Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data

Computers and Structures xxx (2014) xxx–xxx

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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data Stijn Debruyne a,⇑, Dirk Vandepitte a, David Moens b a b

KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300b, 3001 Heverlee, Belgium Lessius Hogeschool, Department of Applied Engineering, KU Leuven Association, J. De Nayerlaan 5, 2860 Sint-Katelijne-Waver, Belgium

a r t i c l e

i n f o

Article history: Received 23 May 2012 Accepted 24 September 2013 Available online xxxx Keywords: Honeycomb beam Random field Epistemic uncertainty Modal analysis Probability density

a b s t r a c t The goal of this paper is to characterise the variability of two important design parameters of thermoplastic honeycomb sandwich beams from the analysis of experimentally determined resonance frequencies and mode shapes of a limited number of test beams, under free boundary conditions. The design parameters are the Young’s modulus of the skin in length direction of the beam and the out-of-plane shear modulus of the honeycomb core. These two independent parameters are expressed as random fields using a Karhunen–Loève series expansion of the covariance matrix, available after updating the finite element models using the experimental vibration data. The random variables of this series expansion are expressed in terms of a Hermite polynomial chaos. The aleatory uncertainty is modelled by the Gaussian random variables while the epistemic uncertainty is described by the randomness of the polynomial chaos coefficients. An estimation of the uncertainty resulting from the experimental and numerical modal analysis is discussed, along with its influence on the two considered stiffness parameters. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Honeycomb panels are geometrically complex but nominally regular structures. Such panels consist of a honeycomb core that is bonded to thin face sheets. The structure of a typical panel is shown in Fig. 1. Their dynamic behaviour, quantified by resonance frequencies and mode shapes, depends on a large number of parameters related to the beam geometry and the elastic properties of the materials used. This paper considers narrow slices of a sandwich panel which are considered to be beam structures. It studies the beam model parameter variability from its experimental structural behaviour. The beams are cut from MonopanÒ sandwich [1] panels. Although there is much repetition and regularity in the geometry of such a panel and in the assembly process, it turns out that local material and geometry characteristics exhibit Abbreviations: PC, polynomial chaos; KL, Karhoenen–Loève; FEM, finite element modelling; FEA, finite element analysis; EMA, experimental modal analysis; RF, random field; PDF, probability density function; FRF, frequency response function; FFT, fast Fourier transform; MC, Monte Carlo; MCMC, Monte Carlo Markov chain; BI, Bayesian inference; SVS, shell volume shell; MAC, modal assurance criterion; CI, confidence interval; COV, coefficient of variation. ⇑ Corresponding author. Address: Zeedijk 101, 8400 Ostend, Belgium. E-mail addresses: [email protected] (S. Debruyne), dirk.vandepitte@mech. kuleuven.be (D. Vandepitte), [email protected] (D. Moens).

a high degree of scatter. Variability exists at two different levels: from one panel to another (inter-variability) and also between different positions in one specific panel (intra-variability or spatial variability). The objective of this research is to quantify scatter based on measurements on commercially available panels, and to derive a relation between the levels of scatter that are observed on the one hand and the expected physical and mechanical properties on the other hand. The random field method is used as tool for the variability analysis of the considered stiffness parameters. The elastic mechanical properties of a typical honeycomb core are described and analytically calculated by Gibson and Ashby [2]. They propose formulas for calculation of the in-plane and out-of-plane elastic moduli and Poisson ratios of the core. The main work on the dynamics of sandwich panels is related to conventional foam – core structures. Nilsson and Nilsson [3] tried to analytically predict natural frequencies of a honeycomb sandwich plate with free boundary conditions using Blevins [4] formula in which the mass per square metre and equivalent bending stiffness are frequency dependent. Another, more practical way to predict natural frequencies and mode shapes of a honeycomb panel is by means of finite element (FE) analysis. In the past few years, different new approaches have been developed which incorporate high order shear deformation of the core. Work in this area has been carried out by Topdar [5] and Liu [6–8]. The latter stated that the

http://dx.doi.org/10.1016/j.compstruc.2013.09.004 0045-7949/Ó 2013 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Debruyne S et al. Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2013.09.004

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S. Debruyne et al. / Computers and Structures xxx (2014) xxx–xxx

Nomenclature Es Gc

qc er H

r Na

c2

Young’s modulus of the homogenized honeycomb beam skin in length direction out-plane-shear modulus of the homogenized honeycomb core material Mass density of the homogenized honeycomb core material normalized Random Error transfer function standard deviation number of averages for frequency response function measurements squared coherence value

shear moduli of the core are important factors in the determination of the values of the natural frequencies and the sequence of mode shapes, especially at high frequencies. At low frequencies, natural frequencies are mostly determined by the bending stiffness of the beam. As frequency increases, the core shear stiffness becomes more and more important. The use of vibration measurement data for the identification of elastic material properties is studied by Lauwagie [9]. His work discusses how Young’s moduli, shear moduli and Poisson ratios of laminated materials can be obtained from modal data such as resonance frequencies and mode shapes. The analysis of variability can be done in various ways. In any statistical analysis the issue of gathering and obtaining a sufficient large set of data is essential [32]. In this study, a population of 22 specimens is used as a basis for statistical analysis. A survey of uncertainty treatment in finite element analysis (FEA) is given by Moens [10]. The focus of the work presented in this article is to make optimal use of the statistical information available from the limited – size experimental data. According to Schuëller [11–14], processes and system behaviour can be regarded as stochastic processes of which the outcome is governed by a set of stochastic random variables. Consequently, this research considers the dynamic behaviour of honeycomb sandwich beams as a stochastic process. In this area a recent approach for quantification of variability describes the quantity or process of interest as a stochastic random field (RF). The random field theory is extensively studied and further developed by Ghanem [15,16]. Soize [17], Desceliers [18], Arnst [19] and Perrin [20] implemented and adapted this theory for inverse problems and for cases where only limited experimental data is available. Mehrez [21,22] adopted this method to describe the variability of the Young’s modulus of composite beams from experimental frequency response functions, by also solving an inverse problem. This paper presents a strategy to identify the variability of structural parameters of honeycomb sandwich beams. These beams are cut from MonopanÒ sandwich panels. The core of this type of sandwich panels consists of cylindrical polypropylene tubes that are arranged in a dense stacking (with each tube having

skin

honeycomb core Fig. 1. A typical honeycomb sandwich panel.

M Nexp kk

uk dkl

number of test samples number of points for realizations of the random field k-th eigenvalue of the covariance matrix of experimental realizations k-th eigenvector of the covariance matrix of experimental realizations Kronecker delta

g(k) l

vector with random coefficients of the k-th KL term truncation number the KL series expansion Fl discretized random field truncated after l terms m, m P l number of independent Gaussian random variables

6 neighbour tubes at 60° positions, see Fig. 15), and that are welded together. After the tubes are stacked and welded, slices are cut before the face skins are attached. The core is welded to the polypropylene skin by fusion welding using a welding foil. The skin consists of a symmetric glass fibre twill weave with a polypropylene matrix and a nominal thickness of approximately 0.7 mm. A polypropylene finishing foil is welded to the skin outer surface to make it smooth and flat. A set of 22 nominally identical MonopanÒ beams are used for this study, each having a length of 850 mm, a width of 50 mm and a thickness of 25 mm. The objective of this research is to quantify scatter on the sandwich material properties using experiments and finite element models. The identification procedure consists of five successive steps, most of which are well known already. However, because each step inevitably involves some degree of uncertainty and/or scatter, each step in the procedure is briefly described, with particular attention to the identification of non-determinism that is relevant at that step. The subsequent discussion covers the steps in the logical order: 1. Experimental modal analysis (EMA): the modal parameters of interest here are resonance frequencies and mode shapes of honeycomb sandwich beams with free boundary conditions. The problem of measurement uncertainty and modal parameter estimation uncertainty is also addressed. It is outlined how this EMA uncertainty may be estimated. 2. Finite element modelling (FEM), complemented with a finite element updating procedure of each individual beam: this model is used to create a database with values of two important design parameters of the considered honeycomb beams. In this research the skin Young’s modulus in length direction (Es) and the out-of-plane shear modulus of the honeycomb core (Gc) are studied. These parameters determine the bending stiffness of a beam section. 3. Estimation of errors due to the EMA and FEM processes on the variability of Es and Gc: the corresponding variability is compared to the estimated physical variability of the elastic parameters. 4. Random field description of the stiffness characteristics of the beam: the probability density functions are estimated for the two parameters at the measurement positions, along with their confidence intervals (CI). The confidence intervals quantify the epistemic uncertainty which arises from the lack of sufficient statistical data. The implementation of the random field model is validated with respect to the number of test samples available. 5. Discussion of the physical relevance of the obtained results: the relation between the estimated PDF’s and the real physical variability of the two design variables is discussed. The so-called aleatory uncertainty includes physically related variability,

Please cite this article in press as: Debruyne S et al. Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2013.09.004

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both in inter-sample and intra-sample variability. This paper focuses on the characterisation of the intra-sample variability of the considered stiffness properties.

1. Glass fibre reinforced Polypropylene skin (thickness ca. 1.2 mm). d

2. Cylindrical Polypropylene honeycomb cells

2. Experimental modal analysis (EMA) Experiments have been conducted on 22 nominally identical beams. On each beam, vibration measurements have been done in 17 evenly spaced points. These measurement locations are identical for all test beams. The beams are elastically suspended to attain free boundary conditions. The beam is excited with an impulse hammer at all 17 locations; the beam’s out-of-plane bending vibrations are measured using light-weight accelerometers of 0.4 grams. Consequently, the vibration motion of only one degree of freedom is being excited and measured. The experimental setup is shown in Fig. 2. From the measured frequency response functions (FRF) [23], modal parameters are extracted using Test.lab from LMS International (resonance frequencies, mode shapes and damping ratios). Natural frequencies and mode shapes are used for further analysis. In the range of 100–1800 Hz, eight vibration modes are identified and used for further analysis. They are shown in Fig. 3. Table 1 gives an overview of the resonance frequencies for all beams.

Fig. 2. Measurement set-up for vibration tests.

Fig. 4. Principle of Monopan honeycomb panels.

In most studies related to modal analysis, experimentally determined frequency response functions are used as true reference data. However, these FRF’s are subject to different kinds of uncertainty. First of all, measurement uncertainty is inherent to both the data acquisition and data analysis methods and to the hardware that is used (transducers, type of excitation, etc.). Secondly, the processes of modal parameter estimation and FRF synthesis are subject to some error as the estimation algorithms used have a finite precision. The third variability is related to the real physical variability of the different beam design parameters that govern a beam’s mass and bending stiffness. 2.1. Measurement uncertainty The frequency variability due to the measurement itself is partially caused by the transducers and the data acquisition system used. Prior to the start of the measurement campaign, the accelerometers and impulse hammer are properly calibrated. A series of 30 calibrations yield a normalised random error on the amplitude of less than 0.007% for all transducers and no traceable frequency error. For all measurements, the fast Fourier transform (FFT) analyser was set at a frequency resolution of 0.25 Hz giving a maximum frequency error of 0.22% for the first mode at approximately 115 Hz. The measurement amplitude error on experimentally determined FRF’s can be estimated from the coherence function [24]. This function expresses the correlation between two signals at a certain frequency. During vibration measurements, one coherence function is determined for every measured frequency response function. It is calculated from the averaged auto-and cross power spectrums of the considered input and output signals, which in this case correspond to the excitation force and response acceleration respectively. A coherence value below 1 indicates a non-perfect correlation between the measured acceleration and the input force signal. A coherence value below 1 leads to an error on the corresponding calculated frequency response function. To reduce the effect of random noise on a signal, each signal is measured a number of times, making the final signal the average of Na measured signals. Eq. (1) expresses the coefficient of variation (COV) er(H) = r/H on the amplitude of the transfer function H as a function of the coherence value c2 and the number of averages Na for one FRF measurement.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  c2 er ðHÞ ¼ 2Na c2

ð1Þ

In this study the number of averages is set to 20. Table 2 gives an overview of the FRF measurement amplitude coefficients of variation, all calculated from obtained coherence functions by expression (1). The COV values are mean values, obtained by evaluating all coherence functions for all test beams at the first eight modes. 2.2. Modal parameter estimation uncertainty

Fig. 3. First 8 vibration modes for honeycomb sandwich beams (free–free).

In this research, the modal parameters of interest are the resonance frequency values and mode shapes of the first 8 vibration modes. They are estimated from the measured frequency response

Please cite this article in press as: Debruyne S et al. Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2013.09.004

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Table 1 Resonance frequencies from experimental modal analysis (EMA), expressed in Hz. Beam/mode number

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

113.5 119.5 118 116.5 115.5 119 117.5 117 120 115.5 118.5 117 117.5 118 118 118.5 118 119.5 118 119 119 119.5

287.5 299 296.5 293.5 291 299 297 295 301.5 292 298 295.5 297 297 298 298 298 300.5 297 299 299 300.5

502.5 520.5 517.5 512 510.5 523.5 522 519.5 528.5 511 522.5 518 522.5 522 524.5 524 525.5 524.5 522 523.5 523.5 524

730.5 757.5 750.5 744 740.5 757 760 761 768.5 744 766 759.5 763 765 770.5 767.5 771 767 762.5 763.5 763 761

975 1009 1000 988.5 984 1002 1005 1009 1018.5 987.5 1015 1008 1013 1011 1024 1013 1024 1014 1012 1013 1012 1009

1225 1260 1248 1237 1235 1252 1260 1263 1270 1239 1268 1265 1265 1268 1278 1269 1277 1271 1264 1260 1260 1264

1474 1519 1504 1484 1488 1508 1522 1522 1528 1488 1524 1520 1524 1524 1539 1534 1535 1531 1521 1522 1519 1523

1715 1758 1747 1720 1719 1744 1769 1768 1776 1728 1763 1767 1771 1765 1788 1765 1778 1783 1763 1761 1761 1766

Table 2 FRF amplitude uncertainty for the first 8 modes, calculated from measured coherence using expression (1) and expressed as coefficient of variation (COV). Mode

1

2

3

4

5

6

7

8

Amplitude COV (%)

0.05

0.04

0.06

0.05

0.07

0.09

0.08

0.07

functions (FRF) by means of the Poly-reference Least Squares Frequency-domain algorithm (p-LSCF) or better known as the PolyMAX algorithm [25]. This algorithm fits a right matrix fraction model on the measured FRF’s using a least squares procedure. The algorithm progress is visualised in a stabilisation chart from which modal parameters such as resonance frequency values and damping ratios can be read. The measured frequency response functions are then reconstructed using the estimated resonance frequencies and damping ratios. Within the whole frequency band of interest used for modal parameter estimation, the relative difference in amplitude of each measured and synthesised FRF is calculated and expressed as one global FRF synthesis error. Considering all measured FRF’s of a beam at a certain resonance frequency, this error on displacement amplitude may be translated to an estimated uncertainty on each of the measured mode shapes. In modal analysis, the modal assurance criterion (MAC) [24] is a mathematical tool to compare vectors, mode shapes in this case. Expression (2) calculates the MAC-value of vectors X and Y.

MACX;Y ¼ 

 t 2 X Y    t X X YtY

ð2Þ

A mode pair with a MAC-value equal to 1 indicates an excellent correlation between the mode shapes. Although the interpretation of MAC-values is highly case-dependent, mode shapes that correlate well have a MAC-value that exceeds 0.8. For each beam, the first eight mode shapes are calculated from a set of 17 synthesized FRF’s.

For each resonance mode this leads to a set of mode shapes. For each set, the mean mode shape (resulting from the set of mean FRF’s) is calculated and considered as a reference. By relating every mode shape of this set of mode shapes to the mean (reference) mode shape, a set of MAC-values is obtained for this set. Each resulting set of MAC-values is assumed to be randomly distributed. Using Expression (2), the mode shape error may consequently be expressed by relative scatter on the MAC-value. This has its physical relevance since measured mode shapes are compared with numerically calculated mode shapes during the process of model updating (see 2.2). The MAC-value is commonly used there. The PolyMAX algorithm needs some parameters to be set before modal parameters can be estimated. These parameters are the number of modes within a certain bandwidth, resonance frequency and damping ratio search tolerances. Any changing of the settings of these parameters alters the obtained synthesised FRF’s. To study the effect of the algorithm parameter changes on these FRF’s, each of the parameters is varied eight times in an interval, centred around the default parameter setting and having a width of 100% of this default setting. Consequently, for each test beam and for each measurement point, a set of 24 (8  3) synthesized FRF’s (and corresponding synthesis errors) is obtained. Consequently, each of the eight vibration mode shapes and corresponding resonance frequencies is estimated within a probability interval which assumed to be normally distributed (see above). As outlined above, the variability of each of the eight mode shapes may be expressed by scatter on the MAC-value and on resonance frequency. Both uncertainties are expressed as a COV. For the set of eight considered modes, Table 3 gives an overview of these. The resonance frequency errors are very small and do not seem significant to take into account in further analysis. The inter-sample resonance frequency variability, presented in Table 1, is much higher than the experienced EMA frequency uncertainty. However, the mode shape errors (MAC variability) are much higher than resonance frequency errors and have to be considered in further statistical analysis.

Table 3 Modal parameter estimation uncertainty, expressed as COV. Mode

1

2

3

4

5

6

7

8

Resonance frequency COV (%) MAC COV (%)

0.02 0.11

0.01 0.13

0.07 0.21

0.03 0.15

0.06 0.14

0.05 0.17

0.04 0.12

0.06 0.18

Please cite this article in press as: Debruyne S et al. Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2013.09.004

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S. Debruyne et al. / Computers and Structures xxx (2014) xxx–xxx Table 4 Cumulated EMA MAC coefficient of variation (COV). Mode

1

2

3

4

5

6

7

8

MAC COV (%)

0.15

0.17

0.25

0.18

0.20

0.23

0.19

0.22

2.3. Conclusions on the EMA uncertainty The two significant error sources in the EMA process are studied for the first 8 resonance frequencies and mode shapes. The error on estimated resonance frequencies is very small and of no importance for further analysis. Table 4 gives an overview of the cumulated estimated mode shape uncertainty, resulting from the whole EMA process. The FRF amplitude uncertainty, originating from measurement errors, is therefore first considered per beam and per mode to express it as a MAC COV (see 1.2). By doing so, the impact of EMA uncertainty on the estimation of the considered honeycomb elastic parameters Es and Gc can be studied in a straightforward way. This is discussed in Section 4. 3. Finite element modeling (FEM) 3.1. The finite element model The honeycomb beams are modelled following the shellvolume-shell (SVS) principle. The honeycomb core is modelled as a homogeneous orthotropic material using volume elements; the face sheets are also modelled as orthotropic materials but using shell elements. Fig. 5 shows the FE model. After a convergence study the mesh size was set at: 100 elements in length direction, 6 elements in width direction and 5 in thickness direction (1 skin + 3 core + 1 skin). The bending stiffness and beam mass are the dominant factors that determine the dynamic behaviour of the beams. The bending stiffness of the beams is mainly governed by the Young’s modulus of the skin Es (in length direction) and the out-of-plane core shear modulus Gc. With the model updating process in mind the FE model is divided into 17 zones, each centred around the corresponding measurement point. Each zone in the FE model has constant values for Es and Gc. Nominal values for these parameters were determined experimentally by performing three point bending tests on each beam sample according to Fan [26]. At this stage of the research the FE model is symmetric with respect to the xy–plane (see Fig. 5), meaning that for a certain beam section the upper and lower skins have the same value for Es. This assumption may be an oversimplification of physical reality but nevertheless the FE model is useful for further variability analysis. Ten modes are calculated under free-free boundary conditions. 3.2. Fem updating: results and related variability The resonance frequencies and mode shapes (from experiments, see Section 1) are used for updating the FE model. Two parameters are considered for updating, Es and Gc. At this stage

Fig. 5. FE model of the honeycomb beams.

of the research these parameters are assumed to be independent which is true for the FE model but may not exactly hold for the physical beam. Each FE model is divided in 17 intervals, with constant values for the three considered parameters in each interval. Consequently, a set of 34 parameters is considered in the whole FE model. The goal of this section is to update these parameters for each individual beam using the resonance frequencies and mode shapes obtained by EMA (see Section 1) as reference data. This process is referred to as FE model updating. Before the updating process, Es and Gc have equal values for all 17 intervals. During the update, the FE software (Siemens NX 8.0) updates the values of Es and Gc to minimise the relative difference between experimental and numerical resonance frequencies. At the same time the MACvalue for corresponding experimental/numerical mode pairs is maximised. For each beam the first 8 resonance frequencies with corresponding mode shapes are used for FE model updating and are called ‘updating targets’. Consequently there are 16 updating targets in this case. A numerical/experimental mode pair is considered as a matching pair if the associated MAC-value is higher than 0.8, which is the case for the considered eight modes. Both target types have equal weights for the updating process. For each beam the objective function, used by the FE software and expressed by (3), has 16 targets with corresponding target errors ei, i = 1, . . . , NT. The number of design variables NDV is 34.

f ðDDV j Þ ¼ min

NT NDV X X Ai jei j þ O Bj jDDV j j i¼1

! ð3Þ

j¼1

The individual target weighting factors Ai and the design variable weighting factors Bj and O (individual and overall) are all set to 1 for each objective function of a design variable DVj. The relative normalised sensitivity coefficients [9] of the considered set of resonance frequencies and mode shapes with respect to the design parameters Es and Gc are given in Tables 5a and 5b. The relative normalised sensitivity of response ri with respect to design variable DVj is denoted by the sensitivity coefficient sij and defined by (4).

sij ¼

@r i DV j @DV j r i

ð4Þ

For each beam and for each measurement interval Figs. 6a and 6b show the Gc and Es values after model updating. As may be expected from the physics of both skin and core, the mean values of Es and Gc do not differ much from sample to sample. The stated variability of Gc is rather high. This is due to the fact that narrow honeycomb sandwich slices are used. By cutting these slices from a panel, core cells, as well as joints between honeycomb core and skin faces, are damaged. The physical meaning of the obtained results is further discussed in Section 5. Table 6 gives the relative differences between EMA and FE resonance frequencies and mode shapes (MAC-values) before and after FE model updating. Mean results of all 22 beams are given. Resonance frequency difference is very low but mode shape differences are higher. The indicator that is used for this purpose is the MAC. This indicator is mainly qualitative, with increasing accuracy of mode shapes when MAC comes close to 100%. A MAC-value of 0.9 is considered to indicate a good mode correlation. The uncertainty on experimentally determined resonance frequencies and mode shapes have been estimated and discussed in Section 1. The order of these errors, related to the EMA, is much smaller than the observed resonance frequency and mode shape differences given in Table 6. Consequently, the errors given in Table 7 are mainly FE modelling errors. However, the goal of this research is not to optimize the FE modelling strategy but to study mainly the intra-sample variability of Es and Gc. The accuracy of the FE mesh partly determines the accuracy of the FE model. A rule of thumb in dynamic FE analysis states that the number of elements

Please cite this article in press as: Debruyne S et al. Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2013.09.004

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Table 5a Frequency and mode shape relative normalised sensitivities for Gc (c1–c17). Label

F/1

F/2

F/3

F/4

F/5

F/6

S/1

S/2

S/3

S/4

S/5

S/6

c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17

0.0003 0.0022 0.0044 0.0057 0.0055 0.0041 0.0022 0.0006 0.0001 0.0007 0.0023 0.0042 0.0055 0.0056 0.0043 0.0021 0.0003

0.0018 0.0095 0.0126 0.0077 0.0014 0.0018 0.0108 0.0223 0.0274 0.0219 0.0103 0.0016 0.0016 0.0080 0.0126 0.0092 0.0016

0.0047 0.0195 0.0136 0.0014 0.0123 0.0362 0.0389 0.0160 0.0015 0.0170 0.0394 0.0355 0.0115 0.0015 0.0140 0.0193 0.0042

0.0087 0.0262 0.0062 0.0153 0.0529 0.0396 0.0043 0.0239 0.0509 0.0226 0.0048 0.0410 0.0522 0.0141 0.0068 0.0266 0.0078

0.0130 0.0269 0.0057 0.0540 0.0510 0.0058 0.0463 0.0454 0.0055 0.0468 0.0448 0.0059 0.0527 0.0526 0.0053 0.0280 0.0119

0.0172 0.0227 0.0232 0.0741 0.0134 0.0482 0.0430 0.0154 0.0663 0.0142 0.0447 0.0464 0.0145 0.0747 0.0217 0.0243 0.0158

0.0000 0.0000 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0000 0.0001 0.0001 0.0000

0.0004 0.0014 0.0001 0.0008 0.0003 0.0002 0.0004 0.0000 0.0009 0.0010 0.0003 0.0000 0.0002 0.0005 0.0004 0.0001 0.0001

0.0002 0.0004 0.0002 0.0001 0.0010 0.0026 0.0008 0.0014 0.0000 0.0017 0.0011 0.0007 0.0014 0.0003 0.0005 0.0015 0.0005

0.0047 0.0125 0.0014 0.0069 0.0102 0.0023 0.0000 0.0028 0.0056 0.0028 0.0012 0.0106 0.0122 0.0015 0.0026 0.0050 0.0009

0.0118 0.0236 0.0053 0.0363 0.0174 0.0006 0.0022 0.0081 0.0020 0.0180 0.0096 0.0024 0.0234 0.0171 0.0016 0.0126 0.0051

0.0240 0.0346 0.0183 0.0795 0.0195 0.0018 0.0258 0.0145 0.0180 0.0057 0.0444 0.0143 0.0207 0.0641 0.0092 0.0248 0.0132

Table 5b Frequency and mode shape relative normalised sensitivities for Es (s1–s17). Label

F/1

F/2

F/3

F/4

F/5

F/6

S/1

S/2

S/3

S/4

S/5

S/6

s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17

0.0000 0.0011 0.0068 0.0208 0.0438 0.0726 0.1010 0.1210 0.1280 0.1210 0.0996 0.0712 0.0425 0.0199 0.0064 0.0010 0.0000

0.0002 0.0057 0.0285 0.0660 0.0961 0.0953 0.0618 0.0200 0.0018 0.0215 0.0638 0.0963 0.0955 0.0646 0.0273 0.0054 0.0002

0.0006 0.0140 0.0547 0.0859 0.0647 0.0166 0.0058 0.0414 0.0653 0.0398 0.0051 0.0182 0.0665 0.0858 0.0533 0.0133 0.0006

0.0012 0.0234 0.0670 0.0576 0.0099 0.0169 0.0514 0.0312 0.0035 0.0326 0.0509 0.0156 0.0110 0.0592 0.0665 0.0225 0.0011

0.0020 0.0311 0.0602 0.0193 0.0121 0.0404 0.0128 0.0145 0.0430 0.0135 0.0138 0.0403 0.0111 0.0208 0.0608 0.0304 0.0018

0.0029 0.0357 0.0416 0.0053 0.0293 0.0123 0.0163 0.0305 0.0048 0.0311 0.0154 0.0131 0.0287 0.0054 0.0430 0.0353 0.0025

0.0000 0.0000 0.0001 0.0002 0.0004 0.0007 0.0008 0.0008 0.0003 0.0003 0.0008 0.0009 0.0007 0.0004 0.0002 0.0000 0.0000

0.0000 0.0011 0.0036 0.0043 0.0026 0.0009 0.0002 0.0004 0.0001 0.0009 0.0026 0.0034 0.0026 0.0009 0.0002 0.0002 0.0000

0.0000 0.0004 0.0010 0.0006 0.0011 0.0016 0.0014 0.0039 0.0013 0.0018 0.0008 0.0023 0.0026 0.0014 0.0036 0.0015 0.0001

0.0002 0.0121 0.0304 0.0217 0.0038 0.0019 0.0028 0.0006 0.0004 0.0067 0.0095 0.0022 0.0040 0.0158 0.0131 0.0031 0.0000

0.0000 0.0278 0.0513 0.0158 0.0018 0.0039 0.0007 0.0034 0.0082 0.0003 0.0080 0.0137 0.0010 0.0125 0.0284 0.0131 0.0008

0.0049 0.0516 0.0696 0.0060 0.0147 0.0163 0.0105 0.0056 0.0018 0.0221 0.0015 0.0187 0.0187 0.0055 0.0485 0.0316 0.0019

Fig. 6a. Obtained values for Gc after FE model update.

per wave length should be higher than 6 to ensure an accurate calculation of natural frequencies and mode shapes. In this study with an element size of 8 mm and a shortest wave length (mode 8) of 260 mm, this ratio is 33, which confirms that the mesh size is sufficient for this frequency range. A next step is an investigation into scatter of the updating process. For each beam, several FE updating runs are performed using different settings for the updating procedure (design variable bounds, type of updating algorithm, number of iterations, initial

values of the design variables, etc. . . .). Different settings cause the update results (calculated resonance frequencies and mode shapes) to vary. By considering all FEM updates, an estimation of the modelling variability is obtained. It is assumed to be random. For the set of eight modes considered, Table 7 expresses the variability of resonance frequencies and MAC-values after FEM update. It is expressed as a COV. For each of the eight modes, its mean experimental counterpart is the reference for expressing MAC-values.

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Fig. 6b. Obtained values for Es after FE model update.

Table 6 Comparison between EMA and FEM resonance frequencies and mode shapes after FEM updating. Mode

Initial frequency difference (%)

Initial MAC (%)

Updated frequency difference (%)

Updated MAC (%)

1 2 3 4 5 6 7 8

2.71 1.23 1.43 2.15 0.73 4.57 3.27 2.89

99.13 97.94 97.33 94.20 91.16 90.25 86.53 81.76

0.13 0.29 0.22 0.09 0.31 0.25 0.41 0.43

98.98 97.77 97.24 95.30 90.33 91.94 90.21 89.42

Table 7 FEM updating variability, expressed as coefficients of variation (COV). Mode

FEM resonance frequency COV (%)

FEM MAC COV (%)

1 2 3 4 5 6 7 8

0.11 0.09 0.23 0.51 0.42 0.17 0.14 0.35

0.59 0.72 1.23 1.13 0.98 0.87 0.73 1.31

the databases. This section discusses the impact of FEM and EMA errors on the databases of Es and Gc. In Sections 1 and 2, the EMA and FE model updating uncertainty is quantified as errors on resonance frequencies and mode shapes (based on MAC-values). The sensitivity matrices of Tables 5a and 5b are therefore used to translate the FEM and EMA errors to variability on Es and Gc. The worst case situation is considered where both resonance frequency and mode shapes errors occur simultaneously. For all 17 measurement points and for all 22 test beams the resulting Es and Gc variability is calculated using Eq. (4) in an inverse sense. Relative scatter on the response ri is hereby translated to relative scatter on the design variable DVj. Table 8 gives an overview of the resulting Es and Gc variability for measurement points 2, 6, 10, 14 and 17, which are chosen randomly. The variability due to EMA and FE model updating uncertainty is very low in comparison to the estimated variability that is due to the honeycomb beam physics. Provided that measurement quality is very good and that a sufficiently large set of modes are considered for model updating, the impact of these errors is very limited. 5. Random fields for skin and core stiffness estimation 5.1. Goal of the method

4. Impact of EMA and fem errors The probability density functions of the two parameters Es and Gc were estimated in the previous section. However, this estimation is based on databases which are obtained through EMA and FE model updating. Consequently, some uncertainty is present in

Table 8 Estimated variability on Es and Gc due to EMA and FEM updating errors at measurement points 2, 6, 10, 14 and 17, expressed as coefficient of variation (COV). Measurement point

2

6

10

14

17

Es mean (GPa) Es COV (%) Gc mean (MPa) Gc COV (%)

7.83 0.63 37.9 0.11

8.11 0.26 28.7 0.43

8.46 0.14 27.8 0.17

7.97 0.21 33.4 0.36

7.49 0.29 28.1 0.58

The quantities of interest in this study exhibit some scatter, illustrated by Figs. 6a and 6b. The dynamic behaviour of the considered honeycomb beams can be regarded as a stochastic process [27–29] governed by a set of stochastic variables, Es and Gc in this case. These stochastic variables cannot be determined exactly, they can only be estimated statistically. Due to the limited size of the available experimental data (only 22 beams) it is difficult to perform an accurate estimation of the distribution type of Es and Gc and of the epistemic uncertainty involved. Advanced techniques should thus be applied to adequately estimate the true variance of the two obtained mixed experimental/numerical databases. The purpose of using the method described in this section is twofold. The first concern is to estimate the true probability density distributions of the two considered databases and to exclude all variability that is not directly related to the real Es and Gc variability. The second purpose is to estimate the epistemic uncertainty caused by the limited statistical data available in this study. Recently special numerical

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Fig. 7. Close-up of a skin showing the variations in glass fibre weave orientation.

techniques are developed to deal with problems of this kind. The two parameters of interest are modelled as independent random fields. The 22 beam samples are cut from one honeycomb panel after they are referenced according to their location in the panel. The goal is to estimate the variability of the considered stiffness parameters Es and Gc at different locations along the length direction of the beams and to describe their spatial variability within a beam sample. The specific location of the measurement points on a beam has no direct physical relevance. Fig. 7 shows a similar honeycomb panel to the one from which the test beams were cut from. The glass fibre weave in the skin clearly shows some pattern deformations that depend on the direction considered. This is a clear physical indication that design parameter variability can alter from one measurement point to another. 5.2. Karhunen–Loève series expansion For each considered design parameter the FE model update results in an experimental database, with dimensions M  Nexp, the sample size M = 22 and the number of measurement points Nexp = 17. The database contains the discrete values of the mathematical random field F describing the specific parameter variability. The covariance matrix [CF] of this database is spectrally decomposed using the Karhunen–Loève (KL) series expansion. Therefore the eigenvalues and corresponding eigenvectors of [CF] need to be calculated, leading to the corresponding eigenvalue system described by (5).

½Ceuk ¼ kk uk ; F

8k : 1 6 k 6 Nexp

ð5Þ

Eq. (6) expresses the KL-series expansion using the components from the covariance matrix and using a finite set of random variables g(k) with zero mean and generally non Gaussian distribution.

F¼ Fe

Nexp X

pffiffiffiffiffi

gðkÞ kk uk

k¼1

ð6Þ

For reasons of computational efficiency this series expansion is truncated after a number of terms that is smaller than Nexp. Based on a convergence study of the normalised sum of the set of eigenvalues, the series can be truncated after l terms. In this study this number is chosen so that the first l eigenvalues of the covariance matrix cover at least 95% of the available statistical data. Quantification of the random field requires the identification of the statistics of the random vector gl = {g(k)} which has a zero mean and the l  l identity matrix as covariance matrix. The joint density of gl however is not known and it has to be estimated from the available random field realisations (experimental data). This estimation process uses a Hermite polynomial chaos (PC) expansion with Bayesian Inference [21,22]. The random variables g(k) are expressed as multi-dimensional Hermite polynomials, evaluated for independent standard Gaussian random variables. Methodologies based on the Maximum likelihood principle and the use of the Bayesian inference algorithm have been proposed to estimate the PC coefficients. A Random Walk Metropolis Hastings Monte Carlo Markov Chain (MH-MCMC) algorithm [27] has been adopted here for the purpose. 5.3. Application of the methodology and results for the databases of Es and Gc In this study the two design parameters of interest are considered as two independent random fields. For both Es and Gc, a database consisting of 22  17 values is obtained through FEM updating. The covariance matrix (dimensions: 17  17) of each database is calculated and shown in Figs. 8a and 8b. After a convergence study, the number of terms in the KL-series expansion is set at 9. In case of the Es database, the largest 9 eigenvalues of the covariance matrix cover over 96% of the available statistical information in the covariance matrix. The corresponding convergence curve is shown in Fig. 9. A third order polynomial chaos expansion is found to be convenient and a population of 100,000 random numbers is used for the evaluation of the Hermite polynomials. Following the procedure that is proposed by Mehrez [22], an MH-MCMC-algorithm estimates the first and second order moments of the PC-coefficient distributions. The random field is evaluated at the measurement points. This procedure yields an estimation of the variability of Es and Gc among the different samples (inter-sample variability) at their corresponding measurement points. In this study the intra-sample variability is identified and quantified in terms of correlation length. From the physics of the skin faces and the honeycomb core it is expected that the variability

Fig. 8a. Covariance matrices of Es database.

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Fig. 8b. Covariance matrices of Gc database.

Fig. 9. Convergence of the normalized eigenvalue sum in case of Es database.

of the core shear modulus Gc is characterised by a smaller correlation length than the Young’s modulus Es. From observations of the skin faces it is expected that Es varies periodically, due to periodical variation of the glass fibre weave yarn orientation. When the covariance matrix of Es (see Fig. 8a) is observed, periodical correlation between the Young’s moduli at some considered measurement points may be observed. This makes sense, knowing that the face sheets have glass fibre woven reinforcements and that originally the test beams were cut from the same honeycomb panel. Consequently, a periodical covariance function is applied to estimate the correlation length of Es and Gc. Paragraph 3.5 discusses the estimation of correlation lengths. The relation between

the physical parameters of the skin faces and honeycomb core on the one hand, and the variability of Es and Gc on the other hand is treated in more detail in Section 5. Figs. 10a and 10b show box plots of the estimated probability distributions for Es and Gc at the 17 corresponding measurement points (for all test beams). The box borders indicate the 25th and 75th percentiles while the median is shown inside a box. Figs. 11a and 11b shows the obtained probability density functions for Es and Gc at measurement locations 2, 6, 10, 14 and 17, which are chosen arbitrarily. The covariance matrices of Es and Gc, estimated through the random field application, are illustrated in Figs. 12a and 12b. One specific goal of this study is to quantify the effect of having observed only 22 test samples. This is referred to as epistemic uncertainty and it is represented by the random character of the estimated coefficients of the polynomial chaos decomposition. In this study the method presented in [20,21] is followed. If enough test samples are observed (M sufficiently large), the distributions of the different estimated PC-coefficients approach a multivariate normal distribution of which the covariance matrix is the inverse of the Fisher information matrix. In [21], an algorithm is proposed to distinguish between the aleatory uncertainty and the epistemic uncertainty, due to M being rather low. The basic idea of this algorithm is to generate a series of Nc sets of mean PC coefficient values. Each set produces a probability distribution of Es or Gc. Consequently, a series of Nc probability distributions is obtained and 95% confidence intervals on these distributions are easily calculated. These confidence intervals, together with the PDF’s for the measurement points 2, 6, 10, 14 and 17, are shown in

Fig. 10a. Box plots of the estimated density of Es at the different evaluation points of the random field.

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Fig. 10b. Box plots of the estimated density of Gc at the different evaluation points of the random field.

Fig. 11a. Probability density functions for Es (GPa) and at measurement points 2, 6, 10, 14 and 17.

Fig. 13a–e for parameter Es and in Fig. 14a–e for Gc. The red and blue curves illustrate the 95% confidence intervals while the green dashed curves represent the earlier estimated mean PDF’s. 5.4. Validation of the method implementation The quality of the constructed stochastic model depends on the size of the experimental database. It is expected that the uncertainty on the estimated eigenvalues and eigenvectors of the covariance matrix [CF] increases when the size of the experimental database is decreases. Referring to [20], sub-sampling is used to study the effect of the database size on the nodal mean values, the eigenvalues and eigenvectors of the covariance matrix. These three quantities are strictly depending on the experimental database and they are the basis of the random field approximation. The sensitivity of the eigenvalues of [CF] to the database composition is studied by randomly picking subsets from the original

database. Fig. 15 shows the first 9 eigenvalues for subsets of sizes 16, 18 and 20 respectively. The black dashed line (–) corresponds to the eigenvalues of the full database. For the subsets of 16 and 18 elements, 50 sets of eigenvalues were calculated while 20 were calculated for the subset of 20 elements. The epistemic uncertainty (see Figs. 12a and b) was estimated using the full database of 22 samples. The selection of various subsets of different sizes leads to different sets of eigenvalues and eigenvectors which are the basis for the KL-decomposition. Consequently, the estimated epistemic uncertainty will vary accordingly. To study this effect, the 95% confidence intervals for the mean PDF’s of Es and Gc are estimated using subsets with sizes 16, 18 and 20. Fig. 16a–d illustrate the increase of epistemic uncertainty as the subset size decreases from 22 to 16. The PDF’s of Es at measurement interval 5 are considered as an example. It is experienced that if the size of the subsets becomes less than 11, the epistemic uncertainty becomes as high as the aleatory uncertainty.

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Fig. 11b. Probability density functions for Gc (MPa) and at measurement points 2, 6, 10, 14 and 17.

Fig. 12a. Reconstructed covariance matrix of Es variability.

In this research the random field method is used to study and describe the variability of Es and Gc at discrete (measurement point) locations on the beam structures, the method is not used yet to its full extent. Any possible relations between parameters Es and Gc have been neglected so far. However, by applying higher-order random fields, it is possible include parameter relations in the stochastic analysis. The next paragraph focuses on the description of the spatial variability of Es and Gc in one beam sample.

5.5. Estimation of correlation length of Es and Gc From direct observations of the skin faces it is expected that the variability of Es is marked by a periodicity ranging from 200 to 400 mm. This implies that macro-scale skin physical effects govern the Es variability. Expression (7) calculates the value of a stationary covariance function K(xi, xj). It is directly related with the elements of the covariance matrix of Es.

  Kðxi ; xj Þ ¼ r2 þ ri rj ekxi xj k=n cos kxi  xj k 2p=p

ð7Þ

Please cite this article in press as: Debruyne S et al. Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2013.09.004

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Fig. 12b. Reconstructed covariance matrix of Gc variability.

Fig. 13. 95% Es (GPa) PDF confidence intervals for measurement intervals 2, 6, 10, 14 and 17.

Fig. 14. 95% Gc (MPa) PDF confidence intervals for measurement intervals 2, 6, 10, 14 and 17.

In (7), the term r2 expresses the ‘noise’ due to the EMA and FEM uncertainty (see Table 8). For simplicity, the mean value of all measurement points is considered. The correlation length n and the period p are estimated using the different covariance matrix coefficients K(xi, xj). Results for this estimation are given in Table 9. The estimated covariance periodicity corresponds with the observed variability of the glass fibre weave yarn orientations. Table 10 presents the estimated correlation length and covariance periodicity for the honeycomb core shear stiffness Gc. A much larger correlation length is estimated here. For Gc, periodicity of its covariance can

originate from the honeycomb panel manufacturing process during the collection phase of the core cylinders. However, no observations are available to validate the estimated periodicity of Gc. 6. Relating the obtained variability of Es and Gc to variability of real honeycomb skin and core parameters In this section, the obtained distributions for Es and Gc are related to the variability of corresponding physical parameters of the skin and core of the honeycomb sandwich beams.

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Fig. 15. Considered covariance matrix eigenvalues for subsets with sizes 16 (top), 18 (middle) and 20 (bottom).

6.1. Relevant design parameters of the skin material Since the considered honeycomb beams are one-dimensional structures the skin is modelled as an isotropic material. The real skin of the tested honeycomb beams is made of glass fibre weave reinforced polypropylene. Its Young’s modulus is governed by a number of design parameters of this composite layer such as the glass fibre volume fraction (vf) and the orientation angles of the warp and weft yarns of the glass fibre weave (see Fig. 9). The orientation angle of the weave yarns in length direction was determined experimentally for all beams around measurement point 5. This was done at 10 evenly spaced locations on a 2.5 cm radius circle around the measurement point. The results of these nondestructive measurements are given in Table 11. It is clear that these small fibre angle variations do not account for the observed scatter on Es. It is not possible to measure scatter on vf in a nondestructive way. However vf has a large influence in governing the mass density of the skin. To have an idea of the scatter on vf, an independent series of weighting tests was performed on 40

honeycomb samples of size 50  50  25 mm, the same size of the measurement point intervals. A mean sample mass density of 184.65 kg/m3 with a relative standard deviation of 1.6% was obtained. As these samples were manufactured from the same honeycomb plate as the 22 beam samples the observed variance is relevant. The mass densities of the homogenised skin and core materials are 1460 and 67.8 kg/m3 respectively. In the assumption that the skin contains all sample mass and with the knowledge that the average glass fibre volume fraction vf in the skin (1.13 mm thickness) is about 12.3%, the variability of vf within one measurement point interval is certainly less than 6.9% (mass densities of glass fibre and polypropylene equal to 2700 and 1100 kg/m3 respectively). These numbers lead to the conclusion that this variability cannot account for the large variability of Es determined earlier. However vf can vary locally, causing local changes of the skin modulus. Another parameter that governs the skin’s elasticity modulus is the crimp of the glass fibre weave. So-called crimp is a measure of the non-straightness of the fibre in the skin. A higher crimp causes a decrease of the elasticity

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Fig. 16. 95% Es PDF confidence intervals in cases of experimental dataset subsets containing 16, 18, 20 and 22 test samples, (a) 95% PDF CI for Es at beam interval 5; subset size = 22, (b) 95% PDF CI for Es at beam interval 5; subset size = 20, (c) 95% PDF CI for Es at beam interval 5; subset size = 18, (d) 95% PDF CI for Es at beam interval 5; subset size = 16.

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S. Debruyne et al. / Computers and Structures xxx (2014) xxx–xxx Table 9 Estimated correlation length and periodicity of Es variability. Parameter

Mean (mm)

Standard deviation (mm)

n p

638 261

74 52

Table 10 Estimated correlation length and periodicity of Gc variability.

6.2. Relevant design parameters of the core material

Parameter

Mean (mm)

Standard deviation (mm)

n p

2520 1280

160 145

modulus. A series of 50 tensile tests was carried out to have an estimation of possible scatter on Es. A mean value of 7310 MPa was obtained along with a relative standard deviation of 4.12%. In this study Es and Gc are the only considered parameters. Since resonance frequencies depend on the bending stiffness of the considered beams also the mean distance between the top and bottom glass fibre weaves of a beam section (distance d on Fig. 4) is an important parameter. Again this is a parameter that cannot be determined experimentally on the existing samples. According to Daniel [30] and Zenkert [1] the bending stiffness D of a sandwich beam can be approximated by Eq. (8) in case of a weak core and if core shear deformation is neglected. 2

D¼

11%. Local variations in fibre volume fraction and fibre orientation angle have more effect on the tensile Young’s modulus as the sample width decreases. In the assumption that the scatter on Es decreases with increasing sample width, the equivalent Es variability from the tensile tests should lie in the 4.25–5.5% range. Again this is of the same order as the estimated variability of the Es database.

Es t3s Es ts d þ 6 2

ð8Þ

In case of thin face sheets the first term is less than 1% of the second term and so it may be neglected. Applying the rules of error propagation [31] on the simplified form of Eq. (8) indicates that a 10% variability of Es has the same effect on D as approximately a 5% variability of distance d. A change in d can be related to a change in beam thickness and/or a change in core thickness. As d cannot be measured directly, two closely related measurements were carried out. Beam thickness measurements are performed on all measurement intervals of all 22 test samples; an overall mean value of 25.15 mm and a relative standard deviation of about 0.16% are obtained. Since d does not differ much from the total beam thickness (approximately 23.98–25.15 respectively) a change of d from 1 to 1.0016 could have the same effect on the bending stiffness as an increase of Es of 0.32% which is very little. The core thickness was also determined experimentally for all beams; a normalized random error of 0.598% and a mean value of 22.85 mm are obtained. Again this may account for an Es variability of 1.2%. To validate the physical relevance of the obtained Es variability multiple series of static tensile tests were performed each on 25 skin samples with dimensions: 200  30 mm. The experienced relative standard deviation from this test varies between 8.5% and

There are several physical design variables of the sandwich core that affect its out-of-plane shear stiffness. The estimated variability of the out-of-plane shear modulus Gc can thus be traced. Dimensional parameters that govern the shear stiffness include the cell wall thickness, cell diameter and the core height. Finite element simulated shear tests were carried out to study the sensitivity of Gc with respect to the related dimensional core parameters mentioned. Fig. 17 shows the FE model of a unit cell used for this purpose. Experimental values for core height, cell wall thickness and cell diameter are obtained using a 3D measurement bench and performing dimensional measurements on 330 core cells. Table 12 gives the normalised relative sensitivity coefficients (Eq. (7)) calculated using the FE core model and the expected relative variability of Gc due to each measured core parameter and its scatter. This calculation involves linearisation, yet from Table 6 it is clear that the observed scatter on the cell wall thickness of the core cylinders may cause a significant scatter on the core shear modulus. During the production process of the honeycomb panels the polypropylene core cylinders are somewhat welded together. Local changes in core cylinder adhesion affect the out-of-plane core shear stiffness strongly. Finite element calculations were carried out to study the effect of core cylinder adhesion. In case of a perfect

Fig. 17. Finite element model of a honeycomb core unit cells.

Table 11 Mean glass fibre orientation angles for measurement point 5, all beams. Beam

Fibre angle (°)

E (Gpa)

DE (%)

Beam

Fibre angle (°)

E (Gpa)

DE (%)

1 2 3 4 5 6 7 8 9 10 11

0.22 0.81 0.16 0.34 0.38 0.046 0.55 0.45 0.08 0.34 0.71

9.000958 8.991891 9.000934 9.000448 8.992636 8.999497 8.997966 8.999461 8.99905 9.000448 8.99465

0.011 0.09 0.011 0.005 0.082 0.0056 0.0226 0.006 0.01 0.005 0.059

12 13 14 15 16 17 18 19 20 21 22

0.34 0.31 0.65 0.41 0.19 0.23 0.30 0.18 0.68 0.034 0.65

9.000448 9.000664 8.995972 8.999846 9.000968 8.996416 9.000723 8.997347 8.995462 8.999632 8.995972

0.005 0.0074 0.045 0.017 0.011 0.04 0.008 0.029 0.05 0.0041 0.045

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Table 12 Normalised relative sensitivity coefficients for Gc with respect to core cell dimensions. Parameter

Mean value from experiments (mm)

COV from experiments (%)

Normalized relative sensitivity

Estimated Gc COV (%)

Cell height Outer cell diameter Cell wall thickness

22.76 8.58

1.1 0.74

0.42 0.83

0.46 0.61

0.57

3.2

1.02

3.26

adhesion of all core cylinders in the unit cell a shear modulus of 43.4 MPa is obtained while in case of zero adhesion of the cylinder walls the shear modulus becomes 20.2 MPa. To obtain a nominal value of Gc dynamic shear tests were carried out on 38 samples with dimensions 50  50 mm, the same size as a measurement interval. A mean value of 30.02 MPa with a relative standard deviation of 17.8% was obtained. Taking the mean value from experiments into account one can conclude that physically there is a partial adhesion between the cylinder walls. 6.3. Conclusions on the physical parameter relations It is impossible to obtain values for Es and Gc from non-destructive measurements on the considered honeycomb beams. However related experiments on other samples than the test beams are carried out to obtain relevant (mean and scatter) reference values for both Es and Gc. The scatter that is found from these experiments corresponds well to the variability estimated from the random field approximation in Section 3. The use of simulated tests gives significant insight in the relation between the experienced variability from the stochastic analysis and the physical variability of the real skin and core design parameters. 7. General conclusions In this study two dominant elastic parameters of honeycomb sandwich beams are treated as stochastic variables governing the vibration behaviour of the beams, representing the process of interest. Experimental vibration data of a limited number of test beams is used as an input database for finite element model updating. The database of Es and Gc values resulting from the EMA and FEM processes is further analysed using a random field methodology to obtain optimal estimations of the variability of the two design parameters in the form of probability density functions. An important issue in this study is that this approach yields an estimation of a confidence interval for each estimated PDF, representing the uncertainty due to a limited number of test specimens. The implementation of the random field methodology and the estimation of the epistemic uncertainty in this case are validated. The objective of the analysis is to quantify the physical scatter on model parameters. This study tries to distinguish the importance and magnitude of each of the successive phases in the identification and estimation procedure. The contribution of the procedures of data acquisition, experimental modal parameter estimation is limited as long as the quality of the EMA process is very good. If the measurement quality is not excellent this leads to high errors on measured mode shapes which may make their use for the model updating phase somewhat doubtful. The uncertain outcome of the FEM updating process has also been addressed. A study was made to relate the estimated PDF’s to plausible variability of physical honeycomb design parameters of core and skin. By performing related tests and by simulating shear tests, these relations were studied successfully.

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Please cite this article in press as: Debruyne S et al. Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2013.09.004

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Please cite this article in press as: Debruyne S et al. Identification of design parameter variability of honeycomb sandwich beams from a study of limited available experimental dynamic structural response data. Comput Struct (2014), http://dx.doi.org/10.1016/j.compstruc.2013.09.004