dynamic stability of uncertain laminated beams subjected

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International Journal of Solids and Structures 45 (2008) 2799–2817

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Dynamic stability of uncertain laminated beamssubjected to subtangential loads

Vijay K. Goyal a,*, Rakesh K. Kapania b,1

a Mechanical Engineering Department, University of Puerto Rico at Mayaguez, P.O. Box 9045 Mayaguez, PR 00681, USAb Aerospace and Ocean Engineering Department, Virginia Polytechnic Institute and State University, MS 0203,

Blacksburg, VA 24061-0203, USA

Received 5 April 2006; received in revised form 29 August 2007

Abstract

Because of the inherent complexity of fiber-reinforced laminated composites, it can be challenging to manufacturecomposite structures according to their exact design specifications, resulting in unwanted material and geometricuncertainties. Thus the understanding of the effect of uncertainties in laminated structures on their static and dynamicresponses is highly important for a reliable design of such structures. In this research, we focus on the probabilisticstability analysis of laminated structures subject to subtangential loading, a combination of conservative and noncon-servative tangential loads, using the dynamic criterion. In order to study the dynamic behavior by including uncertain-ties into the problem, three models were developed: exact Monte Carlo simulation, sensitivity-based Monte Carlosimulation, and probabilistic FEA. These methods were integrated into the existing finite element analysis. Also, per-turbation and sensitivity analysis have been used to study nonconservative problems to study the stability analysisusing the dynamic criterion.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Probabilistic; Finite element; Laminated composites; Dynamic stability; Nonconservative load

1. A computational probabilistic analysis

Laminated composite structures are challenging because their mechanical and physical properties can beuncertain due to changes in various factors like fiber orientations, curing temperature, pressure and time,voids, and impurities among others. The design and analysis using conventional materials is easier than thoseusing composites because for conventional materials both material and geometric properties have either littleor well known variation from their nominal value. On the other hand, the same cannot be said for the design

0020-7683/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijsolstr.2007.11.024

* Corresponding author. Tel.: +1 787 832 4040x2111.E-mail address: [email protected] (V.K. Goyal).

1 Professor at Virginia Polytechnic Institute and State University, USA.

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2800 V.K. Goyal, R.K. Kapania / International Journal of Solids and Structures 45 (2008) 2799–2817

of structures using laminated composite materials. Thus, the understanding of uncertainties in the eigenfre-quency analysis of laminated structures is highly important for an accurate design and analysis of aerospaceand other structures. Uncertainties due to noncognitive sources are studied using a probabilistic approachalthough other approaches exist (Elishakoff, 1998). Here only those uncertainties involving material and geo-metric properties are considered.

Various methods exist to analyze an uncertain structure by integrating probabilistic aspects into the finiteelement modeling (Schueller, 1997). The probabilistic finite element analysis (PFEA) can be classified into twocategories: perturbation techniques and simulation methods. Perturbation techniques are based on seriesexpansion (e.g., Taylor series) to formulate a linear or quadratic relationship between the randomness ofthe material, geometry, or load and the randomness of the response (Nakagiri and Hisada, 1988a,b). Simula-tion methods such as Monte Carlo simulation rely on computers to generate random numbers from the mate-rial, geometry, or load uncertainties and correlate the probabilistic response to it (Shinozuka, 1972; Fang andSpringer, 1993; Vinckenroy et al., 1995).

A considerable amount of research has been done in the field of analysis of structures under stochastic loadsusing the finite element method (Kapania and Yang, 1984; Yang and Kapania, 1984) and in the field of ran-dom structures using the stochastic finite element method to the analysis of static and dynamic problem (Con-teras, 1980; Vanmarcke et al., 1986; Collins and Thompson, 1969; Kiureghian and Ke, 1988; Zhang et al.,1996; Chakraborty and Dey, 1995).

The probabilistic analysis requires the derivatives of the structural matrices as well as the derivatives of theeigenvalues, eigenvectors, and displacements. Lee and Lim (1997) presented an approach for extending sensi-tivity methods to include the structural uncertainty with random parameters using perturbation techniques.Derivatives of eigenvectors with respect to design variables are very useful in certain analyses and design appli-cations (Fox and Kapoor, 1968; Plaut and Huseyin, 1973; Haftka and Adelman, 1986; Liu et al., 1995).

Brenner and Bucher (1995) presented a stochastic finite element-based reliability analysis of large nonlin-ear structures under dynamic loading, involving both structural and loading randomness, with relatively lit-tle computational effort when compared to the traditional Monte Carlo methods. Papadopoulos andPapadrakakis (1998) used a weighted integral method in conjunction with Monte Carlo simulation forthe stochastic finite element-based reliability analysis of space frames. For dynamic analysis, the randomnature of the stiffness matrix, mass matrix, eigenvalues, and eigenvectors can be studied using a Taylor ser-ies expansion up to second order about the mean of each random variable (Zhang and Ellingwood, 1995;Oh and Librescu, 1997).

A number of studies are available on the analysis of composite beams. Kapania and Raciti, 1989a) pre-sented a simple beam element to study vibrations of unsymmetric composite beams. A review of several suchstudies was given by Kapania and Raciti (1989b,c). A recent review is by Yang et al. (2000) and Goyal andKapania (2007). Goyal and Kapania (2007) developed a 21 degree-of-freedom beam element, based on thefirst order shear deformation theory, to study the static and dynamic response of one-dimensional fiber-rein-forced laminated composite structures. Later, the formulation was expanded to take into account for subtan-gential loads (Goyal and Kapania, 2003). Here we integrate the probabilistic approach into the vibrations anddynamic stability analysis using perturbation methods. Because probabilistic models can capture the influenceof these uncertainties, we used three probabilistic theories: probabilistic finite element method, sensitivity-based Monte Carlo simulation, and Monte Carlo simulation.

2. A probabilistic approach

Several probabilistic methods have been used to analyze an uncertain unsymmetrically laminated beam byintegrating uncertain aspects into the finite element modeling such as the perturbation technique using Taylorseries expansion and simulation methods (e.g., the Monte Carlo simulation).

2.1. Function of multiple random variables

In problems where uncertainties are considered, and the information is limited to only the mean values ofthe random variables, perturbation techniques are suggested, among other existing techniques (Ang and Tang,

V.K. Goyal, R.K. Kapania / International Journal of Solids and Structures 45 (2008) 2799–2817 2801

1975; Schueller, 1997). Kapania and Goyal (2002) suggested that the relationship between the independentrandom variables ri’s and a matrix Y can be represented as a function of random variables,

Y ¼ Yðr1; . . . ; rnÞ ð1Þ

Here we consider the randomness in two laminated parameters: each ply’s orientation, and each ply’s axialYoung’s modulus, Exxi . The present analysis will assume that all random variables obey a normal distribution.In most cases the sensitivity derivatives of matrix Y can be obtained. The matrix Y can be expanded usingTaylor series expansion about the mean values:

Yðr1; . . . ; rnÞ ¼ Y0 þXn

i¼1

YIi �i þ

1

2

Xn

i¼1

Xn

j¼1

Y IIij �i�j þ � � �

Y0 ¼ Yjr¼r0 ; YIi ¼

oY

orijr¼r0 ; YII

ij ¼o2Y

oriorjjr¼r0

ð2Þ

where r0 ¼ ðl1; l2; . . . ; lnÞ is a set of mean random variables and �i ¼ ri � li is a set of zero-mean uncorrelatedrandom variables.

2.2. Monte Carlo simulation

Monte Carlo simulation, although computationally expensive, is a quite versatile technique that is capa-ble of handling situations where other methods fail. Here MCS is used to verify the results obtained fromthe perturbation methods. For the present analysis, at least 10,000 realizations of the uncertain beam areperformed, increasing the accuracy of the ply-angle and ply-axial modulus of elasticity distribution fit tothe sample data.

2.3. Probabilistic finite element analysis

Goyal and Kapania (2003) derived the eigenvalue problem for the case of subtangential loading as follows:

½K� PL� kM�/ ¼ 0 ð3Þ

where

L ¼ KG � g KL ð4Þ

where K , M , L are the dimensionless linear stiffness matrix, mass matrix, and loading matrix, respectively. Inthis work, we have included the uncertainties in each matrix and hence converted the deterministic problem toa probabilistic one. We use the subscript k to represent the kth eigenvalue mode, wk as the probabilistic dimen-sionless left eigenvector, and /k as the probabilistic dimensionless right eigenvector. Thus the probabilisticdimensionless eigenvalue problem is expressed as

fwkgT

K � P k L� kk M½ � ¼ 0

K � P k L� kk M½ � /kf g ¼ 0ð5Þ

The presence of structural uncertainties affect the stiffness matrix, K , mass matrix M, and the loading matrix Lare expanded in terms of their mean-centered zeroth-, first-, and second-order rates of change with respect tothe random variables. The buckling loads, eigenfrequencies, and left and right eigenvectors are also affected byuncertainties. The eigenfrequencies and eigenvectors are expressed in terms of their mean-centered zeroth-,first-, and second-order rates of change with respect to the random variables. Further, let the first-order rateof change of the kth mode right eigenvector with respect to the ith random variable evaluated about the meanbe

o /kf gori

��r¼r0

¼ /Iki

� �ð6Þ


and the second-order rate of change of the kth mode right eigenvector with respect to the ith and jth randomvariables evaluated about the mean be

o2 /kf goriorj

��r¼r0

¼ /IIkij

n oð7Þ

Using the same notation for the eigenfrequencies and buckling loads, we can express the following:Eigenfrequencies

kk r1; . . . ; rnð Þ ¼ k0k þ

Xn

i¼1

kIki �i þ

1

2

Xn

i¼1

Xn

j¼1

kIIkij �i�j ð8Þ

Right eigenvectors

/k r1; . . . ; rnð Þf g ¼ /0k

� �þXn

i¼1

/Iki

� ��i þ

1

2

Xn

i¼1

Xn

j¼1

/IIkij

n o�i�j ð9Þ

Buckling loads

Pk r1; . . . ; rnð Þ ¼ P0k þ

Xn

i¼1

PIki �i þ

1

2

Xn

i¼1

Xn

j¼1

PIIkij �i�j ð10Þ

The substitution of these expansions into Eq. (5), results in a probabilistic eigenvalue problem. Since theuncertainties in the random variables are assumed small, in the applied perturbation technique it is sufficientto only consider up to second-order terms. Thus the expansion of the probabilistic eigenvalue problem leads tothree equations which are solved successively:

�0 : K0 � P0kL0 � k0

kM0� �

/0k

� �¼ 0 ð11Þ

�1 : K Ii � PI

kiL0 � kI

kiM0

� �/0

k

� �¼ 0 ð12Þ

�2 : K IIij � PII

kijL0 � kII

kijM0

h i/0

k

� �¼ � K I

i � PIkiL

0 � kIkiM

0� �

/Ikj

n oð13Þ

2.4. Eigenfrequency derivatives

Equating the zeroth-order terms of �i in the eigenvalue expansion, an eigenvalue problem for the mean-val-ued system is obtained. Therefore, the mean-centered zeroth derivative eigenfrequencies and associated eigen-vectors are obtained as follows:

w0k

� �TK0 � P0

kL0 � k0kM0

� �¼ 0

K0 � P0kL0 � k0

kM0� �

/0k

� �¼ 0

ð14Þ

Recall that the loading and mass matrices do not depend on the lamina’s mechanical characteristics. Since thelamina ply angles and axial Young’s modulus are the only random variables considered, the sensitivity deriv-atives of the mass and loading matrix vanish. Thus expressions for the mean-centered first and second-orderderivatives of the eigenfrequencies are found as


kIki ¼

w0k

� �TK I

i � PIkiL

0� �

/0k

� �w0

k

� �TM0� �

/0k

� � ð15Þ

kIIkij ¼

w0k

� �TK II

ij � PIIkijL

0h i

/0k

� �w0

k

� �TM0� �

/0k

� �

þw0

k

� �TK I

i � PIkiL

0 � kIkiM

0� �

/Ikj

n ow0

k

� �TM0� �

/0k

� � ð16Þ

þw0

k

� �TK I

j � PIkjL

0 � kIkjM

0h i

/Iki

� �w0

k

� �TM0� �

/0k

� �
For conservative systems,
L ¼ LT and w0k

� �¼ f/0

kg ð17Þ

Thus for conservative systems, by virtue of Eqs. (12) and (17), the expression for the second-order derivativecan be simplified to the expression derived by Kapania and Goyal (2002):

kIIkij ¼

/0k

� �TK II

ij � PIIkijL

0h i

/0k

� �/0

k

� �TM0� �

/0k

� � ð18Þ

Being the above a special case of the present method. The advantage of this method is that the eigenvalueproblem needs to be solved only once. The sensitivity analysis is done by using results from the mean-valuedeigenvalue problem. This results in great computational saving.

2.5. Eigenvector derivatives

The first-order variations in the eigenvectors have been studied by various researchers (Fox and Kapoor,1968; Murthy, 1986; Bergen and Kapania, 1988; Kapania et al., 1991; Plaut and Huseyin, 1973; and Adhikariand Friswell, 2001). The method employed here is based on the work done by Fox and Kapoor (1968). Wecalculate the left and right eigenvector sensitivities separately and use the fact that the sensitivity derivativesof the mass and loading matrices are zero. Since the right eigenvectors form a complete set of vectors, aneigenvector can be represented by the linear combination of all other right eigenvectors. Thus the derivativeof the kth mode eigenvector with respect to the ith random variable evaluated about the mean is represented asfollows:

o /kf gori

��r¼r0

¼ /Iki

� �¼Xn

j¼1

aðiÞkj /j

� �ð19Þ

where /j

� �is the eigenvector corresponding to the jth mode, and n corresponds to the total number of modes

(dimensions of the stiffness matrix). Thus the problem reduces to calculating the coefficients aðiÞkj . By differen-tiating the following eigenvalue with respect to the ri random variable:

K� P k L� kk M½ �f/kg ¼ 0 ð20Þ

Rearranging and premultiplying the above equation by the transpose of the left eigenvector, wmf gT, we get

Xn

j¼1

aðiÞkj wmf gTK� P k L� kk M½ � /j

� �¼ � wmf gT

K Ii � PI

kiL� kIkiM

� �/kf g ð21Þ

To normalize the eigenvectors we need two independent criteria. Thus let us normalize the eigenvectors suchthat


wj

� �T½M� /j

� �¼ 1 and wj

� �nth nonzero element

¼ /j

� �nth nonzero element

for a selected value of n. As a result,

wj

� �TK� P k L½ � /j

� �¼ kj

Moreover, for distinct eigenfrequencies, the right and left eigenvectors satisfy biorthogonality criteria, i.e.,

wmf gT½M� /j

� �¼ 0 wmf gT

K� P k L½ � /j

� �¼ 0 8 m 6¼ j

Thus Eq. (21) becomes

Xn

j¼1

aðiÞkj wj

� �TK� P k L� kk M½ � /j

� �¼ � wj

� �TK I

i � PIki L� kI

ki M� �

/kf g ð22Þ

Using the biorthogonality criteria, the constants aðiÞkj s for all j 6¼ k are found as

aðiÞkj ¼wj

� �TK I

i � PIki L

� �/kf g

kk � kjð23Þ

To find the constant aðiÞjj ’s, we use the two normalization criteria defined above. Since the right eigenvectorsform a complete set of vectors, an eigenvector can be represented by a linear combination of all othereigenvectors:

/Iki

� �¼Xn

j¼1

aðiÞkj /j

� �ð24Þ

where

aðiÞkj ¼0 j ¼ k

wjf gTKI

i�PIki L½ � /kf g

kk�kjj 6¼ k

8<: ð25Þ

The derivatives of eigenfrequencies depend only on the derivatives of the stiffness and mass matrices. Thederivatives of the stiffness matrix are more involved because they require taking the derivatives of the equiv-alent bending-stiffness matrix (Kapania and Goyal, 2002).

3. Results

In the present analysis, we used 10,000 data points and the results are presented in frequency density dia-grams or histograms, which show the distribution of the eigenfrequencies or buckling loads. The number ofcells used in the frequency density diagram was 16.

3.1. Random variables

In the present study, it is assumed that the beam is composed of identical plies that possess the same geo-metric and mechanical properties, and that the randomness of each ply angle and modulus of elasticity in thex-direction, bExxi , is the same for every ply and are spatially uncorrelated. Let hi and bExxi be the deterministicquantities of the ith lamina. In general, the ply-angle uncertainties are between �2:5�. Thus for the presentstudy we have assumed for hr a Gaussian distribution with a standard deviation of 2.5�. Thus there is a95.3% probability that the ply orientation will have an uncertainty between �5� and 5�, shown in Fig. 1(a).

The randomness in the material properties, with a 95% confidence interval, have an experimental coefficientof variation of 3%. However, in the present work we have assumed a coefficient of variation of 5%. Thus thereis a 95.3% probability that the ply orientation will have an uncertainty between �0.1 and 0.1. The probabilitydensity function for the various cases studied here are shown by Fig. 1(b) and (c). Note that because we have

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

-9.3 -8.0 -6.7 -5.4 -4.0 -2.7 -1.4 0.0 1.3 2.6 3.9 5.3 6.6 7.9 9.2 10.6

Ply Angle, [deg]

Pro

babi

lity

dens

ity

func

tion

[1/

deg] Generated random variable

Normal Distribution

Deterministic: θ = 0.00

Probabilistic: MEAN = 0.03 STD = 2.51

(a) Ply angle.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

-0.19 -0.16 -0.13 -0.11 -0.08 -0.05 -0.03 0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.18 0.21

Dimensionless Axial Modulus of Elasticity, nxx

Pro

babi

lity

dens

ity

func

tion

Deterministic: n xx = 1.00

Probabilistic: MEAN = 0.00 STD = 0.05

(b) Dimensionless axial modulus of elasticityfor isotropic beams.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-2.6 -2.2 -1.8 -1.5 -1.1 -0.7 -0.4 0.0 0.4 0.7 1.1 1.4 1.8 2.2 2.5 2.9

Dimensionless Axial Modulus of Elasticity, nxx

Pro

babi

lity

dens

ity

func

tion

Deterministic: n xx = 13.7088

Probabilistic C.O.V. = 0.05 MEAN = 0.01 STD = 0.69

(c) Dimensionless axial modulus of elasticityfor laminated beams.

Generated random variable

Normal Distribution

Generated random variable

Normal Distribution

Fig. 1. Probability density function for the full generation of possible variation of random variables about with zero mean.


nondimensionalized all quantities in the finite element formulation, we take the variation of the nondimen-sional axial modulus of elasticity:

nxx ¼Exx

Eyy

3.2. Probabilistic finite element analysis

For the uncertain analysis of laminated beams three models are developed: exact Monte Carlo simula-tion (EMCS), sensitivity-based Monte Carlo simulation (SBMCS), and probabilistic finite element analysis(PFEA). For all three methods above described we used the finite element method with five finite ele-ments. The three sets of boundary conditions used are: hinged–hinged, clamped–free, and clamped–clamped. Note that all the analysis was performed using the Goyal–Kapania element (Goyal and Kapania,2007). When studying the effect of uncertainties of random variables on the fundamental natural frequen-cies, it is convenient to study their squared value, i.e., eigenfrequencies, which are given in their nondimen-sional form as

kn ¼ knI0‘

4

Eyyboh3o

~kn ¼ kn12I0‘

4

Eyyboh3o

ð26Þ


The critical loads are given as

Pro

babi

lity

dens

ity

func

tion

Fig. 2.modul

bP n ¼ P n‘2

Eyyboh3o

eP n ¼ P n12‘2

Eyyboh3o

ð27Þ

3.3. Free vibrations

We first study the influence of having an uncertain dimensionless Young’s modulus, nxx, on the free vibra-tions of isotropic beams. Next, we study how the free vibration response of various laminated beams areaffected by uncertainties in nxx. Also, we studied the cases for ply-angle variations.

3.3.1. Isotropic beams: uncertain Young’s modulus

The probability distribution functions of the fundamental dimensionless eigenfrequencies, ~k, are shown inFigs. 2–4. Results show that by randomly generating possible values for the Young’s modulus in the x-direc-tion with a coefficient of variation of 5%, the fundamental dimensionless eigenfrequencies also have a coeffi-cient of variation of 5%. The figures show that a symmetric randomness in the random variables producessymmetric variation in the dimensionless fundamental natural frequency. Also, the sensitivity-based MonteCarlo simulation (SBMCS) when using only 1000 samples are in perfect agreement to those by the exactMonte Carlo simulation method (EMCS) using 10,000 samples.

When isotropic beams have both ends fixed, the variation in Exx cannot be ignored. The main reason is becausethe boundary conditions make the beam stiffer, increasing the fundamental frequency. Although the coefficientof variation is only 5%, it significantly affects the fundamental frequency because of the high frequencies.

For the case of sensitivity-based Monte Carlo simulation, we also studied the influence of the zeroth-, first-,and second-order variation on the dimensionless natural eigenfrequencies, where these orders are understoodas follows:

kk ¼ k0k|{z}

zeroth order

þXn

i¼1kI

ki�i|fflfflfflfflfflffl{zfflfflfflfflfflffl}first order

þ 1

2

Xn

i¼1

Xn

j¼1kII

kij�i�j|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}second order

ð28Þ

Fig. 5(a)–(c) shows that the second order terms have no significant influence on the overall dimensionless fun-damental eigenfrequency.

3.3.2. Laminated beams: uncertain Young’s modulus

Results for a unidirectional laminated beam with a ply angle of 0� are plotted in Fig. 6. Similar trends tothose of the isotropic case can be observed. However, for a 90� unidirectional laminated beam, Fig. 7 shows

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

75.4 77.9 80.5 83.0 85.5 88.1 90.6 93.2 95.7 98.3 100.8 103.3 105.9 108.4 111.0 113.5 116.1

Monte Carlo SimulationNormal Distribution

Dimensionless Eigenfrequency Dimensionless Eigenfrequency

Pro

babi

lity

dens

ity

func

tion

C.O.V. = 0.0508 MEAN = 97.23 STD = 4.86

(a) Monte Carlo Simulation

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

75.00 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

Monte Carlo Simulation

Sensitivity-Based MCS

Probabilistic FEA

Deterministic: 97.40

Probabilistic C.O.V. = 0.0508 MEAN = 97.23 STD = 4.86

(b) All three method sused here

Probability density function of the dimensionless eigenfrequency, ~k, for a simply-supported isotropic beam with uncertain Young’sus in the x-direction.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

9.7 10.0 10.3 10.6 11.0 11.3 11.6 11.9 12.3 12.6 12.9 13.2 13.6 13.9 14.2 14.5 14.9



Pro

babi

lity

dens

ity

func

tion

Pro

babi

lity

dens

ity

func

tion

C.O.V. = 0.050 MEAN = 12.45 STD = 0.6231


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

9.50 10.50 11.50 12.50 13.50 14.50



Probabilistic FEA


Probabilistic: C.O.V. = 0.050 MEAN = 12.45 STD = 0.623


Fig. 3. Probability density function of the dimensionless eigenfrequency, ~k, for a cantilevered isotropic beam with uncertain Young’smodulus in the x-direction.


0.00

0.00

0.00

0.01

0.01

0.01

0.01

0.01

0.02

0.02

390.2 403.3 416.5 429.6 442.8 455.9 469.1 482.2 495.4 508.5 521.7 534.8 548.0 561.1 574.3 587.4 600.6



Pro

babi

lity

dens

ity

func

tion

Pro

babi

lity

dens

ity

func

tion

C.O.V. = 0.050 MEAN = 503.2 STD = 25.14

0.00

0.00

0.00

0.01

0.01

0.01

0.01

0.01

0.02

0.02

375.00 425.00 475.00 525.00 575.00 625.00



Probabilistic FEA




Fig. 4. Probability density function of the dimensionless eigenfrequency, ~k, for a fixed-fixed isotropic beam with uncertain Young’smodulus in the x-direction.


that the Young’s modulus in the x-direction has very little effect on the free vibrational response. The reasonfor this could be that the laminate is stiffer in the x-direction, thus small variations in Exx do not influence thebeam’s fundamental frequency. Results for other boundary conditions were consistent with those of the can-tilevered case.

We also studied several laminated composites such as laminas with a layout of ½0�=h=h=0��, for allh ¼ 30�; 90�. Figs. 8–13 show these results. For all three boundary conditions it can be seen that the variationin the dimensionless fundamental frequency is the smallest for h ¼ 90�. Once again a very good agreementholds for all three methods employed in this study.

3.3.3. Laminated beams: uncertain ply angles

We also considered the cases when the ply orientations may become uncertain. For this case we studiedseveral laminated composites such as sandwiched laminas with a layout of ½0�=h=h=0��, for all h ¼ 30�; 90�.In all three models, the mean values and the coefficient of variations were close. However, the probabilisticfinite element analysis and the sensitivity-based Monte Carlo simulation are conservative in the sense that bothof them overestimate the variation of the natural frequencies. Exact Monte Carlo simulations would have beenthe most accurate approach but also a very expensive one. Therefore, the probabilistic finite element analysiscan be safely used. The sensitivity-based MCS is an alternative approach that produces fairly good results and

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

100.00

120.00

0 100 200 300 400 500 600 700 800 900 1000

Sampling Number

Dim

ensi

onle

ss E

igen

freq

uenc

y

Dim

ensi

onle

ss E

igen

freq

uenc

y

Zeroth Order Variation

First Order Variation

Second Order Variation

(a) Sensitivity derivatives

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

0 100 200 300 400 500 600 700 800 900 1000

Sampling Number




(b) Monte Carlo Simulation

-200.00

-100.00

0.00

100.00

200.00

300.00

400.00

500.00

600.00

0 100 200 300 400 500 600 700 800 900 1000

Sampling Number

Dim

ensi

onle

ss E

igen

freq

uenc

y




(c) All three methods used here

Fig. 5. Effect of the order of the sensitivity derivatives on the dimensionless eigenfrequency, ~k, for various boundary conditions of anisotropic beam with uncertain Young’s modulus in the x-direction.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

130.8 135.2 139.7 144.1 148.5 153.0 157.4 161.9 166.3 170.8 175.2 179.6 184.1 188.5 193.0 197.4 201.9



Pro

babi

lity

dens

ity

func

tion

Pro

babi

lity

dens

ity

func

tion



0.00

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0.04

0.05

0.05

135.00 145.00 155.00 165.00 175.00 185.00 195.00 205.00

Monte Carlo SimulationSensitivity-Based MCS

Probabilistic FEA



(b) All three methods used here

Fig. 6. Probability density function of the dimensionless eigenfrequency for a cantilevered laminated beam (0�) with uncertain Young’smodulus in the x-direction.


0.00

20.00

40.00

60.00

80.00

100.00

120.00

12.34

1

12.34

3

12.34

5

12.34

8

12.35

0

12.35

2

12.35

4

12.35

7

12.35

9

12.36

1

12.36

3

12.36

5

12.36

8

12.37

0

12.37

2

12.37

4

12.37

6

Monte Carlo SimulationNorm al Distribu tion

Prob

abili

ty d

ensi

ty f

unct

ion

Prob

abili

ty d

ensi

ty f

unct

ionPr obab ilistic:

C.O.V. = 0.0003 MEAN = 12.361 STD = 0.004


0.00

20.00

40.00

60.00

80.00

100.00

120.00

12.340 12.345 12.350 12.355 12.360 12.365 12.370 12.375 12.380




Probabilistic FEA

Determ inistic: 12.36

Pr obab ilistic: C.O.V. = 0.0003 MEAN = 12.361 STD = 0.004


Fig. 7. Probability density function of the dimensionless eigenfrequency, ~k, for a cantilevered unidirectional laminated beam (90�) withuncertain Young’s modulus in the x-direction.

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

1.20E-03

1.40E-03

4922 5084 5246 5408 5570 5731 5893 6055 6217 6379 6541 6702 6864 7026 7188 7350 7511



Pro

babi

lity

dens

ity

func

tion

Pro

babi

lity

dens

ity

func

tion



0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

4750.0 5250.0 5750.0 6250.0 6750.0 7250.0 7750.0

Monte Carlo SimulationSensitivity-Based MCSProbabilistic FEA




Fig. 8. Probability density function of the dimensionless eigenfrequency, ~k, for a fixed-fixed laminated beam ([0�/30�/30�/0�]) withuncertain Young’s modulus in the x-direction.

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

1.20E-03

1.40E-03

1.60E-03

4727 4882 5037 5192 5347 5502 5657 5812 5968 6123 6278 6433 6588 6743 6898 7053 7209

Monte Carlo SimulationNorm al Distribu tion

Dimensionless Eigenfrequency

Prob

abili

ty d

ensi

ty f

unct

ion

Prob

abili

ty d

ensi

ty f

unct

ion

Pr obab ilistic: C.O.V. = 0.049 MEAN = 6062.10 STD = 296.67


0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

1.6E-03

4750.0 5250.0 5750.0 6250.0 6750.0 7250.0




Probabilistic FEA

Determ inistic: 6073.016

Pr obabilistic: C.O.V. = 0.049 MEAN = 6062.10 STD = 296.67


Fig. 9. Probability density function of the dimensionless eigenfrequency, ~k, for a fixed–fixed laminated beam ([0�/90�/90�/0�]) withuncertain Young’s modulus in the x-direction.


0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

123 127 131 135 139 143 147 151 155 160 164 168 172 176 180 184 188



Pro

babi

lity

dens

ity

func

tion

Pro

babi

lity

dens

ity

func

tion



0.0E+00

1.0E-02

2.0E-02

3.0E-02

4.0E-02

5.0E-02

6.0E-02

125.0 135.0 145.0 155.0 165.0 175.0 185.0 195.0



Probabilistic FEA




Fig. 10. Probability density function of the dimensionless eigenfrequency, ~k, for a cantilevered laminated beam ([0�/30�/30�/0�]) withuncertain Young’s modulus in the x-direction.


0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

116 120 124 128 132 136 140 144 148 151 155 159 163 167 171 175 179


Pro

babi

lity

dens

ity

func

tion

Pro

babi

lity

dens

ity

func

tion



0. 0E +0 0

1. 0E -0 2

2. 0E -0 2

3. 0E -0 2

4. 0E -0 2

5. 0E -0 2

6. 0E -0 2

120.0 130.0 140.0 150.0 160.0 170.0 180.0






Fig. 11. Probability density function of the dimensionless eigenfrequency, ~k, for a cantilevered laminated beam ([0�/90�/90�/0�]) withuncertain Young’s modulus in the x-direction.

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

8.00E-03

965 998 1030 1062 1094 1127 1159 1191 1223 1255 1288 1320 1352 1384 1416 1449 1481



Pro

babi

lity

dens

ity

func

tion



0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

7.0E-03

900.0 1000.0 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0


Pro

babi

lity

dens

ity

func

tion





Fig. 12. Probability density function of the dimensionless eigenfrequency, ~k, for a simply-supported laminated beam ([0�/30�/30�/0�]) withuncertain Young’s modulus in the x-direction.






0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

8.00E-03

916 947 977 1008 1039 1069 1100 1130 1161 1191 1222 1253 1283 1314 1344 1375 1405


Pro

babi

lity

dens

ity

func

tion


0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

7.0E-03

8.0E-03

900.0 1000.0 1100.0 1200.0 1300.0 1400.0

Pro

babi

lity

dens

ity

func

tion




Fig. 13. Probability density function of the dimensionless eigenfrequency, ~k, for a simply-supported laminated beam ([0�/90�/90�/0�]) withuncertain Young’s modulus in the x-direction.


saves time. This approach produces very good results for only 1000 samples as opposed to 10,000 samplesemployed in the exact MCS.

3.4. Reliability analysis

In the sense of structural stability, a structure is safe only if the actual load applied to the component doesnot exceed the critical load. In the traditional method, the degree of safety is usually expressed by the safetyfactor, SF. A higher value of the safety factor would indicate a safer component. However this is not neces-sarily the case as the inevitable variations must be kept in mind. Let us consider the case of a structure havinga critical load of P cr and the nominal values of P ¼ P cr=SF. Now because of the inherent imperfection in thestructure, let the probability density functions of the dimensionless critical load be f ðkÞ. It is our goal to findthe probability of failure for a structure designed for various safety factors, i.e., k1 ¼ kcr=SF. In the presentwork, we calculate the normal distribution functions by using either the exact MCS, sensitivity-basedMCS, or the probabilistic FEA.

3.4.1. Conservative case

We first study the influence of having an uncertain dimensionless Young’s modulus, nxx, on the buckling ofisotropic beams. Next, we study how the stability of various laminated beams is affected by uncertainties in nxx

and ply angles.For the case of isotropic beams, it was found that the dimensionless Young’s modulus in the x-direction

does affect the dimensionless buckling load, although the variation is small. The variation of the dimensionlessbuckling load is shown in Figs. 14–16. For the case of a fixed–fixed isotropic beam, the variations in the buck-ling load are significant because the variation in the stiffness cannot be ignored. This shows that traditionalmean values maybe misleading and thus the variation should be considered.

We also studied various cases of unidirectional cantilevered laminated beams with a ply angle of 0�. Figs. 17and 18 show that the influence on the buckling load of the variation of Exx is smaller when compared to thevariation in ply angles for unidirectional laminated beams. In other words, the variation in Exx can be ignoredfor such cases. Also, it was found that Exx had no influence whatsoever on the buckling load of unidirectionallaminated beams with a ply angle of 90�. A similar trend was found for all other unidirectional laminatedbeams. The reliability of the structure is shown in Figs. 14–16.

When performing the reliability analysis for all the conservative cases studied here, results showed that, forthe uncertainties considered here, the structure is reliable when it is designed for a safety factor of 1.5, a valuetraditionally used in aerospace design. Thus, when uncertainties in ply orientations and Young’s modulus inthe x-direction affect the laminated beams, the structure can be safely modeled using deterministic approaches.

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

30.00 32.00 34.00 36.00 38.00 40.00 42.00 44.00 46.00 48.00 50.00

Dimensionless Buckling Load

Pro

babi

lity

den

sity

fun

ctio

n



Probabilistic (MCS): C.O.V. = 0.050 MEAN = 40.01 STD = 2.010

Non-Conservativeness:ηη = 0.0

(a) Probability density function

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

45 50 55 60 65 70 75 80 85 90 95 100 105

Reliability (%)

Safe

ty F

acto

r

(b) Structure’s reliability againstvari-ous safety factors

Fig. 15. Probability density function of the dimensionless buckling load, eP , and the structure’s reliability for a fixed-fixed isotropic beamwith uncertain Young’s modulus in the x-direction under a conservative compressive loading.

0.0E+00

5.0E-01

1.0E+00

1.5E+00

2.0E+00

2.5E+00

3.0E+00

3.5E+00

1.80 2.00 2.20 2.40 2.60 2.80 3.00


Pro

babi

lity

den

sity

fun

ctio

n



Probabilistic FEA


Probabilistic (MCS): C.O.V. = 0.05 MEAN = 2.463 STD = 0.125 Non-Conservativeness:

ηηηη = 0.0


1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

45 50 55 60 65 70 75 80 85 90 95 100 105

Reliability (%)

Safe

ty F

acto

r

(b) Structure’s reliability against vari-ous safety factors

Fig. 14. Probability density function of the dimensionless buckling load, eP , and the structure’s reliability for a cantilevered isotropic beamwith uncertain Young’s modulus in the x-direction under a conservative compressive loading.


We must, however, caution that this statement is only true under the uncertainties considered here. Presence ofother uncertainties, e.g., the loads, boundary conditions etc., would make this conclusion invalid. A higherfactor of safety will then be warranted.

3.4.2. Nonconservative case

For the nonconservative problem, we find the critical load such that the first two eigenfrequencies coalesce,known as flutter point. At the onset of flutter we know that

dPdkk¼ 0 ð29Þ

Recall that the zeroth-order eigenvalue problem, given by Eq. (11), is defined as

K0 � P0kL0 � k0

kM0� �

/0k

� �¼ 0 ð30Þ

0.0E+00

1.0E-01

2.0E-01

3.0E-01

4.0E-01

5.0E-01

6.0E-01

7.0E-01

8.0E-01

9.0E-01

7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50

Pro

babi

lity

dens

ity

func

tion



Probabilistic (MCS) C.O.V. = 0.050 MEAN = 9.843 STD = 0.501 Non-Conservativeness:

ηη = 0.0

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

45 50 55 60 65 70 75 80 85 90 95 100 105

Safe

ty F

acto

r


(a) Probability density functionReliability (%)


Fig. 16. Probability density function of the dimensionless buckling load, eP , and the structure’s reliability for a simply-supported isotropicbeam with uncertain Young’s modulus in the x-direction under a conservative compressive loading.

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

25.00 27.00 29.00 31.00 33.00 35.00 37.00 39.00 41.00 43.00


Pro

babi

lity

dens

ity

func

tion



Probabilistic (MCS) C.O.V. = 0.05 MEAN = 33.42 STD = 1.72

Non-Conservativeness:= 0.0


1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

40 45 50 55 60 65 70 75 80 85 90 95 100 105

Reliability (%)

Safe

ty F

acto

r


Fig. 17. Probability density function of the dimensionless buckling load, eP , and the structure’s reliability for a cantileveredunidirectionally laminated beam of a ply of 0� and uncertain Young’s modulus in the x-direction under a conservative compressiveloading.


Now premultiplying the above equation by the left eigenvector and solving for the critical load, we get

P0k ¼

w0k

� �TK0� �

/0k

� �w0

k

� �TL0� �

/0k

� � � k0k

w0k

� �TM0� �

/0k

� �w0

k

� �TL0� �

/0k

� � ð31Þ

By virtue of Eq. (29), at the onset of flutter

w0k

� �TM0� �

/0k

� �¼ 0 ð32Þ

Thus the equations for the sensitivity derivatives for the critical load at the onset of flutter become

PIki ¼

w0k

� �TK I

i

� �/0

k

� �w0

k

� �TL0� �

/0k

� � ; PIIkij ¼

w0k

� �TK II

ij

h i/0

k

� �w0

k

� �TL0� �

/0k

� � ð33Þ

where the left and right eigenvectors, and the stiffness derivatives, are evaluated at the onset of flutter using themean values of the random variables.

Fig. 19 shows the variation in the critical load, which occurs at flutter, for a purely tangential follower load.Results show that isotropic beams under nonconservative loading also have a small probability of failure. For

0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

1.2E-01

1.4E-01

18.00 23.00 28.00 33.00 38.00 43.00 48.00


Pro

babi

lity

den

sity

fun

ctio

n




ηη = 0.0


1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

45 50 55 60 65 70 75 80 85 90 95 100 105

Reliability (%)

Safe

ty F

acto

r


Fig. 18. Probability density function of the dimensionless buckling load, eP , and the structure’s reliability for a cantileveredunidirectionally laminated beam of a ply of 0� and uncertain ply angle under a conservative compressive loading.

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

3.0E-01

3.5E-01

4.0E-01

4.5E-01

15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00 24.00 25.00


Pro

babi

lity

dens

ity

func

tion



Probabilistic (MCS) C.O.V. = 0.052 MEAN = 19.98 STD = 1.71

Non-Conservativeness:ηηηη =1.0


1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

45 50 55 60 65 70 75 80 85 90 95 100 105

Reliability (%)

Safe

ty F

acto

r


Fig. 19. Probability density function of the dimensionless buckling load, eP , and the structure’s reliability for an isotropic cantileveredbeam with uncertain Young’s modulus under a purely tangential follower load.


the laminated beams, shown in Figs. 20 and 21, the structure will be reliable when it is designed for a safetyfactor of 1.5, a value traditionally used in aerospace design. Once again, we reiterate our cautionary noteabout the presence of other uncertainties requiring a higher safety factor.

Thus, as was the case for conservative loading, the laminated beams can be modeled using deterministicapproaches for the type of uncertainties considered here.

4. Final remarks

Monte Carlo simulation has been applied to laminated beams with randomness in ply orientation and themodulus of elasticity in the x-direction to study their effect on the free vibration and stability of the structure.At least 10,000 realizations of the Monte Carlo sampling have been performed to improve the accuracy of theanalysis. A second-order sensitivity-based Monte Carlo simulation (SBMCS) has been developed using pertur-bation methods. Using Taylor series expansion, the eigenvalues have been expressed as probabilistic quanti-ties. The accuracy of the free vibration and stability response has been compared to that obtained by the exactMonte Carlo simulation. A third approach, called the probabilistic finite element approach (PFEA), was alsodeveloped. It gave results that were in good agreement with those given by SBMCS and gave a very good pre-

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

3.0E-01

3.5E-01

4.0E-01

18.00 19.00 20.00 21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00


Pro

babi

lity

dens

ity

func

tion



Probabilistic FEA



ηηηη =1 .0


1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

45 50 55 60 65 70 75 80 85 90 95 100 105

Reliability (%)

Safe

ty F

acto

r


Fig. 20. Probability density function of the dimensionless buckling load, bP , and the structure’s reliability for an undirectional laminatedcantilevered beam (0�) with uncertain Young’s modulus under a purely tangential follower load.

0.0E+00

5.0E-01

1.0E+00

1.5E+00

2.0E+00

2.5E+00

3.0E+00

3.5E+00

4.0E+00

4.5E+00

22.20 22.40 22.60 22.80 23.00 23.20


Pro

babi

lity

dens

ity

func

tion



Probabilistic FEA


Probabilistic (MCS): C.O.V. = 0.005 MEAN = 22.710 STD = 0.093

Non-Conservativeness:ηηηη =1.0


1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

45 50 55 60 65 70 75 80 85 90 95 100 105

Reliability (%)

Safe

ty F

acto

r


Fig. 21. Probability density function of the dimensionless buckling load, bP , and the structure’s reliability for an undirectional laminatedcantilevered beam (0�) with uncertain ply angle under a purely tangential follower load.


diction of the behavior of the fundamental natural frequency in the presence of uncertainties in ply angles andin Young’s modulus in the x-direction.

The two methods employed, SBMCS and PFEA, are advantageous over simulation techniques, such asMCS, because the eigenvalue problem is solved only once. Also, an elegant way to obtain sensitivity deriva-tives was used. Based upon the results, the SBMCS and PFEA result in a great computational saving whenone is interested in predicting the statistics of the fundamental natural frequency of laminated beams. Asan example, the MCS for the case of nonconservative loading took about 9–10 h whereas the other two meth-ods took about 1 min.

For the case of free vibration, it was observed that the eigenfrequencies undergo larger changes when vari-ations in uncertain ply angles than those variations in the modulus of elasticity in the x-direction. Similarbehavior was observed for conservative and nonconservative stability analysis. The reliability analysis showedthat for the types of problems solved here, a deterministic approach, using the traditional safety factor of 1.5,would have been sufficient in the absence of uncertainties not considered here.

In this study, we have restricted ourselves to uncertainties that are small (i.e., sensitivity-based analysis isadequate). For systems with large uncertainties, an eigenvalue analysis using polynomial chaos would be moreappropriate (Mulani et al., 2006; Mulani et al., 2007).


Acknowledgments

This work was performed under the Grant NAG-1-2277 from the NASA Langley Research Center with Dr.Lucas Horta and Dr. Howard Adelman as the grant monitors. The authors gratefully acknowledge the tech-nical discussions with the two grant monitors. The research presented herein is an extension of the work pre-sented at the 44th AIAA/ASME/ACE/AHS/ASC SDM Conference, Norfolk, VA, 7–10 April 2003, AIAA-2003-1914.

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