Estimation of elastic constants of thick laminated plates within a Bayesian framework

www.elsevier.com/locate/compstruct

Composite Structures 80 (2007) 461–473

Estimation of elastic constants of thick laminated plateswithin a Bayesian framework

Federica Daghia, Stefano de Miranda *, Francesco Ubertini, Erasmo Viola

Dipartimento di Ingegneria delle Strutture, dei Trasporti, delle Acque, del Rilevamento, del Territorio (DISTART),

Universita di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

Available online 28 July 2006

Abstract

This paper compares two estimators for the dynamic identification of elastic constants of thick laminated composite plates. Theplate’s response is modeled by finite elements based on Reddy’s third-order theory. The elastic constants are estimated within a Bayesianframework, using two estimators available in the literature. The estimators differ in the way they account for a priori information on theelastic constants to be identified. The resultant estimation procedures are critically examined and compared by paying special attention tothe influence of a priori information on the final estimates. A unified interpretation of the two estimators is given and a modified strategyis proposed in order to improve reliability and convergence of the estimation process. Identification results for various case studies arepresented.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Laminated thick plates; Numerical–experimental techniques; Bayesian estimation

1. Introduction

The aim of this work is to critically examine and com-pare two different estimators for the dynamic identificationof elastic constants of thick laminated composite plates.Estimation is considered in the context of the so callednumerical–experimental techniques, which attempt to min-imize in some sense the difference between the measureddata and the response predicted by the assumed model.

The identification of structural characteristics throughnumerical–experimental techniques based on vibrationdata dates back to the 1970s. Statistical analysis of vibrat-ing structural systems was early carried out by of Collinsand Thomson [1] and Hasselman and Hart [2]. A probabi-listic method to identify the elastic constants of simplestructures starting from natural frequency data was pro-posed by Collins et al. [3]. Since the 1990s, the sameapproach has been applied to laminated composite plates

0263-8223/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2006.06.030

* Corresponding author. Fax: +39 051 2093495.E-mail address: [email protected] (S. de Miranda).

(see for example Viola et al. [4] and the references therein).Laminates are complex structural elements: their behaviorresults from the combined response of fibres and matrixand strongly depends on the number of layers and thestacking sequence, which generally causes couplingbetween in plane and bending behavior. For this reason,it is difficult to determine the elastic constants through sta-tic tests, as for traditional materials, and identificationtechniques based on measuring of natural frequencies arewidely adopted.

Here, only natural frequency measurements are sup-posed to be available for the identification process, sincethey can be evaluated experimentally through a simplemodal test and, in most approaches present in the litera-ture, they are generally found to be sufficient to estimatesatisfactorily the elastic constants. Anyway, the estimationprocess could be improved by adding more information onthe structural response, such as that coming from modeshapes.

Many approaches have been proposed which differ inthe choice of the theoretical and numerical plate model

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462 F. Daghia et al. / Composite Structures 80 (2007) 461–473

and the procedure used to estimate the elastic constants. Inparticular, these procedures aim at minimizing a nonlinearobjective function, which involves the difference betweenmeasured and numerical frequencies, and differ in theobjective function and the algorithm used to solve the non-linear minimization problem.

The simplest objective functions are ordinary leastsquares or weighted least squares (see for example [5]).These functions take into account only the informationcoming from the experiment, and require no a prioriknowledge on the possible parameter values. Althoughgeneral, this approach needs robust and advanced searchalgorithms to obtain reliable results. Optimum design tech-niques were employed in [6], experimental design tech-niques together with the response surface method wereemployed in [7,8] and a feasible directions nonlinear inte-rior point algorithm was employed in [9,10].

Alternatively, more complex objective functions can beadopted by taking into account the a priori informationon the elastic constants (Bayesian framework). This is theapproach used, among others, by Lai and Ip [11] for thinplates and later extended to thick plates by Bartoli et al.[12]. A different choice to take into account the a prioriknowledge on initial values can be found in [13,14]. A com-parison between different methods for the identification ofin plane elastic constants of steel orthotropic plates wascarried out by Lauwagie et al. [15]. In all these cases, theminimum of the objective function can be easily computedthrough a simple iterative algorithm, provided that the ini-tial guesses of the elastic constants are not too far from thetarget values. Indeed, this is often the case for laminatedcomposites, since a reasonable set of initial estimates canbe found through the rule of mixtures, given the geometricproperties of the laminae (fibre and matrix volume frac-tion) and an initial guess on the components’ materialproperties (fibre and matrix elastic constants).

In this work, two well-known estimators which makeuse of a priori information are considered for the identifi-cation of elastic constants of thick plates based on mea-sured natural frequencies. The first estimator stems fromBayes theorem and can be strictly termed as Bayesian.The second estimator is the minimum variance estimator.Although these formulations are very different, the analysisreveals a certain resemblance between the two, which helpsto understand their behavior and suggests an improvedsolution procedure. To complete the estimation process,four nodes finite elements are used to model the thick platewithin a higher order two dimensional theory. In particu-lar, Reddy’s third-order theory is assumed, since it isexpected to accurately capture the first natural modes,which are usually measured for estimation purposes.

The paper is organized as follows. In Section 2, the twoestimators are presented. The laminated plate model isintroduced in Section 3. The analysis and comparison ofthe two estimation procedures are carried out in Section4 and an improved solution strategy is proposed in Section5. Finally, two case studies are discussed in Section 6.

2. Estimation procedures

In a physical problem, the relationship between theproblem data (input) and the outcome of the experiment(output) can be generally represented by

Y ¼ gðbÞ þ e ð1Þwhere Y is the vector containing the output values, g is a func-tion representing the model used to describe the phenomenon,b is the vector of the input parameters and e is the error, whichincludes both modeling and measurement errors.

In structural engineering, b are the geometrical andmechanical properties of the structure and Y representsthe structural response, for example in terms of naturalfrequencies.

An estimation procedure aims at estimating some of theinput parameters, given the outcome of an experiment, byminimizing an objective function. In the following, twowell known estimators are considered: the Bayesian estima-tor (B) and the minimum variance estimator (MVE). Boththe estimators take into account the reliability of the exper-imental data and introduce some a priori knowledge on theparameters to be identified. However, the role played bythe initial estimates in the two procedures is very different.

2.1. Bayesian estimator

The Bayesian estimator [16] is obtained by applyingBayes theorem of conditional probabilities, that states

f ðbjYÞ ¼ f ðYjbÞf ðbÞf ðYÞ ð2Þ

where f(a) is the probability density function of the variablea and f(ajb) stands for the probability of the event a giventhat the event b has occurred. In the B estimator, the aim isto maximize the function (2). The probability densitiesf(Yjb) and f(b) are given by the following functions:

f ðYjbÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffijVY j�1

ð2pÞm

sexp �ðY� gÞTV�1

Y ðY� gÞ2

!ð3Þ

f ðbÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffijVbj�1

ð2pÞp

sexp �

ðb� lÞTV�1b ðb� lÞ

2

!ð4Þ

where VY and Vb are the covariance matrices of the exper-imental error and initial estimates, respectively, l is the vec-tor containing the initial estimates, m is the number ofexperimental observations and p is the number of parame-ters. Eq. (3) represents the information deduced from theexperiment, while Eq. (4) represents the a priori informa-tion on the parameters to be estimated. The matrices VY

and Vb have to be specified by the analyst.Using Eqs. (3) and (4), the natural logarithm of Eq. (2)

takes the form

lnðf ðbjYÞÞ ¼ � 1

2½ðmþ pÞ lnð2pÞ þ ln jVY j þ ln jVbj þ SB�

� lnðf ðYÞÞ

F. Daghia et al. / Composite Structures 80 (2007) 461–473 463

where the only term dependent on the parameters is

SB ¼ ðY� gÞTV�1Y ðY� gÞ þ ðb� lÞTV�1

b ðb� lÞ ð5Þ

thus the B estimator is obtained by minimizing SB. Calcu-lating the derivatives with respect to b and setting themequal to zero, one obtains

rbSB ¼ 2½�STðbÞV�1Y ðY� gðbÞÞ � V�1

b ðl� bÞ� ¼ 0 ð6Þ

where

SðbÞ ¼ rbgðbÞ

is the sensitivity matrix. Notice that in Eq. (6) the parame-ters b appear both explicitly and implicitly.

In order to minimize SB, an iterative procedure is estab-lished. Let b be a vector of initial parameter values. Theterm g(b) is approximated by a truncated Taylor seriesexpansion in the neighbourhood of b

gðbÞ � gðbÞ þ SðbÞðb� bÞ; SðbÞ � SðbÞ ð7Þ

Hence, Eq. (6) takes the form

STðbÞV�1Y ½Y� gðbÞ � SðbÞðb� bÞ� þ V�1

b ðl� bÞ � 0 ð8Þ

and a recurrence relation can be established

ðSkÞTV�1Y ½Y� gk � Skðbkþ1 � lÞ � Skðl� bkÞ�

� V�1b ðbkþ1 � lÞ ¼ 0

where the superscript k denotes the quantity calculated atthe kth iteration, bk are the prior set of estimates andbk+1 are the parameter estimates at the end of the currentiteration.

Solving for bk+1 yields

bkþ1 ¼ lþ PkðSkÞTV�1Y ½Y� gk � Skðl� bkÞ�

Pk ¼ ½ðSkÞTV�1Y Sk þ V�1

b ��1

ð9Þ

2.2. Minimum variance estimator

The estimator presented in the work of Collins et al. [3]is obtained by minimizing the variance associated with theestimator.

The function g(b) is expanded in Taylor series as in (7).The relation (1) between the experimental measurementsand the parameters becomes

Y � gðbÞ þ SðbÞðb� bÞ þ e ð10ÞThen, introducing the following quantities:

�b ¼ ðb� bÞ; Y ¼ Y� gðbÞEq. (10) can be written as

Y ¼ S�bþ e

The variables �b are normally distributed with zero meanand covariance matrix V�b ¼ Vb. The experimental errorse are assumed to be normally distributed, with zero meanand covariance VY, and uncorrelated with �b. Thus, the

mean of Y is null, the covariance of Y and the covarianceof Y and �b are given by

V�Y ¼ SVbST þ VY ; V�Y �b ¼ SVb ð11Þ

The following inverse relation holds:

�b� ¼ GY

where �b� is the vector of the estimates of �b.The aim is to find the estimator G that minimizes the

variance associated with the estimator itself

E½ð�b� � �bÞð�b� � �bÞT� ¼ GV�Y GT � VT�Y �bGT �GV�Y �b þ VY

This yields

G ¼ VT�Y �bV�1

�Y

and, with some algebra, the estimator can be put in theform

b ¼ bþ VbSTðSVbST þ VY Þ�1ðY� gðbÞÞ

By setting b = bk+1 and b = bk, the following iterationscheme can be established:

bkþ1 ¼ bk þ VbðSkÞT½SkVbðSkÞT þ VY ��1ðY� gkÞ ð12Þ

At the end of each iteration it is possible to evaluate thenew covariance matrix of the parameters Vkþ1

b by

Vkþ1b ¼ Vb � VT

�Y �bV�1�Y V�Y �b

¼ Vb � VbðSkÞT½SkVbðSkÞT þ VY ��1SkVb

2.3. Some remarks on the equivalence of the two estimators

The two iteration schemes (9) and (12) coalesce if posingl = bk. In other words, the MVE estimator can be inter-preted as a B estimator with initial parameters not keptfixed, but updated at each iteration. A proof of this state-ment can be obtained through the inversion lemma ofmatrices.

The following matrix identity holds:

ðIP þ ABÞ�1A ¼ AðIN þ BAÞ�1 ð13Þ

where AP·N and BN·P are general matrices, IP and IN arethe identity matrices of dimensions P and N. By posingA ¼ VbSTV�1

Y and B = S, Eq. (13) becomes

ðV�1b þ STV�1

Y SÞ�1STV�1

Y ¼ VbSTðVY þ SVbSTÞ�1

which shows the equivalence between Eqs. (9) and (12)under the hypothesis l = bk.

To sum up, the two estimators coincide in the linearcase, but the iterative corrections needed for nonlinearproblems may lead to very different results. In the B estima-tor, the a priori information enters the objective functionand somehow conditions the final estimates, besides drivingthe estimation process. In the MVE estimator, on the otherhand, the a priori information provides starting values to

Fig. 1. Problem and material coordinate systems.


the solution algorithm and its influence gradually decreasesas the iterations proceed, so that only the experimentalinformation is retained at the end.

2.4. Convergence criterion

The iteration schemes presented in (9) and (12) requirethe definition of a convergence criterion. Here, a criterionbased on the parameter estimates is adopted: the iterationsstop when the updated parameters bk+1 fall within a certaininterval from the previous set of estimates bk. This can bewritten as

bkþ1i � bk

i

bki

�� 6 TOL 8i

where TOL is the prescribed tolerance.

3. Laminated plate model

The estimation procedures discussed in Section 2 are ofgeneral validity and can be applied to any physical phe-nomenon, once a suitable predictive model is chosen to rep-resent it. In this work, the experimental data is composedof a set of natural frequencies of free-free thick laminatedcomposite plates, thus the choice of the model dependson its ability to accurately predict the dynamic behaviorof such structures.

Various theories have been proposed to model compos-ite laminates [17], all of them presenting a trade-off betweenaccuracy and computational complexity. In the two dimen-sional setting, Equivalent Single Layer (ESL) theories rep-resent a heterogeneous laminated plate as a staticallyequivalent homogeneous single layer with a complex con-stitutive behavior. Among ESL theories, the most commonare the Classical Laminated Plate Theory (CLPT) and theFirst-order Shear Deformation Theory (FSDT).

CLPT is the extension to composite laminates of theKirchhoff–Love plate theory. Its kinematic hypothesisneglects the presence of shear strains, thus predictinghigher natural frequencies than the corresponding threedimensional solution. In thick laminated plates significantshear deformation occurs, therefore FSDT or a higherorder theory is required for accurate evaluation of the nat-ural frequencies.

FSDT tends to overcorrect the CLPT predictions byyielding lower natural frequency values than its threedimensional counterpart, both for isotropic [18] and lami-nated composite plates [19]. This effect becomes increas-ingly important when higher modes are considered, as inthe present analysis.

For these reasons, Reddy’s third-order theory [20] hasbeen chosen to model the plate response. It approximatesthe in plane displacements with third-order polynomials,thus allowing for parabolical shear strains, while retainingthe same number of displacement parameters as the FSDT.Here, the usual notation of Reddy’s theory has been

slightly modified, in particular regarding the definition ofthe generalized strains and stresses.

3.1. Displacements and generalized strains

The problem coordinate system (x,y,z) associated to theplate is shown in Fig. 1. This reference system is consideredunless explicitly stated. The displacement components u, v,w are assumed as

u ¼ u0 þ zwx þ z3c wx þ ow0

ox

� �v ¼ v0 þ zwy þ z3c wy þ ow0

oy

� �w ¼ w0

8><>: ð14Þ

where u0, v0 and w0 represent the displacement componentsof the midplane of the plate, wx and wy are the rotationsaround the y and x axes, respectively, c = �4/(3h2) and h

is the laminate thickness. The displacement field (14) is ob-tained from a general third-order theory by introducing thea priori hypothesis of zero shear strains at the top and bot-tom of the laminate, z = ±h/2.

Under the hypothesis of small displacements, the gener-alized strains are given by

ex

ey

cxy

vx

vy

vxy

vhx

vhy

vhxy

cyz

cxz

26666666666666666666664

37777777777777777777775

¼

o=ox 0 0 0 0

0 o=oy 0 0 0

o=oy o=ox 0 0 0

0 0 0 o=ox 0

0 0 0 0 o=oy

0 0 0 o=oy o=ox

0 0 o2=ox2 o=ox 0

0 0 o2=oy2 0 o=oy

0 0 2o2=oxoy o=oy o=ox

0 0 o=oy 0 1

0 0 o=ox 1 0

26666666666666666666664

37777777777777777777775

u0

v0

w0

wx

wy

26666664

37777775

ð15Þor, in compact form, by

e ¼ Du


Notice that, in addition to the classical components of thefirst-order Reissner–Mindlin theory, the higher order cur-vatures vh

x , vhy and vh

xy are introduced.

3.2. Generalized stresses

The generalized stress components can be simply definedas the work-conjugate quantities to the generalized straincomponents

Nx N y Nxy½ � ¼Z h

2

�h2

rx ry rxy½ �dz ð16Þ

Mx My Mxy½ � ¼Z h

2

�h2

rx ry rxy½ �zdz ð17Þ

Mhx Mh

y Mhxy

h i¼Z h

2

�h2

rx ry rxy½ �cz3 dz ð18Þ

Sy Sx½ � ¼Z h

2

�h2

ryz rxz½ �ð1þ 3cz2Þdz ð19Þ

where r• are the components of the stress tensor.

3.3. Constitutive equations of a lamina

The 3D constitutive equations for a single orthotropiclamina in plane stress conditions can be expressed in thematerial coordinate system (1, 2,3) associated to the singlelamina (see Fig. 1) as follows:

r1

r2

r12

264

375 ¼

CL11 CL

12 0

CL12 CL

22 0

0 0 CL66

264

375

e1

e2

e12

264

375

r23

r13

� ¼ CL

44 0

0 CL55

" #e23

e13

�

where e• are the components of the strain tensor in thematerial coordinate system, the superscript L stands forthe Lth lamina and the over bar for the material coordinatesystem. The reduced stiffness coefficients CL

ij are

CL11 ¼

EL1

1� mL12m

L21

; CL22 ¼

EL2

1� mL12m

L21

; CL12 ¼

mL12EL

2

1� mL12m

L21

ð20ÞCL

66 ¼ GL12; CL

44 ¼ GL23; CL

55 ¼ GL13 ð21Þ

being EL1 and EL

2 the normal moduli along the 1 and 2 direc-tions, mL

12 the in plane Poisson’s ratio, GL12, GL

23 and GL13 the

in plane and transverse shear moduli, respectively.The above relations transformed to the problem coordi-

nates (x,y,z) can be written in compact form as

rf ¼ CLf ef

rs ¼ CLs es

ð22Þ

where f and s denote the in plane and shear terms.

3.4. Constitutive equations of the laminate

Using Eqs. (15)–(19) and (22), the constitutive relationsfor the laminate are obtained

ð23Þ

Since the stress–strain relations (22) are defined on the sin-gle lamina, the integrals in (16)–(19) must be split into asum of integrals. The resultant constitutive matrices are

CðqÞf

h i¼XnL

L¼1

Z zL

zL�1

CLijz

q dz; i; j ¼ 1; 2; 6 ð24Þ

CðqÞs

�¼XnL

L¼1

Z zL

zL�1

CLijz

q dz; i; j ¼ 4; 5 ð25Þ

where CLij are the stiffness coefficients of the Lth lamina in

the problem coordinates, zL is the z coordinate of the topsurface of the Lth lamina and nL is the total number oflaminae.

In compact form, Eq. (23) can be written as follows:

s ¼ Ce

3.5. Finite element formulation

The model equations are obtained from Hamilton’s prin-ciple using a finite element formulation. A four nodes quad-rilateral element has been implemented. The displacement u


is approximated over the typical element Xe through shapefunctions in the following way:

u ¼ N�u

where �u is the nodal displacement vector and N is the ma-trix containing the shape functions. In particular, the com-ponents u0, v0, wx and wy are represented through Lagrangelinear shape functions, while w0 is represented throughHermite cubic shape functions. In this way, C1 continuityof w0 can be ensured, as required by the finite elementformulation.

Invoking Hamilton’s principle yields the element stiff-ness matrix

Ke ¼Z

Xe

ðDNÞTCDNdX

and the element mass matrix

Me ¼ I0M0 þ I1M1 þ I2M2 þ I3M3 þ I4M4 þ I6M6

where Mi are the contributions associated to the inertiamoments Ii defined as

I i ¼XnL

L¼1

Z zL

zL�1

qLzi dz

and qL is the density of the Lth lamina.Hence, free vibrations of the laminated plate are gov-

erned by an eigenvalue problem in the following well-known form:

ðK� kMÞa ¼ 0 ð26Þwhere the plate stiffness and mass matrices, K and M, areobtained by standard assembly procedures. The eigen-values k are related to the plate’s undamped natural fre-quencies f by

k ¼ ð2pf Þ2 ð27ÞThe natural frequencies measured experimentally are in-deed the damped natural frequencies. However, experimen-tal post-processing allows to take into account the dampingand evaluate the experimental undamped natural frequen-cies, thus eliminating the need to account for damping inthe plate model.

Note that free-free boundary conditions are assumedbecause they can be easily and accurately realized in anexperimental setup.

3.6. Estimator components

The quantities introduced for a general estimation prob-lem in Section 2 are now defined for the dynamic identifi-cation of laminated plates.

The experimental data is composed of a set ofundamped natural frequencies fexp of the free edge plateand the experimental eigenvalues kexp can be calculatedthrough Eq. (27). The stiffness coefficients of the laminaein the material coordinates (see Eqs. (20) and (21)) are cho-sen as the parameters to be estimated and the elastic con-

stants are simply derived from them. Thus, the vectorswhich appear in the estimator formulae (9) and (12) are

YT ¼ kexp;1 kexp;2 � � � kexp;m½ �gT ¼ k1 k2 � � � km½ �

b ¼ CLij

h iwhere m is the number of experimental frequencies consid-ered and ki are the numerical eigenvalues corresponding tothe current value of the stiffness coefficients b.

The deviation associated to the experimental eigenvaluescan be evaluated as

dkexp ¼okexp

ofexp

� df ¼ 8p2fexpdf

where df is the deviation of fexp. On the other hand, thedeviation associated to the stiffness parameters dependson the deviation associated to the elastic constants throughEqs. (20) and (21).

Based on the above relations, the covariance matricescan be defined

VY ¼ diagðdkexp;iÞVb ¼ diagðdCL

ijÞ

The last term required is the sensitivity matrix S, whichgives the variation of the eigenvalues with respect to theparameters to be estimated. The typical component of S is

Sij ¼oki

objð28Þ

where bj denotes the jth component of b. It can be evalu-ated, as early proposed in [1,21], by

oki

obj¼ 1

aTi Mai

aTi

oK

objai

where ai is the ith eigenvector. Then, the derivative of thestiffness matrix K with respect to the stiffness coefficientsof the laminae can be simply computed by assembling ele-ment contributions of the form

oKe

obj¼Z

Xe

ðDNÞT oC

objDNdX

Finally, taking into account Eq. (23) and using Eqs. (24)and (25), evaluating the derivative of the plate constitutivematrix reduces to compute terms likeXnL

L¼1

Z zL

zL�1

oCLhk

oCLij

zqdz

Notice that the sensitivity coefficients (28) can be normal-ized as follows [22]:

sij ¼oki

obj

bj

kið29Þ

In this way the sensitivity of each mode to different param-eters can be evaluated and compared.


4. Analysis and comparison of the two estimators

In this section the behavior of the B and MVE estima-tors is analyzed and compared through some numericaltests, devised ad hoc to highlight similarities and differ-ences. Two different single-layer orthotropic rectangularplates, whose characteristics are reported in Table 1, aremodeled through Reddy’s third-order theory for a givenset of elastic constants (target). The two plates are discret-ized by uniform meshes of 8 · 4 and 8 · 8 elements, respec-tively. Supposing to operate in ideal conditions, the

Table 3Test 1 on Plate 1: final estimates and number of iterations performed

Deviation d E1 (GPa) E2 (GPa) m12

Initial 100 7 0.33

B 1/3 – – –B 1/6 145.0 10.41 0.2087B 1/10 142.0 10.47 0.2149

MVE 1/3 – – –MVE 1/6 150.1 9.996 0.2539MVE 1/10 149.8 10.01 0.2389

Target 150 10 0.25



Initial 180 12 0.20

B 1/3 152.4 9.870 0.2787B 1/6 156.7 9.663 0.2726B 1/10 159.2 9.630 0.2639

MVE 1/3 150.0 10.00 0.2493MVE 1/6 150.0 10.00 0.2466MVE 1/10 150.3 9.987 0.2626

Target 150 10 0.25

Table 1Characteristics of Plate 1 and Plate 2

Length a

(mm)Width b

(mm)Mean thicknessh (mm)

Densityq (kg/m3)

Stackingsequence

Plate 1 200 100 6.6 1000 [0]Plate 2 100 100 10 1500 [0]

Table 2Input frequencies of Plate 1 and Plate 2

Plate 1 Plate 2

Modeno.

fexp Modeno.

fexp Modeno.

fexp Modeno.

fexp

1 611.23 8 5406.2 1 2555.5 8 13,9852 2002.7 9 6121.1 2 3591.6 9 15,7443 2171.2 10 6389.0 3 6069.2 10 16,3344 2338.5 11 6520.0 4 9146.4 11 16,4405 2478.2 12 7357.6 5 10,677 12 18,0676 3810.3 13 8343.2 6 11,305 13 18,1567 5165.3 14 8809.4 7 11,512 14 18,371

experimental frequencies are taken as equal to the numer-ical ones for the estimation process. This eliminates allthe sources of error (such as modeling and measurementerrors, see Eq. (1)) and the estimators are expected to yieldthe target elastic constants, starting from a set of guessedinitial values. On the other hand, only the first 14 naturalfrequencies are considered in the estimation (see Table 2)since it is well known that in real experimental conditionsit is difficult to measure frequencies associated to highermodes. Moreover, capturing higher modes may requiremore accurate plate models.

Various numerical tests have been carried out for differ-ent initial guesses and parameter covariance. The final esti-mates and number of iterations performed by theestimation procedures are reported in Tables 3–8. To checkconvergence, the tolerance has been set to TOL = 10�3.The deviation of the experimental data was assumed tobe df = 0.01 Æ fexp. In order to establish a common groundfor comparison, the deviation associated to each elasticconstant is taken proportional to the constant itself by afactor d. The sign – indicates that the procedure does notconverge. Since the two plates considered are single-layer,the superscript L referring to the Lth lamina is droppedin the following.

4.1. Sensitivity analysis

Since both the algorithms are based on the sensitivitymatrix, a sensitivity analysis is useful to understand thebehavior of the two estimators. The absolute values of

G12 (GPa) G13 (GPa) G23 (GPa) No. iter.

4 4 1

– – – –3.008 3.554 1.404 153.019 3.924 1.277 7

– – – –3.000 2.995 2.004 873.000 3.015 1.989 85

3 3 2


2 2 4

2.997 2.791 2.261 102.981 2.397 3.442 92.962 2.272 3.801 7

3.000 3.001 1.999 343.000 3.005 1.998 413.000 2.977 2.013 18

3 3 2


Deviation d E1 (GPa) E2 (GPa) m12 G12 (GPa) G13 (GPa) G23 (GPa) No. iter.

Initial 130 11 0.20 3.5 3.5 1.5

B 1/3 148.9 10.12 0.2122 3.002 3.113 1.832 6B 1/6 147.7 10.21 0.2099 3.004 3.234 1.695 6B 1/10 146.5 10.28 0.2104 3.007 3.366 1.602 5

MVE 1/3 150.0 9.999 0.2509 3.000 2.999 2.001 20MVE 1/6 149.9 10.00 0.2458 3.000 3.006 1.997 19MVE 1/10 149.8 10.01 0.2382 3.000 3.015 1.991 56

Target 150 10 0.25 3 3 2



Initial 280 28 0.19 15 15 8

B 1/3 205.7 19.98 0.2626 9.914 10.21 4.090 5B 1/6 198.1 19.68 0.2656 9.872 11.88 4.298 5B 1/10 195.6 19.62 0.2660 9.845 12.67 4.408 7

MVE 1/3 210.2 19.97 0.2393 9.932 9.497 4.159 4MVE 1/6 201.8 19.82 0.5290 9.910 10.24 4.086 139MVE 1/10 207.6 19.99 0.3296 9.925 9.751 4.075 197

Target 207 20 0.25 10 10 4



Initial 160 26 0.31 7 7 6

B 1/3 275.6 20.38 0.5312 11.13 2.988 9.804 11B 1/6 296.3 23.82 0.3084 11.09 2.362 9.393 12B 1/10 197.2 20.48 0.4001 9.861 4.627 7.426 8

MVE 1/3 – – – – – – –MVE 1/6 – – – – – – –MVE 1/10 229.5 19.93 0.2837 9.968 3.869 13.11 82

Target 207 20 0.25 10 10 4



Initial 160 14 0.31 7 7 6

B 1/3 210.8 19.95 0.2260 10.01 9.418 4.077 6B 1/6 215.7 19.82 0.2211 9.927 8.651 4.368 6B 1/10 216.1 19.41 0.2235 9.862 8.316 4.896 5

MVE 1/3 206.9 20.00 0.2364 10.00 10.05 4.002 66MVE 1/6 206.8 20.00 0.2235 10.00 10.08 4.005 13MVE 1/10 210.2 19.97 0.2216 9.933 9.517 4.164 18

Target 207 20 0.25 10 10 4


the normalized sensitivity coefficients (29) are plotted inFigs. 2 and 3 for Plate 1 and Plate 2, respectively.

Both diagrams evidence that the first natural frequenciesare more sensitive to C11, C22 and C66, that is the stiffness

coefficients associated to the two elastic moduli E1 andE2 and the in plane shear modulus G12. This means thatthe experimental data contains more information on C11,C22 and C66 rather than on C44, C55 and C12. As a conse-

Fig. 2. Forward sensitivity analysis for Plate 1.

Fig. 3. Forward sensitivity analysis for Plate 2.


quence, the transverse shear moduli G23 and G13 and Pois-son’s ratio m12 are expected to be more difficult to estimatewith respect to the in plane elastic constants.

The sensitivities are also influenced by the plate aspectratio. Plate 1 (a/b = 2) is more sensitive to C12, whichaccounts for directional effects. On the other hand, Plate2 (a/b = 1) is less sensitive to C11, since the modulus E2 ismuch lower than E1 and, consequently, most low frequencymodes involve the 2 direction. Therefore the geometry ofPlate 1 is preferable as it allows a more effective overallidentification of the elastic constants. This should be takeninto consideration when setting up the experiment. Moredetails on optimal experiment design can be found in [23].

4.2. Choice of covariance matrices and initial values

Some of the quantities which enter the estimation pro-cess should be defined by the analyst. These are the initialparameter guesses l and the covariance matrices Vb andVY. Obviously, convergence of the iterative procedures(9) and (12) strongly depends upon these choices. Thecovariance matrices influence the amount of parametercorrection at each iteration. High confidence on the initialestimates (that is, small entries in Vb) results in small cor-rections, while low confidence on the initial estimatesmay lead to large corrections, especially in the first step.On the other hand, the smaller the covariance VY associ-ated with the experimental values, the larger is the param-

eter correction. Thus, for convergence reasons, Vb shouldnot be too large and VY not too small.

Another important aspect is the combined choice of theinitial estimates for E1 and E2. In particular, for initialvalue sets with E1 overestimated and E2 underestimated,or viceversa, the estimation procedures tend to fail. Anexplanation can be given by considering the sensitivitycoefficients in Eq. (29). Both si1 and si3 are positive forevery mode. As a consequence, initial values for C22 higherthan the real ones result in model frequencies higher thanthe real ones. Analogously, initial values for C11 lower thanthe real ones result in model frequencies lower than the realones. These two opposite effects are both strong, becausethe sensitivity coefficients are high, and may engender anoscillatory behavior of the stepwise corrected parameterswith loss of convergence.

4.3. Comparison of the two estimators

As already pointed out, the difference between the twoestimators, from the analytical point of view, lies on therole of the initial estimates (see Section 2.3). In the B esti-mator, the initial values are fixed, while in the MVE theyserve as starting values of the iterative scheme, which areupdated at each iteration. As a consequence, the depen-dence of the final estimates upon the initial guesses is verydifferent in the two estimators (see Tables 3–8).

If the algorithm converges to the global minimum, theMVE estimates depend only upon the information availablein the experimental data: high sensitivity to the parametersimplies that the target values are obtained. In this case, anydifference between final estimates and target values is due tothe stop criterion. On the contrary, different initial values ordifferent deviations yield significantly different estimatesusing the B algorithm. The bias is stronger as the deviationbecomes smaller and the parameters are forced to remainclose to the initial guesses. A clear example of this behaviorcan be seen by observing Tables 4 and 5: the E1 estimatesare included in the interval between the initial and targetvalues and move closer to the initial guesses as the Devia-tion d becomes smaller. On the other hand, the B estimatoris much more efficient than the MVE estimator, as clearlyrevealed by the number of iterations performed.

5. A modified MVE procedure

Here, a two-stage solution strategy is presented aimingat improving convergence of the MVE estimator. The mod-ified MVE estimator is denoted by MVE-mod. The ideastems from the observation that there are two sets ofparameters with very different influence on the natural fre-quencies. In fact, as noticed in the previous section, theeigenvalues are very sensitive to C11, C22 and C66 and muchless to C12, C44 and C55 (see Figs. 2 and 3). So the correc-tion of the last three stiffness coefficients is much slower.This suggests to split the MVE estimation procedure intotwo stages as follows.


• In the first stage, the estimation procedure is carried outusing small values for the parameter deviations, in orderto avoid the risk of non convergence. The tolerance is setto a value larger than the prescribed one: TOL = 10�2.This stage is designed for the parameters most sensitiveto the frequency data (typically E1, E2 and G12), whichquickly converge.

Table 9Test 1 on Plate 1: MVE-mod final estimates and number of iterations perform


Initial 100 7 0.33

Stage 1 1/6 151.2 9.868 0.3129Stage 2 1/3 150.0 10.00 0.2507

Stage 1 1/10 149.2 10.13 0.2080Stage 2 1/3 150.0 10.00 0.2492

Target 150 10 0.25



Initial 180 12 0.20

Stage 1 1/3 149.9 10.01 0.2416Stage 2 1/2 150.0 10.00 0.2498

Stage 1 1/6 149.7 10.02 0.2234Stage 2 1/3 150.0 10.00 0.2492

Stage 1 1/10 150.4 9.978 0.2635Stage 2 1/3 150.0 10.00 0.2506

Target 150 10 0.25



Initial 130 11 0.20

Stage 1 1/3 150.1 9.990 0.2598Stage 2 1/2 150.0 10.00 0.2503

Stage 1 1/6 150.0 10.00 0.2407Stage 2 1/3 150.0 10.00 0.2491

Stage 1 1/10 149.5 10.08 0.2214Stage 2 1/3 150.0 10.00 0.2491

Target 150 10 0.25



Initial 280 28 0.19

Stage 1 1/3 210.1 19.97 0.2394Stage 2 1/2 210.2 19.97 0.2392

Stage 1 1/6 201.7 19.81 0.3299Stage 2 1/3 201.7 19.82 0.5467

Stage 1 1/10 206.7 19.99 0.2569Stage 2 1/3 208.0 19.99 0.4095

Target 207 20 0.25

• In the second stage, the initial values are taken as the finalestimates of the first stage, but the deviations associatedto the parameters are increased. This is expected to accel-erate convergence of the remaining parameters, withoutinterfering with the estimation of the parameters whichhave already reached convergence during the first stage.The tolerance is set to the prescribed value: TOL = 10�3.

ed


4 4 1

2.998 2.892 2.196 413.000 2.999 2.001 16

3.002 3.083 1.815 93.000 3.001 1.999 17

3 3 2

ed


2 2 4

3.000 3.011 1.994 243.000 3.000 2.000 7

3.001 3.024 1.987 93.000 3.001 1.999 15

2.999 2.958 2.030 143.000 2.999 2.000 11

3 3 2

ed


3.5 3.5 1.5

3.000 2.987 2.008 93.000 3.000 2.000 7

3.000 3.004 1.996 53.000 3.001 1.999 10

3.001 3.054 1.893 73.000 3.001 1.999 17

3 3 2

ed


15 15 8

9.932 9.500 1.459 39.932 9.497 4.159 1

9.905 10.87 4.148 249.911 10.19 4.080 30

9.920 10.06 4.083 89.927 9.523 4.056 61

10 10 4

Table 13Test 2 on Plate 2: MVE-mod final estimates and number of iterations performed


Initial 160 26 0.31 7 7 6

Stage 1 1/10 227.7 20.16 0.3298 9.994 3.897 10.74 24Stage 2 1/3 230.0 19.93 0.2795 9.968 3.869 13.15 12

Target 207 20 0.25 10 10 4

Table 14Test 3 on Plate 2: MVE-mod final estimates and number of iterations performed


Initial 160 14 0.31 7 7 6

Stage 1 1/3 206.5 20.00 0.2140 9.998 10.15 4.003 5Stage 2 1/2 207.0 20.00 0.2452 10.00 10.02 4.001 44

Stage 1 1/6 207.4 20.00 0.2223 10.00 9.981 4.011 8Stage 2 1/3 206.9 20.00 0.2393 10.00 10.04 4.002 44

Stage 1 1/10 212.1 19.96 0.2199 9.944 9.254 4.184 8Stage 2 1/3 210.0 19.97 0.2227 9.931 9.547 4.161 4

Target 207 20 0.25 10 10 4

Table 16Input frequencies for cross-ply laminate

Mode no. fexp Mode no. fexp

1 1369.7 8 10,1032 2335.6 9 11,7793 3598.6 10 12,7554 6255.1 11 12,8725 7390.8 12 16,0716 8372.0 13 17,2807 8755.9 14 18,504

Table 17Final estimates and number of iterations performed for cross-ply laminate

E1

(GPa)E2

(GPa)m12 G12

(GPa)G13

(GPa)G23

(GPa)No.iter.

Initial 150 17 0.25 8 8 5

Deviation 17.5 4 0.05 1.5 1.5 1.5


To show the advantages of the MVE-mod procedure,the same numerical tests discussed in the previous sectionare considered. The results are shown in Tables 9–14. Asit can be seen from the last column, the final estimatesare very similar to the ones predicted by the classicalMVE procedure, but convergence is faster.

6. Case studies

Three case studies are solved using the B, MVE andMVE-mod procedures. A cross-ply and an angle-ply lami-nates are taken as representative examples of multi-layerplates; in these examples, the input frequencies are assumedas equal to the numerical frequencies, so to disregard anysources of error. The third case study deals with a single-layer carbon–epoxy plate studied by Frederiksen [5]. In thiscase, the input frequencies are real experimental measure-ments, so that both modeling and measurement errorsare present.

In stage 2 of the MVE-mod procedure, the deviationsare twice those in the first stage.

6.1. Cross-ply symmetric laminate

The specimen characteristics are reported in Table 15and the input natural frequencies computed by a uniformmesh of 20 · 20 elements are reported in Table 16. Table 17

Table 15Characteristics of cross-ply laminate

Lengtha (mm)

Widthb (mm)

Thicknessh (mm)

Densityq (kg/m3)

Stackingsequence

100 50 3.3 1500 [(0/90)80(90/0)8]

collects the results for the three procedures. The bias pres-ent in the B estimator is still evident in this case study. Boththe MVE and MVE-mod estimators converge to the targetvalues, but the second is faster.

6.2. Angle-ply nonsymmetric laminate

The specimen characteristics are reported in Table 18and the input natural frequencies computed by a uniform

B 120.1 13.69 0.2812 5.989 6.206 3.206 3MVE 125.0 9.458 0.3058 6.000 5.996 3.000 39

MVE-modStage 1 124.9 9.449 0.3306 5.999 5.983 2.985 10Stage 2 125.0 9.491 0.3013 6.000 5.999 3.000 13

Target 125 9.5 0.30 6 6 3

Table 18Characteristics of angle-ply laminate

Lengtha (mm)

Widthb (mm)

Thicknessh (mm)

Densityq (kg/m3)

Stackingsequence

100 50 3.3 1500 [(0/45/90/�45)4

0(45/90/�45/0)4]

Table 19Input frequencies for angle-ply laminate


1 2074.4 8 10,6572 2250.7 9 11,6413 4956.8 10 13,4594 5602.9 11 15,4915 7456.6 12 17,1826 8609.7 13 19,1137 8613.2 14 19,121

Table 21Characteristics of carbon–epoxy plate

Lengtha (mm)

Widthb (mm)

Meanthicknessh (mm)

Densityq (kg/m3)

Stackingsequence

Fibre/volumefraction

100.1 53.0 3.36 1537 [0] 60%

Table 22Measured frequencies for carbon–epoxy plate


1 1121.4 8 7671.42 2795.4 9 8073.03 2971.5 10 8431.84 3590.6 11 10,2125 3633.1 12 10,2816 5947.5 13 14,0077 7655.1 14 14,048


mesh of 20 · 20 elements are reported in Table 19. Table 20collects the results for the three procedures. The angle-plylaminate has the same geometrical and mechanical charac-teristics as the cross-ply plate considered in the first casestudy, but the stacking sequence changes. The more com-plicated stacking sequence increases the difficulties in theidentification of the in plane elastic constant. In the B esti-mator, this is reflected in final estimates that are more influ-enced by the initial values. In the MVE and MVE-modprocedures, the number of iterations required increases,but the accuracy of the final estimates is not significantlymodified.

6.3. Single-layer carbon–epoxy plate

A single-layer unidirectional carbon–epoxy (T300 car-bon fibre) composite plate is considered. The elastic con-stants were estimated by Frederiksen [5] using a two stepapproach. Firstly, a thin specimen was considered for theestimation of the four in plane elastic constants. Subse-quently, a second thicker specimen was taken in order toestimate the two transverse shear moduli. In this work,only the second specimen is taken into consideration inorder to simultaneously estimate all the six elastic con-

Table 20Final estimates and number of iterations performed for angle-ply laminate

E1

(GPa)E2

(GPa)m12 G12

(GPa)G13

(GPa)G23

(GPa)No.iter.

Initial 150 17 0.25 8 8 5

Deviation 17.5 4 0.05 1.5 1.5 1.5

B 119.0 12.84 0.3231 6.943 6.178 3.179 3MVE 124.9 9.306 0.3233 6.103 5.998 2.996 94

MVE-modStage 1 124.3 8.653 0.4175 6.504 5.983 2.986 9Stage 2 125.0 9.469 0.3036 6.017 6.000 2.999 34

Target 125 9.5 0.30 6 6 3

stants. The known specimen characteristics are reportedin Table 21 and the measured natural frequencies arereported in Table 22.

The initial set of estimates needed for the estimationprocess are obtained through the rule of mixtures, with afibre/volume fraction vf of 60%. Producers’ technical sheetsprovide the following data:

Carbon fibre T300 : Ef ¼ 230� 240 GPa

mf ¼ 0:27

Epoxy resin : Em ¼ 7 GPa

mm ¼ 0:36

where the subscripts f and m stand for fibre and matrixelastic constants. The values of E and m allow to evaluateGf and Gm, then the rule of mixtures can be applied

E1 ¼ Efvf þ Emvm ¼ 144 GPa

E2 ¼EfEm

Efvm þ Emvf

¼ 16:5 GPa

m12 ¼ mfvf þ mmvm ¼ 0:30

G12 ¼ G13 ¼Gf Gm

Gfvm þ Gmvf

¼ 6 GPa

where vm is the matrix/volume fraction. The initial valuefor G23 is taken as 4 GPa. The plate is analyzed using a

Table 23Estimates for carbon–epoxy plate

E1

(GPa)E2

(GPa)m12 G12

(GPa)G13

(GPa)G23

(GPa)

Initial 144 16.5 0.30 6 6 4

Deviation 15 3.5 0.05 1.5 1.5 1.5

B 113.5 8.47 0.361 4.44 4.31 3.39MVE 113.3 8.52 0.323 4.45 4.42 2.91

Frederiksen [5] 113.0 8.50 0.323 4.45 4.43 2.97


50 · 50 element mesh, which has been verified to satisfacto-rily resolve the first 14 natural modes.

The results of the estimation process are collected inTable 23, together with the assumed deviations. As it canbe observed, they are in good agreement with the valuesfound in [5].

Comparing the estimators, one notices once again thatthe B estimator exhibits some bias in the identification ofthe constants m12, G13 and G23, because of the less sensitiv-ity of the natural frequencies. On the contrary, this is notobserved in the MVE estimator, which yields quite goodestimates.

7. Concluding remarks

Two well-known estimators for the identification of elas-tic constants of thick laminated plates have been analyzedand compared. The role played by initial knowledge hasbeen critically discussed and some similarities hidden bythe very different formulations have been highlighted. TheB estimator has been shown to be very efficient and robust,but the final estimates tend to be biased by the a priori infor-mation. On the contrary, the MVE estimator is potentiallymore effective, but has been shown to be less efficient androbust. In particular, it is more sensitive to local minimaand convergence is generally slower. These drawbacks havebeen alleviated by an improved solution procedure.

Acknowledgements

This research was supported by the Italian Ministry ofEducation, University and Research MIUR (PRIN 2003– ‘‘Parameter estimation of fibre reinforced compositematerials using Bayesian sensitivity analysis and optimiza-tion techniques’’). The research theme is one of the subjectsof the Centre of Study and Research for the Identificationof Materials and Structures (CIMEST) – M. Capurso. Thecomputational facilities were provided by the Laboratoryof Computational Mechanics of the University of Bologna(LAMC).

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Estimation of elastic constants of thick laminated plates within a Bayesian framework

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