, or the internal

layers, have the same failure probabilities. This is clearly

indicated in Fig. 20. This figure also shows that the

reliability of the internal layers is much lower than that

of external layers. Consequently, the probability of

failure of the plate, considered as a weakest-link system,

will be close to the probability of failure of its internal

layers (i.e. layer 2 or 3).

Fig. 18 (continued )

Fig. 19. Four-layer laminate plate under uniaxial tension.


5. Conclusions

Based on the results presented in this paper, the

following conclusions can be made.

1. Concepts and methods of probability have to be used for

the reliability evaluation and development of acceptance

and design criteria for FRCs.

2. There are some studies describing the failure process by

using statistical models for strength and failure of FRCs.

However, most of the studies neglect the randomness in

loading.

3. The reliability of a composite laminate plate is not easy

to evaluate. Each layer has a different strength due to its

fibers’ orientation and, therefore, a different reliability.

For reliability computations, a laminate plate is

Fig. 20. Probability of failure of the layers of the four-layer laminate plate shown in Fig. 19. (a) Arithmetic scale and (b) logarithmic scale for failure

probability.


considered as a series system of different layers. The

Tsai–Wu failure criterion can be used in conjunction

with Monte Carlo simulation.

4. The presence of an additional layer in a composite

laminate plate does not necessarily increase the

reliability of the plate. In fact, it is possible that this

additional layer increases the probability of failure of

other layers. For this reason, it is necessary to consider

layer-interaction effects in evaluating the reliability of

FRCs.

5. It would be dangerous to assume that the reliability of FRC

plates increases with the increase in the thickness of one or

more layers. The thickness ratio of layers has to be chosen

rationally in order to increase the reliability of the plate.

6. The orientation of fibers has a significant effect on the

reliability of FRC plates.

7. The probability of failure of a fiber-reinforced

composite laminate plate can be significantly reduced

if the number and thickness of layers and orientation

of fibers under prescribed load combinations are

optimized.

8. The reliability experts and FRCs experts must work in

synergy to develop evaluation, acceptance, and design

criteria for FRCs.

Acknowledgements

The present study is a part of the project dealing with

acceptance test specifications and guidelines for fiber-

reinforced polymeric bridge decks, funded by the Federal

Highway Administration to Georgia Institute of Technology

(Prof. Abdul Zureick, Principal Investigator). Part of this

project was subcontracted to the University of Colorado at

Boulder (Prof. Dan M. Frangopol, Principal Investigator).

The first author also acknowledges partial support for the

US National Science Foundation under NSF Grant No.

CMS-9912525 to the University of Colorado at Boulder.

Thanks are due to Prof. Abdul Zureick, Georgia Institute of

Technology, for some stimulating discussions. The second

author spent six months at the University of Colorado at

Boulder as a visiting student from the French Institute for

Advanced Mechanics, IFMA, to study reliability of

composite materials. Thanks are also due to Mr David

Guillot, visiting student from IFMA, who contributed to the

final version of this paper. The opinions and conclusions

presented in this paper are those of the authors and do not

necessarily reflect the views of the Federal Highway

Administration, the National Science Foundation, and/or

the French Institute for Advanced Mechanics.

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Reliability of fiber-reinforced composite laminate plates