modeling of viscoelastic dampers

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Computers and Structures 88 (2010) 1–17

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Computers and Structures

journal homepage: www.elsevier .com/locate /compstruc

Identification of the parameters of the Kelvin–Voigt and the Maxwellfractional models, used to modeling of viscoelastic dampers

R. Lewandowski *, B. Chora _zyczewskiPoznan University of Technology, ul. Piotrowo 5, 60-965 Poznan, Poland

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 June 2008Accepted 3 September 2009Available online 14 October 2009

Keywords:Parameters identificationFractional rheological modelsViscoelastic dampers

0045-7949/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.compstruc.2009.09.001

* Corresponding author. Tel.: +48 61 6652 472; faxE-mail address: [email protected]

Fractional models are becoming more and more popular because their ability of describing the behaviourof viscoelastic dampers using a small number of parameters. An important difficulty, connected withthese models, is the estimation of model parameters. A family of methods for identification of the param-eters of both the Kelvin–Voigt fractional model and the Maxwell fractional model are presented in thispaper. Moreover, the equations of hysteresis curves are derived for fractional models. One of the methodspresented used the properties of hysteresis curves. The validity and effectiveness of procedures have beentested using artificial and real experimental data.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Viscoelastic (VE) dampers have often been used in controllingthe vibrations of aircrafts, aerospace and machine structures.Moreover, VE dampers are used in civil engineering to reduceexcessive oscillations of building structures due to earthquakesand strong winds. A number of applications of VE dampers in civilengineering are listed in [1]. The VE dampers could be dividedbroadly into fluid and solid VE dampers. Silicone oil is used to buildthe fluid dampers while the solid dampers are made of copolymersor glassy substances.

Good understanding of the dynamical behaviour of dampers isrequired for the analysis of structures supplemented with VEdampers. The dampers behaviour depends mainly on the rheolog-ical properties of the viscoelastic material the dampers are made ofand some of their geometric parameters. These characteristics de-pend on the temperature and the frequency of vibration. Temper-ature changes in dampers can occur due to environmentaltemperature fluctuations and also on the internal temperature ris-ing due to energy dissipation. A classic shift factor approach iscommonly used to capture the effect of temperature changes.Therefore, only the frequency dependence of damper characteris-tics is considered in this paper.

In a classic approach, the mechanical models consisting ofsprings and dashpots are used to describe the rheological proper-ties of VE dampers [2–7]. A good description of the VE dampers re-quires mechanical models consisting of a set of appropriatelyconnected springs and dashpots. In this approach, the dynamic

ll rights reserved.

: +48 61 6652 059.l (R. Lewandowski).

behaviour of a single damper is described by a set of differentialequations (see [5,6]), which considerably complicates the dynamicanalysis of structures with dampers because the large set of motionequation must be solved. Moreover, the cumbersome, nonlinearregression procedure, described for example in [8,9], is proposeto determine the parameters of the above mentioned models.

The rheological properties of VE dampers are also describedusing the fractional calculus and the fractional mechanical models.Currently, this approach has received considerable attention andhas been used in modeling the rheological behaviour of linear vis-coelastic materials [10–15]. The fractional models have an abilityto correctly describe the behaviour of viscoelastic material usinga small number of model parameters. A single equation is enoughto describe the VE damper dynamics, which is an important advan-tage of the discussed models. In this case, the VE damper equationof motion is the fractional differential equation. The fractionalmodels of VE fluid dampers are proposed in [16,17]. The parame-ters of the model proposed in [17] are complex numbers whichhas added to the complexity of the dynamic analysis of structureswith VE dampers, especially when the in time domain analysismust be performed. However, fractional models of which theparameters are real numbers are also proposed, for example, in[10,21,22]. The dependence of the above mentioned model param-eters on excitation frequency could lead to major complications ofanalysis of structures with VE dampers in a time domain. Fortu-nately, efficient time-domain approaches have been proposed re-cently. The methods, based on Prony series and Biot model, areproposed in papers [4,29,30]. Additionally, the method which usedthe generalized Maxwell model and the Laguerre polynomialapproximation is suggested in [5]. Very recently, seismic analysisof structures with VE dampers modeled by the Kelvin chain and

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2 R. Lewandowski, B. Chora _zyczewski / Computers and Structures 88 (2010) 1–17

the Maxwell ladder is presented in [6]. Moreover, the non-viscousstate-space model, described in [31], could be used in this context.The analysis in the frequency domain is possible and presented, forexample, in [3].

An important problem, connected with the fractional rheologi-cal models, is an estimation of the model parameters from exper-imental data. In the past, different methods have been tested forthe estimation of model parameters from both static and dynamicexperiments [4,8,9,16–19,21]. The process of parameter identifica-tion is an inverse problem which is overdetermined and can be illconditioned (see, for example [8,19]) because of noises existing inthe experimental data. The mathematical difficulties may be over-come by the regularization method which is described, for exam-ple, in [8,18].

It is the aim of this paper to describe a few methods of identifi-cation of the parameters of VE dampers. The parameters are esti-mated using the results obtained from dynamical tests. Thefractional rheological models are used to describe the dynamicbehaviour of dampers. The Kelvin–Voigt model and the Maxwellmodel are discussed. Three kinds of identification methods are sug-gested. Equations of hysteresis curves are derived for both frac-tional models and some of the properties of these curves areused to develop the first-kind identification procedure. The sec-ond- and third-kind identification procedures are based on timeseries data. Each identification procedure consists of two mainsteps. In the first step, for the given frequencies of excitation,experimentally obtained data are approximated by simple har-monic functions while model parameters are determined in thesecond stage of the identification procedure. The results of a typicalcalculation are presented and discussed.

Three symbols will be used in the paper to describe the damperforce. The symbol ueðtÞ denotes the results of experimental mea-surements, ~uðtÞ is the function which is an analytical approxima-tion of experimental results while uðtÞ is the solution to thedamper equation of motion. The symbols qeðtÞ; ~qðtÞ and qðtÞ areused in the same way to describe the damper displacement.

2. General explanation of fractional rheological models

Due to their simplicity, the simple rheological models, such asthe Kelvin–Voigt model or the Maxwell model, are used very of-ten to describe the dynamic behavior of VE dampers installed onvarious types of civil structures. For example, the Kelvin–Voigtmodel is used in papers [2,3,22–24], while the Maxwell modelis used in papers [2,3,25–28]. The Kelvin–Voigt model consistsof the spring and the dashpot connected in parallel, while theMaxwell model is built from the serially connected spring anddashpot. The discussed simple rheological models have a smallnumber of parameters but do not have enough parameters toaccurately capture the frequency dependence of damper parame-ters. More sophisticated classical rheological models which, how-ever, contain many parameters, can be used to correctly describethe dynamic behavior of VE dampers. Models of this kind aredeveloped in [6,7].

The next group of models used to describe the behaviour of VEdampers are the fractional derivative models. Using the fractionalcalculus, a number of rheological models, e.g., the fractional Kel-vin–Voigt model [32], the fractional Zener model [10,11], the frac-tional Jeffreys model [12], or the fractional Maxwell model [17] canbe developed. It has been proved in [4,15] that the fractional deriv-ative models can better capture the frequency dependent proper-ties of VE dampers. The simple fractional models discussed in[4,13,15] are sufficient to correctly capture the VE dampers proper-ties. These models contain only a few parameters but the dynamicanalysis requires knowledge of the fractional calculus.

The simple Kelvin–Voigt model seems to be more appropriateto describe the solid dampers behavior, while the simple Maxwellmodel is mainly used to describe the liquid dampers behavior. Re-cently, it has been shown in [6] that both the generalized Kelvinmodel and the generalized Maxwell model are also useful modelsof the solid VE damper.

One important problem, considered here and connected withthese models, is how to determine model parameters from exper-imental data in an efficient way. This problem is solved in this Sec-tion for the fractional Kelvin–Voigt model and the fractionalMaxwell model.

2.1. Fractional models equation of motion and the steady state solutionto the motion equation

In order to construct the fractional models equation of motion,we introduce the fractional element called the springpot whichsatisfies the constitutive equation:

uðtÞ ¼ ~caDat qðtÞ ¼ cDa

t qðtÞ; ð1Þ

where c ¼ ~ca and a;0 < a 6 1, are the springpot parameters andDa

t qðtÞ is the fractional derivative of the order a with respect to timet. There are a few definitions of fractional derivatives which coin-cide under certain conditions. Here, symbols such as Da

t qðtÞ denotethe Riemann–Liouville fractional derivatives with the lower limit at–1 (see [34]). Some valuable information about fractional calculuscan be found in [34].

The springpot element, also known as the Scott–Blair’s element(see [33]), is schematically shown in Fig. 1a. The considered ele-ment can be understood as an interpolation between the springelement ða ¼ 0Þ and the dashpot element ða ¼ 1Þ.

The fractional Kelvin–Voigt model consists of the springand the springpot connected in parallel, while the fractionalMaxwell model is built of the serially connected spring and spring-pot. These models are shown schematically in Fig. 1b and c,respectively.

The motion equation of the above mentioned fractional Kelvin–Voigt model, obtained in a usual way, is in the following form:

uðtÞ ¼ kqðtÞ þ ksaDat qðtÞ; ð2Þ

where sa ¼ ~ca=k ¼ c=k.The following relationships can be written for the fractional

Maxwell model (Fig. 1c)

uðtÞ ¼ kq1ðtÞ; uðtÞ ¼ ~caDat ðqðtÞ � q1ðtÞÞ: ð3Þ

Eliminating q1ðtÞ from the above relationships, we get the equationin the form:

uðtÞ þ saDat uðtÞ ¼ ksaDa

t qðtÞ: ð4Þ

It is easy to recognize that both of the considered fractional modelshave three real and positive value parameters: k, c, and a.

In the case of a harmonically excited damper, i.e. when

qðtÞ ¼ q0 expðiktÞ; ð5Þ

the steady state solution of the motion equation of both of the frac-tional models is assumed in the form:

uðtÞ ¼ u0 expðiktÞ: ð6Þ

Taking into account that in our case (see [34, p. 311])

Dat expðiktÞ ¼ ðikÞa expðiktÞ; ð7Þ

for k > 0, we obtain the following equations

u0 ¼ k½1þ ðiskÞa�q0; u0 ¼ kðiskÞa

1þ ðiskÞaq0; ð8Þ

Fig. 1. Diagram of fractional rheological models.

R. Lewandowski, B. Chora _zyczewski / Computers and Structures 88 (2010) 1–17 3

from (2) and (4), respectively. Moreover, after introducing the fol-lowing formula ia ¼ cosðap=2Þ þ i sinðap=2Þ, we can rewrite rela-tionships (8) in the form

u0 ¼ ðK 0 þ iK 00Þq0 ¼ K 0ð1þ igÞq0; ð9Þ

where gðkÞ ¼ K 00ðkÞ=K 0ðkÞ is the loss factor, K 0ðkÞ is the storagemodulus and K 00ðkÞ is the loss modulus. These quantities are definedas:

K 0 ¼ k½1þ ðskÞa cosðap=2Þ�; K 00 ¼ kðskÞa sinðap=2Þ; ð10Þ

g ¼ ðskÞa sinðap=2Þ1þ ðskÞa cosðap=2Þ

; ð11Þ

for the fractional Kelvin–Voigt model and

K 0 ¼ kðskÞa ðskÞa þ cosðap=2Þ1þ ðskÞ2a þ 2ðskÞa cosðap=2Þ

;

K 00 ¼ kðskÞa sinðap=2Þ1þ ðskÞ2a þ 2ðskÞa cosðap=2Þ

; ð12Þ

g ¼ sinðap=2ÞðskÞa þ cosðap=2Þ

; ð13Þ

for the fractional Maxwell model.If the classical Kelvin–Voigt model (i.e. a ¼ 1Þ is used as the VE

dampers model the above mentioned quantities are given by

K 0ðkÞ ¼ k; K 00ðkÞ ¼ ksk; gðkÞ ¼ K 00ðkÞ=K 0ðkÞ ¼ sk ð14Þ

while for the classical Maxwell model we have

K 0ðkÞ ¼ ks2k2

1þ s2k2 ; K 00ðkÞ ¼ ksk

1þ s2k2 ; gðkÞ ¼ 1sk: ð15Þ

The dependence of the above mentioned model parameters on exci-tation frequency could significantly complicates in time domainanalysis of structures with VE dampers.

According to the classical Kelvin–Voigt model the storage mod-ulus is the constant function of k, while the loss modulus and theloss factor linearly increases with k. Behaviour of the fractional Kel-vin–Voigt model is substantially different in comparison with theclassic Kelvin–Voigt model. The storage modulus is increased sig-nificantly when the non-dimensional frequency increases andwhen the parameter a decreases. Moreover, the loss modulusand the loss factor decrease for an increasing non-dimensional fre-quency and the decreasing values of the parameter a.

The properties of the fractional Maxwell model, for differentvalues of the parameter a are shown in Figs. 2 and 3. In the figuresthe non-dimensional frequency is defined as sk. The calculation ismade using the value of the k parameter equal to 1,00,000.0 N/m.

The storage modulus grows with non-dimensional frequency forall values of the parameter a. However, for 0 < sk < 1 the functionK 0ðaÞ increases for decreasing values of a. An opposite tendency isevident for sk > 1. The function of the loss modulus could be flat(for instant when a ¼ 0:4Þ. Moreover, gð0Þ ¼ tgðap=2Þ whichmeans that the loss factor has a finite value for k ¼ 0. This is animportant difference in comparison with the loss factor functionof the classic Maxwell model of which the values approach infinityif k approaches zero and in comparison with the both versions ofthe Kelvin–Voigt model for which the loss factor is equal to zerofor sk ¼ 0.

According to results presented by Lion in [35] the fractional rhe-ological models fulfill the second law of thermodynamics whenvalues of the storage modulus K 0ðkÞ and the loss modulus K 00ðkÞ, gi-ven by formulae (10)–(13), are positive for all possible values offrequency of excitation k. It can easily be demonstrated that bothmodels fulfill the second law of thermodynamics when0 6 a 6 1; s > 0 and k > 0.

If the damper displacement varies harmonically in time and isdescribed using the trigonometric functions, i.e.:

qðtÞ ¼ qc cos kt þ qs sin kt; ð16Þ

the steady state solution to the fractional Kelvin–Voigt model equa-tion of motion (2) is given by

uðtÞ ¼ uc cos kt þ us sin kt; ð17Þ

where

uc ¼ u1qc þu2qs; us ¼ �u2qc þu1qs; ð18Þu1 ¼ kþ cka cosðap=2Þ; u2 ¼ cka sinðap=2Þ: ð19Þ

The steady state solution to the equation of motion (4) of the frac-tional Maxwell model is also given by (16) and (17) and the coeffi-cients qc; qs;uc and us are interrelated in the following way:

qc ¼ /1uc � /2us; qs ¼ /2uc þ /1us; ð20Þ

where

/1 ¼1

kðskÞaðskÞa þ cos

ap2

h i; /2 ¼

1kðskÞa

sinap2: ð21Þ

2.2. Hysteresis loops of the damper models

The equation of the hysteresis loops of the fractional Kelvin–Voigt model could be derived if the damper’s kinematical excita-tion is given by

qðtÞ ¼ q0 sin kt: ð22Þ

Fig. 2. The storage modulus of the fractional Maxwell model for different values of a parameter.

Fig. 3. The loss modulus of the fractional Maxwell model for different values of a parameter.


Taking into account that (see [34, p. 311])

Dat qðtÞ ¼ kaq0 sin½kt þ ðap=2Þ�; ð23Þ

and introducing (22) and (23) into Eq. (2) we can write

uðtÞ ¼ k½1þ ðskÞa cosðap=2Þ�q0 sin kt þ kðskÞaq0 sinðap=2Þ cos kt:

ð24Þ

Next, using (22) and the identity sin2ðap=2Þ þ cos2ðap=2Þ ¼ 1, wecan rewrite (24) in the form of the following equation

uðtÞ � k½1þ ðskÞa cosðap=2Þ�qðtÞkðskÞaq0 sinðap=2Þ

� �2

þ qðtÞq0

� �2

¼ 1; ð25Þ

which describes the first version of the hysteresis loop of the frac-tional Kelvin–Voigt model.

Derivation of the second version of the hysteresis loop startedwith an assumption that the damper’s displacement is describedby

qðtÞ ¼ q0 sin½kt � ðap=2Þ�; ð26Þ

which means that

Dat qðtÞ ¼ kaq0 sin kt: ð27Þ

Introducing relationship (26) into the motion Eq. (2) we can write

uðtÞ � ksaDat qðtÞ ¼ kq0½sin kt cosðap=2Þ � cos kt sinðap=2Þ�: ð28Þ

Using (27) it is possible to write the second version of the hysteresisloop equation in the following form:

uðtÞ � kk�a½ðskÞa þ cosðap=2Þ�Dat qðtÞ

kq0 sinðap=2Þ

� �2

þ Dat qðtÞkaq0

� �2

¼ 1: ð29Þ

In a case of the classical Kelvin–Voigt model we have:

uðtÞ � kqðtÞkcq0

� �2

þ qðtÞq0

� �2

¼ 1;uðtÞ � c _qðtÞ

kq0

� �2

þ_qðtÞkq0

� �2

¼ 1; ð30Þ

instead of (25) and (29).


Let us now proceed to developing the equation describing thehysteresis loop of the fractional Maxwell model. The excitation isdescribed by

uðtÞ ¼ u0 sin kt; ð31Þ

what means that

Dat uðtÞ ¼ kau0 sin½kt þ ðap=2Þ�: ð32Þ

Introducing (32) into the motion Eq. (4) we obtain the followingrelationship

uðtÞ � ksaDat qðtÞ ¼ �ðskÞau0½sin kt cosðap=2Þ þ cos kt sinðap=2Þ�;

ð33Þ

which could be rewritten as

ksaDat qðtÞ � ½1þ ðskÞa cosðap=2Þ�uðtÞ

ðskÞau0 sinðap=2Þ

� �2

þ uðtÞu0

� �2

¼ 1: ð34Þ

This is the first version of the hysteresis loop equation of the frac-tional Maxwell model.

To obtain the second version of the hysteresis loop equation themotion Eq. (4) must be fractionally integrated. This operation isformally defined in [34] and here it will be denoted using the sym-bol D�a

t uðtÞ if the function uðtÞ is integrated. Moreover, it can bedemonstrated that D�a

t ðDat uðtÞÞ ¼ uðtÞ.

After the fractional integration of Eq. (4) we obtain

D�at uðtÞ þ sauðtÞ ¼ ksaqðtÞ: ð35Þ

Now, if uðtÞ is given by (31) then

D�at ðu0 sin ktÞ ¼ u0k

�a sin½kt � ðap=2Þ�: ð36Þ

Introducing (31) into (4) we can write Eq. (35) in the form:

Fig. 4. Hysteresis loops of the fractional Kelv

kðskÞaqðtÞ � ðskÞauðtÞ ¼ u0½sin kt cosðap=2Þ� cos kt sinðap=2Þ�: ð37Þ

Next, with the help of relationships (31) and sin2ðap=2Þþcos2ðap=2Þ ¼ 1 we can transform the above equation to

kðskÞaqðtÞ � ½ðskÞa þ cosðap=2Þ�uðtÞu0 sinðap=2Þ

� �2

þ uðtÞu0

� �2

¼ 1; ð38Þ

which is the second version of the hysteresis loop equationsearched for.

It is obvious that, for the classical Maxwell model ða ¼ 1Þ weobtain:

c _qðtÞ � uðtÞsku0

� �2

þ uðtÞu0

� �2

¼ 1; ðskÞ2 kqðtÞ � uðtÞu0

� �2

þ uðtÞu0

� �2

¼ 1;

ð39Þ

instead of (34) and (38), respectively.The hysteresis loops of the fractional Kelvin–Voigt model and

the fractional Maxwell model are shown in Figs. 4 and 5, respec-tively, for different values of a and for k = 1,00,000.0 N/m, c =1,00,000.0 Ns/m, k = 10.0 rad/s. It is evident that for both modelsthe damper’s damping abilities decrease as the values of adecrease.

3. General remarks concerning identification methods

The problem of determination of the parameters of the frac-tional derivative rheological models is, more or less extensively,discussed in papers [10,13,17,20,21,37–40]. In two papers[10,13], Pritz describes the method of parameters identificationof two fractional derivative rheological models with four and fiveparameters, respectively. The method utilizes some asymptoticproperties of the storage and loss modulus functions, experimentally

in–Voigt model for different values of a.

Fig. 5. Hysteresis loops of the fractional Maxwell model for different values of a.


obtained over a certain range of excitation frequencies. However,no systematic procedure of identification is presented. Similarmethods are developed in [20,38] where more detailed descriptionof identification procedure is also presented. The above mentionedmethods require experimental data from a large range of excitationfrequencies. Very recently, in paper [39], the identification proce-dure of parameters of fractional derivative model of viscoelasticmaterials is also presented. The method uses an optimization pro-cedure together with the experimentally and numerically obtainedfrequency response functions to determine the parameters of theconsidered model VE materials. The differences between the mea-sured and calculated frequency response functions are minimizedin order to estimate the true values of the searched parameters.Moreover, the least squares method, in which the error betweenthe model and the experimental complex modulusK�ðkÞ; ðK�ðkÞ ¼ K 0ðkÞ þ iK 00ðkÞÞ is minimized in order to obtain theparameters of the fractional model of viscoelastic materials is sug-gested in [21,37,40]. Unfortunately, details of the used leastsquares method are presented in [40] only.

There are many papers containing a description of the methodof identification of parameters of the classic rheological modelsof VE materials and VE dampers. In paper [4] Park used the Pronyseries together with the least squares method to determine param-eters of the generalized Maxwell model of the VE solid and the VEliquid damper. The normalized error between the experimentallyobtained and the calculated complex modulus is minimized. Theoptimization problem is linear because the number of Prony seriesand the relaxation times are fixed. The last square method is alsoused in [6] to determine the parameters of two mechanical modelsof VE dampers. The models are the Kelvin chain and the Maxwellladder. Some information concerning identification of the parame-ters of the so-called GHM model and the ADF model used for mod-eling viscoelastic materials are presented in [41]. The advancednumerical methods for parameter identification of VE materialsare presented in [8,9,42] where the least squares method together

with the regularization technique is used to solve the consideredidentification problem.

In this paper, the parameters of the fractional Kelvin model andthe fractional Maxwell model are determined using results of dy-namic tests. The proposed identification methods differ substan-tially in comparison with the previously mentioned ones. First ofall, instead of using the experimentally obtained complex modulusthe suggested methods utilize the measured steady state responsesof damper. The previous methods, presented in [10,13,20,38], re-quire experimental data from a large range of excitation frequencywhile the suggested methods could be also used when data areavailable only from a narrow range of excitation frequency. More-over, the error functional minimized in the least squares methodand adopted in our identification methods is different from oneused in previous papers. The results of calculation show that theproposed methods are not sensitive to measurements noises and,therefore, it is not necessary to use regularization techniques.Additionally, a detailed description of the proposed identificationmethod and the identification procedure are presented.

Three kinds of identification methods are suggested in this pa-per to determine three parameters of the fractional Kelvin–Voigtmodel and the fractional Maxwell model. Each identification pro-cedure consists of two main steps. In the first step, for the givenfrequencies of excitation, experimentally obtained data areapproximated by simple harmonic functions while model parame-ters are determined in the second stage of the identificationprocedure.

4. Identification of model parameters for fractional Kelvin–Voigt model

4.1. Identification procedures based on hysteresis loop (first method)

First of all, the first kind method based on the hysteresis loopand applied to the fractional Kelvin–Voigt model will be described.


Eq. (25) seems to be more useful in comparison with (29). If, for thegiven frequency of excitation and t ¼ t1, we have qðt1Þ ¼ q0 > 0and uðt1Þ ¼ u1 > 0 then from Eq. (25) it follows

k½1þ ðskÞa cosðap=2Þ� ¼ kþ cka cosðap=2Þ ¼ u1=q0: ð40Þ

Moreover, for t ¼ t2 when qðt2Þ ¼ 0 and uðt2Þ ¼ u2 > 0 from (25) weobtain

kðskÞa sinðap=2Þ ¼ cka sinðap=2Þ ¼ u2=q0: ð41Þ

For a given frequency k, relationships (40) and (41) constitute a setof two nonlinear equations with three unknowns: k; c;a or k; s;a.The quantities u1;u2 and q0 have clear physical meanings and theirvalues can easily be obtained from the experimental data. Alterna-tively, these quantities could be calculated from trigonometric func-tions (A1) and (A6) used for the approximation of experimentaldata. Details are given in Appendix A. For a given a, the above setof equations are linear with respect to k and c.

During experiments the damper is several times harmonicallyexcited and in each case the excitation frequency, denoted hereas ki, ði ¼ 1;2; ::;nÞ, is different. The steady state response of thedamper is measured, which means that the experimental damperdisplacements qeiðtÞ and the experimental damper forces ueiðtÞand functions ~qiðtÞ ~uiðtÞ which approximate the experimental dataare known for each excitation frequency ki. Moreover, the abovementioned quantities, such as u1;u2 and q0 can easily be deter-mined for each excitation frequency. These quantities obtainedfrom experimental data are denoted as u1i;u2i and q0i.

Now it is assumed that resulting quantities u1i;u2i and q0i

approximately fulfill relationships (40) and (41). For each excita-tion frequency we can write

ri ¼ kþ ckai cosðap=2Þ � u1i=q0i � 0; ð42Þ

si ¼ ckai sinðap=2Þ � u2i=q0i � 0; ð43Þ

where i ¼ 1;2; ::;n. Symbols ri and si denote residuals obtained afterintroducing u1i;u2i and q0i into (42) and (43).

The above equations constitute a set of overdetermined nonlin-ear equations with respect to k; c and a. A pseudo-solution to theabove system of equation is chosen in such a way that it minimizesthe following functional

eJKV ðk; c;aÞ ¼Xn

i¼1

ðr2i þ s2

i Þ: ð44Þ

If we assume that the parameter a is known, then stationaryconditions:

@eJKV ðk; c;aÞ@k

¼ 0;@eJKV ðk; c;aÞ

@c¼ 0: ð45Þ

give us the following set of equations, which are linear with respectto k and c

knþ cXn

i¼1

kai cosðap=2Þ ¼

Xn

i¼1

u1i

q0i;

kXn

i¼1

kai cosðap=2Þ þ c

Xn

i¼1

k2ai ¼

Xn

i¼1

kai

u1i

q0icosðap=2Þ þ u2i

q0isinðap=2Þ

� �:

ð46Þ

The right value of a is obtained using the systematic searchingmethod. The set of values of a, denoted as ajðj ¼ 1;2; ::;mÞ, whereaj ¼ aj�1 þ Da is chosen from a given range of a. For each aj the cor-responding values of k and c (denoted as kj and cjÞ are determinedfrom (46) and the value of functional (44) is calculated. These valuesof aj; kj and cj for which the functional (44) has a minimum valueare the searched parameters of the fractional Kelvin–Voigt dampermodel.

4.2. Identification procedures based on time series data (second andthird method)

The identification procedure based on time series data is alsodeveloped. The second-kind method will be described first. Theexperimentally measured damper displacement is approximatedusing the harmonic function (A1). The parameters ~qc and ~qs are ob-tained from the set of Eq. (A3).

Moreover, it is assumed that the experimentally obtained stea-dy state solution represented by the harmonic function (A1)approximately fulfills the steady state Eq. (18) of the fractionalKelvin model where ~qc and ~qs are introduced in a place of qc andqs, respectively.

Now, for a given excitation frequency k, the quantities u1 andu2, appearing in (18), are determined as described below. The rightvalues of u1 and u2 are the ones which minimize the functional

JKV ðu1;u2Þ ¼1

t2 � t1

Z t2

t1

½ueðtÞ � uðtÞ�2dt: ð47Þ

considered here as the functional of u1 and u2. Stationary condi-tions of the above-mentioned functional give us the following linearequations with respect to u1 and u2

a11u1 þ a12u2 ¼ b1; a21u1 þ a22u2 ¼ b2; ð48Þ

where coefficients a11; a12; a21; a22; b1 and b2 are given by

a11 ¼ ~q2c Icc þ 2~qc~qsIcs þ ~q2

s Iss; a12 ¼ a21 ¼ ~qc~qsðIcc � IssÞ þ ð~q2s � ~q2

c ÞIcs;

ð49Þa22 ¼ ~q2

c Iss � 2~qc~qsIcs þ ~q2s Icc; b1 ¼ ~qcIcu þ ~qsIsu; b2 ¼ �~qcIsu þ ~qsIcu;

ð50Þ

Icu ¼Z t2

t1

ueðtÞ cos ktdt; Isu ¼Z t2

t1

ueðtÞ sin ktdt; ð51Þ

Icq ¼Z t2

t1

qeðtÞ cos ktdt; Isq ¼Z t2

t1

qeðtÞ sin ktdt: ð52Þ

The solution to Eq. (48) is

u1 ¼b1a22 � b2a12

a11a22 � a12a21; u2 ¼

�b1a21 þ b2a11

a11a22 � a12a21: ð53Þ

During experiments, the data are measured for different excitationfrequencies ki; ði ¼ 1;2; ::;nÞ and subsequently the values of u1 andu2, denoted now as u1i and u2i, respectively, are calculated from(48) or (53). For each particular excitation frequency ki the follow-ing equations, very similar to (42) and (43), could be written:

ri ¼ kþ ckai cosðap=2Þ �u1i ¼ 0; si ¼ cka

i sinðap=2Þ �u2i ¼ 0:

ð54Þ

The solutions to a set of overdetermined equations (i.e. Eq. (54)written for i ¼ 1;2; ::;nÞ are found in a similar way as in the firstidentification procedure described above. The searched pseudo-solution minimizes the functional (44) with ri and si defined by for-mulae (54). For a given value of a, the parameters k and c fulfill theequations

knþ cXn

i¼1

kai cosðap=2Þ ¼

Xn

i¼1

u1i;

kXn

i¼1

kai cosðap=2Þ þ c

Xn

i¼1

k2ai ¼

Xn

i¼1

kai ½u1i cosðap=2Þ þu2i sinðap=2Þ�;

ð55Þ

and the parameter a is obtained with the help of the systematicsearching method described earlier in this paper. One can see thatEq. (55) are very similar to Eq. (46).


Please note, in this procedure only the data concerning thedamper displacement qeðtÞ are approximated using the trigono-metric function.

The second identification procedure based on in-time data ser-ies (called here the third-kind method) will also be formulated forthe fractional Kelvin–Voigt model. In this procedure, experimen-tally measured variations of both the damper force and the damperdisplacement are approximated using the harmonic functions(A1) and (A6), respectively. Moreover, it is assumed that Eq. (18)are approximately fulfilled when ~qc; ~qs; ~uc and ~us are introducedin a place of qc; qs; uc and us, respectively. Now, from Eq. (18) weobtain

u1 ¼~qc ~uc þ ~qs~us

~q2c þ ~q2

s; u2 ¼

~qs~uc � ~qc ~us

~q2c þ ~q2

s: ð56Þ

The quantities u1 and u2 obtained for a particular frequency ofvibration ki, will be denoted as u1i and u2i, respectively. For a givenset of excitation frequencies ki; ði ¼ 1;2; ::;nÞ, the overdeterminedset of Eq. (54) could be written.

The values of model parameters k, c and a are chosen in such away that the functional (44) will be minimized (ri and si are de-fined by relationships (54)). The procedure described in this sub-section could be used without being modified at all. The rightvalue of a is obtained using the systematic searching method andthe values of the associated parameters k and c are determinedfrom the set of Eq. (55).

It is easy to notice that both of the methods described above dif-fer only in the way u1i and u2i are calculated. The first identifica-tion procedure based on in-time series approximate theexperimental data concerning the damper displacement with thehelp of the trigonometric function while the third-kind methodused this approximation for both experimental data.

5. Identification of model parameters for fractional Maxwellmodel

5.1. Identification procedures based on hysteresis loop (first method)

A similar method to the one described in Section 4.1 is devel-oped for the fractional Maxwell model. For a given excitation fre-quency k and t ¼ t1, when uðt1Þ ¼ u0 > 0 and qðt1Þ ¼ q1 > 0, fromEq. (38) we obtain:

kðskÞaq1 � ðskÞau0 � u0 cosðap=2Þ ¼ 0; ð57Þ

which could also be written in the form:

ckaq1 � ~skau0 � u0 cosðap=2Þ ¼ 0: ð58Þ

Moreover, at t ¼ t2 when the state of the damper is described byuðt2Þ ¼ 0 and qðt2Þ ¼ q2 < 0, from the hysteresis loop Eq. (38) itfollows:

kðskÞaq2 þ u0 sinðap=2Þ ¼ 0; ð59Þ

which subsequently could be rewritten in the form:

ckaq2 þ u0 sinðap=2Þ ¼ 0: ð60Þ

If we have experimental data for a given set of excitation frequen-cies ki; ði ¼ 1;2; ::;nÞ then the following set of equations (with re-spect to c; ~s and aÞ

ri ¼ ckai q1i � ~ska

i u0i � u0i cosðap=2Þ ¼ 0;si ¼ cka

i q2i þ u0i sinðap=2Þ ¼ 0; ð61Þ

could be written.As in the previous model, a pseudo-solution to the above set of

equations is chosen as one which minimized the functional:

eJMðc; ~s;aÞ ¼Xn

i¼1

ðr2i þ s2

i Þ: ð62Þ

If we assume that the parameter a is known, the stationary condi-tions of (62) give us the following system of equations:

cXn

i¼1

k2ai ðq2

1i þ q22iÞ � ~s

Xn

i¼1

k2ai u0iq1i

¼Xn

i¼1

kai u0i½q1i cosðap=2Þ � q2i sinðap=2Þ�;

� cXn

i¼1

k2ai u0iq1i þ ~s

Xn

i¼1

k2ai u2

0i ¼ �Xn

i¼1

kai u2

0i cosðap=2Þ: ð63Þ

For particular values of a, the values of damper parameters result-ing from (61) could be negative. These solutions do not have phys-ical meaning and must be rejected.

As described in Section 4.1, the right value of a is obtained usingthe method of systematic searching. The values of a, ~s and c forwhich the functional (62) has a minimum value are the searchedparameters of the fractional Maxwell model.

5.2. Identification procedures based on time series data (second andthird method)

In the second-kind method, for the given frequency of excita-tion the experimentally measured damper force ueðtÞ is approxi-mated using function (A6) where ~uc and ~us are determined fromEq. (A8). The steady state solution to the equation of motion (4)is given by (16) and (17) and the coefficients qc; qs;uc and us areinterrelated as it is given by relation (20).

Next, ~uc and ~us, obtained from the experimental data and for thegiven frequency of excitation k, are introduced in relationships (20)in the place of uc and us and the functional

JMð/1;/2Þ ¼1

t2 � t1

Z t2

t1

½qeðtÞ � qðtÞ�2dt; ð64Þ

is minimized with respect to /1 and /2. The stationary conditions ofthis functional give us the following set of equations

a11/1 þ a12/2 ¼ b1; a21/1 þ a22/2 ¼ b2; ð65Þ

where

a11 ¼ Icc ~u2c þ 2~uc ~usIsc þ ~u2

s Iss; a22 ¼ ~u2c Iss � 2~uc ~usIsc þ ~u2

s Icc; ð66Þa12 ¼ a21 ¼ ð~u2

c � ~u2s ÞIsc þ ~uc ~usðIss � IccÞ; ð67Þ

b1 ¼ ~ucIcq þ ~usIsq; b2 ¼ ~ucIsq � ~usIcq: ð68Þ

The values Iss; Isc; Icc are defined by the relationships (A4) and Isq; Icq

by (A5).The values of /1 and /2, determined from the set of Eq. (65), is

given by

/1 ¼b1a11 � b2a12

a11a22 � a12a21; /2 ¼

�b1a21 þ b2a11

a11a22 � a12a21: ð69Þ

The model parameters c, k and a will be determined using the least-square method. As previously, the error functional (62), where

ri ¼1kþ 1

ck�a

i cosap2� /1i; si ¼

1c

k�ai sin

ap2� /2i; ð70Þ

is minimized with respect to model parameters. The residuals ri andsi are obtained from relationships (21) after introducing ki instead ofk and taking into account that (21) could only approximately be sat-isfied. As previously, the index i of such quantities as /1i emphasizestheir dependence on the i-th frequency of excitation used in theexperiments.

Start

for i=1,...,nApproximate q (t) and u (t) using Eqns (A1) and A(6)

e e

Kelvin-Voigt model Maxwell model

Determine q , u , u 0i 1i 2i

Determine k, c and using Eqns (46) and the searching method

α

Stop

for i=1,...,nDetermine u , q and qoi 1i 2i

Determine c, and using Eqns (63) andthe searching method

τ α

for i=1,...,n

Fig. 6. Flowchart of the procedure of the first-kind identification method.

Start

for i=1,...,nApproximate q (t) using Eqn (A1)

e e


1i 2i


α

Stop

for i=1,...,nCalculate and using Eqn (69)

1i

Determine k, c and using Eqns (71) andthe searching method

α

for i=1,...,n

φφϕϕ

Approximate u (t) datausing Eqn (A6)

Calculate andusing Eqn (53)

for i=1,...,n2i

Fig. 7. Flowchart of the procedure of the second-kind identification method.

Start

for i=1,...,nApproximate q (t) and u (t) using Eqns (A1) and A(6)

e e


Determine and 1i 2i


α

Stop

for i=1,...,nDetermine and 1i 2i

Determine k, c and using Eqns (71) andthe searching method

α

for i=1,...,nφφϕϕ

Fig. 8. Flowchart of the procedure of the third-kind identification method.


The parameter a is determined with the help of the searchingmethod described earlier, while the values of c and k are obtainedfrom the following set of equations:

n�kþ �cXn

i¼1

k�ai cosðap=2Þ ¼

Xn

i¼1

/1i;

�kXn

i¼1

k�ai cosðap=2Þ þ �c

Xn

i¼1

k�2ai

¼Xn

i¼1

k�ai ½/1i cosðap=2Þ þ /2i sinðap=2Þ�; ð71Þ

where �k ¼ 1=k and �c ¼ 1=c.The third-kind method, applied to the fractional Maxwell mod-

el, utilizes the functions ~qðtÞ and ~uðtÞ, described, for k ¼ ki, by rela-tionships (A1) and (A6). The parameters ~uci; ~usi; ~qci and qsi

introduced now in a place of ~uc; ~us; ~qc and qs approximately fulfillthe following equations

~qci ¼ /1~uci � /2~usi; ~qsi ¼ /2i~uci þ /1i~usi; ð72Þ

resulting from Eq. (20). From the equations above we have

/1i ¼~qci~uci þ ~qsi~usi

~u2ci þ ~u2

si

; /2i ¼~qsi~uci � ~qci~usi

~u2ci þ ~u2

si

: ð73Þ

Model parameters are determined with the help of the proceduredescribed above, where the error functional (62) and the identicalsearching procedure together with relationships (70) and (71) areused.

For the fractional Maxwell model both of the methods de-scribed in this subsection differ only in the way /1i and /2i arecalculated.

The reciprocal comparison of all proposed identification meth-ods makes it possible to formulate the following remarks:

(a) as proved in Section 6, none of the proposed methods is sen-sitive to measurements noises,

(b) for both rheological models the second-kind method and thethird-kind method differ only in the way in which /1i and /2i

or u1i and u2i are calculated,(c) the second-kind method analytically approximates experi-

mental data obtained for the damper displacement (the Kel-vin–Voigt model) or only for the damper force (in the case ofthe Maxwell model) while the third-kind method analyti-cally approximates the experimental data for both the dam-per displacement and the damper force,

(d) similar calculation efforts are needed when each of the pre-sented methods is used to identify the dampers parameters,

(e) any existing differences between the discussed methodshave no significant influence on the effectiveness of themethods and on the results of identification.

6. Algorithms of identification methods and results of typicalcalculation

6.1. Algorithms of identification methods

All of the identification procedures have some similaritieswhich were mentioned, to some extent, in previous Subsections.Generally speaking, all algorithms consist of three steps. Approxi-mation of the experimental data using the trigonometric functionsis made in the first step. In the second step, some quantities, forexample, u1iu2i;/1i and /2i are calculated for all frequencies ofexcitation. Finally, the model parameters are determined in thethird step. For the readers’ convenience, the flowcharts of all ofthe algorithms are presented in Figs. 6–8.


6.2. Results of typical calculation – the fractional Kelvin–Voigt model

The identification procedure described above is applied to arti-ficial data. The artificial experimental steady state solutions aregenerated on the basis of the solution to the motion Eq. (2). Thesesolutions are given by functions (16) and (17). If, for example,qc ¼ 0, then

qðtÞ ¼ qs sin kt; u1 ¼ qsk½1þ ðskÞa cosðap=2Þ�;u2 ¼ qskðskÞa sinðap=2Þ: ð74Þ

The artificial data obtained from relationships (74) are modified byapplying small, randomly inserted perturbations. The noises of 1%intensity of the damper force or the damper displacement areapplied.

The following data are used to generate artificial experimentaldata: qs = 0.001 m, qc = 0, k = 2,90,000.0 N/m, c = 68,000.0 Ns/m,a = 0.6, n = 9. The values of excitation frequencies ki chosen in thisexample and the values of u1i;u2i and q0i determined from the arti-ficial data are given in Table 1.

After application of the identification procedure based on thehysteresis loop the following results are obtained: a = 0.601,k = 2,89,936.0 N/m, c = 67,646.8 Ns/m, which is in good agreementwith exact values. Results of calculation performed for differentnoise levels are presented in Fig. 9. It is easy to notice that the rel-

Table 1Artificially generated experimental data.

Frequency (Hz) Fractional Kelvin–Voigt model

q0 (m) u1 (N) u2 (N)

0.5 0.991010e�3 366.564 108.521.0 0.991010e�3 414.464 165.182.0 1.004310e�3 471.856 250.974.0 1.005330e�3 564.810 382.756.0 1.000410e�3 637.696 489.898.0 0.990801e�3 702.286 580.12

10.0 0.997754e�3 761.647 653.9212.5 1.007520e�3 834.779 753.7415.0 0.991348e�3 902.471 836.23

Fig. 9. Relative errors of the a parameter (small rhombs), the k parameter (small cross midentification method applied to the Kelvin–Voigt fractional model).

ative errors of values of a, k and c parameters are of the order ofnoises.

The third-kind method is also applied to artificial experimentaldata described above. The following results are obtained: a = 0.609,k = 2,92,115.0 N/m, c = 64,984.0 Ns/m when 3% noises are ran-domly introduced to artificial data. Relative errors of damperparameter values performed for different noise levels are shownin Fig. 10.

Very good results are obtained using the second-kind method.Following the method presented in [36], random noises is addedto the artificial data uðtÞ and qðtÞ using the formulae:

ueðtÞ ¼ uðtÞ þ exðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivarðuðtÞÞ

p;

qeðtÞ ¼ qðtÞ þ ezðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivarðuðtÞÞ

p; ð75Þ

where e is the noise level, xðtÞ and zðtÞ are the standard distributionvectors with zero mean ant unit deviation, varðuðtÞÞ and varðqðtÞÞare the variation of uðtÞ and qðtÞ, respectively. Results of calculationperformed for different noise levels are presented in Fig. 11. In theconsidered case, the influence of noises on damper parameters isvery small.

The discussed fractional Kelvin–Voigt model was also used todetermine the parameters of the small size VE damper. The damperconsists of three steel plates and two layers made from viscoelasticmaterial VHB 4959 manufactured by 3M. The thickness of each vis-coelastic layer is 3 mm. A harmonically varying excitation with dif-

Fractional Maxwell model

u0 (N) q1 (m) q2 (m)

0 297.306 0.232195e�2 �0.177817e�22 302.356 0.191155e�2 �0.119442e�27 300.441 0.160286e�2 �0.787455e�30 299.111 0.140054e�2 �0.517613e�33 298.085 0.131743e�2 �0.401366e�37 297.083 0.127103e�2 �0.338493e�31 300.699 0.125184e�2 �0.295670e�37 301.900 0.121994e�2 �0.261395e�30 301.531 0.121300e�2 �0.234501e�3

arks) and the c parameter (small triangles) versus the level of noises (the first-kind

Fig. 10. Relative errors of the a parameter (small rhombs), the k parameter (small cross marks) and the c parameter (small triangles) versus the level of noises (the third-kindidentification method applied to the Kelvin–Voigt fractional model).

Fig. 11. Relative errors of the a parameter (small rhombs), the k parameter (small cross marks) and the c parameter (small triangles) versus the level of noises (the second-kind identification method applied to the Kelvin–Voigt fractional model).

Table 2Stiffness and damping factors of small damper.

Frequency (rad/s) k (N/m) c (Ns/m)

3.14 73303.1 14676.56.28 86615.5 9659.612.56 98416.0 5927.025.12 106055.0 3362.037.68 106111.0 2295.050.24 105254.0 1702.062.80 101254.0 1373.078.50 98644.0 1085.094.20 86823.0 851.0


ferent excitation frequencies taken from the range 0.5–15.0 Hz isused during the experiments.

The Kelvin–Voigt model is adopted to describe the VE damper’sbehaviour. At the beginning, the classical Kelvin–Voigt model, i.e.the fractional model with a ¼ 1, was used to determine the damp-ers parameters. In this case, the values of k and c parameters can bedetermined independently for each frequency of excitation fromEqs. (54). The results of calculation based on a times series are pre-sented in Table 2 where values of the parameters k and c in depen-dence on the values of excitation frequency are given. As expected,the model parameters strongly depend on excitation frequency. Acomparison of the experimentally obtained hysteresis curve withones resulting from the identification procedure is made inFig. 12 where the comparison is given for k ¼ 1:0 Hz. The smallcross marks show the experimental data while the solid line pre-sents identification results. The very good agreement betweenthe both curves is obvious.

The results obtained using the fractional Kelvin–Voigt modeland the third-kind method will be briefly described. The followingvalues of model parameters: a = 0.3755, k = 2603.1 N/m i c =

Fig. 12. A comparison of the experimentally obtained hysteresis curve (small cross marks) and the hysteresis curve resulting from the identification procedure (solid line).

Fig. 13. Storage modulus – comparison of experimental data (small cross marks) with results obtained from the identification procedure (solid line) – the fractional Kelvin–Voigt model.


30,333.9 Ns/m are obtained after application of the identificationprocedure. A comparison of the storage modulus resulting fromthe identification procedure (the solid line) with the storage mod-ulus obtained experimentally (the small cross marks) is presented

in Fig. 13. A similar comparison for the loss modulus is shown inFig. 14. The presented results indicate that the three parameterfractional Kelvin–Voigt model could reasonably well describe thebehaviour of solid VE dampers.

Fig. 14. Loss modulus – comparison of experimental data (small cross marks) with results obtained from the identification procedure (solid line) – the fractional Kelvin–Voigtmodel.


6.3. Results of typical calculation – the fractional Maxwell model

As previously, the identification procedure described above isapplied to artificial data. The artificial experimental steady statesolutions are generated on the basis of the solution to Eq. (4). If,for example, uc ¼ 0 and u0 ¼ us then

q2 ¼ qc ¼ �u0

kðskÞasin

ap2;

q1 ¼ qs ¼u0

kðskÞaðskÞa þ cos

ap2

h i: ð76Þ

Fig. 15. Relative errors of the a parameter (small rhombs), the k parameter (small cross midentification method applied to the Maxwell fractional model).

The artificial data are modified by applying small, randomly in-serted perturbations.

The following data are used to generate artificial data: uc = 0,us = u0 = 300.0 N, k = 2,90,000.0 N/m, c = 68,000.0 Ns/m, a = 0.6,n = 9. The values of excitation frequencies chosen in this exampleand values of u0i; q1i and q2i are given in Table 1.

After application of the identification procedure of first-kindthe following results are obtained: a = 0.601, k = 2,90,469.0 N/m,c = 67,786.2 Ns/m which is in good agreement with exact values.The results of calculation performed for different noise levels(taken in the range of 0–5 %) are presented in Fig. 15. It is easy

arks) and the c parameter (small triangles) versus the level of noises (the first-kind

Fig. 16. Relative errors of the a parameter (small rhombs), the k parameter (small cross marks) and the c parameter (small triangles) versus the level of noises (the third-kindidentification method applied to the Maxwell fractional model).

Fig. 17. Error functional eJM of the fractional Maxwell model versus the a parameter for artificial data with noise intensities of 3%, 5% and 10%.


to notice that the relative errors of values of a, k and c parametersare of the order of noises.

The third-kind method is also applied to the artificial experi-mental data. The following results are obtained: a = 0.610,k = 2,84,543.0 N/m, c = 68,096.0 Ns/m when 3% noises are ran-domly introduced to artificial data. The results of calculationsperformed for different noise levels are shown in Fig. 16. Addi-tionally, in Fig. 17 the plot functional eJM (given by relationship(62)) of the fractional Maxwell model versus the parameter a ispresented for three levels of noises. In the range of values ofinterest of the parameter a we have one minimum of the func-tional. Very similar results to those presented above are obtainedusing the second-kind method. Moreover, a similar plot to theone presented in Fig. 17 is obtained for the fractional Kelvin–Voi-gt model.

The results of calculations performed for artificial data showedthat if the noise levels are not too high then all of the suggested

methods of parameters identification are not sensitive to thenoises. Errors in the values of the parameters obtained in the iden-tification procedures are of the order of the noise levels.

The next step is to apply the identification procedure to realexperimental data. The experimental data presented by Makrisand Constantinou [17] are chosen and used in this example. Intheir investigations, Makris and Constantinou were using a dampermanufactured by GERB Schwingungsisolierungen GmbH & Co. KG.The following values of parameters of the fractional Maxwell mod-el are determined: a = 0.77, k = 503.350 kN/m, c = 13.823 kNs/m. InFigs. 18 and 19 a comparison of the experimental and approxi-mated storage modulus K 0 and the loss modulus K 00 is presented.The values of K 0 and K 00 resulting from the identification procedureare calculated from relationships (12) and (13). The presentedmodel works satisfactorily. The identification procedure of theparameters of the fractional Maxwell model is simple, well appli-cable and efficient.

Fig. 18. Storage modulus – fluid damper.

Fig. 19. Loss modulus – fluid damper.


7. Concluding remarks

A family of parameters identification methods for the Kelvin–Voigt and the Maxwell fractional models are proposed in the paper.Three kinds of identification methods are suggested to determinethree parameters of the fractional Kelvin–Voigt model and the frac-tional Maxwell model. Each identification procedure consists oftwo main steps. In the first step, for the given frequencies of exci-tation, experimentally obtained data are approximated by simpleharmonic functions while model parameters are determined inthe second stage of the identification procedure. The parameters

of fractional models are determined using results from dynamicaltests. The suggested identification methods differ substantially incomparison with previously proposed method. Rather than usingthe experimentally obtained complex modulus, the discussedmethods directly utilize the measured steady state responses ofdamper. The methods could be also used when data are availableonly from a narrow range of excitation frequency. The results ofcalculation show that the proposed methods are not sensitive tothe measurement noises and, therefore, it is not necessary to useregularization techniques. Similar calculation efforts are neededwhen each of the presented methods are used to identify the dam-


per parameters and the existing differences between the discussedmethods have no significant influence of the effectiveness of themethods and on results of identification. All of the proposed iden-tification procedures are simple, well applicable and efficient.Moreover, hysteresis loop equations are derived for both models.Some properties of the hysteresis curves are used to develop oneof the identification procedures. The validity and effectiveness ofthe identification procedures have been successfully tested usingboth artificially generated and real experimental data. It was foundthat they are real VE dampers to which this model can be fittedwith a satisfactory accuracy.

Acknowledgments

The authors wish to acknowledge the financial support receivedfrom the Poznan University of Technology (Grant No. DS 11-018/08) in connection with this work. The authors would like to expresstheir appreciation to the reviewers for their valuable suggestions.

Appendix A. Approximation of experimental data bytrigonometric functions

In this Appendix A we assume that, during the experimentaltests, two output functions ueðtÞ (function of force in a time do-main) and qeðtÞ (function of displacement in a time domain) whichrepresent the steady state behavior of damper were obtained. Bothfunctions could be approximated using simple trigonometricfunctions, as described below. Two procedures of the parametersidentification based on time series data are presented in thisSubsection.

Experimentally measured displacements of the damper areapproximated using the function:

~qðtÞ ¼ ~qc cos kt þ ~qs sin kt: ðA1Þ

Using the least-square method, parameters ~qc and ~qs of function(A1) are determined. This method requires minimization of the fol-lowing functional:

J1ð~qc; ~qsÞ ¼1

t2 � t1

Z t2

t1

½qeðtÞ � ~qðtÞ�2dt; ðA2Þ

where the symbols t1 and t2 denote the beginning and the end ofthe time range in which the damper’s displacements were mea-sured. Part of the measuring results related to steady state vibrationare used as data in this identification procedure. From the station-ary conditions of the functional (A2), the following system of equa-tions with respect to parameters ~qc and ~qs are received:

Icc~qc þ Isc~qs ¼ Icq; Isc~qc þ Iss~qs ¼ Isq; ðA3Þ

where

Icc ¼Z t2

t1

cos2 ktdt; Iss ¼Z t2

t1

sin2 ktdt; Ics ¼ Isc

¼Z t2

t1

sin kt cos ktdt; ðA4Þ

Icq ¼Z t2

t1

qeðtÞ cos ktdt; Isq ¼Z t2

t1

qeðtÞ sin ktdt: ðA5Þ

In a similar way the experimentally measured damper force isapproximated using the following harmonic function:

~uðtÞ ¼ ~uc cos kt þ ~us sin kt: ðA6Þ

The parameters ~uc and ~us are determined with the help of the last-square method. The functional and the system of equations with re-spect to ~uc and ~us are:

J2ð~uc; ~usÞ ¼1

t2 � t1

Z t2

t1

½ueðtÞ � ~uðtÞ�2dt; ðA7Þ

Icc ~uc þ Isc ~us ¼ Icu; Isc ~uc þ Iss~us ¼ Isu; ðA8Þ

where Icc; Iss and Ics ¼ Isc are given by relationships (A4). Moreover,

Icu ¼Z t2

t1

ueðtÞ cos ktdt; Isu ¼Z t2

t1

ueðtÞ sin ktdt: ðA9Þ

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journal homepage: www.elsevier .com/locate/compstruc

Comparison on numerical solutions for mid-frequency responseanalysis of finite element linear systems

Jin Hwan Ko a, Doyoung Byun b,*

a School of Mechanical and Aerospace Engineering, Seoul National University, Daehak-dong, Gwanak-gu, Seoul 151-742, South Koreab Department of Aerospace Information Engineering, Konkuk University, 1 Hwayang-dong, Kwangjin-Gu, Seoul 143-701, South Korea

a r t i c l e i n f o

Article history:Received 28 October 2008Accepted 26 September 2009Available online 28 October 2009

Keywords:Mid-frequency response analysisAlgebraic substructuringFrequency sweep algorithmModal acceleration method


* Corresponding author. Tel.: +82 2 450 4195; fax:E-mail address: [email protected] (D. Byun).

a b s t r a c t

Mid-frequency response analysis often faces computational difficulties when a conventional modalapproach is used for a finite element linear system. In this paper, the computational burden is relievedby frequency sweep algorithm or mode acceleration method for a reduced-order system constructedby algebraic substructuring, a variant of model order reduction. The two methods are compared withthe help of numerical experiments and their computational complexity. As demonstrated by the finiteelement simulations, in which proportional damping is assumed, of a turbo-prop aircraft and a ring res-onator, the frequency sweep algorithm for reduced-order systems shows the best performance among allconsidered numerical solutions, including the conventional approach.


1. Introduction

Frequency response analysis is a crucial tool in characterizingthe dynamic behavior of a mechanical system; it provides not onlythe resonance frequencies of the system, but also the amplitudes ofthe responses under unit force excitations. Responses for a low fre-quency range have conventionally been computed using a deter-ministic approach such as the finite element method, and thosein the high-frequency range have been solved by an energy ap-proach such as the Statistical Energy Method. However, the devel-opment of numerical methods for solving the responses in themid-frequency range has remained a frontier area yet to be con-quered. In this paper, numerical solutions of finite element linearsystems are considered for the mid-frequency response analysis.

The discretized model of a structure for a continuous single-in-put single-output second-order system can be written as follows:

M€xðtÞ þ D _xðtÞ þ KxðtÞ ¼ buðtÞ;yðtÞ ¼ lT xðtÞ;

(ð1:1Þ

where xð0Þ ¼ x0 and _xð0Þ ¼ v0. Here, t is the time variable, xðtÞ 2 RN

is a state vector, and N is the order of the system. uðtÞ is the inputexcitation force and yðtÞ is the output measurement function.b 2 RN and l 2 RNare the input and output distribution vectors,respectively, and M;K, and D 2 RN�N are the system mass, stiffnessand damping matrix, respectively. It is assumed that M and K are

ll rights reserved.

+82 2 444 6670.

symmetric positive-definite. The input–output behavior of themodel (1.1) is characterized by the frequency response function:

HðxÞ ¼ lT �x2M þ ixDþ K� ��1

b ¼ lT GðxÞ�1b; ð1:2Þ

where x is the frequency, i is the imaginary unit, and GðxÞ is thesystem dynamic matrix. Mathematically, the mid-frequency re-sponse analysis pertains to the computation of HðxÞ for x in therange ½xmin;xmax�. A conventional numerical solution for the fre-quency response analysis has been solved by the mode superposi-tion method on a finite element linear system.

However, the mode superposition method faces two types ofcomputational difficulties in regard to mid-frequency responseanalysis. First, a finely discretized model is required to representthe modes for which the natural frequencies are close to themid-frequency range, leading to a dramatic increase in computa-tional cost. The computational burden of a large order systemcan be relieved by model order reduction. So far, substructuring-based model order reductions have been developed for improvingefficiency in large systems. Automatic multilevel substructuring[1] and algebraic substructuring [2] are included in this type.Moreover, a variant of algebraic substructuring has recently beendeveloped for solving only the modes of the interior eigenvalues[3], which contribute more to the mid-frequency responses thanthe extreme eigenvalues do.

The second difficulty is due to an increase in the number of re-tained modes. Modal superposition requires two or three times max-imum frequency for a given range to achieve a desired accuracy;therefore, when the frequency range under consideration shifts tomid-range, the number of modes increases significantly. Residual





J. Hwan Ko, D. Byun / Computers and Structures 88 (2010) 18–24 19

flexibility modes play a role in decreasing the number of retainedhigh-frequency modes, considering the static response to a givenload [4], but all of the low frequency modes are still required. There-fore, numerical techniques that consider low- and high-frequencymode truncations are also required. There have been two recently-introduced methods to compensate for low- and high-truncations;the first is the frequency sweep algorithm and the second is the mod-al acceleration method. The frequency sweep algorithm was initiallyintroduced by Bennighof and Kaplan [1], and its convergence wasverified by Ko and Bai when it was applied to compensate for theboth truncation errors [5]. The modal acceleration method, as intro-duced in Ref. [6], was also used to compensate for both errors for asystem on FE subspace. In this paper, modal superposition is modi-fied for solving a reduced system on the algebraic substructuringsubspace, referred to as AS subspace.

In the present study, the performances of frequency sweep andmodal acceleration in AS subspace are compared in detail in termsof their computational complexity and numerical experimentsdone on the finite element model of a turbo-prop aircraft and a ringresonator. Comparison of both methods with a conventional meth-od on the FE subspace also included.

2. Frequency response analysis

Assuming Rayleigh damping D ¼ aM þ bK , and the introductionof the shift r, which is used for obtaining the modes of the interioreigenvalues, the frequency response function HðxÞ on the FE sub-space, of column dimension N, can be written thus:

HðxÞ ¼ lT c1Kr þ c2M½ ��1b ¼ lT GðxÞ�1b; ð2:1Þ

where Kr ¼ K � rM, c1 ¼ c1ðxÞ ¼ 1þ ixb and c2 ¼ c2ðx;rÞ ¼�x2 þ rþ ixðaþ rbÞ. The AS subspace is composed of columnvectors in Am, constructed by retained substructure modes in thealgebraic substructuring (AS), and typically has a dimension m thatis much smaller than N [5]. Projecting Eq. (1.1) onto the AS subspaceby xðtÞ ¼ AmzðtÞ yields the following equation:

Mm€zðtÞ þ Dm _zðtÞ þ KmzðtÞ ¼ bmuðtÞ;yðtÞ ¼ lT

mzðtÞ;

(ð2:2Þ

where Km ¼ ATmKAm;Mm ¼ AT

mMAm; lm ¼ ATml and bm ¼ AT

mb; The cor-responding frequency response function is

HmðxÞ ¼ lTm c1Kr

m þ c2Mm� ��1bm ¼ lT

mpmðxÞ; ð2:3Þ

where Krm ¼ Km � rMm. pmðxÞ is solved by the parameterized linear

system of order m:

GmðxÞpmðxÞ ¼ bm; ð2:4Þ

where GmðxÞ ¼ c1Krm þ c2Mm. In this paper, HmðxÞ is used as an

approximation for HðxÞ.

2.1. Modal superposition method

In the AS method, the order m typically remains too high for adirect method of computing frequency responses to be applied.Therefore, the modal superposition method is used. Here, thepnðxÞ values occur in the subspace spanned by the retained globalmodes Un from a reduced eigensystem Kr

m/ ¼ hrMm/ and theptðxÞ values occur in the subspace spanned by the low- andhigh-frequency truncated modes. Given pnðxÞ ¼ UngnðxÞ for somecoefficient vector gnðxÞ, the Eq. (2.4) then follows:

GmðxÞ UngnðxÞ þ ptðxÞð Þ ¼ bm: ð2:5Þ

The modal mass and stiffness matrices becomes:

UTnMmUn ¼ I and UT

nKrmUn ¼ Hr

n ; ð2:6Þ

where I is a diagonal unit matrix and Hrn is a diagonal matrix be-

cause the global modes satisfy the mass and stiffness orthogonalityconditions. In this paper, the global modes are normalized so thatthe modal mass /T

i Mm/i ¼ 1, where i ¼ 1; . . . ;n. When Eq. (2.5) is

pre-multiplied by UTn;U

TnGmðxÞptðxÞ, where ptðxÞ is expressed by

a linear combination of the truncated modes, is eliminated due tothe orthogonality and the vector pnðxÞ is then given by the n uncou-pled equations:

pnðxÞ ¼ Un UTnGmðxÞUn

� ��1UT

nbm ¼ Un c1Hrn þ c2I

� ��1UTnbm

¼ UngnðxÞ: ð2:7Þ

From (2.3), the frequency response function is solved by

HmðxÞ ¼ HnðxÞ ¼ lTmpnðxÞ: ð2:8Þ

HnðxÞ contains errors due to the truncated modes in the low-and high-frequency ranges, which are compensated for by the nexttwo methods.

2.2. Frequency sweep algorithm

An algorithm known as the frequency sweep (FS) algorithm wasintroduced [1] to compensate for the error of the truncated modes.The FS algorithm is an iterative scheme that includes the modalsuperposition method. Eq. (2.5) can be written as a parameterizedlinear system for ptðxÞ:

GmðxÞptðxÞ ¼ bm � GmðxÞpnðxÞ: ð2:9Þ

Using the Galerkin subspace projection based on the orthogo-nality of Un and Ut , and the Taylor expansion of G�1

m aroundðKr

mÞ�1, the following iteration is derived [5] for computing the vec-

tor ptðxÞ:

p‘tðxÞ ¼ p‘�1t ðxÞ þ

1c1

Krm

� ��1 �Un Hrn

� ��1UTn

h ir‘�1

m ðxÞ; ð2:10Þ

where r‘�1m ðxÞ ¼ bm � GmðxÞðpnðxÞ þ p‘�1

t ðxÞÞ, and is the ð‘� 1Þ-thresidual vector for ‘ ¼ 1;2; . . ., with an initial guess of p0

t ðxÞ.p0

t ðxkÞ is determined by a linear extrapolation of the computed vec-tor at a previous frequency if k > 2; otherwise, p0

t ðx0Þ ¼ 0 andp0

t ðx1Þ ¼ ptðx0Þ. A practical stopping criterion is set to test the rel-ative residual error:

krmðxÞk2=kbmk2 6 �; ð2:11Þ

for a given tolerance �. The iteration (2.10) guarantees its conver-gence based on the derivation in [5] by satisfying the condition thatthe contraction ratio n is smaller than one when the global cutoffvalues are determined by

krmin ¼ �dmax=n and kr

max ¼ dmax=n; ð2:12Þ

where dðx;rÞ ¼ j � c2=c1j, and dmax ¼maxfdðxk;rÞ; 1 6 k 6 nf g, inwhich nf is the number of sampling frequencies required to obtainthe frequency response curve.

In order to compare the performance of the frequency sweepwith that of the modal acceleration method, the computationalcomplexities of each are analyzed in detail. First, the computa-tional complexity of the FS is shown in Table 1, in which niter de-notes the total number of frequency sweep iterations. Here,Kr

m;Mm;Un;Hrn , and bm are real, but the other vectors or matrices

are complex. The value of niter depends on the tolerance of thestopping criterion and the values of the variable n, which standsfor its convergence rate. Complex–real multiplications are countedtwice and complex–complex multiplications are counted fourtimes when compared to real–real multiplications. A real–complexdivision is considered as six times as expensive as a real–real

Table 1Cost of operations for the frequency sweep algorithm.

Step Task Cost

(i) UTnbm m � n

for k ¼ 1; nf

(ii-1) gnðxkÞ ¼ ðc1Hrn þ c2IÞ�1 � ðiÞ 10n

(ii-2) pn ¼ UngnðxkÞ 2m � n(ii-3) ifðk > 2Þ; p0

t ðxkÞ by linear extrapolation 2m(iii-1) r0

mðxkÞ ¼ bm � ðc1Krm þ c2MmÞp0

mðxkÞ; 2m � c þ 10m

where p0mðxkÞ ¼ pnðxkÞ þ p0

t ðxkÞ(iii-2) ifðkr0

mðxkÞk2=kbmk2 6 �Þ break 5m

for ‘ ¼ 1;2; . . .

(iv-1) Un Hrn

� ��1UTnr‘�1

m ðxkÞ 4m � nþ 2n

(iv-2) ðKrmÞ�1r‘�1

m ðxkÞ 2m

(iv-3) p‘t ðxkÞ ¼ p‘�1t ðxkÞ þ 1

c1½ðiv-2Þ—ðiv-1Þ� 4m

(v-1) r‘mðxkÞ ¼ bm � ðc1Krm þ c2MmÞp‘mðxkÞ; 2m � c þ 10m

where p‘mðxkÞ ¼ pnðxkÞ þ p‘t ðxkÞ(v-2) if kr‘mðxkÞk2=kbmk2 6 �

� �break 5m

end(vi) HmðxkÞ ¼ lTmpmðxkÞ 2m

end

Main cost nf �m � ð2nþ 2cÞ þ niter �m � ð4nþ 2cÞ

20 J. Hwan Ko, D. Byun / Computers and Structures 88 (2010) 18–24

multiplication. Scalar–scalar operations, such as 1=c1, are not con-sidered in the table.

Note that the (iv-2) value takes only 2m operations because Krm

is a diagonal matrix. This is another big advantage of using the ASsubspace. The multiplication of Mm to a complex vector in (iii-1)and (v-1) can be estimated by 2m � c, where c is an integer deter-mined by the sparsity of Mm. The main cost in Table 1 is split intotwo parts: an nf -dependent part from (ii), (iii), and (vi), and an ni-

ter-dependent part from (iv) and (v). Generally, c;n, and nf aremuch smaller than m. Moreover, the convergence rate of the FSis known to be very fast [5], implying that niter is expected tobe much smaller.

2.3. Modal acceleration method

The modal acceleration (MA) method, another iterative schemewhich is used in addition to the modal superposition method, wasalso developed to compensate for the errors caused by the trun-cated modes. Here, the method introduced in [6] is adopted. Whenit is applied to solve HmðxÞ, the following equation is used:

pmðxÞ � p1ðxÞ þ p2ðxÞ

¼ c�11 Kr

m

� ��1XL‘¼0

�c2

c1MmðKr

mÞ�1

� �‘

bmðxÞ

þXn

j¼1

�c2

c1hrj

!Lþ1

gnðxÞð Þj/j; ð2:13Þ

where p1ðxÞ is the first term, p2ðxÞ is the second term, andðgnðxÞÞj is the j-th component of gnðxÞ. If L ¼ �1;pm is equal topn in Eq. (2.7) of the modal superposition method. As L increases,accuracy is expected to improve. The value of L is typically smallerthan 5; it is given an identical value at every frequency [6]. Notethat p1 is computed from the vectors in Krylov subspace

bm;Arbm; . . . ;ALr bm

n o, where Ar ¼ �c2

c1Mm Kr

m

� ��1. Because the vectorstend very quickly to become almost linearly dependent during thecomputation, methods relying on Krylov subspace frequently in-volve some orthogonalization scheme, but the conventional MAdoes not involve it; thus, this may cause bad behavior in conver-gence of the MA, which will be shown in Section 3.3.1.

In this paper, Eq. (2.13) is reformulated into an alternativerecursive form allowing us to use the same stopping criterion(2.11) and to compare it with the FS. The equation where L ¼ 0

is considered in the initial step, which is consistent with the linearextrapolation of the FS. The first and second terms in this casebecome:

p01ðxÞ ¼ c�1

1 ðKrmÞ�1b0

mðxÞ; p02ðxÞ ¼ Ung0

nðxÞ; ð2:14Þ

where b0mðxÞ ¼ bmðxÞ and g0

nðxÞ ¼�c2c1

Hrn

� ��1gnðxÞ. Next, the fol-lowing refinement iterations for computing the two terms are refor-mulated from (2.13)

p‘1ðxÞ ¼ p‘�11 ðxÞ þ c�1

1 Krm

� ��1b‘mðxÞ; p‘2ðxÞ ¼ Ung‘nðxÞ; ð2:15Þ

where b‘mðxÞ ¼�c2c1

Mm Krm

� ��1b‘�1m ðxÞ and g‘nðxÞ ¼

�c2c1

Hrn

� ��1g‘�1n ðxÞ

for ‘ ¼ 1;2; . . .,. At each frequency xk, iterations are run untilp‘mðxkÞ ¼ p‘1ðxkÞ þ p‘2ðxkÞ satisfies (2.11), in which rmðxkÞ ¼bmðxkÞ � GmðxkÞp‘mðxkÞ. The convergence of the modal accelerationmethod is guaranteed so long as the condition of (2.12) is satisfiedaccording to the convergence condition in Ref. [6].

The computational complexity of MA is described in Table 2, inwhich only Mm;K

rm; k

rr ;/r , and bm are real, but the other vectors or

matrices are complex. Scalar–scalar operations such as 1=c1 andc2=c1 are not considered.

The main cost is also split into two parts, the nf -dependent partfrom (ii), (iii), and (vi), and the niter-dependent part from (iv) and(v). nf is given by the user, and c and m are determined after con-structing the projected system. Consequently, the number of globalmodes n and the convergence rate are the most important factorsin the performance comparison between the MA and the FS. Theconvergence rate of the MA is also known to be very fast [6].According to the main costs in Tables 1 and 2, the first terms areequal and the second terms are expected to be close to each other,provides the convergence rates of both methods are similar, as aren and c. Detailed comparisons are represented in the next section.

3. Numerical experiment

3.1. Numerical methods

The previously presented methods on the AS subspace areimplemented based on ASEIG [2]. The multilevel partitioning isdone by METIS [7]. The eigenpairs of the sparse matrices are com-puted by the shift-invert Lanczos method of ARPACK [8] withSuperLU [9], and those of the dense matrices are solved by LAPACK.The methods for the experiments are listed below:

Table 2Cost of operations for the modal acceleration method.

Step Task Cost

(i) UTnbm m � n

for k ¼ 1;nf

(ii-1) gnðxkÞ ¼ ðc1Hrn þ c2IÞ�1 � ðiÞ 10n

(ii-2) g0nðxkÞ ¼ �c2

c1Hr

n

� ��1gnðxkÞ 6n

(ii-3) p01ðxkÞ ¼ c�1

1 ðKrmÞ�1b0

mðxkÞ; 6m

where b0mðxkÞ ¼ bmðxkÞ

(ii-4) p02ðxkÞ ¼ Ung0

nðxkÞ 2m � n

(iii-1) r0mðxkÞ ¼ bm � c1Kr

m þ c2Mm� �

p0mðxkÞ, 2m � c þ 10m

where p0mðxkÞ ¼ p0

1ðxkÞ þ p02ðxkÞ

(iii-2) if kr0mðxkÞk2=kbmk2 6 �

� �break 5m

for ‘ ¼ 1;2; . . .

(iv-1) g‘nðxkÞ ¼ �c2c1

Hrn

� ��1g‘�1n ðxkÞ 6n

(iv-2) b‘mðxkÞ ¼ �c2c1

Mm Krm

� ��1b‘�1m ðxkÞ 2m � c þ 6m

(iv-3) p‘1ðxkÞ ¼ p‘�11 ðxkÞ þ c�1

1 Krm

� ��1b‘mðxkÞ 6m

(iv-4) p‘2ðxkÞ ¼ Ung‘nðxkÞ 2m � n

(v-1) r‘mðxkÞ ¼ bm � c1Krm þ c2Mm

� �p‘mðxkÞ, 2m � c þ 10m

where p‘mðxkÞ ¼ p‘1ðxkÞ þ p‘2ðxkÞ(v-2) if kr‘mðxkÞk2=kbmk2 6 �

� �break 5m

end(vi) HmðxkÞ ¼ lTmpmðxkÞ 2m

end

Main cost nf �m � ð2nþ 2cÞ þ niter �m � ð2nþ 4cÞ


� Direct: This computes the frequency responses from HðxkÞ usingSuperLU as a direct sparse solver.

� BL + RFM: This computes the frequency responses from HðxkÞusing mode superposition with residual flexible modes [4] andglobal modes from a Block Lanczos method of BLZPACK [10],which is the package of choice for computing a relatively largenumber of eigenvectors. BLZPACK uses the subroutines ofMA47 [11], which is a direct sparse solver in multi-frontalscheme. The maximum cutoff value is given bykmax ¼ ð1:2xmaxÞ2 and the minimum limit is zero.

� AS + FS: This computes the frequency responses from HmðxkÞusing the frequency sweep algorithm of Table 1.

� AS + MA: This computes the frequency responses from HmðxkÞusing the modal acceleration method of Table 2.

3.2. Numerical examples

3.2.1. Turbo-prop aircraftA turbo-prop aircraft, such as the Bombardier Dash 8, has a

main excited frequency of 60Hz; the noise due to the excitationhas been analyzed and controlled by experimental approaches[12]. Computation of the interior noise requires a mid-frequencyresponse analysis for an unconstrained structure.

Fig. 1. A simplified turb

The analysis employs the FE model with shell elements of thesimplified aircraft shown in Fig. 1, of which the order is 75,918.Matrices, K and M, from the finite element method are symmetricand positive-definite. Assume that region A in Fig. 1 is under an ex-cited load and region B is for sensing the dynamic responses. Thediameter of the fuselage is 2.5 m, which is close to the geometryof the Bombardier, and it has a thickness of 3 mm. Material prop-erties of aluminum are used. The eigenvalue of the mode near60 Hz in Fig. 1b is 580-th, which is located in the mid-frequencyrange. Frequency responses are computed at 201 frequencieswhich are discretized in [60 � 2, 60 + 2] Hz: nf ¼ 201. For propor-tional damping, Craig defines the damping factor to bef ¼ 0:5ðaxþ b=xÞ [13]. Damping coefficients, which makef ¼ 0:2% at f = 60 Hz, are used.

3.2.2. Micro-scale resonatorThe FE simulation of a micro-scale ring resonator is used for

designing a high-frequency band-pass filter, e.g. surface acousticwave devices in a cell phone. Recently-introduced ring resonatorsemploy the so-called extensional wine-glass mode [14], which isdepicted in Fig. 2b. The corresponding eigenvalue of the mode isthe 306-th of the finite element model in Fig. 2a, which is locatedin the mid-frequency range. The corresponding natural frequency

o propeller aircraft.


is 630 MHz and the order of the FE model with solid elements is92,220. The frequency responses are computed at 201 frequencieswhich are discretized in [630 � 3, 630 + 3] MHz: nf ¼ 201. Assumethat region A in Fig. 2 is under an excited load and region B is forsensing the dynamic responses. The ends of four beams are set tothe clamped boundary condition. The geometry and materials arefrom the 634.6 MHz resonator of Ref. [14]. The damping coeffi-cients are set for a 0.01% damping factor at 630 MHz due to thehigh sensitivity of the resonator.

All experiments were conducted on a platform, that has 4 GB ofphysical memory, utilized 2.66 GHz Intel Xeon processor, and usesthe Red Hat Linux operating system.

3.3. Numerical results and discussion

3.3.1. Convergence and computational cost

(1) Convergence vs. ToleranceFirst, the convergence rate is explored by varying the toler-ance � of the stopping criterion with the contraction ration fixed at 0.5. The results of these variations are depictedin Fig. 3 with 10�1;10�2;10�3;10�4; and 10�5 tolerances.As seen in Fig. 3, the convergence rates of AS + FS are muchfaster than those of AS + MA. Moreover, AS + FS can obtainmore accurate results because AS + MA diverges at 10�3 forthe turbo-prop aircraft and at 10�4 for the ring resonator,but AS + FS converges at all given tolerances.

Fig. 2. A micro-scale

Fig. 3. Influence on the number of itera

(2) Convergence vs. Contraction ratioNext, the effect of n-variation is explored in a situation wherethe tolerance is fixed at � ¼ 10�3. To explore the effect of n,the dimension of the AS subspace, m, is kept constant at2,362 for the turbo-prop aircraft and 1,345 for the ring reso-nator. The effects on the number of the iterations, niter, andthe number of global eigenmodes, n, are depicted in Fig. 4.As shown in Fig. 4, while n increases in both methods, nbecomes smaller and niter becomes larger. Fig. 4 also indi-cates that the rate of increase of niter with the FS is smallerthan with the MA, as n increases. Moreover, the divergencethat occurs at n less than 1 for the MA, but not for the FS. It isspeculated that this divergence is attributed mainly to the lin-ear dependency of the vectors in p1 computation of Eq. (2.13).

(3) Computational cost vs. Contraction ratioIt is clear that the computational cost should increase as thetolerance decreases. However, the effect of n-variation oncomputational cost is complex. The time periods for com-puting the global modes and the iterations are depicted inFig. 5.

The sparsity of Mm; c, is 352 for the turbo-prop aircraft and 146for the ring resonator. Thus, n in Fig. 4 is smaller than c except 0.1with regard to the turbo-prop aircraft and then the computationalcost per iteration for the FS is a little smaller than that for the MA,according to the niter-dependent parts in Tables 1 and 2, in casethat n is smaller than c. Moreover, the FS requires a much smaller

ring resonator.

tions when varying the tolerance.


number of iterations than the MA. Subsequently, the computingtime periods of the MA iteration exceed those of the FS as seenin Fig. 5. Fig. 5 also indicates that as n increases, the rate at whichtime for the global modes decreases is faster than the rate at whichthe time for the iterations increases, especially in the case of theturbo-prop aircraft. Subsequently, larger n, which makes n smaller,is advantageous. Thus, the following values are adopted in the nextcomparison: 0.9 for the FS and 0.2 for the MA of the turbo-prop air-craft, and 0.9 for FS and 0.6 for MA of the ring resonator.

Fig. 4. Influence on n and niter wh

Fig. 5. Influence on computing time w

Fig. 6. Frequency re

3.3.2. Performance comparisonIn cases such as the previous section where the parameters of

the AS-based methods are given, the frequency response curvesof the two methods are depicted in Fig. 6, in which those ofBL + RFM and the direct method are included. As shown, the fre-quency responses of AS + FS and AS + MA are in good agreementwith those of the direct method. The frequency responses ofBL + RFM for the turbo-prop aircraft have some discrepancies, butare very close to those of the direct method near the regions of

en varying the contraction ratio.

hen varying the contraction ratio.

sponse curves.

Table 3The numbers of iterations and global modes, the elapsed time in seconds and speedup over BL + RFM.

Methods niter n TAS TGM TSOL TTOT Speedup

Turbo-prop AS + FS 226 54 77.32 8.94 3.94 90.20 8.4

Aircraft AS + MA 338 232 77.32 49.14 6.76 133.22 5.7BL + RFM – 759 – 752.69 4.99 757.68 –

Ring resonator AS + FS 165 7 85.36 1.04 0.84 87.24 8.5AS + MA 643 10 85.36 1.40 2.44 89.20 8.3BL + RFM – 397 – 704.03 1.46 741.49 –


resonance. For the ring resonator, those of BL + RFM also are ingood agreement.

Next, the number of iterations and global modes, the computingtime, and speedup over BL + RFM for performance comparison arelisted in Table 3 where TAS is the construction of the AS subspace,TGM is computation of the global modes, TSOL is the time needed tosolve the frequency responses, and TTOT is the total time. Table 3shows that the time for the FS or the MA iteration is short whencompared to the total time. AS-based methods retain a smallernumber of global modes than BL + RFM; the order is reduced from75,918 to 2,362 for the turbo-prop aircraft and from 92,220 to1,345 for the ring resonator. Subsequently, the speedup overBL + RFM is higher than 5.5 for the turbo-prop aircraft and 8 forthe ring resonator. The timings in the comparison with BL + RFMcan be slightly changed if difference direct linear system solversare used. When comparing two AS-methods, AS + FS needs a smal-ler number of global modes and iterations than AS + MA, due to itsbetter convergence behavior, detailed in the previous section. Con-sequently, AS + FS shows a better performance among the consid-ered numerical methods in this paper. Meanwhile, the MA hasadvantages in terms of extending applications, as it has been ap-plied to sensitivity analysis etc. [6], whereas the FS has not. Hence,it is critical to improve the convergence of the MA and it is alsoessential to extend the FS to these applications.

4. Conclusion

Solving mid-frequency responses has remained a frontier re-search area in structural dynamics. A conventional numerical solu-tion has been obtained by a modal approach to a finite elementlinear system, which often faces computational difficulties due toa large increase in the order of the system and a high number ofretained modes. The computational burden of such a large ordercan be relieved by using the model order reduction, and a variantof this, based on algebraic substructuring, was employed in this pa-per. Meanwhile, the number of modes can be reduced by numeri-cal methods that use the modes of the interior eigenvalues, such asthe frequency sweep algorithm and the modal acceleration meth-od. Comparisons between these two methods were made in thispaper through numerical experiments that provided a detailedanalysis of their computational complexity. It might be expectedthat the two methods would show similar performance if theirconvergence rates were similar. However, for the finite elementsimulations, in which proportional damping is assumed, of a tur-bo-prop aircraft and a ring resonator, the frequency sweep algo-rithm showed a faster convergence rate and was more robust

than the modal acceleration method. Consequently, the frequencysweep algorithm for a reduced-order systems showed the best per-formance among the considered numerical solutions includingconventional modal approach. Future work may involve the useof the frequency sweep algorithm, along with the algebraic sub-structuring, for sensitivity, transient, vibro-acoustic analysis andalso the structural problems considering nonproportionaldamping.

Acknowledgment

This research was supported by Basic Science Research Programthrough the National Research Foundation of Korea funded by theMinistry of Education, Science and Technology (Grant No. 2009-0083068 and 2009-0074875).

References

[1] Bennighof JK, Kaplan MF. Frequency sweep analysis using multi-levelsubstructuring, global modes and iteration. In: Proceedings of the 39th AIAA/ASME/ASCE/AHS structures, structural dynamics and materials conference;1998.

[2] Gao W, Li S, Yang C, Bai Z. An implementation and evaluation of the AMLSmethod for sparse eigenvalue problems. ACM Trans Math Software2008;34(4).

[3] Ko JH, Jung SN, Byun DY, Bai Z. An algebraic substructuring using multipleshifts for eigenvalue computations. J Mech Sci Technol 2008;22:440–9.

[4] Thomas B, d Gu RJ. Structural-acoustic mode synthesis for vehicle interiorusing finite-boundary elements with residual flexibility. Int J Vehicle Des2000;23:191–202.

[5] Ko JH, Bai Z. High-frequency response analysis via algebraic substructuring. IntJ Numer Meth Eng 2008;76(3):295–313.

[6] Qu ZQ. Accurate methods for frequency responses and their sensitivities ofproportionally damped system. Comput Struct 2001;79:87–96.

[7] Karypis G, Kumar V. METIS: unstructured graph partitioning and sparse matrixordering system. Technical Report 1995. Department of Computer Science,University of Minnesota, <http://www-users.cs.umn.edu/karypis/metis/metis/index.html>; 1995.

[8] Lehoucq R, Sorensen DC, Yang C. ARPACK user’s guide: solution of large-scaleeigenvalue problems with implicitly restarted Arnoldi methods. Philadelphia:SIAM; 1998.

[9] Demmel JW, Eisenstat SC, Gilbert JR, Li XS, Liu JWH. A supernodal approach tosparse partial pivoting. SIAM J Matrix Anal Appl 1999;20(3):720–55.

[10] Marques OA, BLZPACK users guide, <http://crd.lbl.gov/~osni/>; 2001.[11] Duff IS, Reid JK. MA47, a Fortran code for direct solution of indenite symmetric

systems of linear equations. Report RAL 95-001. Oxfordshire, England:Rutherford Appleton Laboratory; 1995.

[12] Yonsefi-Koma A, Zimcik DG. Applications of smart structures to aircraft forperformance enhancement. Can Aeronaut Space J 2003;49(4):163–72.

[13] Craig RR. Structural dynamics: an introduction to computer methods. JohnWiley and Sons; 1981.

[14] Xie Y, Li SS, Lin YW, Ren Z, Nguyen CTC. UHF micromechniacal extensionalwine-glass mode ring resonators. Technical digest. In: IEEE internationalelectron devices meeting, Washington DC; 2003.

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http://crd.lbl.gov/~osni/





Aluminium foams structural modelling

M. De Giorgi *, A. Carofalo, V. Dattoma, R. Nobile, F. PalanoDipartimento di Ingegneria dell’Innovazione, Università del Salento, Via per Arnesano, 73100 Lecce, Italy


Article history:Received 24 February 2009Accepted 3 June 2009Available online 30 June 2009

Keywords:Aluminium foamsMechanical characterisationFEMMicrostructural model


* Corresponding author.E-mail addresses: [email protected] (M. De Gi

(A. Carofalo), [email protected] (V. Dattoma), [email protected] (F. Palano).

a b s t r a c t

The aim of this work is the development of microstructural numerical models of metallic foams. In par-ticular, attention is focused on closed cell foam made of aluminium alloy. By means of a finite elementscode, the material cellular structure was shaped in different ways: firstly the Kelvin cell, with both planeand curved walls; finally an ellipsoidal cell defined by random dimensions, position and orientation hasbeen adopted as base unit. In order to validate the foam numerical models, static tests were performed toobtain the typical stress–strain curves and then compared with the numerical analysis results.


1. Introduction

Metallic foams are a new class of materials that turn out to bepromising in different engineering fields [1–7]. They show a cellu-lar structure consisting in a solid and a gaseous phase and theirclassification is based on the cells morphology. A certain numberof manufacturing processes are currently used and other innova-tive methods are being developed in order to obtain a more reliableand uniform production. Technological process determines also ifthe morphology of the component is made of open or closed cells.

Metallic foams show an interesting combination of physical andmechanical properties that make them particularly versatile: thelow apparent density, for example, allows obtaining a high stiff-ness/specific weight ratio, the presence of cavities and the essentialin-homogeneity provide them acoustic and thermal insulationproperties, besides the possibility to absorb impact loads and todamp vibrations. Finally, the metallic structure potentially givesthose good electromagnetic shielding properties and inalterabilityin time.

The combination of these physical properties makes metallicfoams able to be competitive in terms of performances and costsin several unusual advanced applications. For example, the realisa-tion of structures able to shield efficaciously the electromagneticfields is very interesting. Within this context, one of the most fre-quently used solutions consists in holding the volume to beshielded in a metallic surface. In some situations, it is required thatthe capability to support relevant load levels without excessively

ll rights reserved.

orgi), [email protected]@unile.it (R. Nobile),

resting on the whole structure weight. Furthermore, electromag-netic shielding device transparent to light is sometimes required.In these applications, metallic foams seem to have interesting util-isation perspective, because their mechanical properties guaranteethe structural auto-supporting.

The absence of well-established procedure for the structural ver-ification of innovative materials represents a relevant obstacle totheir effective use, even if they could assure the best performances.The use of components made of metallic foams or their use as a fil-ler for hollow structures, in fact, requires a deeper knowledge ofstructural behaviour. Designer needs not only experimental data,but also reliable and relatively simple analytical or numerical meth-ods for calculation of applied stress and for prediction of failure ofmetallic foam components. In perspective, these design tools mustguarantee a reliability level that would be comparable to the onesused for traditional materials. Intrinsic inhomogeneities of metallicfoam represent a serious problem for calculation needed in designphases. In particular, it would be desirable to establish a link be-tween microstructural and macroscopic behaviour of this kind ofmaterials. It is beyond doubt that microscopic geometry and prop-erties affect macroscopic structural behaviour, which is essentialfor the design of components, but it is very difficult to derive mac-roscopic properties from microstructure.

The problem of deriving macroscopic from microscopic proper-ties is very complex and probably its resolution will lead to partialsolutions that will be applicable in well-defined and restricted con-text. Nevertheless, this step will represent an important advancingin this research field.

This paper is inserted in this general perspective, since the mainaim is to try to evaluate macroscopic mechanical behaviour start-ing from a micro-structural approach for particular metallic foam.The purpose is using different numerical methodologies to model a








26 M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35

specific class of metallic foams in order to evaluate advantages,limitations and simplicity of use. The metallic foam class thatwas selected for this study is the closed cell aluminium foam, sinceits high relevance in industrial context.

In the literature, there are some studies regarding the way toforesee the foams macroscopic behaviour by cell geometric data.The references [8–14] represent some significant examples ofstructures formed by closed cells.

Hanssen et al. [8] present results from an experimental pro-gramme for the general validation of constitutive models for alu-minium foam. The obtained database is applied in order toinvestigate how different existing constitutive models for alumin-ium foam cope with quasi-static and low-velocity dynamic load-ing. Compared with the experiments in the validationprogramme, none of the models managed to represent all load con-figurations with convincing accuracy. One reason for this is thatfracture of the foam is a very likely local failure mechanism, whichwas not taken into account in any of the models considered.

Belingardi et al. in [9] show the results of some finite elementssimulations of a set of experimental compression tests, both staticand dynamic, in order to verify the quality of the material models.For the load phase, they found that the numerical results agreedwith the experimental ones, both in the static and dynamic casesfor which the strain rate effects are taken into account. Moreover,the authors develop a new material model for the unload phase.

Roberts and Garboczi [10] have computed the density andmicrostructure dependence of the Young’s modulus and Poisson’sratio for several different isotropic random models based on Voro-noi tessellations and level-cut Gaussian random fields. The results,which are best described by a power law E / qn (1 < n < 2), showthe influence of randomness and isotropy on the properties ofclosed cell cellular materials.

In [11], Lu et al. analyse the deformation behaviour of two dif-ferent types of aluminium alloy foam (with closed and semi-opencells) under tension, compression, shear and hydrostatic pressure.The influence of relative foam density, cell structure and cell orien-tation on the stiffness and strength of foams is studied; the mea-sured dependence of stiffness and strength upon relative foamdensity is compared with analytical predictions. Loading pathsare compared with predictions from a phenomenological constitu-tive model. It is found that the deformations of both types of foamsare dominated by cell wall bending, attributed to various processinduced imperfections in the cellular structure. The closed cellfoam is found to be isotropic, whereas the semi-open cell foamshows strong anisotropy.

Another possibility is to use X-ray tomography to produce adata file containing all the geometrical information on a relativelylarge foam specimen. In this case the model is realised throughbrick elements associated to each voxel identified by X-ray scan[12,13]. Numerical results show a large influence of scan spatialresolution, which influence directly computational time. Finally,an interesting approach is presented in [14], where a spectral anal-ysis carried out on X-ray tomography of open cell foams allows toderive statistical parameters used to build solid finite elementmodels.

Although the variety of approaches that haves been briefly pre-sented, all the model could be classified into two different catego-ries. The first one tries to identify a simple geometrical feature thatcan constitute the repetitive cell used to built a foam model [8–11].The unit cell, which is not necessarily close to the real foam geom-etry, is chosen on the basis of energy or morphology considerationand leads to models generally based on shell elements. On the con-trary, the efforts of the other works are concentrated to obtain anexact reproduction of foam geometry by means of experimentaltechniques [12–14]. In this case, solid models are generallyobtained.

In this work the first approach has been followed. The use ofelements like beam or shell, in fact, presupposes an idea of foamstructural behaviour at an intermediate scale, being in authorsopinion the better compromise nowadays possible to study foamstructural properties both at microscopic and macroscopic scale.

The problems to solve in the micro-structural model definitionof closed cell metallic foams can be resumed as follows:

– choice of a cell morphology to be employed as a repetitive unitfor the model construction;

– definition of geometric dimensions, which characterise the cellform;

– selection of cells assembly criteria;– metallic foam block numerical model realisation through the

repetition of several cells in the three directions of the space.

In this paper, two different cell morphologies were consideredand numerical results were compared to experimental data. In par-ticular, the model adequacy was evaluated by comparison of thepredicted and actual compressive elastic modulus.

2. Survey of proposed models

The first model considered was referred to the ideal Kelvin cells(tetrakaidecahedron) with planar walls. This cell morphology hasbeen already used by other authors, since it corresponds to theminimum surface energy for a fixed volume [1,15].

Models based on Kelvin cells were various: the most simpleconsidered regular Kelvin cell having planar faces, while morecomplex models were obtained introducing the cells walls curva-ture and thickness variation. These modifications could improvethe mechanical behaviour approximation. Further refinements ofthis approach could give a better approximation, but limitationdue to the regularity of the resulting cellular structure was notremovable and constituted an important starting error. Accordingto other authors [10,16–18] the structure regularity could be con-sidered as the origin of the higher model stiffness with respect tothe real foam behaviour.

For this reason a different approach was used to develop a sec-ond model that considered the real cellular structure irregularity.In order to reproduce the pores feature and to consider its randomvariability, an elementary ellipsoidal feature was chosen as unitarycell. The assembly of a fixed number of ellipsoidal cells, havingdimensions, position and orientation varying parametrically in arandom way, could constitute a better approximation of the foamreal structure.

3. Foam mechanical characterisation

Experimental data used for comparison between numericalmodels were obtained through compressive tests. These data rep-resented the necessary experimental term of comparison used un-til now for the numerical model improvement.

The uniaxial compression tests represented the easiest andmost efficacious way to determine the metallic foams mechanicalbehaviour under static conditions. The resulting data, particularlythe applied load versus the overall strain trend, defined the macro-scopic behaviour of the foam cellular structure. Then, r�e com-pressive curve was the term of comparison used to evaluate thestudied numerical model efficiency and reliability.

The analysed foam was made of AlSi10Mg aluminium alloy andproduced by Alulight Company. Table 1 resumes the general prop-erties of this aluminium foam. The material was available in formsof 10-mm-thick sheets that were cut to obtain the specimens. Itwas observed that sheets showed a central region with a foam

Table 1Mechanical properties of AlSi10Mg base alloy.

Aluminiumalloy

Solidus densityqS (kg/m3)

Elastic modulusES (GPa)

Yield strengthry (MPa)

Poissonmodulus

AlSi10Mg 2860 69 250 0.33

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35 27

density quite uniform, while external zones were characterised byhigher values of density. It was probably due to the different initialdistribution of the gas particles in the matrix during manufacturingprocess. Therefore, specimens having different density were easilyobtained from the same sheet. Mechanical experiments wereplanned in the following way:

� initial tests were carried out to establish the influence of thespecimen dimension on mechanical properties: 40 � 40 mm(D40), 70 � 70 mm (D70), 100 � 100 mm (D100) specimenshaving a similar density were considered;

� extensive experimental tests were then carried out on D40 spec-imens having different density, with the aim to highlight theinfluence of apparent density on the stress–strain curve; thesecompressive tests were executed into two steps, as describedin the following, in order to evaluate constitutive behaviour ofaluminium foam;

Tests were carried out on a servo-hydraulic MTS 810 machinehaving 100 kN load capacity equipped with two planar plates,according to the indication reported in [1]. Since the crossheadspeed was fixed at 0.005 mm/s and foam thickness was 10 mm,the initial compressive deformation rate of the test was0.5 � 10�3 mm/mm s. Strain was calculated dividing the cross-head displacement by the specimen thickness.

As a first result, the specimen dimensions seemed to be not rel-evant with regard to mechanical foam properties, as it was evidentby comparing the plots reported in Fig. 1a and obtained on speci-mens D40, D70 and D100 having different dimensions and similardensities. On the basis of this observation, further tests were car-ried out on specimen D40.

Stress–strain curves deduced from the experimental tests weresimilar to those reported in the literature [1–5,17,19] and showedthe existence of three different and characteristic zones:

– a first part, corresponding to little deformations, in which thematerial presented an elastic behaviour and the cells walls chan-ged their curvature;

– a second very wide zone, characterised by a plateau with arather constant stress, phase in which the cells underwent phe-nomena of local instability, yielding and rupture;

Fig. 1. Uniaxial compression tests: (a) effect of sp

– finally a third region in which the material showed a stiffnessincrement due to the remarkable material densification and tothe strain hardening phenomena.

It is also important to notice that aluminium foam showed a dif-ferent behaviour between the first load application and the subse-quent ones: in fact, the material seemed to undergo an initialarrangement, which gave rise to a different elastic modulus whencomparing the first and the following loading cycles. This fact isgenerally explained invoking an initial rearrangement and local-ised plastic strain of cell walls that is realised in the first compres-sion phases and responsible of the subsequent drastic Young’smodulus increment. The Young modulus measured in the load cy-cles subsequent to the first one is assumed to be the most impor-tant constitutive parameter of the metallic foam.

As a consequence, compressive test were executed into twosteps:

– a first loading step up to about 75% of plateau stress: this stepleaded to the determination of the first cycle Young modulus(Eload), followed by an unloading phase that allowed to deter-mine a characteristic Young modulus (Eunload);

– a second loading step up to foam densification.

Fig. 1b represents the trends obtained by testing three sampleshaving different densities and allows drawing attention to the den-sity influence on the foam mechanical behaviour. The curvesshowed that, when the specimen density decreased, also the elasticmodulus, the compressive strength and the curve slope reduced,while the strain corresponding to the densification beginningraised.

Fig. 2 reports the linear interpolation of experimental data ofthe loading and unloading phases that was used for the evaluationof the compressive Young’s modulus Eload and Eunload, respectively.Data reported in Table 2 allowed to evaluate the difference existingin the load and unload phases. The stress–strain curve reported inFig. 2 could be taken as reference in order to describe the generalbehaviour of the examined foam and it was achieved imposingfirstly two subsequent loading cycles and then loading the sampleuntil its compacting.

Testing some specimens with only one loading cycle, we notedthat the compacting phase trend was not influenced by the pres-ence of a possible unloading step. The reference loading curve em-ployed for the numerical results verification was then obtainedwith a first loading phase until 75% of plateau stress and a follow-ing unloading step. In fact, such a curve was repeatable and inde-pendent from initial arrangement phenomenon, which ischaracteristic of cellular materials.

ecimen size; (b) effect of specimen density.

Table 2Experimental elastic modulus in loading and unloading phases.

Specimennumber

Specimentype

Maximum stresssmax (MPa)

Density(kg/m3)

Eload

(MPa)Eunload

(MPa)

1 D70 3.9 423.2 109 Loading only2 D100 4.7 473.7 783 D100 4.9 445.3 1114 D40 4.1 433.1 1035 D40 12.5 771.2 2436 D40 5.0 432.5 146 5777 D40 7.8 546.8 229 4368 D40 4.9 432.9 155 4339 D40 4.4 435.6 152 48110 D40 9.6 568.2 256 45311 D70 4.8 412.3 97 21512 D40 4.2 555.6 222 49313 D70 5.1 418.1 181 522

Fig. 3. Tetrakaidecahedral geometry used in the Kelvin cell model.

Fig. 2. Loading and unloading behaviour for the determination of elastic modulus.


4. Numerical analysis

4.1. Kelvin cell model

A simple microstructural model reproducing a cellular materialcan be based on the regular space repetition of a unit cell withproper geometry and dimensions. The tetrakaidecahedron or Kel-vin cell is the most utilised in literature as unit cell to model thefoam material [1–2,15], since it is the lowest surface energy unitcell known consisting of a single polyhedron. It is defined by sixplanar square faces and eight hexagonal faces that could be planar,with a mean curvature equal to zero (Fig. 3), or convex, with a po-sitive mean curvature.

In order to simulate the actual foam structure, FEM 3D modelbased on the Kelvin unit cell was refined in the course of the workintroducing the following improvements:

1. planar faces model2. non-zero faces curvature3. thickness faces variability

FEM analysis was performed using the Ansys software adoptinga discretisation in shell element named Shell93 having eight nodesand a quadratic formulation. The study was performed in the largedisplacements range. The material model was isotropic with elasticperfectly plastic stress–strain behaviour. The material properties,reported in Table 1, were the actual mechanical properties mea-sured directly on Alulight foam by micro-hardness characterisation[17].

The unit cell apparent density was assumed equal to the actualdensity of the foam, of about 500 kg/m3. On the contrary, the actualcell size was assumed based on a statistical analysis reported in

[18]. In this work, the authors affirm also that the cell size is inde-pendent from direction. The used foam properties are summarisedin Table 3.

A sample load of 1 MPa was used to perform a mesh sensitivityanalysis, considering that the model was used to determine macro-scopic properties of the foam. In particular the evaluation of theupper surface cell displacement was directly used to calculatefoam Young modulus. For this reason, the Von Mises stress in thecentral node of the upper face and the mean displacement of thecentral upper face were considered in the sensitivity analysis.Onthe basis of the results resumed in Fig. 4, authors decided to divideeach cell edge in four parts. In this manner, the unit cell consistedin 352 elements. The global model was obtained replicating theunit cell in all the space directions. In order to optimise the solu-tion efficiency, the model presented nine cells on the basis and fouralong the thickness. The uninfluence of the cells number on themacroscopic mechanical properties of the cellular structure waspreliminarily verified through a model having a number of 16 cellson the basis.

Boundary conditions were applied on the lower face, imposing azero value to z-direction displacement and rigid body boundaries;pressure load was applied to nodes of the square faces on theupper surface and it was set to 4 MPa.

Fig. 5 shows the node displacements in the load direction and alimited edge effect due to absence of adjacent cells can be ob-served. However, the displacement of the middle node of the cen-tral face is considered as representative of the global behaviour ofthe foam. Foam macroscopic strain, calculated as displacement/model thickness ratio, was recorded at each load increment to dis-play the stress–strain curve.

Young’s modulus value of this model resulted unacceptablyhigher than experimental one. This discrepancy was due to theexcessive stiffness associated with the idealised structure, whichresulted unrealistically ordered. In fact, the real structure presentsimperfections originated by the production process and that aretypical of the foam.

The case study based on Kelvin cell was refined trying to reduceits inadequacy: the face curvature of the unit cell was then intro-duced in the model. The face curvature can be calculated frommeasurement of the chord length L and the triangular area A indi-cated in Fig. 6 [17]. Considering the angle h between the normals atthe end of the cell wall, it can obtain the relationship:

Table 3Geometrical properties and results of Kelvin cell models.

Average celldimension (mm)

Averagecurvature L/2R

Edge lengthL (mm)

Density(kg/m3)

Wall thicknesstf (mm)

Edge thicknesste (mm)

Edge volumefraction U

Young’s modulus(MPa)

Alulight (experimental) 418.1–555.6 – – – 433–577Planar faces model 500 0.1360 0.1360 – 3287Curved faces model 2.6 0.37 0.9192 500 0.1178 0.1178 – 1944

432.5 0.113 0.113 – 1265Curved faces with variable

thickness model432.5 0.0973 0.2350 50 2340

0.0876 0.2465 55 21980.0778 0.2575 60 19470.0681 0.2680 65 1714

Fig. 4. Mesh sensitivity analysis of Kelvin cell model.


h ¼ 4 tan�1 4A

L2

� �ð1Þ

and the normalised cell curvature, L/2R, results [17]:

L2R¼ sin

h2

� �ð2Þ

In order to keep constant the density value, the face thickness wasrecalculated referring to the following equation:

AC � tC ¼ A � t ð3Þ

where AC and tC were, respectively, the area and the thickness of thecurved wall, A and twere, respectively, the area and the thickness ofthe planar wall. Table 3 reports the geometric parameters of theunit cell with curved walls.

The curved walls model presented a less evident edge effect re-spect to the planar walls one. Analysing the data reported in Table3, it could be observed that the new model showed stiffness 40%lower than the initial model with the same density. This behaviour

could be explained considering that the curved wall was less stiffthan the planar face; moreover, cell faces were characterised by alower thickness that involved a further decrease of the stiffness.

In the more realistic model with curved walls, the effect of thedensity on the Young’s modulus has been analysed. The compari-son of models with planar faces and curved faces having the samedensity showed that the stiffness of the two models were compa-rable if the density was higher than 560 kg/m3. On the contrary, ifthe density approached 500 kg/m3 or less the difference of the twomodels became very significant. For example, in the interest den-sity range of 432.5–500 kg/m3, because of the introduction of thewalls curvature curved face model stiffness decreased at about60% of the planar cell model.

The last refinement of the model consisted in the introductionof the wall thickness variability. In the real structure, in fact, duringthe foaming process the liquid foam drains toward the cell edgescausing a thickening of these. According to [2], the solid fractionat the edge results:

/ ¼ t2e

te þ Zf�n tf l

ð4Þ

where �n is the average number of the edges of each face, Zf the num-ber of faces converging in each edge, te and tf the edge and the facethickness, respectively, and l the edge length.

The relative foam density is:

q�

qS¼ f

C4

�n2Zf

t2e

l2þ 1

2tf

l

� �ð5Þ

where C4 is the constant related to the cell volume and f the numberof faces per cell. For the most foam, Zf = 3, n � 5, f � 14 and C4 � 10,the face and edge thickness related to the chosen solid fractioncould be obtained with good accuracy [2]:

q�

qS¼ 1:2

t2e

l2 þ 0:7tf

l

� �ð6Þ

tf

l¼ 1:4ð1� /Þq

�

qSð7Þ

te

l¼ 0:93ð/Þ

12

q�

qS

� �12

ð8Þ

In order to assign a different thickness value to edge and face, themodel was divided in different areas. Fig. 7 shows the mesh per-formed on the model with thickening edges. Cell dimensions werethe same of the previous model but the mesh resulted denser andconsisting in 792 elements per cell. The U effect was evaluated sim-ulating the compressive test using different U values and an appliedload of 4 MPa.

Results are reported in Fig. 8, where Young’s modulus was dia-grammed versus solid fraction parameter U. Elastic modulus wasgenerally higher respect to the model with the curved face only.Probably, the greater tendency to instability of the cell walls, dueto thickness face decrease, was compensate by the stiffening

Fig. 5. Map of the z-displacement for the Kelvin cell planar faces model.

Fig. 6. Parameters of a curved cell wall.

Fig. 7. Mesh of Kelvin cell model with variable thickness cell face.

Fig. 8. Dependence of elastic modulus against solid fraction U.


originated by solid concentration at the edge, at least for the U val-ues considered in this analysis. Therefore, considering the solidconcentration, it resulted a useless complication.

4.2. Ellipsoidal cell model

The Kelvin cell model, discussed in the previous section, re-sulted inadequate to describe the foam structure. In particular,Young’s modulus of the foam was overestimated and the typicalplateau region of the foam compression curve was not reproducedby Kelvin cell model. This difference was originated by the regulardisposition of Kelvin cell into the model, which was far from thereal aspect of the metallic foam. For this reason, the efforts werefocalised to obtain a model that would be able to conjugate thesimplicity of a geometrical elementary cell and the possibility tobuild a random assembly. In order to define a schematisation withhighly random cell morphology, authors’ attention focused on asimple ellipsoidal unit cell, having the same length for two of thethree axes. It was possible and very easy to build a parametricmodel of this elementary cell (Fig. 9) by the definition of the fol-lowing parameters:

Fig. 9. Geometrical parameters of ellipsoidal cell unit.

Fig. 10. Aluminium foam specimen.


� xc, yc, zc: cartesian coordinates of the ellipsoid centre;� r: length of the major semi-axis;� t: ratio of major/minor semi-axis;� h, u: orientation angles of major semi-axis respect to a global

coordinates system;

By this way, the dimension and the position of the elementarycell could be easily defined using an algorithm for the generationof set of random real number. The choose of a probability densityfunction for the random parameter generation is the object of alarge number of papers in the literature. Readers can refer to fun-damental work of Kaminski [20] for an overview about probabilis-tic approach in computational mechanics. For the purpose of thisstudy, an algorithm based on a random uniform distribution intothe parameter range variability was considered adequate. Uniformrandom distribution is characterised by a mean value and a vari-ance equal respectively to:

mean value ¼ aþ b2

ð9Þ

variance ¼ ðb� aÞ2

12ð10Þ

where a and b are the interval minimum and maximum values.It is important to observe that the random generation of dimen-

sion, position and orientation of the different cells produces inmost cases the co-penetration of the cells. In this manner, poresof various shape, which will exalts the irregularity of the model,are obtained.

Range variability parameter could be defined on the basis ofsimple considerations.

Cartesian coordinates of the ellipsoid centre were a direct con-sequence of the dimension chosen for the numerical model: in or-der to reduce number of nodes and elements of the model to anacceptable dimension, the choice was to model a metallic foam re-gion constituted by a 10 mm side cube. Therefore, variability rangeof xc, yc, zc was fixed to �1 to 11 mm in order to consider also ellip-soid having centre beyond the cubic region but a consistent surfaceinterior to model.

Length of major axis r was related, but not equal, to the half ofthe average cell dimension. Variability range of r was assumed tobe 2–3 mm. In this manner the single ellipsoidal cell has dimen-sions that were twice with the respect to the mean pore dimensionof the foam. When ellipsoidal cell were assembled to build themodel, interceptions that occurred between cells produced pores.It means that dimension would be near the half of ellipsoidal celldimension.

Ratio of major/minor semi-axis t was chosen in the range 1–2.Lower limit of t = 1 corresponded to a spherical shape. For thechoice of upper limit, some trials suggested that values of ratiohigher than 2 originated some troubles in the discretisation ofellipsoid cells in elements.

Finally, orientation angles h, u of major semi-axis varied in therange 0–180� in order to obtain all the possible orientation in thespace.

The complete definition of the model required also the choice oftwo other parameters that were not completely independent.These parameters were the number of cells used to build the modeland the thickness associated to cell walls that was considered con-stant for all the cells. The number of cells had to be sufficient toregularly fill the model and not excessive in order to obtain poreshaving dimension similar to foam morphology. On the other hand,the number of cells determined also the amount of areas that con-stitutes the model. Since the volume of material involved was pro-portional to the foam density, an increase of cell number requires adecrease of the cell wall thickness. Also in this case, several trailswere considered and the best solution for the reference cubic vol-ume 10 mm side was obtained considering a number of 35 cellsand a corresponding thickness of 0.11 mm, which was not far fromreality.

Finally, since the foam was delimited by two solid surfaces(Fig. 10), two planes corresponding to the upper and lower surfacesof the cubic volume of the foam were added.

On the basis of the previous observations, the ellipsoidal cellmodel has been built following this operative steps:

– definition of a parameter set assigning to each factor a randomvalue in their range variability: a random uniform distribution,provided by a Matlab routine, was used for this purpose; theparameter set generated are reported in Table 4, with the indi-cations of the material volume and the foam density corre-sponding to the amount of all the ellipsoidal cells;

– modelling of a single ellipsoid corresponding to each parameterset: a CAD parametric software, in particular PRO-E� WildFire2.0 modeller, used the input data to generate the ellipsoid sur-face feature in a neutral format that is easily recognised by sev-eral FEM software;

Table 4Geometrical parameters used for the generation of ellipsoidal cell model.

Thickness (mm) Parameters

0.11 xc yc zc r t h / Volume (mm3)Cell number [�1;11] [�1;11] [�1;11] [2;3] [1;2] [0;180] [0;180]

1 0.11 5.09 1.78 2.86 1.59 14.56 81.31 4.3032 0.27 6.76 5.48 2.38 1.21 139.9 109.77 5.1053 0.55 9.15 6.43 2.59 1.3 162.92 10.69 5.2574 0.56 5.78 0.98 2.6 1.47 96.08 56.85 4.1445 0.74 1.84 9.29 2.39 1.23 19.65 139.09 4.9826 1.62 9.41 9.87 2.67 1.84 148.65 125.36 2.7927 1.79 7.28 4.03 2 1.19 60.86 22.56 3.6948 1.89 2.8 4.72 2.7 1.23 52.92 23.43 6.3939 2.3 1.49 4.81 2.58 1.17 134.34 16.62 6.43910 2.36 4.04 3.2 2.75 1.23 1.86 1.41 6.63711 2.93 3.84 4.11 2.35 1.44 120.22 76.16 3.51212 3.19 9.62 7.38 2.24 1.31 108.62 118 3.84613 3.69 2.88 1.44 2.02 1.92 124.66 130.13 1.44814 3.78 9.99 2.37 2.73 1.43 127.31 95.62 4.83815 4.18 0.78 4.76 2.33 1.18 140.43 19.59 5.13916 4.19 9.62 3.91 2.4 1.9 100.2 113.72 2.10617 4.24 9.99 6.73 2.89 1.98 131.35 22.77 2.83418 4.34 1.42 8.6 2.68 1.44 133.43 24.17 4.59419 4.73 6.56 2.06 2.4 1.11 140.65 17.75 6.17120 4.76 0.27 4.07 2.7 1.26 8.72 25.56 6.09221 4.98 4.55 5.61 2.59 1.41 11.09 30.29 4.46922 5.03 1.86 8.31 2 1.59 132.18 35.32 2.06923 5.14 8.8 9.03 2.99 1.26 94.7 57.15 7.50124 5.16 4.15 2.54 2.49 1.6 109.42 56.96 3.20225 5.65 4.91 6.61 2.47 1.71 123.07 39.16 2.75826 5.66 3.77 3.28 2.66 1.22 87.34 45.19 6.30327 5.68 2.83 7.65 2.64 1.12 51.84 160.73 7.36528 5.78 1.49 7.83 2.96 1.3 143.81 126.58 6.90329 5.83 8.17 5.04 2.54 1.32 98.92 100.03 4.90030 6.16 7.75 8.66 2.37 1.42 71.37 33.2 3.67531 6.4 1.19 6.19 2.83 1.51 132.29 38.17 4.66932 6.44 1.81 1.11 2.24 1.09 15.93 13.92 5.55633 6.5 4.9 1.25 2.73 1.26 18.87 164.48 6.23134 6.64 5.65 5.07 2.3 1.8 21.15 127.21 2.15135 6.8 2.91 3.13 2.84 1.03 60.77 100.4 10.107

Total volume (mm3) 168.2Foam density (kg/m3) 481.0


– import of all the ellipsoidal cells in the chosen FEM code by neu-tral .iges format and subsequent assembly in FEM assemblymodule (Fig. 11a): since ellipsoidal cell model was more com-plex than Kelvin cell model, the possibility to use parallel com-puting resources advised authors to prefer Abaqus software inthis case;

– generation of planes delimiting the cubic volume of the model:only the top and the bottom face of the cube are consideredmaterial rigid planes (Fig. 11b), while the remaining were usedas trimmer planes (Fig. 11c);

– trim and elimination of the surface regions that reverted outsidethe cube (Fig. 11d): this operation reduced the amount of sur-faces that constitutes the model, therefore volume materialand foam density reported in Table 4 decreased; however theuse of a wall thickness of 0.11 mm allowed to obtain a modelcharacterised by a final density of 438 kg/m3 that is very closeto experimental mean value;

– application of boundary condition and pressure load on thelower and upper plane, respectively.

Fig. 12a reports the assembly of the ellipsoidal cells and the ri-gid surfaces. Fig. 12b, which reports the z-displacement map, al-lows to evaluate the discretisation utilised for the mesh. Inparticular, the elements were dominant quadratic shell S8R, havingeight nodes and reduced integration. The mesh consisted in133,428 nodes and 93,228 elements. The geometrical complexityof the structure in correspondence of intersection line between

ellipsoidal cells did not allow meshing the model without admit-ting the presence of degenerate elements.

Material model was isotropic with elastic perfectly plasticstress–strain behaviour. The material properties were the sameof the previous model (Table 1). Boundary conditions were appliedon the lower face, imposing a zero value to z-direction displace-ment and rigid body boundaries; load was applied to the uppersurface as a 10 MPa pressure load distributed in three load stepsto overcome convergence troubles.

Displacement map in Fig. 12b was quite uniform in the z-direc-tion and the upper nodes presented the highest values of displace-ment since they were in proximity of the loaded surface.Displacement distribution that has been obtained in the model re-flected the expected behaviour, confirming the approach validity.

The ratio of upper surface displacement to model thickness al-lowed to calculate the global strain of the foam and, then, thestress–strain curve was obtained dividing the displacement bymodel thickness (Fig. 13). In order to carry out a reliable compari-son of numerical and experimental behaviour, not only the vari-ability of foam density but also the scatter of experimental datamust be considered. For this reason, mean values of elastic modu-lus and plateau stress of the specimens, having the same apparentdensity of the numerical model, have been calculated and used fora direct comparison (Table 5). It is important to observe that thismodel was capable to predict the first stage of the stress straincurve plateau. Besides, experimental and numerical maximumstress were practically coincident, while a difference still existed

Fig. 11. Model assembly steps: (a) import of ellipsoidal cells, (b) generation of upper and lower planes, (c) generation of trimmer planes and (d) removing of the exceedingsurfaces.

Fig. 12. (a) Random model based on ellipsoidal cells. (b) Mesh and z-displacements map.


in the elastic modulus. This difference, that was however the bestapproximation that has been obtained, seemed to be quite large,but the direct comparison of numerical and experimental stress–strain curve revealed a good agreement. In particular, the graphin Fig. 14, where both the numerical curve and the experimentalone of Specimen 6 are plotted, suggests that a simple comparisonof elastic modulus values could be not adequate to evaluate thereal accuracy of the model that is instead evident through the di-rect comparison of the two curves.

However, this model presented some limitation during the sim-ulation of compaction phase at strain higher than 3.5%. In thesephysical conditions the cells begin to collapse and a large number

of internal contact region is created. Due to its complexity, it wasnot possible to accomplish this phenomenon into the model. As aconsequence, model accuracy rapidly decreased and finally conver-gence problems appeared

Ellipsoidal cell model could be also used to evaluate the influ-ence on the numerical results when limited changes of apparentdensity were considered. It was possible, in fact, to consider thesame cell geometry but different values of wall cell thickness. Thischange produced a variability of the foam apparent density, with-out the need to change the whole model. Four different values ofapparent density have been considered in the range438–550 kg/m3, obtaining the stress–strain curves reported in

Fig. 13. Numerical stress–strain curve for the model with ellipsoidal cells.

Table 5Comparison of experimental and numerical results of ellipsoidal cell model.

Specimennumber

Specimentype

Maximum stresssmax (MPa)

Density(kg/m3)

Eunload

(MPa)

6 D40 5.0 432.5 5778 D40 4.9 432.9 4339 D40 4.6 435.6 481Experimental mean values 4.8 433.7 497Ellipsoidal cell model 5.3 438 906

Fig. 14. Comparison between numerical and experimental stress–strain curves.

Fig. 15. Initial behaviour of stress–strain curves for different density values.


Fig. 15. As expected, density increase produced an increase of elas-tic modulus and plateau stress, even if there is not a directrelationship between them. Further and minor changes appearedin the shape of the curves.

5. Conclusions

A comparative study of different metallic foams micro-struc-tural modelling was carried out. Two different approaches wereconsidered, which differed for the cell morphology.

The easiest model was based on the regular repetition of a Kel-vin cell. This cell has been largely used [1,15] because it corre-sponds to the minimum surface energy for a constant volume.The simple planar walls model was refined introducing the wallscurvature and material thickening along the faces edges. The cur-vature introduction remarkably improved the initial elastic regionapproximation, but the model was still too rigid with respect to theexperimental data. Moreover, it was impossible to recognise theplateau region characteristic of the foam compression curve. Then,when the solid volume fraction concentrated in the edges ex-ceeded the 60%, the wall thickness variation further improvedthe approximation, but not in a significant way.

The comparison with the experimental data obtained for com-mercial AlSi10Mg foam produced by Alulight showed the excessivestiffness of the Kelvin cell models. The elastic modulus calculatedthrough the last model resulted more than the double respect tothe experimentally one.

The best results were obtained with the model realised byassembling the ellipsoidal cells having random dimensions, orien-tation and position. The model was able to reproduce also the pla-teau region, where cells collapsed and densification phenomenabecame evident. Even if the elastic modulus was overestimated,agreement between experimental and numerical stress–straincurves was excellent.

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[20] Kaminski MM. Computational mechanics of composite materials. London:Springer-Verlag; 2005.





Metamodel-based lightweight design of B-pillar with TWB structurevia support vector regression

Feng Pan, Ping Zhu *, Yu ZhangThe State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China


Article history:Received 28 February 2009Accepted 24 July 2009Available online 20 August 2009

Keywords:B-pillarTailor-welded blankCrashworthinessMetamodelingSupport vector regressionLightweight


* Corresponding author. Tel./fax: +86 21 34206787E-mail address: [email protected] (P. Zhu).

Vehicle lightweight design becomes an increasingly critical issue for energy saving and environment pro-tection nowadays. Optimum design of B-pillar is proposed by using tailor-welded blank (TWB) structureto minimize the weight under the constraints of vehicle roof crush and side impact, in which support vec-tor regression (SVR) is used for metamodeling. It shows that prediction results fit well with simulationresults at the optimal solution without compromising the crashworthiness performance, and the weightreduction of B-pillar reaches 27.64%. It also demonstrates that SVR is available for function approxima-tion of highly nonlinear crash problems.


1. Introduction

Continuous strict standards for fuel efficiency and gas emissionhave been a great boost to vehicle lightweight design. As the auto-body structure weighs about 30% of whole vehicle, the weightreduction in auto-body structure plays a rather important role indecreasing the mass of full vehicle. Generally, mass savings canbe achieved by using high-strength steels and other lightweightmaterials, or utilizing optimization techniques to develop a morerobust and lightweight structure. However, what is noteworthy isthat the lightweight vehicle should maintain, or even improvethe required performance compared to the original one, basicallyincluding stiffness performance, crashworthiness performance,noise, vibration, harshness (NVH) performance as well as durabil-ity performance etc. In other words, the main target of lightweightdesign is to minimize the weight of auto-body without compro-mising the performance of vehicle.

Besides alternative materials and optimization techniques, tai-lor-welded blank (TWB) is also one of the promising approachesto achieve weight reduction and to meet vehicle performance tar-gets since TWB can combine different thicknesses or differenttypes of steels into a single one prior to the forming process, whichhas contributed to a decrease in body-in-white weight and thenumber of components [1]. The use of TWB in automobile industryis generally known. An inner door panel using TWB was developedthrough topology, size and shape optimization [2], while Song and

ll rights reserved.

.

Park [3] conducted multidisciplinary design optimization of anautomotive front door considering stiffness, natural frequency,and side impact performance with a TWB structure, in whichresponse surface method (RSM) was utilized to approximate thecomplicated nonlinear problems. Recently, optimum design of aninner door panel using TWB was proposed integrating finite ele-ment analysis, artificial neural network (ANN) and genetic algo-rithm, aiming at reducing the weight and enhancing occupantsafety in the case of side impact collision [4]. TWB structure canimprove the local performance of component such that no addi-tional reinforcements are needed. Besides front door, B-pillar alsoplays a critical role in protecting occupant during side impact acci-dent. Additionally, the proposed new rollover regulations are forc-ing manufacturers to significantly strengthen the B-pillar. While B-pillar can be strengthened either by heavier materials (increasingthe gauges) or by stronger but lighter materials (high-strength, ex-tra high-strength or ultra high-strength steel), other advancedmanufacturing technique like TWB can also be implemented. Usu-ally, the B-pillar contains some reinforcement parts to enhancestiffness and rigidity in order to resist deflection in side impactor roof crush. So it is available to utilize TWB to improve the per-formance of B-pillar, while weight reduction can be achievedthrough optimal distribution of thicknesses in different areas con-sidering its contributions to roof crush and side impact after thesereinforcements removed.

Meanwhile, there is a growing interest in using metamodel toapproximate the complicated highly nonlinear problems to dealwith analysis and optimization process, and the computational costcan also be reduced. This method is referred to as metamodel-based





F. Pan et al. / Computers and Structures 88 (2010) 36–44 37

optimization or surrogate-based optimization [5]. In this work, sur-rogate model is synonymous with metamodel. There are variousmetamodeling techniques like response surface method (RSM),artificial neural network (ANN), radial basis functions (RBF), kriging(KR) and support vector regression (SVR) available for engineeringdesign. A number of the literatures can be found dealing with engi-neering optimization on vehicle crashworthiness problems by RSM,KR and RBF etc. [6–10], while application has been found rarely forapproximation on these problems by using SVR. Metamodeling isan approximation method so that there is always the issue of modelaccuracy. It is found RBF and KR has their own characters forapproximating crashworthiness problems or slightly nonlinear re-sponses [9,11]. However, SVR achieves more accurate and robustfunction approximations than RSM, KR and RBF through many casesvalidation in their study [12]. And it is also concluded that SVR per-forms very well in highly nonlinear vehicle crash problems accord-ing to our researches, both on accuracy and on efficiency [13].Hereby, it is utilized as an alternative technique to approximatefunctions of responses of interest in vehicle crashworthinessproblems.

In this study, optimum design of B-pillar is performed with aTWB structure subjected to roof crush and side impact. Finite ele-ment (FE) models for vehicle roof crush and side impact were mod-eled firstly and validated with physical test. Then design ofexperiment is performed to obtain the response values of interest,where Latin hypercube sampling (LHS) is selected for sampling.And metamodels are constructed through SVR. The optimal solu-tion is solved by the sequential quadratic programming (SQP) algo-rithm. Finally, the analysis of optimization results is compared tothe original one, confirming the availability and feasibility of opti-mal solution.

2. Methodology and theory

2.1. Methodology of metamodel-based optimization

Metamodel-based optimization is an effective approach forengineering design [14–16]. This approach involves the followingsteps typically, also shown in Fig. 1.

1. Define the optimization problem and targets, includingobjective(s), constraints and design variables, etc.

Convergence

Refine design space

stopYes

No

Definition of optimization problem

Sampling of the design space

Numerical simulation at sampled points

Construction of metamodel

Model validation Perform optimization

Fig. 1. Metamodel-based optimization procedure.

2. Generate a sample of the design variables space as a trainingdata set, also called Design of Experiment (DOE). Theseinclude Latin hypercube design, Uniform design, Orthogonalarrays, Central composite designs, face-centered cubicdesign, factorial designs.

3. Conduct numerical simulations using the sampled pointsfrom step 2 and extract the values of responses of interest.

4. Construct metamodel based on SVR or other metamodelingtechniques, using the data obtained previously. The basictheory of SVR will be discussed later in this section.

5. Assess the predictive capabilities of the surrogate model inunsampled design space. Usually it is measured by someerror metrics, including R square, relative average absoluteerror, root mean square error, etc.

6. Solve the constructed metamodes, to make sure the result isconvergence in the design space. If achieved, stop; or else,refine design space by increasing additional sample pointsto update the metamodel.

7. Iterate until convergence.

2.2. Theory of support vector regression

SVR is derived from support vector machine (SVM) technique,which is a method from the statistical learning disciplinary. TheSVM has been introduced for dealing with classification (patternrecognition) and regression (function approximation) problems.The algorithmic principle of SVM is to create a hyperplane, whichseparates the data into two classes by using the maximum marginprinciple. The linear separator is a hyperplane which can be writ-ten as

f ðxÞ ¼ hw � xi þ b ð1Þ

where w is the parameter vector that defines the normal to thehyperplane and b is the threshold, and hw � xi is the dot productof w and x.

In order to produce a prediction which generalizes well, the ba-sic two aims of SVR are to find a function f(x) that has at most edeviations from each of the targets of the training inputs, and tohave this function to be as flat as possible. So it is available to min-imize the model complexity by minimizing the vector norm |w|2,that is, the flatter the function the simpler it is, the more likely itis to generalize well. Therefore, the optimization problem can beformed as

Min12jwj2

s:t:yi � hw � xii � b 6 ehw � xii þ b� yi 6 e

� ð2Þ

An assumption above is that the function f(x) can approximate allthe yi training points within e precision. However, such a solutionmay not actually exist, and two slack variables could be introducedto yield a modified formulation to obtain better predictions. Thus,the optimization problem can be described as

Min12jwj2 þ C

Xl

i¼1

ðni þ n�i Þ

s:t:yi � hw � xii � b 6 eþ ni

hw � xii þ b� yi 6 eþ n�ini; n

�i P 0

8><>:

ð3Þ

where C determines the tradeoff between the flatness and toler-ance. This is referred as e-insensitive loss function, which enablesa sparse set of support vectors to be obtained for regression. Afterapplying Lagrangian principle and substituting Karush–Kuhn–Tucker

Fig. 2. Components of B-pillar assembly.

38 F. Pan et al. / Computers and Structures 88 (2010) 36–44

conditions into Lagrangian function, we can write the optimizationproblem in dual form as

Max� 1

2

Pl

i;jðai � a�i Þðaj � a�j Þhxi � xji

� � ePl

i¼1ðai þ a�i Þ þ

Pl

i¼1yiðai � a�i Þ

8>>><>>>:

s:t:Pl

i¼1ðai � a�i Þ ¼ 0

ai � a�i 2 ½0;C�

8><>:

ð4Þ

Finally, the weights w and the linear regression f(x) can be calcu-lated through

wXl

i¼1

ai � a�i� �

xi

f ðxÞ ¼Xl

i¼1

ai � a�i� �

hxi � xi þ b

ð5Þ

For the nonlinear regression, approximation can also be achieved byreplacing the dot product of input vectors with kernel function.Typical choices for the kernel function include linear nonlinearpolynomial, Gaussian and Sigmoid, etc. After applying the kernelfunction to the dot product of input vectors, we can obtain

Max� 1

2

Pl

i;jðai � a�i Þ aj � a�j

� �kðxi � xÞ

�ePl

i¼1ai þ a�i� �

þPl

i¼1yiðai � a�i Þ

8>>><>>>:

s:t:Pl

i¼1ai � a�i� �

¼ 0

ai � a�i 2 ½0;C�

8><>:

ð6Þ

And the final result of SVR model for function estimation is obtained

f ðxÞ ¼Xl

i¼1

ai � a�i� �

kðxi � xÞ þ b ð7Þ

In this study, a Gaussian function is chosen for kernel function sinceit is used for dealing with engineering problem mostly.

Kðx; x0Þ ¼ exp � x� x0k k2

2r2

!ð8Þ

Fig. 3. Finite element model of vehicle.

3. Finite element models

The B-pillar of vehicle is the structure located between the frontand rear doors of compartment. It houses electrical wiring and con-nections spots for the passenger seatbelts, also does provide struc-tural support for the compartment in the case of lateral or roofimpact. This work focuses on optimum design of the B-pillar,which has reinforcement in it. It consists of four components,namely, B-pillar inner, B-pillar reinforcement, seatbelt fixing sup-port, and B-pillar outer, which is shown in Fig. 2. For simplicity,B-pillar outer is called B-pillar later in this study. It is availableto apply TWB technique into B-pillar and remove the reinforce-ments in order to reach mass saving by optimal distribution ofthickness of B-pillar. Compared with B-pillar’s contributions tobending or torsion stiffness and frequency performance of body-in-white (BIW), its effect on crashworthiness is more critical andimportant. The B-pillar can hold the rigidity of the compartmentgreatly so that structural intrusion into the compartment can bereduced in roof crush or side impact [17,18]. Therefore, the optimi-zation for B-pillar should consider these crashworthiness perfor-

mances primarily. Hence, FE model of vehicle roof crush and sideimpact is presented firstly, and validation experiments are carriedout to ensure the availability of the finite element (FE) model.

In order to improve the simulation accuracy, a detailed full-scale FE model of vehicle has been established, including bodyassembly, engine, drive system and tires. The FE model, containing520,205 shell elements, 4302 solid elements, 5479 beam elementsand 3723 mass elements, consists of 564 parts, which is shown inFig. 3. The basic material model is type 24 (piecewise linear plastic-ity), and the major contact algorithms are automatic single surfaceand automatic surface to surface. Usually, reduced integration isemployed to save the computational cost in an explicit crash anal-ysis code like LS-DYNA, whereas it may cause the elements withspurious zero energy models, which should not be used in practical[19]. Hourglass control is used to avoid these phenomena, conse-quently. Among shell elements, nearly 97.6% is quadrangle ele-ments with average mesh size of 10 mm. Some other meshquality indexes are also checked, including the degree of warping,aspect ratio, skew, and the maximum and minimum interior anglesof quadrangle and triangle elements. And stricter indexes are em-ployed to some critical parts, such as front side rail. The crashmode or deformation mode of front side rail can be influenced byits meshing quality largely. These indexes include mesh refinementwith smaller size, meshing lines perpendicular or parallel withforce passing direction as much as possible, and different triangleelements not sharing with the same nodes etc. In addition, aneffective meshing method, transition region, is applied to somelarge-size components such as floor panel, roof panel, and wind-screen, which deform largely on one side and slightly on the otherside simultaneously in the case of roof crush or side impact colli-sions so that computational time can be cut down obviously dueto the number of grids decreases largely. And the front and rearend of floor and roof panels should be with the average mesh sizeaccording to their characters in frontal and rear crash. For example,Fig. 4 shows the FE model of windscreen, in which two layers oftransition regions are used to connect meshes of different sizes.

Fig. 4. Finite element model of windscreen using two transition regions (3137nodes).


The number of shell elements is 2936 for this windscreen model,whereas the number is 10,255 if average mesh size of 10 mm isused only. More detailed information about this meshing methodcan be found in Shi et al. [20].

Vehicle roof crush intends to enhance passenger protection dur-ing rollover accident. The test procedure is based on the updatedFederal Motor Vehicle-Safety Standards 216 (FMVSS 216), whichis a quasi-static test. The auto-body is located on a horizontalground and a rectangular plate with 1829 mm by 762 mm is addedon the driver’s side top of the body as specified by the FMVSS 216with roll angle (a = 25�) and pitch angle (b = 5�). The lower surfaceis tangent to the surface of the vehicle and initial contact point ison the longitudinal centerline of the lower surface of the plateand 254 mm from the forward most point of the centerline. Theforce–displacement relationship of the plate is deemed as roofstrength [21]. Fig. 5 is the FE model of roof crush, where the plate

Fig. 5. Finite element model of roof crush.

Fig. 6. Finite element m

is assumed to be rigid. The physical experiment is performed bycrushing the roof slowly, at about 8.9 mm/s and for about 10–30 s [22], so that numerical simulation is time-consuming if LS-DYNA is adopted for this case. Some potential approaches are pro-vided in LS-DYNA to reduce the computational time by using massscaling or increasing the velocity of crushing plate. While the ana-lytical result through mass scaling is influenced by the choice ofscaling parameter mostly and it is still time-consuming anyway,another approach is employed by increasing the velocity of platein this work, consequently. Bathe et al. [22] proposed that a speedof 0.5 mph can obtain the reasonable results compared with theone of 10 mph for vehicle roof crush, and also pointed out thatthe ratio of kinetic energy to strain energy of the model shouldnot be too large in their study. However, the velocity of 0.5 mphstill has computational burden for optimization in which the mod-el is calculated many times for design of experiment. Recently, itwas reported that the velocity of 5 mph (2235.2 mm/s) is alsoavailable through comparing the simulation results with physicaltest [23,24]. In addition, we concludes that the speed is reasonablefor roof crush if the ratio of kinetic energy to strain energy is lessthan 15% based on our benchmark study on Ford Taurus providedby National Crash Analysis Center. Hence, from the viewpoint ofcomputational efficiency and accuracy, the speed of 5 mph(2235.2 mm/s) is used for roof crush simulation. The normal vectorof plate is related with roll angle (a) and pitch angle (b), which canbe written as

~n ¼ fcos a sin b; sina;� cosa cos bg ð9Þ

For side impact protection, the National Crash Legislation side im-pact test configuration (GB20071-2006) is considered, which is veryclose to European Enhanced Vehicle-Safety Committee side impactprocedure. According to GB20071-2006, a side impact simulationwas performed with a movable barrier impacting the vehicle per-pendicularly at a speed of 50 km/h. Fig. 6 shows the FE model ofside impact. LS-DYNA version 971 is used to solve the model. Thephysical crash test was also carried out, and comparison of struc-ture deformation on left side between test and simulation is illus-trated in Fig. 7. Occupant safety is the main concern in sideimpact, which includes head injury criterion (HIC), chest viscouscriterion (VC), rib deflection criterion (RDC) of upper, middle andlower ribs, abdomen load and pubic symphysis force. The last twoindexes are not considered in this study, however. Fig. 8 comparesthe simulation results of head acceleration, lower rib deflection andVC of dummy with test results, which indicate that they fit wellboth on peak values and on changing trends. The difference of peakvalues on head accelerations is due to lack of interior trim parts in

odel of side impact.

Fig. 7. Comparison of side structure deformation. (a) simulation; (b) test.


the FE model. Anyway, the FE models of roof crush and side impactare available for further studies.

Fig. 8. Performances comparison between test and simulation. (a) head accelera-tion; (b) lower rib deflection; (c) lower rib viscous criterion.

4. Lightweight design of B-pillar via metamodels

4.1. Partitions of B-pillar and design targets

The B-pillar with tailor-welded blank (TWB) technique canmeet requirements in all partial regions as described before, suchthat no additional individual parts are required. In this case, B-pil-lar reinforcement which is described in Fig. 2 can be removed, andTWB technique can be applied into B-pillar not only to ensure thecrashworthiness performance, but also to reduce structural weightand manufacturing cost. Probably, there may be small reinforce-ment plates welding with B-pillar for local rigidity at the placesof door hinges and locks, and these small reinforcements are notincluded and discussed here, however. The partitions of B-pillarfor TWB design is based on its functions in roof crush and side im-pact. The B-pillar must have a high level of resistance and rigidityin the region of door hinges and door locks where the intrusionwould reduce the survival space and hurt the occupant directlyin the event of a lateral crash, and also have enough strength to re-sist deformation under roof crush. Another principal for partition isthat the welding line should not be located at the areas with com-plex structure, such as beads, because tearing near the weldingseam often occurs in forming process induced by the materialproperties in the heat-affected zone in welding areas. Hence,TWB structure of B-pillar can be simply divided into three seg-ments according to the aforementioned requirements and distribu-tion of the original reinforcement, which is illustrated in Fig. 9. Thesolid lines represent the positions of the weld lines. The verticallengths l1, l2 and l3 of the three segments are 290 mm, 560 mm,and 300 mm, respectively. The rest work is to find the optimalthickness of three segments, provided that the type of steel is thesame with the original one. Therefore, the thickness of three seg-ments are defined as design variables, denoted by t1, t2 and t3,respectively (1.0 mm 6 ti 6 2.5 mm, i = 1, 2, 3).

The objective of optimization is to minimize the weight of B-pil-lar while structural crashworthiness and occupant safety should beguaranteed primarily subjected to roof crush and side impact. For

roof crush, the increase of gauge design variables tends to get ahigher resistance force, whereas the increase of B-pillar’s weightis undesirable, which is also the same for the case of side impact.As defined on the updated FMVSS216, the force generated by vehi-cle resistance must be greater than 2.5 times the unloaded vehi-cle’s weight with a maximum allowable displacement of 112 mmin this study such that it won’t touch or hurt the male dummy(50%) with seatbelt placed in the compartment. The resistant force(PRoof) of roof crush, which is the peak force in the force–displace-ment relationship of the crushing plate, was set to be 27 kN, wherethe unloaded vehicle’s weight M = 1100 kg. In side impact, re-sponses of HIC, RDC and VC of upper, middle and lower rib areconsidered. Besides, the intrusion velocity of B-pillar at middle

l1=290mm

l2=560mm

l3=300mm

Fig. 9. Partition of B-pillar with tailor-welded blank.


point (VB-pillar) is also included. The design targets are listed in Ta-ble 1.

4.2. Metamodeling by support vector regression

Metamodel is widely used to approximate the highly nonlinearproblems, such as vehicle crashworthiness problem. In this study,support vector regression (SVR) is employed to construct meta-models for approximating responses of interest. Latin hypercubesampling (LHS) is adopted for selecting the initial design pointsin the exploratory design space. It can provide an efficient estimate

Table 1Regulation requirements and design targets.

Index Symbol

Roof resistance force (kN) PRoof

Roof crush displacement (mm) DRoof

Head injury criterion HICViscous criterion (m/s) VCUpper

VCMiddle

VCLower

Rib deflection Criterion (mm) RDCUpper

RDCMiddle

RDCLower

Intrusion velocity of B-pillar (m/s) VB-pillar

Table 2Design matrix and responses.

No. Design variables Responses

t1 (mm) t2 (mm) t3 (mm) PRoof (kN)

1 2.1485 2.4700 1.8968 31.372 2.0031 2.0647 1.0026 30.313 1.2647 1.4851 1.9700 26.004 1.3968 2.1771 1.1771 27.195 1.4177 1.1770 1.6636 26.476 2.2851 1.5485 2.2674 30.347 1.0674 1.9636 1.3972 24.988 2.3026 1.2177 1.7647 29.589 1.6700 1.6026 2.3031 28.3510 1.7972 1.8031 1.4769 29.1611 1.5769 2.3738 2.1738 28.4812 1.8738 1.0968 2.0485 28.2913 1.1636 1.7972 1.5851 25.5114 1.9304 1.3304 1.2177 28.8715 2.4771 2.2674 2.4304 32.15

of the overall mean of the response than the estimated based onrandom sampling [25]. LHS is one of ‘‘space-filling” methods,which can treat all regions of design space equally and the sampledpoints should be chosen to fill the entire design space for computerexperiments [26]. Meanwhile, the accuracy of metamodel is alsorelated with the number of design points in the training data set.Too less of design points may not represent the relationship be-tween inputs variables and outputs functions accurately, whiletoo many of design points would increase the computational bur-den. Based on the research in [27], 5N sample points generated byLHS are used in this study, where N is the total number of designvariables. That is, a training data set with 15 sample points is gen-erated as there are three design variables. It is noted that the num-ber of sampling points can vary according to the availability ofcomputing capability.

Since the lower rib was always found to experience the largestdeflection and highest VC in side impact compared to other tworibs and the values of middle and upper ribs (RDCMiddle, RDCUpper,VCMiddle, VCUpper) are much lower than the design targets, the re-sponses of RDCLower and VCLower are considered in the optimizationprocess only. In addition, because displacement of the plate at peakforce is always much less 112 mm, it is also unconsidered in theoptimization process. Therefore, metamodels for five responsesshould be constructed only, namely PRoof, HIC, RDCLower, VCLower,and VB-pillar. The design matrix and values of the responses ob-tained from FE simulations are given in Table 2.

Then Matlab toolbox LSSVM was used to approximate the func-tions of responses obtained from the training data set. And it iswell known that SVR generalization performance depends on agood setting of hyper-parameters C, e, and the kernel parameters.Gaussian kernel function was selected due to its ‘‘good featuresand strong learning capability” [28]. And e is the default value of0.001, while C is1 .The radius of Gaussian kernel is automatically

Requirement Original Target

P26.95 25.52 P276112 65.2 611261000 327 6330

0.27861.0 0.333 60.53

0.53121.38

642 23.02 63637.07

– 8.16 67.9

HIC RDCLower (mm) VCLower (m/s) VB-pillar (m/s)

326 34.86 0.503 7.858351 36.93 0.527 7.638326 33.97 0.528 8.011315 35.98 0.534 7.936315 35.98 0.535 7.986368 35.37 0.538 8.059291 36.32 0.535 8.083340 35.33 0.541 8.128360 34.61 0.563 8.029331 34.80 0.520 7.940299 37.85 0.518 7.991318 33.67 0.607 8.375293 38.93 0.504 8.063387 32.94 0.548 7.944433 37.19 0.527 7.886

Table 4Accuracy assessment.

Criterion R2 RAAE RMSE RMSE (%)

PRoof 0.9603 0.1649 0.0197 4.0811HIC 0.9681 0.1456 0.0070 1.9670RDCLower 0.9113 0.2746 0.0100 2.2268VCLower 0.8470 0.3387 0.0041 1.1400VB-pillar 0.9072 0.2499 0.0258 5.7233


optimized based on the given training data, aiming at minimizingthe generalized mean square error using leave-one-out cross-vali-dation. The parameters tuning process can be achieved in LSSVMtoolbox. All the design variables and responses are normalizedquantitatively according to

xn ¼ 0:2þ 0:6xi � xmin

xmax � xminð10Þ

where xn is the normalized value, xi is the actual value, and xmax andxmin are the maximum and minimum values of that data column,respectively.

The vector of Lagrange multipliers ðai � a�i Þ, offset b, and param-eter r for five responses are listed in Table 3. With these results,the explicit analytical formulas can be expressed according toEqs. (7) and (8) for each response.

For accuracy, the goodness of fit obtained from training data setis not sufficient to assess the accuracy of newly predicted points[26]. For this reason, another set of 8 sample points generated ran-domly by Matlab are used to verify the accuracy of the approxima-tion functions. Three different prediction metrics are used to assessthe accuracy: R Square (R2), Relative Average Absolute Error(RAAE), and Root Mean Square Error (RMSE) [29]. The equationsfor these three measures are given below, respectively.

R2 ¼ 1�Pn

i¼1ðyi � yiÞ2Pni¼1ðyi � �yÞ2

ð11Þ

RAAE ¼Pn

i¼1jyi � yijPni¼1jyij

ð12Þ

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1ðyi � yiÞ2

n

sð13Þ

where yi is the corresponding predicted value for the tested valueyi; �y is the mean of the tested values. The larger the value of R2,the more accurate the metamodel is; the larger the value of RAAEor RMSE, the less accurate the metamodel is.

Validation results of selected error metrics for different meta-models are shown in Table 4. The model fit of PRoof, HIC, RDCLower

Table 3Results of parameters obtained by support vector regression.

Index FRoof HIC RDCLower VCLower VB-pillar

ai � a�i 0.9176 �0.0638 �0.3040 �0.9900 2.90760.6492 0.3343 0.7989 �0.1383 �11.555�0.6648 0.0027 �0.7793 �0.0906 �2.8199�0.7144 �0.1562 0.2744 0.1242 3.2276�0.6497 �0.3058 0.3357 0.1842 �5.6302

0.4898 0.6238 �0.0254 0.2712 �1.6382�0.0081 �0.4964 0.4595 0.5566 4.1215

0.2611 0.0541 �0.0284 0.3643 6.1206�0.0551 0.5221 �0.4101 1.1815 2.1558

0.3463 �0.1428 �0.3170 �0.3798 2.0121�0.3445 �0.6083 1.2850 �0.4587 �1.6418�0.2340 �0.3663 �0.9264 0.0228 0.1119�1.1063 �0.4296 0.0102 �1.1565 1.6173�0.0926 1.0269 �1.3076 0.6336 2.9576

1.2024 0.0055 0.9346 �0.1246 �1.9475b 0.2329 �0.1252 �0.1449 �0.1939 �0.8398r 1.2576 1.0434 0.3645 0.3721 12.1195

Table 5Comparisons of predicted solutions to finite element analysis (FEA).

Index PRoof (kN) HICPredicted optimum 28.15 329.96FEA validation 27.86 326% error 1.04% 1.21%

and VB-pillar are very good with high values of R2 (P0.9). Althoughthe metamodel of VCLower is constructed with the R2 value of0.8470, it is still acceptable with the RMSE% value of 1.14%. There-fore, metamodels for PRoof, HIC, RDCLower, VCLower, and VB-pillar arereasonable to be used for design optimization.

4.3. Optimization formulation and results

In this case, the objective is to minimize the weight of B-pillarunder constraints of roof crush and side impact. According to thedesign targets in Table 1, the problem can be formulated as follows

Min MðtÞs:t: FRoof ðtÞP 27 kN

HICðtÞ 6 330RDCLowerðtÞ 6 36 mmVCLowerðtÞ 6 0:53 m=sVB-pillarðtÞ 6 7:9 m=s1:0 mm 6 ti 6 2:5 mm; i ¼ 1;2;3

ð14Þ

where M(t) is the weight of B-pillar, which is a linear function withthe thickness of three segments. And M(t) can be formulated asfollows:

MðtÞ ¼ 0:518 t1 þ 0:8733 t2 þ 1:0413 t3 ð15Þ

It is to be noted that M(t) should also be normalized according to Eq.(10) for optimization further.

RDCLower (mm) VCLower (m/s) VB-pillar (m/s)35.43 0.526 7.8935.83 0.518 7.81�1.12% 1.54% 1.02%

Fig. 10. Force–displacement relationship of roof crush before and afteroptimization.


For this simple optimization problem with three design vari-ables, sequential quadratic programming (SQP) (function, fminconin MATLAB) is used to search for optimal point. Because the SQPis dependent on the initial guess of the optimal point, a series oftrials with 10 points have been performed before getting the finaloptimal point.

Finally, the optimal point of design variables and objective withde-normalized form are as follows:

t1 = 1.615 mm, t2 = 1.822 mm, t3 = 1.276 mm, Mmin = 3.77 kg.

Fig. 11. Comparison of time histories of rib deflection and visco

The original mass of B-pillar (including B-pillar reinforcement)is 5.21 kg, while the mass of B-pillar with TWB structure is3.77 kg, with mass saving of 1.44 kg, due to the use of TWB tech-nique and the removal of reinforcement. It achieved 27.64% weightsaving less than the original structure. Not only raw material costcan be saved, but also manufacturing cost (i.e., welding, assembly)would be reduced.

In order to validate the feasibility of the optimal results, finiteelement (FE) simulations for roof crush and side impact were also

us criterion for (a) upper rib; (b) middle rib; (c) lower rib.


carried out based on the optimal results obtained via metamodelsconstructed through SVR. The results are given in Table 5. The dif-ferences between the predicted responses and FE analytical resultsare all less than 2%; this indicates that the metamodels fit well too.The maximum error (VCLower) is due to the less accuracy of surro-gate model than others partially, where R2 value is 0.8470.

The comparison of force–displacement relationship before andafter optimization shows in Fig. 10. The optimum solution by usingTWB gives an increase of 9.2% on resistance force in the case of roofcrush. For vehicle side impact, the solution gives reductions of 4.3%on velocity of B-pillar (VB-pillar), while keeping the same level on re-sponse of HIC. The rib deflection and VC of upper, middle, and low-er ribs were also compared to those of the original design, whichare shown in Fig. 11. It can be observed that only few increaseson rib deflection and VC of upper and middle ribs, whereas de-creases on lower ribs is also tiny. The results satisfy the design tar-gets, also indicate the optimization can meet the requirements oftargets and standards, including the updated FMVSS216 andGB20071-2006. Therefore, it can be confirmed that the optimiza-tion results are reasonable and practicable, which can reduce theweight of B-pillar, while guaranteeing the structural crashworthi-ness and occupant safety in the cases of side impact and roof crush.All the results above demonstrate the successful implementationof TWB structure on B-pillar in this study. And it also indicates thatSVR shows great potential for metamodeling applications for vehi-cle structural crashworthiness design.

5. Conclusions

Aiming at reducing the structural weight, lightweight design ofB-pillar by using TWB is performed in this paper. A full-scale finiteelement model was used in performing vehicle crash simulationson roof crush and side impact. The B-pillar is divided into threesegments according to its contributions and the position of the ori-ginal reinforcement. Response functions of the dummy’s hurt in-jury criterion, rib deflection and viscous criterion of lower rib,intrusion velocity of B-pillar, and vehicle’s roof crush force wereapproximated by using DOE and SVR, where Latin hypercube sam-pling was employed for sampling. And SQP is used to search foroptimum. The optimal solution shows that the weight of B-pillarcan be reduced by 27.64%, while other constraints, including struc-tural crashworthiness and occupant safety are either improved orkept in comparison with the original design. Metamodels used inthe optimization process fit well according to the low error be-tween the predicted responses and FE analytical results at the opti-mal point. It can also be concluded SVR is a promisingmetamodeling technique for function approximation of vehiclecrash problems with high accuracy, and can be further applied tostructural design optimization.

Acknowledgments

The work presented in this paper was supported by NationalNatural Science Foundation of China (Grant # 50875164). Theauthors also acknowledge Altair for providing the educational li-cense of Hyperworks for this work. We acknowledge the anony-mous reviewers for their erudite comments and constructivecriticism that immensely helped us improve the paper.

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http://dx.doi.org/10.1243/09544070JAUTO1045





A practical method for proper modeling of structural damping in inelasticplane structural systems

Farzin Zareian a,*, Ricardo A. Medina b

a Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, United Statesb Department of Civil Engineering, University of New Hampshire, Durham, NH 03824, United States


Article history:Received 5 March 2009Accepted 3 August 2009Available online 28 August 2009

Keywords:Rayleigh dampingInelastic responsesStructural dampingDynamic analysisSeismic responseSeismic evaluation


* Corresponding author. Tel.: +1 949 824 9866; faxE-mail address: [email protected] (F. Zareian).

a b s t r a c t

This study addresses some of the pitfalls of conventional numerical modeling of Rayleigh-type dampingin inelastic structures. A practical modeling approach to solve these problems is proposed. Conventionalmodeling of Rayleigh-type damping for inelastic structures generates responses in which unrealisticdamping forces are present that results in underestimation of peak displacement demands, overestima-tion of peak strength demands, and underestimation of buildings’ collapse potential. The approach pro-posed in this paper avoids these problems by modeling each structural element with an equivalentcombination of one elastic element with stiffness-proportional damping, and two springs at its two endswith no stiffness proportional damping.


1. Introduction

One component that can arguably be considered overlooked incurrent methods for estimation of response of structural systems isthe appropriate modeling of structural damping. This shortcomingbecomes even more apparent when compared with the advancesmade in modeling of the inelastic response of structural and non-structural components during the past decades. These advance-ments include computational and modeling techniques that canreasonably capture material and geometric nonlinearities, as wellas the evolution of damage in a structure. In this context, the termstructural damping refers to energy dissipation mechanisms pres-ent in a structure due to structural and nonstructural componentresponses to dynamic excitation other than the energy dissipatedin inelastic excursions. Traditionally, linear viscous damping hasbeen utilized to model the energy dissipation characteristics of astructural system exposed to dynamic excitation. In such model,the effect of damping is accounted for at a global scale – the energydissipated through friction and slippage in joints for structural andnonstructural components is represented by means of equivalentlinear viscous damping. However, the use of a linear viscous damp-ing model in many cases produces inaccurate estimates of dis-placements and internal forces in members. These inaccurateestimates of internal forces are related to nodes in the structuralmodel in which unrealistic damping forces are generated.

ll rights reserved.

: +1 949 824 2117.

The failure in proper modeling of structural damping is com-pounded by (1) the lack of reliable experimental data to validatethe structural damping models used to represent the energy dissi-pation characteristics of structural systems in inelastic regimes;and (2) the decreased identification accuracy for damping ratioswhen compared to that of elastic natural frequencies and modeshapes. Results from full-scale tests show that changes in dampingare much greater than those for frequency over a similar amplituderange [1]. In the absence of more accurate structural dampingmodels, linear viscous damping is usually utilized for convenience.The most common model of viscous damping used in modeling ofmulti-degree-of-freedom (MDOF) systems is the Rayleigh-typedamping where the damping matrix, C, is composed of the super-position of a mass-proportional damping term (i.e., aM) and a stiff-ness-proportional damping term (i.e., bK) [2].

C ¼ aMþ bK ð1Þ

Physically, mass-proportional damping (MPD) is equivalent tohaving externally supported dampers attached to the dynamic(inertial) degrees of freedom while stiffness-proportional damping(KPD) implies the presence of viscous dampers (i.e., dashpots) thatjoin two adjacent dynamic degrees of freedom. MPD has the effectof having modal damping ratios that are inversely proportional tothe frequencies of vibration of the system, while KPD producesmodal damping ratios directly proportional to the modal frequen-cies of the structure. In both cases, very little experimental verifi-cation exists for such model, particularly for structural systemsthat undergo significant levels of inelastic deformations.





46 F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

In the last few years, several researches have studied and iden-tified important limitations of Rayleigh-type damping as it appliesto inelastic systems. As shown by Bernal [3], Medina and Krawin-kler [4], and Hall [5], when Rayleigh-type damping based on theinitial stiffness matrix is used, unrealistic damping forces are gen-erated at joints in which structural elements undergo abruptchanges in stiffness. This is due to the tendency of degrees of free-dom with small inertias to undergo abrupt changes in velocityonce the stiffness of the element changes during the inelastic re-sponse [3]. Thus, unrealistic damping forces develop at these de-grees of freedom resulting in an underestimation of the peakdisplacement demands in the structure and an overestimation ofinternal forces for elements in the system that do not undergochanges in stiffness [4]. Moreover, as shown in this paper, inappro-priate model of structural damping may also cause underestima-tion of the collapse potential of buildings.

The objective of this paper is to present a more appropriate andeasy-to-apply numerical modeling approach for implementingRayleigh-type damping in structures. This approach eliminatesthe presence of unrealistic damping forces in inelastic time historyresponses. The implication is that the proposed approach will pro-vide improved inelastic dynamic response predictions. An illustra-tion of the benefits of using this new approach in the context ofseismic performance assessment is presented.

2. Viscous damping, Rayleigh damping, and inelastic responses

Although damping is considered the primary energy dissipationmechanism for elastic structures, the focus on inelastic structuresis warranted by the presence of unrealistic damping forces whenRayleigh-type damping based on initial stiffness is assigned tostructural elements that experience inelastic responses. In addi-tion, for relatively large levels of inelastic behavior, the energy dis-sipated via structural damping, as predicted by numerical modelswith Rayleigh-type damping, still constitutes a significant percent-age (e.g., 25%) of the total dissipated energy. Fig. 1 shows the en-ergy dissipated due to linear viscous damping relative to thetotal input energy for a 4-story MDOF model, which represents anon-ductile reinforced-concrete building, at various levels ofground motion intensity measure (gray line with diamond mark-ers). The MDOF model has fundamental period T = 0.4 s, and theglobal strength of the structure corresponds to a response modifi-cation factor of 6 assuming an over-strength factor of 2 for this

Relative Dissipated Energy to Input Energy Mean IDA curves N=4, T1=0.6, Rμ = 3.0 , ξ=0.05, Peak-Oriented model, Northridge EQ

0

1

2

3

4

5

0 0.25 0.5 0.75 1Relative Dissipated Energy to Input Energy

Sa/g

Damping Energy Hysteretic Energy

Fig. 1. Total damping and hysteretic energy dissipated relative to the input energyfor the 4-story case study building.

structure (i.e., R = 6, X = 2, Rl = 3). Rayleigh-type damping basedon initial stiffness is used and damping ratios at the first and thirdmode are set to 5%. On the same plot, the energy dissipated due tohysteretic action of structural components that enter the inelasticregime relative to the total input energy is shown with a black linewith square markers. The vertical axis shows the level of groundmotion intensity in terms of Sa/g (spectral acceleration at the firstmode period of the structure). The ground motion recording usedbelongs to the 1994 Northridge earthquake. This building has beenmodeled with a concentrated plasticity approach in which the hys-teretic response at the end of beams and columns exhibits moder-ate levels of monotonic and cyclic deterioration based on themodel developed by Ibarra et al. [6].

As illustrated in Fig. 1, when the structural system begins toexperience inelastic deformations (i.e., Sa/g > 0.75), the fraction ofthe total input energy dissipated through viscous damping is re-duced and the energy dissipated in structural components hyster-esis loops is increased. However, the energy dissipated throughviscous damping has a lower-bound of approximately 27% of thetotal input energy. In this case, even when cyclic deteriorationand relatively large levels of inelastic behavior are present (aroundSa/g = 4.0), the contribution of viscous damping in dissipating theinput energy to the structural system is increased as the hystereticenergy dissipation capacity of structural components is exhausteddue to damage and deterioration. Therefore, inappropriate model-ing of structural damping has the potential to provide erroneousdemand prediction for structures exposed to strong dynamic exci-tations. This example utilized the most commonly used linear vis-cous damping model, which is the Rayleigh-type damping basedon initial stiffness, i.e., a time invariant stiffness matrix. Mutoand Beck [7] have also highlighted the importance of appropriatemodeling of viscous damping when applying system identificationtechniques to estimate hysteretic structural parameters of systemssubjected to earthquake loading. They showed that excluding vis-cous damping from identification models will significantly modifythe value of the identified hysteretic parameters if viscous damp-ing is present in the structure.

Studies conducted by Medina and Krawinkler [4] on regularmoment-resisting frame structures exposed to far-field groundmotions have shown that improper modeling of 5% critical struc-tural damping using the Rayleigh model based on initial stiffnessresults in the underestimation of peak-drift demands, on average,in the order of 10%. However, the overestimation in peak strengthdemands can be in the order of 30% depending upon the structuralproperties and ground motion characteristics. This phenomenon isillustrated in Fig. 2, which presents representative resultscorresponding to a numerical model of a moment-resisting frame

0%

10%

20%

30%

40%

50%

60%

0 1 2 3 4 5 6 7 8 9

(Mc

-Mp,

b)/M

p,b

[Sa/g] / γγ

NR94cnp-5% damping ratioNR94cnp-10% damping ratioLP89cap-10% damping ratio

Fig. 2. Relative difference between maximum column moment (Mc) and beamplastic moment (Mp,b) at the top floor of a single-bay, moment-resisting frame.

Model A Model B

(a) (b)

Fig. 4. Idealized SDOF structures (a) flexible beam with elastoplastic moment–rotation relationship at both ends and (b) flexible beam + semi-rigid rotationsprings to represent elastoplastic behavior.

-3

0

3

6

Dis

plac

emen

t Duc

tility

Rat

io

Time (sec.)

Displacement Ductility Ratio, SDOF, Model AMass Proportional Damping

Initial-Stiffness Proportional Damping

-1

-0.5

0

0.5

1

1.5

0 5 10 15

Mco

lum

n/ M

p,be

am

Column-end Moment Demand, SDOF, Model A

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53 47

structure exposed to two different ground motion records. One ofthe records is from the 1994 Northridge Earthquake – Canoga ParkStation (NR94cnp) and the other one from the 1989 Loma PrietaEarthquake – Capitola Station (LP89cap). The frame was designedsuch that plastic hinges have the potential to form at the end ofeach beam and at the bottom of the first-story columns. All col-umns are infinitely strong. Modal damping ratios were variedand inelastic dynamic analyses were conducted to quantify the rel-ative difference between the maximum column moment and thebeam plastic moment at the top floor. In concept, this relative dif-ference must be negligible. This is clearly not the case when initialstiffness proportional damping is assigned to elements whose stiff-ness changes during the time history (i.e., inelastic beam ele-ments). As expected, the relative difference between columnmoment and beam moment is negligible in the elastic range (seeFig. 3). In addition, as both damping ratio and level of inelasticbehavior increase, the relative difference between the maximumcolumn moment and the beam plastic moment increases. The levelof inelastic behavior is quantified by the ratio of the pseudo-spec-tral acceleration at the fundamental period of the structure to thebase shear strength coefficient, i.e., [Sa/g]/c. Such errors in overes-timation of demand forces can lead to over-designed structuralmembers that are sized using force-based methods. Consequently,these elements have the potential to attract larger forces into thejoint, which may cause the structure to be more vulnerable todamage and collapse once deterioration of structural componentsis modeled. Similarly, underestimation of deformation demandsmay lead to unsafe structural designs that use displacement-basedmethods such as performance-based design.

A better understanding of the consequences of modeling vis-cous damping based on initial stiffness can be obtained with themodel in Fig. 4a, which depicts a Single-Degree-Of-Freedom(SDOF) structure in which two rigid columns are joined by a flexi-ble beam (Model A). The beam is modeled based on a concen-trated-plasticity approach in which elastoplastic moment–rotation relationships are assigned to the beam ends. The periodof the system is 0.4 s and the plastic moment capacity of the beamis 3000 kip.-in. (3518.7 N m). To better show the inaccuracy of Ray-leigh damping modeling based on initial stiffness, a damping ratioof 10% is assumed, and damping in the SDOF system is modeledbased on mass-proportional damping (Case 1) and initial stiffnessproportional damping (Case 2). Both models are subjected to a re-corded ground motion that is scaled such that a displacement duc-tility ratio of 4 (l = 4) is attained. Fig. 5 shows the results of thisanalysis in terms of displacement ductility ratios and normalizedcolumn-end moments, i.e., column-end moment divided by thebeam plastic moment. Fig. 5a demonstrates that by using Model

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

0 5 10 15

Mom

ent (

kip

-in.)

Time (s)

Column MomentBeam Moment

Fig. 3. Moment time history at the top floor of the moment-resisting frame of Fig. 2exposed to ground motion LP89cap; damping ratio = 10%; [Sa/g]/c = 8.

-1.50 5 10 15

Time (sec.)

Mass Proportional DampingInitial-Stiffness Proportional Damping

Fig. 5. Comparison between response histories of the idealized SDOF system(Model A) whose 10% critical damping is modeled using ‘‘mass-proportionaldamping” and ‘‘initial stiffness proportional damping” subjected to a ground motionrecording from Northridge Earthquake: (a) displacement ductility ratio, and (b)normalized column-end moment.

A the displacement ductility ratio response is identical no matterwhich method was used to model structural damping, i.e., Cases1 and 2. However, Fig. 5b shows that the column-end momentsfor Case 2 (i.e., initial stiffness proportional damping) exceeds thebeam plastic moment of 3000 kip.-in. (3518.7 N m). Once again,this is due to the presence of unrealistic damping moments inthe response. In this particular case, these unrealistic dampingmoments do not affect the displacement response significantly


because they act for relatively short periods of time (spikes inFig. 5a) as compared to the total strong motion duration.

3. Current approaches to overcome the limitations of Rayleigh-type damping based on initial stiffness

Solutions to this problem have been proposed by Bernal [3],Hall [5], and Charney [8]. Bernal [3] proposed a solution in whichCaughey-type damping should be used and the damping matrixshould be assembled by restricting the exponents of the Caugheyseries (l terms in Eq. (2)) to zero or negative. Rayleigh-type damp-ing is a special case of Caughey-type damping.

C ¼MX

l

al½M�1K�l ð2Þ

This proposed solution has the effect of not assigning stiffness-proportional damping to degrees of freedom without mass. Forexample, if values of l = 0 and �1 are used in Eq. (2), the corre-sponding Caughey-type matrix becomes:

C ¼ a0Mþ a�1MK�1M ð3Þ

It is evident from Eq. (3) that for typical structural models usedin earthquake engineering practice for which masses are lumped atfloor levels, the mass matrix will be diagonal, and hence, the damp-ing matrix will only have non-zero terms for degrees of freedomassociated with the inertial masses. Thus, the potential for unreal-istic damping forces at rotational degrees of freedom is eliminated.However, this approach does not avoid the presence of unrealisticdamping forces at rotational degrees of freedom when masses areassigned to them. The implementation of such Caughey-typedamping compromises the numerical efficiency of the solution ofthe equations of motion because for l < 0 the calculation of the in-verse of the stiffness matrix K will be required. In addition, as dem-onstrated by Oliveto and Greco [9], Caughey-type damping withl < 0 does not keep the same Caughey coefficients once a changein the support conditions of the structure takes place.

An alternative could be to use only mass-proportional damping(l = 0) in Eq. (3), i.e., Eq. (4). Although this approach will eliminatespurious damping forces, the displacement response of the multi-degree-of-freedom structure will exhibit significant higher-fre-quency content that is not present in the response of real struc-tures. This is due to the presence of small damping ratios at thehigher modes once this solution is devised. Studies such as thoseconducted by Otani [10] demonstrate that damping models thatincorporating stiffness-proportional terms provide a better corre-lation with experimental results.

C ¼ a0M ð4Þ

Hall [5] suggested the elimination of mass-proportional damp-ing contribution and the incorporation of an artificial cap (orbound) to the stiffness proportional damping component. In theauthors’ opinion, this approach would require modifications tothe numerical solution of the equations of motion. Moreover, Char-ney [8] proposed an extension to Bernal’s approach in which thestiffness-proportional component of the Rayleigh-type dampingmatrix is based only on the diagonal terms of the initial stiffnessmatrix, i.e., terms that correspond to the dynamic degrees of free-dom, in order to avoid assigning stiffness-proportional damping todegrees of freedom without mass.

Alternatively, one can assemble a Rayleigh-type damping ma-trix based on the tangent stiffness of the system, i.e., the dampingmatrix is updated at each time step, Eq. (5), in which Kt(t) is thetangent stiffness matrix. Petrini et al. [11], based on test resultsof reinforced concrete bridge piers, showed that using tangentstiffness proportional damping is more appropriate and results in

an increase in estimates of displacement demands compared withpredictions based on initial stiffness or mass-proportional damp-ing. However, the application of this approach may cause numeri-cal solution instabilities once significant changes in stiffness valuesoccur, e.g., changes due to material strength and stiffness deterio-ration. This approach is also computationally more expensive thanthat in which the initial stiffness matrix is used.

CðtÞ ¼ aMþ bKtðtÞ ð5Þ

Leger and Dassault [12] proposed a solution in which Rayleigh-type damping with variable coefficients and the tangent stiffnessmatrix are used, Eq. (6). Leger and Dassault argue that this partic-ular model provides a more rational control of the amount of en-ergy dissipated by viscous damping in nonlinear seismicanalyses. However, the calculation of the scalar coefficients at eachtime step, while preserving modal orthogonality, is computation-ally expensive. In addition, the application of this approach is ques-tionable for structural systems that experience significantdegradation in stiffness because of material strength and stiffnessdeterioration.

CðtÞ ¼ aðtÞMþ bðtÞKtðtÞ ð6Þ

4. Proposed approach for proper modeling of Rayleigh-typedamping in inelastic structures

The approach proposed in this study deals with a formulation ofa Rayleigh-type matrix with a time invariant stiffness matrix thatis assembled by assigning zero stiffness-proportional damping tostructural elements that have the potential to experience inelasticdeformations. This approach requires an increase of the stiffnessproportional damping term to those elements that remain in theelastic range throughout the response to enforce damping energyconservation. The implication is that the structural model will becomposed of a combination of elastic and inelastic elements, whichis a common approach in current earthquake engineering simula-tion studies, but Rayleigh damping is solely applied to the elasticelements. As it will be shown in this section, this modeling ap-proach provides results that are consistent with those obtainedwhen Rayleigh-type damping based on the tangent stiffness ofthe system is used.

The examples presented in this paper will incorporate modelsin which concentrated (localized) plasticity is used. The applicationof these concepts to other types of models, e.g., fiber models, is thesubject of current research by the authors. In the first step, we pro-pose an approach for proper modeling of Rayleigh damping in theform explained here for beam elements whose moment gradient istime invariant. Next, a general approach for elements whose mo-ment gradient is time variant will be presented.

4.1. Structural elements with time invariant moment gradient

For structural elements with time invariant moment gradient, atwo-dimensional, prismatic beam element with six degrees of free-dom (see Fig. 6) is to be replaced with a two-dimensional, pris-matic beam element composed of semi-rigid rotational springs atthe ends and an elastic beam element in the middle (see Fig. 6and Model B in Fig. 4b). The 6-degree-of-freedom beam elementis referred to as the original beam element and the 8-degree-of-freedom beam element as the modified beam element. If the rota-tional stiffness at the end of the original beam element withouttransverse loads is denoted as K0 = 6EI/L (where E is the modulusof elasticity, I the moment of inertia, and L the length of the beam),and the ratio of the rotational spring stiffness, KS, to the elastic

Ordinary beam element

Equivalent elastic beam element with end springs

Modeling Degrees of freedom

Modeling Degrees of freedom

12

34 5

6

123

4 5 6 7 8

i j

j

is js

i

Fig. 6. Beam element and equivalent model that consists of an elastic beam element with springs at both ends.

-3

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3

6

Dis

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Rat

io

Time (sec.)

Displacement Ductility Ratio, SDOF, Model B

Mass Proportional DampingInitial-Stiffness Proportional Damping (Proposed Approach)Tangent-Stiffness Proportional Damping

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15

0 5 10 15

Mco

lum

n/ M

p,be

am

Time (sec.)

Column-end Moment Demand, SDOF, Model B

Mass Proportional DampingInitial-Stiffness Proportional Damping (Proposed Approach)Tangent-Stiffness Proportional Damping

Fig. 7. Comparison between response histories of the idealized SDOF system(Model B) whose 10% critical damping is modeled using ‘‘mass-proportionaldamping”; ‘‘initial stiffness proportional damping” based on the proposed approachin this paper; and ‘‘tangent stiffness proportional damping” subjected to a groundmotion recording from Northridge Earthquake: (a) displacement ductility ratio and(b) normalized column-end moment.


beam stiffness, Ke, of the modified beam element is defined asn = KS/Ke then:

K0 ¼KSKe

KS þ Ke¼ n

nþ 1

� �Ke ð7Þ

In this approach, when stiffness-proportional damping is used,zero stiffness-proportional damping is assigned to the semi-rigidsprings and the stiffness-proportional damping multiplier (seeEq. (1)) of the modified elastic beam element is varied from b tob0. The value of b

0is calculated by equating the damping work done

by the elastic beam of the modified element plus the dampingwork done by the rotational springs with the damping work doneby the original elastic beam, i.e., Eq. (8). In this equation, WD is thetotal damping work done by the element; WD

e is the damping workdone by the elastic beam; WD

s is the damping work done by therotational springs; M is the bending moment at the end of the ele-ment; he is the rotation at the end of the elastic beam; hs is thespring rotation; _he is the rotational velocity at the end of the elasticbeam; and _hs is the spring rotational velocity. Given that hs = 1/nhe

and _hs ¼ 1=n _he one can utilize Eq. (9) for calculating b0.

WD ¼ 2WDe þ 2WD

s ¼ 212

Mhe

� �þ 2

12

Mhs

� �¼ 2

12

bKehe_he

� �

þ 212

bKshs_hs

� �ð8Þ

b0 ¼ ½ð1þ nÞ=n�b ð9Þ

It is important to note that in this approach the semi-rigidspring of the modified beam element has a post-yield stiffnesstuned to provide the target hysteretic response at the end of thebeam. Evidently, one may advocate the use of a fully rigid springat the end of the element, which will make the estimation of b

0a

moot issue, i.e., b = b0. However, this solution is not recommended

in order to avoid numerical instabilities in the response, especiallywhen piece-wise linear hysteretic models are used.

The requirement that the moment gradient on the beam ele-ment be time invariant guarantees that the behavior of the modi-fied beam element is identical to the behavior of the original beamelement. Another implication of this approach is that the computerprogram used to conduct the numerical studies should have thecapability of assigning a stiffness-proportional damping multiplierto individual structural elements. This is a capability common tomany computer programs currently available.

By using the approach proposed hereby for modeling Rayleighdamping of elements, the displacement response is significantlydifferent from that of initial stiffness proportional damping or

mass-proportional damping. Model B in Fig. 4b is an SDOF systemmodeled using the proposed approach and Fig. 7 shows the re-sponse of this system exposed to the same ground motion usedin Fig. 5. It can be seen that when the normalized column-end mo-ment is plotted, unrealistic damping moments are no longer pres-


ent in responses corresponding to the proposed approach andthose corresponding to damping based on the tangent stiffness.The conclusion is that a conventional formulation with initial stiff-ness proportional damping in elements that undergo changes ofstiffness throughout the response overestimates the amount ofdamping when the response is inelastic. This constitutes a majordrawback of the current implementation of Rayleigh-type dampingbased on initial stiffness. This conclusion is consistent with theobservations made by Petrini et al. [11], who suggest that numer-ical models based on initial stiffness proportional damping tend tounderestimate inelastic displacement demands obtained from testconducted with reinforced-concrete piers.

4.2. Structural elements with time variant moment gradient

An extension to the approach for proper modeling of viscousdamping using the Rayleigh model that can be applied to beam/col-umn elements whose moment gradient is time variant is proposed.The proposed approach is useful in modeling elements that experi-ence transverse loads, and elements for which the location of thepoint of inflection is expected to change considerably during theirinelastic dynamic responses. The varying moment gradient resultsin varying elastic stiffness of the structural element making the stiff-ness ratio of the end springs to elastic element, n, a variable. The solu-tion presented for structural components with time invariantmoment gradient can be interpreted as a special case of this moregeneral one that is applicable to structural elements for whichchanges in the location of the point of inflection are not expected.

The approach for proper modeling of Rayleigh-type damping instructural elements with time variant moment gradient involvesmodifying the stiffness matrix of the elastic internal beam elementexplained previously such that the effect of fixed stiffness of thesprings at its two ends is compensated. The modified stiffness matrixof the internal elastic beam element is obtained by equating the stiff-ness matrix of a general prismatic beam element with the condensedform of the stiffness matrix of an assembly that consists of an elastic

beam element with the two end springs. The upper portion of Fig. 6shows the prismatic beam element with 6 degrees of freedom whosestiffness matrix, K0, can be expressed as shown in Eq. (10). In thisequation, A is the cross sectional area, and L is the length of the beamelement. The gray numbers around the stiffness matrix show theassociated degrees of freedom in Fig. 6.

ð10Þ

The lower portion of Fig. 6 shows the assembly consisting of anelastic beam element and two end springs (i.e., 8 degrees of free-dom). The stiffness matrix of the assembly can be expressed by Ka:

Ka ¼Kbb Kbc

Kcb Kcc

� �ð11Þ

In Eq. (11), Kbb is a 2 � 2 stiffness matrix that corresponds to thedegrees of freedom #3 and #6 (i.e., to be eliminated through staticcondensation), Kcc is a 6 � 6 stiffness matrix that corresponds to de-grees of freedom to be kept after condensation, and Kbc and Kcb are2 � 6 and 6 � 2 stiffness matrices generated through partitioningof Ka. Eqs. (12)–(14) express the components of Ka as a function ofstiffness coefficients Sii, Sjj, and Sij; and the elastic element’s momentof inertia Ie. The parameters A and E are those defined for the pris-matic beam element in Eq. (10). The gray numbers around the stiff-ness matrix show the associated degrees of freedom in Fig. 6.

ð12Þ

ð13Þ

ð14Þ


The objective is to find the values of Sii, Sjj, and Sij such that thecondensed form of the stiffness matrix of the assembly, Kcc, shownin Eq. (15), is equal to K0.

Kcc ¼ Kcc � Kcb � K�1bb � Kbc ð15Þ

By assuming predefined values for Ks, and Ie shown in Eqs. (16)and (17), respectively, as a function of n, one can find the values ofSii, Sjj, and Sij identified in Eqs. (18) and (19).

Ks ¼ n6EIe

Lð16Þ

Ie ¼nþ 1

nI ð17Þ

Sij ¼ Sji ¼6ð1þ nÞ2þ 3n

ð18Þ

Sii ¼ Sjj ¼1þ 2n1þ n

Sij ð19Þ

This implies that, in the elastic range of response, the original beamelement could be modeled with an assembly that consists of an

Stiffness coefficients for equivalent elastic beamsprings at both ends of elastic element

2

2.5

3

3.5

4

0 5 10 15 20 25n

stiff

ness

coe

ffici

ent S ii = S jj

S ij = S ji

Fig. 8. Variation of stiffness coefficients Sii and Sij with n for equivalent elastic beamelement.

yM

Basic Parameters

eK

yM

c

y

M

M

pc

p

θθ

pθ

Initial Stiffness

Yield Moment

Capping moment ratio

Plastic Hinge Rotation Capacity

Post-Capping Rotation Capacity Ratio

Derived Parameters

c y pθ θ θ= +

M

Capping Rotation

cM

yy

e

M

Kθ = Yield Rotation

θ

Yielding Point

ME

Fig. 9. Component backbone c

elastic beam element with moment of inertia obtained from Eq.(17), stiffness coefficients obtained from Eqs. (18) and (19), and twoend springs with initial stiffness obtained from Eq. (16). The variationof the stiffness coefficients Sii and Sij with respect to n is plotted inFig. 8. Values of Sii and Sij asymptotically reach 4.0 and 2.0 for largevalues of n. The equivalent stiffness proportional damping coefficient,b0, for the elastic beam element is found by using Eq. (9).

The aforementioned modification of stiffness coefficients Sii, Sjj,and Sij for the elastic beam element in the assembly guarantees theresponse of the assembly is identical to the elastic response of itsequivalent prismatic beam. The inelastic parameters of the endsprings in the assembly are tuned such that the responses of thetwo equivalent systems are identical once the springs in theassembly go inelastic. The backbone characteristics of the endsprings are shown in Fig. 9. The constitutive models consideredfor these nonlinear springs not only include strength and stiffnessdegradation (represented by hp, hpc/hp, and Mc/My in Fig. 9) but alsogradual deterioration of strength and stiffness under cyclic loading(represented by the parameter k), considering a peak-oriented hys-teretic model, based on the energy dissipated in each cycle [13]. Bytuning the values of hp, hpc/hp, and k of end spring elements in orderto obtain the target inelastic behavior of the assembly, one can ob-tain proper modeling of inelastic behavior with proper modeling ofdamping in the component.

The general approach proposed in this paper for modeling Ray-leigh damping of elements has the advantage that it can be appliedto beam/column elements whose moment gradient varies withtime. In addition, this approach includes the use of a constantdamping matrix, which avoids the additional computational effortrequired to calculate damping forces based on the tangential stiff-ness of inelastic members.

5. Implications for seismic performance evaluation

Improper modeling of structural damping has significant impli-cations in terms of the reliability of seismic design and assessmentprocedures. Inadequate estimates of force demands in current de-sign methods as a result of improper modeling of structural viscousdamping can lead to inefficient designs. Similarly, proper estima-tion of deformation and acceleration demands in a building dueto seismic excitation is fundamental to improve the reliability of

yθ cθ θeK

c

y

M

M

cc y

y

MM

M= Capping Moment

pθ pcθ

uθ

u c pcθ θ= +

Capping Point

Post-CappingPost-Yielding Pre-Cappinglastic

urve and its parameters.


building loss estimates that are part of performance-based assess-ment methods [14,15]. Appropriate estimation of the collapse po-tential of buildings is another important component of currentperformance-based assessment methods. In this section, inconsis-tencies in estimation of seismic demands and collapse potential ofstructures with conventional modeling of structural damping areevaluated and compared to values obtained from the viscousdamping modeling methodology proposed in this paper.

An evaluation of the displacement response of a MDOF systemwith Rayleigh-type damping based on the initial stiffness modeledusing the approach proposed in this paper, as well as conventionalRayleigh-type damping, provides a more comprehensive illustra-tion of the differences between both approaches. Fig. 10 depictsmean values of 1st story drift ratio response for the 4-story rein-forced-concrete frame used in Fig. 1 exposed to a set of 40 groundmotion recordings. Modal damping ratios of 5% are applied to thefirst and third modes. Each record was scaled and the peak 1ststory drift ratio responses were plotted as a function of the inten-sity of the records, Sa/g. Scaling was conducted until the limit stateof collapse was imminent. It is evident from this figure that thestructural damping model significantly influences the estimatesof peak drift ratio demands.

The influence of the proposed modeling approach for Rayleigh-type damping based on initial stiffness can also be evaluated in the

1st Story Drift Mean IDA curvesN=4, T1=0.6, Rμ = 3.0 , ξ=0.05, Peak-Oriented model

0

1

2

3

4

5

6

0 5 10 15 20 25 301st story drift (inch)

Sa/g

Conventional damping model Improved damping model

Fig. 10. Mean 1st story drift IDA curves for a 4-story moment-resisting framemodeled with conventional and improved Rayleigh stiffness proportional damping.

Effect of Damping Model on Collapse Fragility CurvesN = 4, T1 = 0.6, Rμ = 3.0, ξ = 0.05, Peak Oriented Model

0

0.25

0.5

0.75

0 1 2 3 4 5Sa(T1)/g

Prob

abili

ty o

f Col

laps

e

Collapse fragility curve from Conventional damping model: datapoints

analytical fragility curve

Collapse fragility curve from Improved damping model: datapoints

analytical fragility curve

Fig. 11. Effects of damping model on collapse fragility curves.

context of seismic collapse assessment by estimating collapse fra-gility curves [16]. A collapse fragility curve expresses the probabil-ity of collapse as a function of Sa. In Fig. 11, the data points showthe smallest value of Sa at which the nonlinear response historysolution of the building subjected to a given ground motion hasconverged, i.e., collapse capacity. The solid squares show this valuefor a building whose damping is modeled using the modeling ap-proach proposed in this paper whereas the diamonds show the col-lapse capacities obtained using a conventional Rayleigh-typedamping model based on initial stiffness. The cumulative distribu-tion function, assuming a lognormal distribution, of these spectralacceleration values that correspond to structural collapse is de-fined as the ‘‘collapse fragility curve” and is shown with heavy linesin Fig. 11. It can be seen that in this case conventional modeling ofRayleigh damping based on the initial stiffness of the system canlead to underestimation of the median of collapse capacity by 30%.

6. Summary and conclusions

This study demonstrates that conventional modeling of linearviscous damping via the implementation of a Rayleigh-dampingmatrix with initial stiffness proportional damping results in inelas-tic dynamic responses that exhibit unrealistic damping forces. Thepresence of these unrealistic damping forces is more prevalentwhen both the damping ratio and the level of inelastic behaviorof the structural system increase. For relatively small levels ofinelastic behavior, deformation demands are not significantly af-fected by conventional modeling of Rayleigh-type damping basedon initial stiffness. This is not the case for strength demands, aspeak column moment demands may increase considerably forthe case of moment-resisting frames.

For levels of inelastic behavior consistent with structural sys-tems approaching the limit state of collapse, the collapse capacityof a system with Rayleigh-type damping based on initial sitffness isoverestimated, i.e., the probability of collapse for a given groundmotion intensity is underestimated. This is due to an overestima-tion of the energy dissipation capacity of the system once conven-tional modeling of Rayleigh-type damping based on initial stiffnessis implemented. This latter issue is significant because contrary tothe common notion that damping energy dissipation in structuralmodels may be de-emphasized in the inelastic range, the numeri-cal results from this study showed that even when cyclic deterio-ration and relatively large levels of inelastic behavior are present,the contribution of viscous damping in dissipating the input energyto the structural system is significant (on the order of 25%) as thehysteretic energy dissipation capacity of structural components isexhausted due to damage and deterioration. Therefore, inappropri-ate modeling of structural damping has the potential to provideerroneous demand prediction for structures exposed to strong dy-namic excitations.

This study proposed a modeling approach developed by theauthors to mitigate these effects and obtain more reasonable esti-mates of seismic demands and seismic collapse capacities for planestructural systems modeled with localized (concentrated) plastic-ity approaches, as well as Rayleigh-type damping based on initialstiffness. The proposed approach proved to be viable even for casesin which cyclic strength and stiffness deterioration was present inthe response.

Acknowledgements

This study originated from discussion among the authors whilethe first two authors of the paper were doctoral students at Stan-ford University working under the supervision of Prof. Helmut Kra-winkler. His guidance and helpful suggestions are appreciated. Thefinancial support provided by the National Science Foundation


through the Pacific Earthquake Engineering Research (PEER) Centeris gratefully acknowledged. The authors would also like to givespecial thanks to Dr. Luis Ibarra for sharing his ideas and providingmuch valuable input at the beginning stages of this study.

References

[1] Jeary AP, Wong J. The treatment of damping for design purposes. In:Proceedings, international conference on tall buildings and Urban Habitat,Kuala Lumpur; 1999.

[2] Chopra AK. Dynamics of structures. 2nd ed. Prentice Hall: New Jersey; 2001.[3] Bernal D. Viscous damping in inelastic structural response. ASCE J Struct Eng

1994;120(4):1240–54.[4] Medina RA, Krawinkler H. Seismic demands for nondeteriorating frame

structures and their dependence on ground motions. PEER Report 2003/15.Berkeley (CA): Pacific Earthquake Engineering Research Center; 2004.

[5] Hall JF. Problems encountered from the use (or misuse) of Rayleigh damping.Earthquake Eng Struct Dynam 2005;35(5):525–45.

[6] Ibarra L, Medina RA, Krawinkler H. Hysteretic models that incorporate strengthand stiffness deterioration. Earthquake Eng Struct Dynam 2006;34(12):1489–511.

[7] Muto M, Beck JL. Bayesian updating and model class selection for hystereticstructural models using stochastic simulation. J Vibr Control 2008;14(1–2):7–34.

[8] Charney FA. Unintended consequences of modeling damping in structures.ASCE J Struct Eng 2008;134(4):581–92.

[9] Oliveto G, Greco A. Some observations on the characterization of structuraldamping. J Sound Vibr 2002;256(3):391–415.

[10] Otani S. Nonlinear dynamic analysis of reinforced concrete building structures.Canadian J Civil Eng 1980;7(2):333–44.

[11] Petrini L, Maggi C, Priestley N, Calvi M. Experimental verification of viscousdamping modeling for inelastic time history analyses. J Earthquake Eng2008;12(1):125–45.

[12] Leger P, Dussault S. Seismic-energy dissipation in MDOF structures. ASCE JStruct Eng 1992;118(5):1251–69.

[13] Zareian F, Krawinkler H. Simplified performance-based earthquakeengineering. Report No. 169, John A. Blume Earthquake Engineering Center,Department of Civil and Environmental Engineering, Stanford University,Stanford, CA.

[14] Krawinkler H, editor. Van Nuys hotel building tested report: exercising seismicperformance assessment. Report No. PEER 2005/11, Pacific EarthquakeEngineering Research Center. Berkeley, California: University of California atBerkeley; 2005.

[15] ATC-58 35% draft guidelines for seismic performance assessment of buildings.Redwood City, CA: Applied Technology Council; 2005.

[16] Zareian F, Krawinkler F. Assessment of probability of collapse and design forcollapse safety. Earthquake Eng. Struct. Dynam. 2007;36(13):1901–14.





Structural finite element model updating using transfer function data

A. Esfandiari a,b, F. Bakhtiari-Nejad a,*, M. Sanayei b, A. Rahai a

a Amirkabir University of Technology, 424, Hafez Ave., Tehran, Iranb Tufts University, Department of Civil and Environmental Engineering, 200 College Ave, Medford, MA, 02155 USA


Article history:Received 16 March 2009Accepted 11 September 2009Available online 21 October 2009

Keywords:StructureStructural health monitoringFinite element model updatingParameter estimationFrequency Response Function

0045-7949/$ - see front matter � 2009 Published bydoi:10.1016/j.compstruc.2009.09.004

* Corresponding author. Tel.: +9821 64543417; faxE-mail address: [email protected] (F. Bakhtiari-Ne

A new method is presented for the finite element model updating of structures at the element level uti-lizing Frequency Response Function data. Response sensitivities with respect to the change of mass andstiffness parameters are indirectly evaluated using the decomposed form of the FRF. Solution of thesesensitivity equations through the Least Square algorithm and weighting of these equations has beenaddressed to achieve parameter estimation with a high accuracy. Numerical examples using noise pol-luted data confirm that the proposed method can be an alternative to conventional model updatingmethods even in the presence of mass modeling errors.

� 2009 Published by Elsevier Ltd.

1. Introduction

Identifying structural damage by using nondestructive test datahas been investigated by many researchers during the past twodecades. This identification plays an important role in betterunderstanding structural behavior and management. Monitoringof structural responses and characteristic parameters such as nat-ural frequencies, mode shapes, Frequency Response Function (FRF),and static displacements provides a great index to assess and iden-tify structural integrity and is addressed by many researchers[1–4]. In this paper, ‘‘damage” is defined as an actual damage, dete-rioration, or errors in initial section stiffness and mass properties.

Many researchers are interested in using dynamic response formodel updating, which can be classified into modal-based and re-sponse-based methods. The modal-based model updating tech-nique relies on the modal characteristics data obtained from anexperimental modal analysis that is extracted from the measuredFRF data indirectly. This numerical extraction process inherentlyintroduces errors and inaccuracies over and above those alreadypresent in the measured data. Sestieri and D’Ambrogio [5] empha-size that the numerical procedures used for modal identificationusing experimental vibration data can introduce errors exceedingthe level of required accuracy to update FE models. In particular,if the tested structure exhibits close modes or regions of high mod-al density, traditional updating tools will fail to give reliable resultsas the extracted modal properties are associated with high levels ofmeasurement errors as well as processing errors.

Elsevier Ltd.

: +9821 66419736.jad).

In response-based finite element model (FEM) updating meth-ods, the measured FRF data are directly utilized to identify the un-known structural parameters. In this class of model updatingmethods the FE models are updated in view of the fully damped re-sponse along a frequency axis and not an estimated set of modalproperties. Also, the amount of available test data is not limitedto a few identified eigenvalue and eigenvectors, and FEM updatingcan be performed using many more data points in the FRF. Natke[6] and Cottin et al. [7] state the benefits of FRF-based over mod-al-based updating algorithms.

Most of the FRF-based model updating techniques are used tominimize a residual error between analytical and experimental in-put force or output response [8–11]. The input error (nodal errorforce) formulations are characteristically different compared tomany other FRF model updating formulations since the linear de-sign parameters (such as material properties and section geometricproperties) remain linear in the updating. Cottin et al. [7] showthat these formulations have a tendency to result in more biasedparameter estimations than those done by output residualmethods.

Larsson and Sas [12] developed a model updating techniqueutilizing an exact dynamic condensation in which the objectivefunction did not require the computation of the impedance matrix.They emphasized that the desired frequency range, which can beupdated, is inherently limited by the condensation procedure.Incomplete measurements and their implication on the FRF modelupdating formulation seem to severely restrict the method’s abilityto update larger FEMs.

Modak et al. [13] tuned structural models by a direct modelupdating method and an iterative model updating method based





A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64 55

on the FRF data through computer and laboratory experiments. Thepredictions by the directly updated model are reasonably accuratein the lower frequency range but the predictions contain a signifi-cant error in the higher frequency range.

Araujo dos Santos et al. [14] proposed a damage identificationmethod based on the FRF sensitivities. They indicate that betteridentification results can be obtained in the lower frequency rangeusing excitation points where there are no modal nodes. It is dem-onstrated in this research that for small damage, the measurementerrors are the main influence in the identification quality, whereasfor large damage the incompleteness becomes the most importantfactor.

In contrast to using objective functions based on input residualmethods, minimization of the output-residuals can be used for fi-nite element model updating [15–18]. Using output-based meth-ods, the difference between the measured and predictedstructural response is minimized. The algebraic nonlinearity ofmodel updating algorithm makes this approach more challenging.Lin and Ewins [19] developed a model updating method by directlyusing measured FRF data. They state that providing much moreinformation in a desired frequency range is a major advantage ofusing FRF data over using modal data in model updating.

Conti and Donley [20] updated a full finite element model usingresponse data by minimizing the residual error between the ana-lytical and experimental response. In order to overcome coordinateincompatibility, the analytical system is reduced to the size of theexperimental data set using a reduction technique. Wang et al. [21]derived a damage detection algorithm that can be used to deter-mine a damage vector by indicating both location and magnitudeof damage from perturbation equations of FRF data. An iterativescheme and an effective weighting technique has been introducedto overcome incomplete measurements and reduce the adverse ef-fects of measurement errors.

Park and Park [22] proposed a damage detection technique thatutilizes an accurate analytical finite element model based on theincomplete FRF data in numerical and experimental environments.Authors also discuss frequency regions where the suggested meth-od works satisfactorily. De Sortis et al. [23] investigated the dy-namic behavior at low vibration levels of an existing masonrybuilding subjected to forced vibration sine dwell or sine sweepusing an output error equation. Possibly, on account of weak non-linearities of the tested building, the measurements obtained withsine dwell tests seemed more suited than those obtained by sinesweep tests for the application of identification techniques.

Hwang and Kim [24] used FRF data to find the location andseverity of damage in a structure by considering only a vector sub-set of the full set of FRFs. Zimmerman et al. [25] extended theirprevious developments in minimum rank perturbation theory(MRPT) for damage detection using experimental and numericalFRF data. The FRF-based results are shown to be less sensitive tonoise if proper frequency points are used, and provide a damageassessment similar to that obtained using identified modal param-eters, but at a substantially reduced level of effort.

Model updating techniques ordinarily assume that damage isuniformly distributed along element body and consider a stiffnessreduction factor for the whole element. This is an effective cross-sectional property leading to the same effective stiffness. It is anidealized assumption while damage might be partially distributedalong an element body or appears as a localized stiffness reductionby a crack or a notch. At such cases, model updating techniques re-sult in an equivalent stiffness reduction factor for the partially dis-tributed damage. For a more accurate prediction of the damagelocations and severity, an appropriately fine finite element meshshould be adopted. This will increase the number of unknownparameters for the model updating. As an alternative, parametergrouping can be used to decrease the number of unknown param-

eters. Also by a two stage method, a coarse mesh can be used tofind the region of damage and recognizing undamaged parts, andthen an accurate prediction can be done by using a fine mesh forthe damaged components [18].

A relatively fine mesh is necessary for reliable predictions of theanalytical response for comparison with the modal responses inthe measured frequency range. After determining the mesh size,the number of analytical degrees of freedom (DOF) always exceedsthe number of measured coordinates and consequently FRF formu-lations based on both the input-residuals and the output-residualscannot be solved directly. Expansion of the measurement vector orreduction of the full finite element is necessary due to incompati-bility between the number of measured DOFs and the number ofanalytical DOFs. No matter how sophisticated is it, the expansionor reduction schemes may introduce numerical errors FRF modelupdating as an inevitable consequence. While using a full set ofmeasurements at all degrees of freedom, the model updating prob-lem is algebraically linear and can be solved in one iteration.Reducing the model or expanding the data converts model updat-ing to an algebraically nonlinear problem due to matrix inversioncontaining unknown parameters. Although the Taylor seriesexpansion of the objective function can be used without modelreduction or data expansion, model updating using the Taylor ser-ies linearization will still be nonlinear.

In this study, a structural model updating technique is presentedusing FRF data and measured natural frequencies of the damagedstructure without any expansion of the measured data or reductionof the finite element model. To decrease nonlinearity of the modelupdating process, the change of FRF of a structure is correlated tothe change of stiffness, mass and damping through sensitivity equa-tions, which have been derived using the change of eigenvectorsand measured natural frequencies of the damaged structure. Thechange of eigenvector is also expressed as the linear combinationof the original eigenvectors, while eigenvector participation factoris a function of the perturbation of the stiffness and mass. The sen-sitivity equation set is solved by the Least Square method through aproper weighting procedure. The effect of excitation frequency andweighting factor on finite element model updating has been suc-cessfully addressed through a truss model example.

2. Theory

Using a finite element modeling, equation of motion of a linearelastic time-invariant structure with n degree of freedoms is givenby:

M€xðtÞ þ C _xðtÞ þ KxðtÞ ¼ f ðtÞ ð1Þ

where M, C and K are the mass, damping and stiffness matrices ofthe structure, respectively. f(t) is the vector of applied forces andx(t) is the vector of structural response to the applied force. For aharmonic input, as an external force, displacement response canbe expressed as:

f ðxÞ ¼ FðxÞejxt and xðxÞ ¼ XðxÞejxt ð2Þ

where x is the frequency of the excitation load. Substituting (2)into (1) yields the input–output relationship using transfer functionH(x):

XðxÞ ¼ HðxÞFðxÞ ð3Þ

where transfer function H(x) (or the impedance matrix) in terms ofsystem properties, can be defined as:

HðxÞ ¼ ð�x2M þ jxC þ KÞ�1 ð4Þ

by a spectral decomposition [3], the response of the structure to aunit harmonic load can be expressed as:

56 A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

HilðxÞ ¼Xn

r¼1

/ir/lr

x2r �x2 þ 2jfrxrx

ð5Þ

where Hil(x) is the displacement of the ith degree of freedom whensubjected to the applied unit force at the lth degree of freedom.Hij(x) represents the entire impedance matrix where i representsthe measurement points and j is the excitation point. The rth modeshape (eigenvector), natural frequency and damping loss factor arerepresented as ur, xr and fr, respectively. The higher mode shapeswhose natural frequencies are far from the excitation frequency ofthe applied load x can be excluded from (5) with some impactdue to using incomplete measurements.

Due to damage, the rth mode shape of the structure changes bythe amount of dur, therefore:

/rd ¼ /r þ d/r ð6Þ

where the index d indicates a damage case. Substituting (6) into (5)and using the first nm measured natural frequencies of the damagedstructure; the response (displacement) of the damaged structurecan be evaluated as:

Hild ffiXnm

r¼1

/ird/lrd

x2rd �x2 þ 2jfrdxrdx

þXn

r¼nmþ1

/ir/lr


ð7Þ

The approximation given in (7) is realistic because we can measurenatural frequencies with high accuracy. The second term is relatedto the unmeasured part of natural frequencies and damping lossfactors and is used to alleviate the effects of incomplete measure-ments. Numerical simulation shows this part increases the accuracyof (7) and its convergence rate. The second part of (7) corrects itselfas the optimization process updates the parameters of the struc-ture. Substituting (6) into (7) and neglecting the second order termyields:

HildðxÞ ffiXnm

r¼1

/ir/lr


þXnm

r¼1

/ird/lr


þXnm

r¼1

d/ir/lr


þXn

r¼nmþ1

/ir/lr


ð8Þ

The first term of this equation can be evaluated using the eigenvec-tor of the intact structure, the measured natural frequencies, anddamping loss factors of the damaged structure.

Eigenvectors of a structure introduce an orthogonal space vec-tor that can be used to represent any vector of the same order bytheir linear combination. Therefore, the modal method [26] makesuse of the assumption that, the rate of change of modal vector forany mode shape can be approximated as a linear combination ofeigenvectors for all modes. Therefore, change of mode shapes ofthe structure due to the damage, can be evaluated using a first or-der series as [26]:

d/i ffiXn

q¼1

aiq/q ð9Þ

where

aiq ¼/T

q dK �x2i dM

� �/i

x2i �x2

q

� � for q – i; aii ¼ �/T

i dM/i

2for q ¼ i

ð10Þ

For cases such as a large structure where not all eigenvalues andeigenvectors are available, truncated forms of eigenvector deriva-tion can be used [27]. In comparison to using a Taylor series expan-sion of the transfer function, the first order expansion in (9) isweakly nonlinear. This is due to the fact that (9) does not requirederivation of the denominator in (5). Using the formulations givenin (9) and (10), the measured response is correlated to the perturba-tion of the stiffness and mass matrices as:

HildðxÞ ¼Xnm

r¼1

/ir/lr


þXn

r¼nmþ1

/ir/lr


þ 2Xnm

r¼1

Xn

q¼1

/ir /TqdK/r

� �/lr


� �x2

i �x2q

� �

� 2x2i

Xnm

r¼1

Xn

q¼1

/ir /TqdM/r

� �/lr


� �x2

i �x2q

� � ð11Þ

The stiffness and mass matrices of an individual structural elementsuch as a bar or beam can be described as follows [28]:

Ke ¼ ASePSeATSe and Me ¼ AMePMeAT

Me ð12Þ

where PSe and PMe contain nonzero eigenvalues of stiffness and massmatrices as their diagonal entries and ASe and AMe contain corre-sponding eigenvectors of nonzero eigenvalues of the stiffness andmass matrices. The stiffness and mass matrices of the structureare constructed by assembling the elemental stiffness and massmatrices as:

K ¼Xne

i

TTeiASeiPSeiA

TSeiTei ¼ ASPSAT

S and

M ¼Xne

i

TTeiAMeiPMeiA

TMeiTei ¼ AMPMAT

M ð13Þ

where Tei is the transformation matrix of the ith element from thelocal coordinate to the global coordinate. Matrices AS(n � nPS) andAM(n � nPM) are defined as the stiffness and mass connectivitymatrices, and the diagonal matrices PS and PM have the same ele-mental stiffness and mass parameters as their diagonal entries.Numbers of parameters are dependent on element type: while fortruss elements there is only one axial rigidity parameter, forbeam-column elements there are two parameters for axial and flex-ural rigidity.

Since the global stiffness matrix is a linear function of the ele-mental stiffness parameters, (13) can be perturbed to get:

K þ dK ¼ ASðPS þ dPSÞATS and M þ dM ¼ AMðPM þ dPMÞAT

M ð14Þ

where dPS and dPM are the changes of elemental stiffness and massparameters caused by the damage. Expanding (14) and subtracting(13) from it yields a parameterized form of the perturbed globalstiffness and mass matrices as:

dK ¼ ASdPSATS and dM ¼ AMdPMAT

M ð15Þ

Using these definitions for dK and dM in (15) and substituting into(11) and by a mathematical manipulation, the change in the FRFof the structure is correlated to the damage as:

HilðxÞ ¼ SSilðxÞdPS þ SM

il ðxÞdPM ð16Þ

where

HilðxÞ ¼ HildðxÞ � eHildðxÞ ð17Þ

and

Fig. 1. Geometry of truss model.

Table 1Cross-sectional area of truss members.

Member Area (cm2)

1–6 187–12 1513–17 1018–25 12

Fig. 2. Degrees of freedom of truss model.


eHilðxÞ ¼Xnm

r¼1

/ir/lr


þXn

r¼nmþ1

/ir/lr


ð18Þ

SSilðxÞ and SM

il ðxÞ are (1 � np) row vectors of the stiffness and massparameters sensitivity for ith degree of freedom subjected to the ap-plied unit load at the lth degree of freedom as:

SSilðxÞ ¼ 2

Xnm

r¼1

Xn

q¼1

/ir /TqAS � diag AT

S /r

� �� /lr


� �x2

r �x2q

� �and

SMil ðxÞ ¼ 2

Xnm

r¼1

Xn

q¼1

/ir /TqAM � diag AT

M/r

� �� /lr


� �x2

r �x2q

� � ð19Þ

Operator diag represents the entries of a diagonal matrix as a vectorand vice versa. dPS and dPM are the vectors of stiffness and massparameters. Sensitivity equations of all measurements at the inter-ested frequency ranges can be represented by a set of equations as:

HðxÞ ¼ ½ SS SM � dPS

dPM

" #¼ SðxÞdP ð20Þ

where S(x) is the total sensitivity matrix for all measurements anddP is the vector of all stiffness and mass parameters changes.

One source of problems in the sensitivity-based model updatingmethod is the noise-induced measurement error in the sensitivitymatrix that will cause a convergence to a local minimum. In theproposed method, measured natural frequencies and damping lossfactors are necessary to construct sensitivity equations. Usingmodern electronic transducers, measured natural frequencies arevery accurate and some researchers have assumed noise-free mea-surement of natural frequencies. Also, any inevitable error does noteffect results if the excitation frequency of the applied load is not inthe vicinity of the nearest measured natural frequencies. For un-damped or lightly damped structures, the denominators of (7),and consequently S(x), are dominated by x2

id �x2. If the excita-tion frequency is selected close to the resonance frequency a smallerror in measured frequency introduces a significant change inx2

id �x2, causing large changes in the response. By moving awayfrom the resonance frequency, this error sensitivity is significantlyreduced, resulting in less vulnerability to measurement errors.

Although damping loss factors measurement is not as accurateas natural frequency measurement, appropriate selection of excita-tion frequency can alleviate measurement error. Moving the exci-tation frequency away from resonance frequency will result inresponses that are less sensitive to damping and errors in dampingmeasurement.

Eq. (20) can be solved by several methods such as Least Squaremethod (LS), Non-Negative Least Square method (NNLS) and Sin-gular Value Decomposition method (SVD). The quality of predicteddamage by expression (20) depends on the quality of the measuredFRF data and on a weighting technique which needs to be appliedto the equations. This weighting technique uses a weighting factorwithout which the Least-square solution would be dominated byequations with the largest coefficients. This means that some equa-tions would overshadow the information from other equations.

Several methods have been suggested in the literature forweighting the equation. One may normalize each equation by itssecond norm so all equations have the same weight in parameterestimation. If an equation associated with the frequency x has val-ues of Hd(x) and HdðxÞ of similar magnitudes, the measurementerrors may be significantly magnified after the weighting. To over-come this problem, such an equation should be removed [29].

Kwon and Lin [30] state that weighting the sensitivity equationby x�1 decreases inaccuracy of the finite element modeling athigher frequencies [30]. In this study, sensitivity of the FRF (20)is derived using first order derivation of the mode shapes using(9). Inaccuracies in mode-shape changes using first order approxi-mations increase in higher frequency ranges, causing less accuratesensitivity equations. However, the sensitivity of FRFs increases inhigher frequency ranges. From a sensitivity point of view, there-fore, it is necessary to amplify the sensitivity equation in higherfrequency ranges. Due to more approximations at higher frequen-cies, the weight of these sensitivity equations should be decreased.Numerical simulation shows that sensitivity equation weightingby x�1 creates a balance between the use of higher sensitivity ofFRF to change of parameters and the lower accuracy of first orderderivation. The following example shows a successful applicationof the proposed method using a bowstring truss.

3. Bowstring truss example

The presented damage detection algorithm was applied to atruss structure as shown in Fig. 1. The structure is modeled numer-ically using the finite element method with basic structural ele-ments, i.e. bar elements. The finite element model of thestructure consists of 25 elements and 21 DOFs. Truss elementsare made from steel martial with Young’s modulus of 20 MPa,and cross-sectional areas are given in Table 1. Kinematics degreesof freedom of this truss model are shown in Fig. 2.

The unknown parameters are axial rigidity of elements, EAwhere A is the cross-section area of truss element and E is theYoung’s modulus. Several damage cases are considered to investi-gate the influence of location, severity and number of the damagedelements on the results. Table 2 shows specifications of these

Table 2Percent of stiffness and mass reduction of elements.

Case number Element number and percent of damage

1 Element number 4 10 – – –Damage 30%(K) 50%(K) – – –

2 Element number 3 9 22 – –Damage 40%(K) 50%(K) 60%(K) – –

3 Element number 3 9 20 25 –Damage 20%(K) 30%(K) 30%(K) 20%(K) –

4 Element number 5 10 13 20 24Damage 40%(K) 40%(K) 50%(K) 40%(K) 30%(K)

5 Element number 5 11 17 – –Damage 20%(M) 30%(M) 30%(M) – –

6 Element number 3 14 19 23 –Damage 20%(M) 30%(M) 30%(M) 30%(M) –

7 Element number 4 8 12 24 –Damage 30%(K) 40%(K) 30%(K) 40%(K) –Element number 8 16 19 – –Damage 30%(M) 30%(M) 30%(M) – –

8 Element number 4 12 19 – –Damage 30%(K) 30%(K) 40%(K) – –Element number 8 16 23 – –Damage 30%(M) 30%(M) 30%(M) – –

9 Element number 4 12 16 25 –Damage 30%(K) 30%(K) 30%(K) 40%(K) –Element number 9 20 23 – –Damage 30%(M) 30%(M) 30%(M) – –

-35

-30

-25

-20

-15

-10

0 50 100 150 200 250 300 350

f (Hz)

log(

FR

F)

Exact

nm =15

nm =10

Fig. 3. FRF of the intact structure at 10th degree of freedom.


damage cases. In reality, for the structure shown in Fig. 1, the FRFdata should be extracted from an experiment setup. For this re-search, representative FRF data were simulated numerically usingthe finite element method with superimposed measurement errors.

Some sources of errors exist in the proposed damage detectionalgorithm. One is caused by adopted linear approximation to eval-uate the change of mode shape in (9). The adverse effects of thisapproximation can be reduced by an iterative model updating tomove from the intact model to damaged model of the structure.Another source of error in the presented method is the incompletemeasurements of the natural frequencies of the damaged struc-ture. The proposed method does not work well over all frequencyregions; therefore, the selection of excitation frequencies is veryimportant to successful identification. Adverse effects can be de-creased by a proper selection of the excitation frequencies. As anexample, damage case #4 is considered to describe the procedurefor selection of the excitation frequency points in detail.

Due to practical limitations, it is assumed that only a few of thenatural frequencies are measurable, since amplitude of oscillationat higher mode shapes decreases and accurate measurements be-come impractical. This incompleteness is a source of error in themathematic of sensitivity equation (20). To investigate this impact,by a numerical simulation the FRF of the truss structure at DOF #10subjected to the applied load at DOF #15 is plotted in Fig. 3. TheseFRFs are evaluated using responses at DOFs 10 and 15, and alsousing all 21 structural mode shapes data.

As Fig. 3 shows, the impact of incomplete measurement of nat-ural frequencies on the calculated FRF of the structure depends onexcitation frequency of the applied harmonic load. As mentionedby Dos Santos et al. [14], to achieve an accurate evaluation of theFRF of the structure it is necessary that the natural frequency ofthe included mode shapes be large enough compared to the fre-quency of excitation. For excitation frequency in low mode shapefrequency ranges, the evaluated FRF is very accurate even by afew measured natural frequencies. Acceptable evaluation of theFRF using a truncated set of the measured natural frequencies ispossible for excitation frequencies close to the natural frequencies

of the structure even at higher modes frequency range. This is dueto the fact that for the excitation frequencies close to the naturalfrequencies, H(x) is dominated by the nearest resonances.

In this paper a set of sensitivity equations is derived to relatethe change of structural parameters to the subtraction of the mea-sured response of the structure Hd(x) and the calculated value ofeHdðxÞ. Fig. 4, illustrates the calculated value of eHildðxÞ, using 10,15 and 21 mode shapes along with exact Hd(x) for the 10th DOFsubjected to the harmonic loads at the 15th DOF. eHildðxÞ is approx-imated in (18) using natural frequencies of the damaged structureand mode shapes of the intact structure.

It is very important that the differences between the responseof the damaged structure Hd(x) in (11) and eHðxÞ in (19) be largeenough to increase the chance of successful prediction of the dam-age location and severity. Fig. 4 shows, regardless of the excitationfrequency of the applied load, change of the structural responsedue to damage at lower frequency range is small, and this changemay be overshadowed by measurement errors. Note that the main

-35

-30

-25

-20

-15

-10

0 50 100 150 200 250 300 350f (Hz)

log

(FR

F)

FRF of Damaged Case

, nm=21

, nm=15

, nm=10

~ωH~ωH

)()()(

~ωH

Fig. 4. FRF of the damaged structure and eHildðxÞ at 10th degree of freedom.

Table 4Selected frequency ranges for model updating.

Damage case 1, 2, 3, 4 5, 6 7, 8, 9

Frequency range 80–90 230–245 80–90100–120 290–315 100–120185–200 335–345 185–200230–245 385–400 230–245280–295 – 280–295– – 325–335– – 385–390– – 420–435


difference between the formulation of eHðxÞ and the exact FRF ofthe damaged structure Hd(x) is the used mode shapes. This isdue to the fact that in eHdðxÞ, mode shapes of the intact structuresare used instead of the mode shapes of the damaged structures asdone in Hd(x). Low discrepancy of value of Hd(x) and eHdðxÞ indi-cates that the change of mode shapes of the structure due to dam-age is small. As mentioned before, when using close value of Hd(x)and eHdðxÞ, noise-induced errors in the model updating process be-come more significant and parameter estimation process show lessstability and robustness. Therefore, the excitation frequency of ap-plied loads must be moved to a higher frequency range with a sig-nificant difference between Hd(x) and eHdðxÞ.

Although large relative difference of Hd(x) and eHðxÞ also occursat anti-resonance frequencies, these regions should not be selectedfor model updating. At these regions amplitude of vibration is verylow and therefore the response is easily contaminated. Anti-reso-nance frequencies are also a local characteristic of the structuresand may not be sensitive to damage in a complex and large struc-ture. Excitation frequencies were selected at a region close to thenatural frequencies of the damaged structure and not exactly atits natural frequencies. Natural frequencies of the damaged casesand frequency ranges of model updating are given in Tables 3and 4. In these frequency ranges, model updating is applied withfrequency increments of 1 Hz. A single harmonic excitation is ap-plied at DOFs 3, 9, 15 and 19 for each load case, and DOFs 1, 8,14, 15, 20 and 21 are selected as six response measurementlocations.

Table 4 shows that for damage cases 5–9, where there are somechanges in mass matrices, selected frequency ranges for modelupdating should be adopted at higher ranges, because at theseranges contribution of x2M become large in compare to K and

Table 3Natural frequencies of the damaged truss.

Mode number Intact Damage case

1 2 3 4

1 30.3 29.2 28.1 29.32 69.0 67.4 65.3 67.23 96.3 93.1 93.2 95.74 181.8 178.7 168.9 176.2 15 223.2 210.2 215.7 215.7 26 275.6 270.9 252.2 265.4 27 321.6 320.0 313.9 318.2 38 352.0 349.5 322.1 340.9 39 357.7 357.5 342.1 355.1 3

10 373.0 369.7 362.1 361.5 311 414.7 409.2 399.9 410.7 412 460.5 443.8 438.4 446.0 413 465.7 463.5 452.5 457.6 4

the structural response is more sensitive to the mass changes. Anincrease of natural frequencies at these ranges indicates that thestructural response is dominated by mass reduction. It should benoted that the model updating process diverges for cases 5–9 ata frequency range same as cases 1–4. At these frequency ranges,changes of the FRF are controlled by stiffness and are less sensitiveto mass changes.

In addition to the error present in mathematical modeling, mea-surement and data processing errors are also existed in the exper-iment. To simulate these experimental inaccuracies, measurementerror is considered by adding random error to the simulated finiteelement data. A 5% uniform random error has been added to theexact simulated FRF by the finite element method and natural fre-quency measurement is considered to be noise-free. Figs. 5–13show results of the proposed finite element model updatingalgorithm.

Figs. 5–13 show that the proposed method is capable of detectingthe damage location, and also quantify the severity using the incom-plete noisy measured FRF data. In order to quantify the accuracy ofthe predicted results, some indices are used to evaluate the confi-dence level of the results. The Mean Sizing Error defines an averagevalue of the absolute discrepancies between the parameters truedamage values dPt and the predicted damage values dPp [31]:

MSE ¼ 1ne

Xne

e¼1

jdPt � dPpj for 0 � MSE � 1 ð22Þ

the relative error,

RE ¼Pne

e¼1jdPt j �Pne

e¼1jdPpjPnee¼1jdPt j

for 0 � RE � 1 ð23Þ

and closeness index,

CI ¼ 1� kdPt � dPpkkdPtk

for 0 � CI � 1 ð24Þ

5 6 7 8 9

28.8 30.7 31.2 29.0 30.2 30.466.9 70.7 70.1 65.5 69.0 69.093.0 98.2 98.6 97.0 96.5 95.968.0 186.0 185.2 178.6 178.2 180.013.6 225.9 228.2 214.3 213.9 212.363.9 281.0 282.1 275.1 272.6 273.310.5 324.5 333.7 321.5 330.5 314.824.9 354.3 358.9 336.7 340.8 347.136.8 362.4 366.9 365.3 360.2 365.261.6 377.9 383.6 375.5 381.2 376.610.7 421.3 418.3 412.3 410.2 403.122.6 465.6 471.8 436.8 452.3 453.029.1 471.1 481.7 470.0 471.1 473.3

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent

Actual DamagePredicted Damage

Fig. 5. Actual and predicted damage of case 1 using noisy data.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent



-0.1

0

0.1

0.2

0.3

0.4

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent



-0.1

0

0.1

0.2

0.3

0.4

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent



-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent



-0.1

0

0.1

0.2

0.3

0.4

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent




gives an index of the distance between the true and estimated dam-age parameter vectors. An element is identified as damaged ifjdPpj � 2�MSE [31]. The smaller values of the MSE and RE and lar-ger values of the CI indicate better results. The damage indices ofpredicted results are given in Table 5.

As Figs. 5–13 and Table 5 show, the proposed finite elementmodel updating method is capable of localization and quantifica-tion of damage for all elements. Additionally, a few elements ap-peared as false positive slight severity of damage. This is due tothe presence of noise in the FRF data, and sometimes to selected


excitation frequency. Comparison of predicted damages shows thestability and success of detection despite increasing severity ofdamage. Increasing the severity of damage and difference of FRFresponses of the intact and damaged elements smeared undesir-able affects of noisy measurements.

-0.1

0

0.1

0.2

0.3

0.4

0.5

Element No.

Dam

age

Perc

ent


1 3 5 7 9 11 13 15 17 19 21 23 25

()a(

Fig. 11. Actual and predicted damage of case 7

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 3 57 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent


b()a(


-0.1

0

0.1

0.2

0.3

0.4

0.5

Element No.

Dam

age

Perc

ent


1 3 5 7 9 11 13 15 17 19 21 23 25

)a(


Averages of the estimated parameters do not reflect the robust-ness and confidence of the parameters estimation process. Toinvestigate robustness of a method, it is necessary to evaluate thestandard deviation and/or coefficient of variation (COV) of the pre-dicted unknown parameters in the Monte Carlo simulations. Low

-0.1

0

0.1

0.2

0.3

0.4

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.D

amag

e Pe

rcen

t


)b

using noisy data: (a) mass; (b) stiffness.

-0.1

0

0.1

0.2

0.3

0.4

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent


)


-0.1

0

0.1

0.2

0.3

0.4

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent


)b(


Table 5Comparison of damage indexes.

Case number Index

MSE RE CI

1 0.009 �0.220 0.852 0.011 �0.169 0.893 0.016 �0.310 0.814 0.040 �0.381 0.775 0.003 �0.066 0.966 0.005 �0.070 0.947 0.012 0.456 0.868 0.016 0.302 0.799 0.018 0.397 0.80

0

0.1

0.2

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

CO

V

Fig. 14. Coefficient of variations of the estimated parameters at case 7.

Table 6Frequency ranges for stiffness deterioration case.

Set 1 Set 2

Frequency ranges 80–93 80–93100–120 100–120165–177 165–177185–200 185–200210–220 210–220228–235 228–235– 305–317– 325–335

-0.025

0

0.025

0.05

0.075

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent

Actual Damage

Predicted Damage

()a(

Fig. 15. Actual and predicted stiffness for 5% random deterioration of truss using frparameters.


standard deviation and COV indicate less scatter in the predictedparameters. As a template, COVs of the estimated unknown param-eters for the seventh damage scenario are plotted in Fig. 14.

Except for element number 13, COVs of all predicted stiffnessparameters of elements are low and indicate a robust solution.High COV of element 13 can be interpreted as a low observabilityof this element compared to other elements. This issue can be re-solved by rearranging excitation and response measurementstations.

Accuracy and confidence in the predicted parameters heavilydepends on the ratio of measurement error to the observedchange of the monitored structural response. Monitored struc-tural response can be natural frequencies, mode shapes, or FRFs.For a low change of response which can be related to the adoptedstructural response for monitoring, or level of damage, modelupdating even using low error contaminated measurement canyield a low confidence and high scattered prediction. As statedby Ren and Beards [29], weighting of equations related to themeasurement points with a low level of change in response canadversely affect predicted results. At such cases using higher fre-quency ranges can improve the confidence level in parameterestimation, since; generally higher frequency ranges and modeshapes are more localized and might be able to observe smallchange. To investigate low level damage scenarios a 5% randomstiffness variation is added to all truss members as a deteriorationcase and also a 10% random stiffness variation of the bottomchord (elements 7–12) is considered as another deteriorationcase. Model updating is done at frequency set 1 given by Table6. The predicted parameters and their COV are plotted in Figs.15 and 16, respectively.

Figs. 15 and 16 show that the proposed method is generallycapable of detection of light distributed damage cases, althoughsome elements are detected by low accuracy. Elements 13 and 7have high coefficients of variations and show false stiffnesschanges. This might be due to low observability of these elementsusing the frequency set 1 in Table 6. Parameters estimation resultscan be improved by adopting model updating frequency points athigher frequency ranges. For verification, the above deteriorationcases are redone using higher frequency ranges as given by set 2of Table 6. The estimated parameters and COVs are plotted in Figs.17 and 18.

Fig. 17 shows that overall 5% random stiffness variations arewell captured with COVs well below 5%. Fig. 18 shows that 10%random stiffness variations are well captured with COVs well be-low 15%. Both cases show that measurement set 2 can better ob-serve the overall changes in the stiffness. Better parametersestimation and lower COV of the predicted parameters prove that

0

0.05

0.1

0.15

0.2

0.25

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent

)b

equency set 1: (a) stiffness estimation; (b) coefficient of variations of estimated

-0.1

-0.05

0

0.05

0.1

Element No.

Dam

age

Perc

ent

Actual Damage

Predicted Damage

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 3 5 7 9 11 13 15 17 19 21 23 25

1 3 5 7 9 11 13 15 17 19 21 23 25Element No.

Dam

age

Perc

ent

)b()a(

Fig. 16. Actual and predicted stiffness for 10% random deterioration of bottom chord using frequency set 1: (a) stiffness estimation; (b) coefficient of variations of estimatedparameters.

1 3 5 7 9 11 13 15 17 19 21 23 25-0.1

-0.075

-0.05

-0.025

0

0.025

0.05

0.075

0.1

Element No.

Dam

age

Perc

ent

Actual Damage

Predicted Damage

0

0.05

Element No.

Dam

age

Perc

ent

)b()a(

3 21 5 7 9 11 13 15 17 19 1 23 25

Fig. 17. Actual and predicted stiffness for 5% random deterioration of truss using frequency set 2: (a) stiffness estimation; (b) coefficient of variations of estimatedparameters.

-0.1

-0.05

0

0.05

0.1

Element No.

Dam

age

Perc

ent

Actual Damage

Predicted Damage

0

0.05

0.1

0.15

1 3 5 7 9 11 13 15 17 19 21 23 25

1 3 5 7 9 11 13 15 17 19 21 23 25Element No.

Dam

age

Perc

ent

)b()a(

Fig. 18. Actual and predicted stiffness for 10% random deterioration of bottom chord using frequency set 2: (a) stiffness estimation; (b) coefficient of variations of estimatedparameters.


using higher frequency ranges can improve the quality of predic-tions. These results and the low number of iteration illustratesthat this is a highly robust method for parameter estimation.

Although in most real cases, the mass matrices of the structuresare not usually changed by damage, some deviation in stiffnessidentification is possible due to an inaccurate assumption regard-ing the mass of the intact and damaged elements. This inaccuracyintroduces some errors in eigenvector of the undamaged structure

that will be used to construct sensitivity equations. In this study,numerical simulation proves that, considering up to a 10% randomerror in mass matrices, the parameter estimation process is still ro-bust. Results of such a case of stiffness parameters estimation aregiven in Fig. 19 and the coefficient of variation of the predictedparameters is given in Fig. 20. Low COVs of the estimated parame-ters indicates stability and robustness of the method against massmodeling errors.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

Dam

age

Perc

ent


Fig. 19. Actual and predicted damage considering mass modeling errors.

0

0.1

1 3 5 7 9 11 13 15 17 19 21 23 25

Element No.

C.O

.V

Fig. 20. Coefficient of variations of the estimated parameters considering massmodeling errors.


4. Conclusions

A structural damage detection method is presented using Fre-quency Response Function (FRF) and measured natural frequen-cies. In a decomposed form, change of the FRF of the structuredue to damage is evaluated by measured natural frequencies andderivation of mode shapes with respect to mass and stiffnessmatrices. The change of mode shapes is expressed as a linear com-bination of original eigenvectors of the intact structures. Elementallevel sensitivity of the FRF of the structure to occurrence of damageis characterized as a function of stiffness, mass, and dampingparameters change. The sensitivity is weighted by x�1 to improvethe stability of the method and obtain more confidence results.Sensitivity equations are solved by the Least Square method toachieve change of structural parameters. Results of truss modelshow the ability of this method to identify location and severityof parameters change at the elemental level in a structure. It wasalso shown that, the finite element parameter estimation resultsare improved using higher excitation frequencies.

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[6] Natke HG. Einfuehrung in Theorie und Praxis der Zeitreihen- undModalanalyse, 3rd ed. Wiesbaden: Vieweg Verlag; 1983.

[7] Cottin N, Felgenhauer HP, Natke HG. On the parameter identification ofelastomechanical systems using input and output residuals. Ing Arch1984;54(5):378–87.

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[10] Friswell MI, Mottershead J. Finite element model updating in structuraldynamics. Solid mechanics and its application, vol. 38. Dordrecht/Boston/London: Kluwer Academic Publishers; 1995.

[11] D’Ambrogio W, Fregolent A, Salvini P. Updateability conditions ofnonconservative FE models with noise on incomplete input–output data. In:International conference structural dynamic modelling test, analysis andcorrelation, Milton Keynes, UK; 1993 [organised by DTA, NAFEMS with thesupport of DTI].

[12] Larsson PO, Sas P. Model updating based on forced vibration testing usingnumerically stable formulations. In: Proceedings of the 10th internationalmodal analysis conference, San Diego, USA; 1992.

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[14] Araujo dos Santos JV, Mota Soares CM, Mota Soares CA, Maia NMM. Structuraldamage identification in laminated structures using FRF data. Compos Struct2005;67:239–49.

[15] Fritzen CP. Localization and correction of errors in finite element models basedon experimental data. In: Proceedings of the 17th international seminar onmodal analysis, Katholieke Universiteit Leuven, Belgium; 1992.

[16] Imregun M, Sanliturk KY, Ewins DJ. Finite element model updating usingfrequency response function data-II. Case study on a medium size finiteelement model. Mech Syst Sig Process 1995;9(2):203–13.

[17] Imregun M, Visser WJ, Ewins DJ. Finite element model updating usingfrequency response function data-I. I. Theory and initial investigation. MechSyst Sig Process 1995;9(2):187–202.

[18] Pothisiri T, Hjelmstad KD. Structural damage detection and assessment frommodal response. J Eng Mech 2003;129(2):135–45.

[19] Lin RM, Ewins DJ. Model updating using FRF data. In: Proceedings of the 15thinternational seminar on modal analysis, Leuven, Belgium; 1990.

[20] Conti P, Donley M. Test/analysis correlation using frequency responsefunctions. In: Proceedings of the 10th international modal analysisconference; 1992. p. 724–9.

[21] Wang Z, Lin RM, Lim MK. Structural damage detection using measured FRFdata. Comput Method Appl Mech Eng 1997;147:187–97.

[22] Park NG, Park YS. Damage detection using spatially incomplete frequencyresponse functions. Mech Syst Sig Process 2003;17(3):519–32.

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[24] Hwang HY, Kim C. Damage detection in structures using a few frequencyresponse measurements. J Sound Vib 2004;270(1–2):1–14.

[25] Zimmerman DC, Simmermacher T, Kaouk M. Model correlation and systemhealth monitoring using frequency domain measurements. Struct HealthMonitor 2005;9(4-3):213–27.

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[28] Doebling SW, Peterson LD, Alvin KF. Experimental determination of localstructural stiffness by disassembly of measured flexibility matrices. J VibAcoust 1998;20:949–57.

[29] Ren Y, Beards CF. Identification of joint properties of a structure using FRF data.J Sound Vib 1995;186(4):567–87.

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Upper and lower bound limit analysis of plates using FEM and second-ordercone programming

Canh V. Le a,*, H. Nguyen-Xuan b, H. Nguyen-Dang c

a Department of Civil and Structural Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdomb Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science, VNU-HCM, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Viet Namc LTAS, Division of Fracture Mechanics, University of Liège, Bâtiment B52/3 Chemin des Chevreuils 1, B-4000 Liège 1, Belgium


Article history:Received 17 March 2009Accepted 7 August 2009Available online 12 September 2009

Keywords:Limit analysisUpper and lower boundsDisplacement and equilibrium modelsCriterion of meanSecond order cone programming


* Corresponding author.E-mail addresses: [email protected] (C.V.

(H. Nguyen-Xuan), [email protected] (H. Nguy

This paper presents two novel numerical procedures to determine upper and lower bounds on the actualcollapse load multiplier for plates in bending. The conforming Hsieh–Clough–Tocher (HCT) and enhancedMorley (EM) elements are used to discrete the problem fields. A Morley element with enhanced momentfields is used. The constant moment fields is added a quadratic mode in which the pressure is equilibratedby corner loads only, ensuring that exact equilibrium relations associated with a uniform pressure can beobtained. Once the displacement or moment fields are approximated and the bound theorems applied,limit analysis becomes a problem of optimization. In this paper, the optimization problems are formu-lated in the form of a standard second-order cone programming which can be solved using highly effi-cient interior point solvers. The procedures are tested by applying it to several benchmark plateproblems and are found good agreement between the present upper and lower bound solutions andresults in the literature.


1. Introduction

The yield line theory has been proved to be an effective methodto perform plastic analysis of slabs and plates [1,2]. This well-known method can predict very good upper-bound of the actualcollapse multiplier for many practical engineering problems. How-ever, this hand-based analysis method encounters difficulties inproblems of arbitrary geometry, especially in the problems involv-ing columns or holes. Consequently, over last few decades variousnumerical approaches based on bound theorems and mathemati-cal programming have been developed [3–10]. Numerical limitanalysis generally involves two steps: (i) numerical discretization;and (ii) mathematical programming to enable a solution to be ob-tained. The finite element method, which is one of the most popu-lar numerical methods, is often employed to discrete velocity orstress fields. Of several displacement and equilibrium elementsthat have been developed for Krichhoff plates in bending, the con-forming Hsieh–Clough–Tocher (HCT) [11] and equilibrium Morleyelements [12] are commonly utilized in practical engineering. Theoriginal HCT element will be used in the paper without any mod-ification while the Morley element will be modified by adding acomplementary field. Once the stress or displacement fields are

ll rights reserved.

Le), [email protected]).

approximated and the bound theorems applied, limit analysisbecomes a problem of optimization involving either linear ornonlinear programming. Problems involving piecewise linearyield functions or nonlinear yield functions can respectively besolved using linear or non-linear programming techniques[13,14,5,15,16]. However, difficulty exists in the upper-bound opti-mization problem is that the objective function is convex, but noteverywhere differentiable. One of the most efficient algorithms toovercome this singularity is the primal-dual interior-point methodpresented in [17,18] and implemented in commercial codes suchas the Mosek software package [19], such as second-order coneprogramming. The algorithm is also suitable for solving lower-bound limit analysis since most of yield conditions can be cast asa conic constraint [20]. These limit analysis problems can then besolved by this efficient algorithm [21–23].

In this paper two numerical procedures for upper and lowerbound limit analysis of rigid-perfectly plastic plates governed bythe von Mises criterion are proposed. A second degree moment fieldproposed by Debongnie and Nguyen-Xuan [24–26] is added toMorley moment fields to achieve exact equilibrium relations whenapplying a uniform pressure to plates. The enhanced Morley (EM)element will be adopted in the lower-bound limit analysis of plateproblems. Attention is also focused on treating the performance ofyield condition in numerical limit analysis. The criterion of meanproposed in [27] will be used instead of the exact criterion whichis required to strictly satisfy. Due to this weakness of the yield







66 C.V. Le et al. / Computers and Structures 88 (2010) 65–73

condition we expect to obtain only an approximation of lower-bound in the statically admissible limit analysis. Attempts are alsomade by formulating both upper and lower bound limit analysisproblems in terms of a standard second-order cone programming(SOCP). To illustrate the method it is then applied to a series ofplate bending problems, including those for which solutions al-ready exist in the literature.

2. Limit analysis formulations

2.1. Limit analysis duality theorems

Consider a rigid-perfectly plastic body of volume X 2 R3 withboundary C. Let Cu and Cg denote, respectively, an essentialboundary (Dirichlet condition) where displacement boundary con-ditions are prescribed and a natural boundary (Neumann condi-tion) where stress boundary conditions are assumed, Cu [ Cg ¼ C.The external loads which are denoted by g and f, respectively sub-ject to surface and volume of the body. Let _u be a plastic velocity orflow field that belongs to a space Y of kinematically admissiblevelocity fields and r be a stress field belonging to an appropriatespace of symmetric stress tensor X. The mathematical formulationsfor limit analysis will be briefly described in this section. More de-tails can be found in [28,22,23].

The external work rate of forces ðg; f Þ associated with a virtualplastic flow _u is expressed in the linear form as

Fð _uÞ ¼Z

Xf _udXþ

ZCg

g _udC: ð1Þ

The internal work rate for sufficiently smooth stresses r and veloc-ity fields _u is given by the bilinear form

aðr; _uÞ ¼Z

XrT _�ð _uÞdX; ð2Þ

where _�ð _uÞ are strain rates.The equilibrium equation is then described in the form of vir-

tual work rate as follows:

aðr; _uÞ ¼ Fð _uÞ; 8 _u 2 Y and _u ¼ 0 on Cu: ð3Þ

Furthermore, the stresses r must satisfy the yield condition for as-sumed material. This stress field belongs to a convex set, B, obtain-ing from the used field condition. For the von Mises criterion,

B ¼ fr 2 Xjsijsij 6 2k2g; ð4Þ

where sij denotes stress deviator tensor and k is a parameterdepending on material properties.

If defining C ¼ f _u 2 YjFð _uÞ ¼ 1g, the exact collapse multiplierkexact can be determined by solving any of the following optimiza-tion problems

kexact ¼maxfkj9r 2 Baðr; _uÞ ¼ kFð _uÞ;8 _u 2 Yg ð5Þ¼max

r2Bmin

_u2Caðr; _uÞ ð6Þ

¼min_u2C

maxr2B

aðr; _uÞ ð7Þ

¼min_u2C

Dð _uÞ; ð8Þ

where Dð _uÞ ¼maxr2Baðr; _uÞ is the plastic dissipation rate. Problems(5) and (8) are knows as static and kinematic principles of limitanalysis, respectively. The limit load of both approaches convergesto the exact solution. Herein, a saddle point ðr�; _u�Þ exists such that

both the maximum of all lower bounds k� and the minimum of allupper bounds kþ coincide and are equal to the exact value kexact .

2.2. Formulations for plates

Considers a plate bounded by a curve enclosing a plane area Awith kinematical boundary Cw [ Cwn and static boundaryCm [ Cmn , where the subscript n stands for outward normal. Thegeneral relations for limit analysis of thin plates associated withKirchhoff’s hypothesis are given as follows.

Equilibrium: Collecting the bending moments in the vectormT ¼ ½mxx myy mxy�, the equilibrium equations can be written as

ðr2ÞT mþ kp ¼ 0; ð9Þ

where p is the transverse load and the differential operatorr2 is de-

fined by r2 ¼ @2

@x2@2

@y2 2 @2

@x@y

h iT.

Compatibility: If w denotes the transverse displacement, the cur-vature rates can be expressed by relations

_j ¼ �½ _jxx _jyy 2 _jxy�T ¼ �r2 _w: ð10Þ

Flow rule and yield condition: In framework of a limit analysis prob-lem, only plastic strains (curvatures) are considered and are as-sumed to obey the normality rule _j ¼ _l @w

@m, where the plasticmultiplier _l is non-negative and the yield function wðmÞ is convex.In this study, the von Mises failure criterion in the space of momentcomponents is used

wðmÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimT Pmp

�mp 6 0; ð11Þ

where mp ¼ r0t2=4 is the plastic moment of resistance per unitwidth of a plate of uniform thickness t;r0 is the yield stress and

P ¼ 12

2 �1 0�1 2 00 0 6

264

375: ð12Þ

The dissipation rate: The internal dissipation power of the two-dimensional plate domain A can be written as a function of curva-ture rates as

Dð _jÞ ¼Z

A

Z t=2

�t=2r0

ffiffiffiffiffiffiffiffiffiffiffi_�T Q _�

pdz dA ¼ mp

ZA

ffiffiffiffiffiffiffiffiffiffiffiffi_jT Q _j

pdA; ð13Þ

where

_� ¼_�xx

_�yy

_cxy

264

375 ¼ z _j; ð14Þ

Q ¼ P�1 ¼ 13

4 2 02 4 00 0 1

264

375: ð15Þ

Details on the derivation of the dissipation for plate problems canbe found in [6,29].

3. Finite element discretization

3.1. Lower-bound formulation

In numerical lower-bound limit analysis problem, a staticallyadmissible stress or moment field for an individual element is cho-sen so that equilibrium equations and stress continuity require-ments within the element and along its boundaries are met. Thewell-known equilibrium Morley element with constant varyingmoment is the simplest model for practical engineering. It is,therefore, advantage to extent the use of the element to lower-

C.V. Le et al. / Computers and Structures 88 (2010) 65–73 67

bound limit analysis problem in this paper. The moment field m isassumed to vary constantly within an element and expressed as

m ¼ Ib; ð16Þ

where I is an identity matrix and b ¼ ½b1 b2 b3�T is an unknown

vector.The generalized loads comprise three corner loads Z1; Z2; Z3

and three normal moments bending along edges m12; m23; m31

as shown in Fig. 1. All generalized loads can be expressed in termsof moment parameters, if G denotes the generalized vector, therelations are written as

G ¼ Cb; ð17Þ

where

G ¼ Z1 Z2 Z3 m12 m23 m31½ �T ; ð18Þ

C ¼

c3s3 � c1s1 c1s1 � c3s3 c21 � s2

1 � c23 þ s2

3

c1s1 � c2s2 c2s2 � c1s1 c22 � s2

2 � c21 þ s2

1

c2s2 � c3s3 c3s3 � c2s2 c23 � s2

3 � c22 þ s2

2

c21L12 s2

1L12 c1s1L12

c22L23 s2

2L23 c2s2L23

c23L32 s2

3L32 c3s3L32

2666666664

3777777775

ð19Þ

in which the direction cosines of the outward normal to the elementboundary ðci; siÞ are determined as

ci ¼yj � yi

Lij; si ¼

xi � xj

Lij; ij ¼ 12;23;31 ð20Þ

Fig. 1. Morley equilibrium element.

Fig. 2. Relations between global syste

and Lij is the length of edge ij.It is important to note that, in the case when a uniform pressure

is applied, the Morley element does not result in an exact equilib-rium relation. This is because Eq. (9) does not hold with the use ofthe constant moment fields. It is, therefore, necessary to add to theconstant moment fields by a particular higher degree solutionwhich has to be such chosen such that side loads are compatiblewith the original element. A second degree moment field whichcan be added to equilibrium elements of either degree one or de-gree zero has been proposed by [24–26] and can be expressed as

mc ¼ kpaeT; ð21Þ

where ae is the area of an element and T ¼ ½Txx Tyy Txy�T and is gi-ven as

T ¼ �13

� X3Y3

k1 þ X3�X2Y3

k2 � X3ðX3�X2ÞY3X2

k3 þ 12ae

X2 � X22k2 � X2

3k3

� �� Y3

X2k3 þ 1

2aeY2 � Y2

3k3

� �� 1

2 k1 þ 12 k2 � 2X3�X2

2X2k3 þ 1

2aeðXY � X3Y3k3Þ

266664

377775:

ð22Þ

This complementary mode is constructed based on a particular sys-tem of axes as shown in Fig. 2, in which the side 1–2 is chosen to bethe X axis and Y must go through node 1 and is orientated so that Y3

is positive. Three area coordinates are denoted by k1ðX;YÞ; k2ðX;YÞand k3ðX; YÞ. The modified Morley element was called as enhancedMorley (EM) element by [26].

Similarly, the three generalized loads at corners of the triangu-lar element are added by � aep

3 . The equilibrium equation Eq. (17) isthen rewritten as

G ¼ C�b; ð23Þ

where

�b ¼ b1 b2 b3 k½ �;

C ¼

c3s3 � c1s1 c1s1 � c3s3 c21 � s2

1 � c23 þ s2

3 � pae3

c1s1 � c2s2 c2s2 � c1s1 c22 � s2

2 � c21 þ s2

1 � pae3

c2s2 � c3s3 c3s3 � c2s2 c23 � s2

3 � c22 þ s2

2 � pae3

c21L12 s2

1L12 c1s1L12 0c2

2L23 s22L23 c2s2L23 0

c23L32 s2

3L32 c3s3L32 0

2666666664

3777777775: ð24Þ

The overall equilibrium for the structure can be obtained by assem-bling all local equilibrium equations of elements and expressed as

Csbs ¼ 0 ð25Þ

m (Oxy) and local system (OXY).


with bs ¼ ½b1 b2 � � � b3�nele k�; nele is the number of elements.Notes that boundary conditions are also imposed here in the assem-ble scheme.

Furthermore, the modified moment field �m is not allowed toviolate the yield condition

wð �mÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�mT P �m

p�mp 6 0; ð26Þ

where

�m ¼ bþ kpaeT: ð27Þ

However, in numerical analysis it is not always possible to satisfythis requirement since the yield condition is commonly fulfilled atGauss points or nodes. Instead of strictly satisfying the exact crite-rion, Nguyen–Dang proposed the criterion of mean [27,30] which issatisfied locally within element domains. For plate problem the cri-terion of mean can be expressed as

1ae

Zae

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�mT P �m

pda�mp 6 0: ð28Þ

Introducing the smoothed value of �m the Eq. (28) can be rewrittenas

wðqÞ ¼ffiffiffiffiffiffiffiffiffiffiffiqT Pq

p�mp 6 0; ð29Þ

where q is the smoothed version of �m and given by

q ¼ 1ae

Zae

�mda ¼ bþ kpZ

ae

Tda ¼ bþ kpS ð30Þ

in which S is the exact integration ofR

aeT da in the local coordinate

OXY.If defining Bi ¼ fqijwðqiÞ 6 0g is the set of admissible discrete

moments for each element, the lower-bound limit analysis (5)can be now written in terms of discrete moment space as

k� ¼max k

s:tCsbs ¼ 0;qi ¼ bi þ kpSi;

qi 2 Bi; i ¼ 1;2; . . . ;nele

8><>: ð31Þ

and accompanied by appropriate boundary conditions.

3.2. Upper-bound formulation

In numerical upper-bound limit analysis of plate problem, thevelocity field with an element is represented by a continuous func-tion expressed in terms of spatial coordinates and nodal values. ForKrichhoff plates, an element of class C1 should be employed toapproximate the velocity field. The conforming Hsieh–Clough–To-cher (HCT) triangular element will be utilized and briefly summa-rized in this section. A triangular element is subdivided into threesub-elements using individual cubic expansions over each sub-ele-ment as shown in Fig. 3. The element has 12 degrees of freedom:

Fig. 3. HCT e

the transverse displacements and 2 the rotation components ateach corner node ðwi; hxi ¼ @wi=@xji; hyi ¼ @wi=@yji; i ¼ 1;2;3Þ andnormal rotations at three mid-side nodes ðhi ¼ @wi=@nji; i ¼ 4;5;6Þ.

The displacement expansion wðkÞ can be expressed in terms ofarea coordinates f ¼ ðf1; f2; f3Þ over each sub-triangle as

wðkÞðfÞ ¼ NðkÞe ðfÞ þNðkÞ0 ðfÞF� �

qe; k ¼ 1;2;3; ð32Þ

where the partitions NðkÞe ðfÞ and NðkÞ0 ðfÞ, respectively, represent theinterpolation functions associated with element displacements qe

and internal nodal displacements and F is the matrix of eliminationobtained by applying compatible requirements at internal nodes 7,8, 9.

The plastic dissipation for a sub-element is now formulated as

DðkÞðjðkÞÞ ¼ mp

ZAse

ffiffiffiffiffiffiffiffiffiffiffiffi_jT Q _j

pdA ¼ mp

Xng

j¼1

nj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_jTðfjÞQ _jðfjÞ

q; ð33Þ

where ng ¼ 3 is the number of Gauss integration points in each sub-element AðkÞ; nj is the weighting factor of the Gauss point fj andjðkÞðfjÞ are curvatures at the Gauss point fj

_jðkÞðfjÞ ¼

_jðkÞxx ðfjÞ_jðkÞyy ðfjÞ_jðkÞxy ðfjÞ

2664

3775 ¼

NðkÞe;xxðfjÞ þNðkÞ0;xxðfjÞF

NðkÞe;yyðfjÞ þNðkÞ0;yyðfjÞF

NðkÞe;xyðfjÞ þNðkÞ0;xyðfjÞF

2664

3775 _qe: ð34Þ

By summing all dissipations of all sub-elements and elements, theplastic dissipation of the whole plate is

D ¼ mp

Xnele X3 Xng

j¼1

nj


q: ð35Þ

Similarly, the work rate of applied loads can be expressed as

F ¼Xnele X3 Xng

j¼1

njp _wðkÞðfjÞ: ð36Þ

The upper-bound limit analysis of plate bending is now written as

kþ ¼min mp

Xnele X3 Xng

j¼1

nj


q

s:tPnele P3 Png

j¼1njp _wðkÞðfjÞ ¼ 1

_q ¼ 0 on Cw:

8><>: ð37Þ

4. Second-order cone programming

4.1. Conic programming

The general form of a Second-Order Cone Programming (SOCP)problem with N sets of constraints is written as follows:

lement.


min fT x

s: t: kHixþ vik 6 yTi xþ zi for i ¼ 1; . . . ;N; ð38Þ

where x 2 Rn are the optimization variables, and the problem coef-ficients are f 2 Rn;Hi 2 Rm�n;vi 2 Rm; yi 2 Rn, and zi 2 R. For optimi-zation problems in 2D or 3D Euclidean space, m ¼ 2 or m ¼ 3. Whenm ¼ 1 the SOCP problem reduces to a linear programming problem.In framework of limit analysis problems, the two most commonsecond-order cones are the quadratic cone

Cq ¼ x 2 Rkþ1jx1 P

ffiffiffiffiffiffiffiffiffiffiffiffiffiXkþ1

j¼2

x2j

vuut ¼ kx2!kþ1k

8<:

9=; ð39Þ

and the rotated quadratic cone

Cr ¼ x 2 Rkþ2jx1x2 PXkþ2

j¼3

x2j ¼ kx3!kþ2k2

; x1; x2 P 0

( ): ð40Þ

0 50 100 150 200 250 30046

48

50

52

54

56

58

60

number of elements

SQP MatlabSOCP Mosek

Fig. 5. Comparison the perfor

Fig. 4. Square plate clamped along edges and loaded by a uniformly pressure.

4.2. Lower-bound programming

Since the matrix P is a positive definite matrix, the constraint(29) can be cast in terms of a conic quadratic constraint as

q 2 Cq; Cq ¼ q 2 R4jq4 P kJT1q1!3k;q4 ¼ mp

n o; ð41Þ

where J1 is the so-called Cholesky factor of P

J1 ¼12

2 0 0�1

ffiffiffi3p

00 0 2

ffiffiffi3p

264

375: ð42Þ

The lower-bound limit analysis of plates is then cast in the form of asecond-order cone programming as

k� ¼max k

s:t

Csbs ¼ 0;qi ¼ bi þ kpSi;

qi 2 Ciq; i ¼ 1;2; . . . ;nele

8><>: ð43Þ

and accompanied by appropriate boundary conditions.

4.3. Upper-bound programming

In order to cast the optimization problem (37) in the form of astandard second-order cone programming, its objective function isfirstly formulated in a form involving a sum of norms as

mp

Xnele X3 Xng

j¼1

nj


q¼ mp

Xnele X3 Xng

j¼1

njkJT2 _jðfjÞk; ð44Þ

where J2 is the Cholesky factor of Q

J2 ¼1ffiffiffi3p

2 0 01

ffiffiffi3p

00 0 1

264

375: ð45Þ

By introducing auxiliary variables t1; t2; . . . ; tnele�3�ng the presentupper-bound optimization problem can be rewritten in the formof a standard SOCP problem as

0 50 100 150 200 250 30028

30

32

34

36

38

40

42

number of elements

SQP MatlabSOCP Mosek

mance of SQP and SOCP.

0 500 1000 1500 2000 2500 3000 3500 400025

30

35

40

45

50

55

60

number of elements

******

(*) Upper−bound in [6](**) Mixed approach [5](***) Lower−bound in [3]

λ+ (HCT element)

λ− (EM element)The average value

Fig. 6. Bounds on the collapse multiplier vs. number of elements using SOCP.

0 500 1000 1500 2000 2500 3000 3500 400023

23.5

24

24.5

25

25.5

26

26.5

number of elements

***

(*) Upper−bound in [6]

(**) Lower−bound in [3]

λ+ (HCT element)


Fig. 7. Bounds on the collapse multiplier vs. number of elements using SOCP.


kþ ¼min mp

Xnele�3�ng

k

nktk

s:t

Pnele P3 Png

j¼1njp _wðkÞðfjÞ ¼ 1;

_q ¼ 0 on Cw;

ri ¼ JT2 _j;

krik 6 ti; i ¼ 1;2; . . . ;nele� 3� ng

8>>>>>><>>>>>>:

ð46Þ

in which krik 6 ti expresses quadratic cones and ri are additionalvariables, where every ri is a 3� 1 vector. The total number of vari-

Fig. 8. Mesh refinements for a q

ables of this optimization problem is sdof þ 4� 3� ng � nele; sdofis the degrees of freedom of system.

5. Numerical examples

The numerical performance of the procedures are illustrated byapplying it to uniformly loaded plate problems for which, in mostcases, solutions already exist in the literature (the method is appli-cable to problems of arbitrary geometry). For all the examples con-sidered the following was assumed: length L ¼ 10 m; platethickness t ¼ 0:1 m; yield stress r0 ¼ 250 MPa. Quarter symmetrywas assumed when appropriate. Note that, solutions obtained in

uarter of the circular plate.

0 500 1000 1500 2000 2500 3000 3500 40009

10

11

12

13

14

15

number of elements

λ+ (HCT element)

λ− (EM element)The exact solution

Fig. 9. Bounds on the collapse multiplier vs. number of elements using SOCP(circular plate).

free edge

simply supported

q

L / 2 L / 2

L / 2

L / 2

Fig. 10. L-shaped geometry.


the static problems are approximations of lower-bound due to cri-terion of the mean was used. However, as the discretization is suf-ficiently fine, increasingly close approximations of the true plasticcollapse load multiplier can be expected to be obtained.

The first examples is a square plate with clamped supports andsubjected to uniform out-of-plane pressure loading. This problemwas solved by the top-right quarter of the plate and uniform meshgeneration was used, see Fig. 4. Matlab optimization toolbox 3.0and Mosek version 5.0 optimization solvers were used to obtainsolutions (using a 2.8 GHz Pentium 4 PC running Microsoft XP).

Fig. 11. Mesh refinemen

The efficacy of various optimization algorithms was firstly con-sidered. The limit analysis problems (31) and (37) are typicallynon-linear optimization problems and it can be solved using a gen-eral non-linear optimization solver, such as a sequential quadraticprogramming (SQP) algorithm (which is generalization of New-ton’s method for unconstrained optimization) [31]. Fig. 5 showsthat solutions obtained using SQP and SOCP algorithms are in verygood agreement. However, the SOCP algorithm produced solutionsvery much more quickly and somewhat more accurate, despite thefact that the number of variables involved was much greater

t for L-shape plate.


(sdof þ 4� 3� ng � nele cf. sdof when using SQP). To computesolutions for a mesh of 288 elements, the SOCP algorithm typicallytook only 5–30 s, compared with 1280–7000 s when using SQP.Moreover, the SOCP algorithm is able to solve problems up to152,148 of variables with less than 400 s CPU time (for the meshof 4050 elements). It is also important to note that the SOCP algo-rithm can be guaranteed to identify globally optimal solutions,whereas SQP cannot.

The performance of the presented numerical limit analysis pro-cedures is further investigated in convergence analysis as shown inFig. 6. It can be observed that both upper and lower bounds con-verge to the actual collapse multiplier when the size of elementstends to zero. A upper-bound of 45.12 was achieved by presentmethod, which is slightly smaller than the solution previously ob-tained in [6]. In comparison with previously obtained lower-boundsolution, the present method provides higher solutions than in [3]where quadratic moment fields were used, by 0.6%.

The next example comprises a square plate with simply sup-ported on all edges. Convergence analysis of collapse load multipli-ers is shown in Fig. 7. It can be seen from the figure that the upper-bound converges to the actual collapse multiplier when relativelysmall number of elements was used; and the gap between upperand lower bound is considerably smaller than the clamped case.This may be explained by the fact that the displacement filed inthis problem does not exhibit a singularity in the form of aso-called hinge along boundary. The solutions obtained by theproposed method are in good agreement with previously achievedbounds. Considering previously obtained upper-bound solutions,the present method provides lower solutions than in [3,6], by6.16% and 0.01%, respectively. Furthermore, a computed lower-bound of 24.93 was found, which is 0.3% higher than the best low-er-bound found in [3] where quadratic moment fields were used.

In the two examples examined above, the computed upper-bounds are slightly higher than solution in [29] where theElement-Free Galerkin method was used to approximate thedisplacement filed. However, the presented method can providevery tight lower-bound solutions and based on the computedbounds the actual collapse multiplier can be estimated, e.g. takingthe mean value of the obtained upper and lower bounds. For theseexamples, the computed mean values are in excellent agreementwith solutions in [5].

Further illustration of the method can be made by examining aclamped circular plate, for which the exact solution exists [32],k ¼ 12:5 mp

R2 where R is the radius. Mesh refinements for a quarterof the plate are shown in Fig. 8.

0 1000 2000 3000 4000 50005.2

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2

number of elements

λ+ (HCT element)


Fig. 12. Bounds on the collapse multiplier vs. number of elements using SOCP(L-shape plate).

Fig. 9 shows the improvement in the computed collapse load asthe problem is refined uniformly. Due to the singularity of the dis-placement field along the boundary of the plate, the displacementmodel (HCT) results in a slower convergence than when using theequilibrium model (EM). When 4050 elements were used, the low-er-bound was found to be 12.42, just 0.64% different to the exactsolution.

Finally, an L-shape plate subject to a uniform load was consid-ered. The plate geometry and uniform mesh refinements areshown in Figs. 10 and 11, respectively. Collapse load multipliersfor various numbers of elements are plotted in Fig. 12. The L-shapeplate problem exhibits both stress and displacement singularitiesat the re-entrant corner. This evidently results in a slow conver-gence and the gap between upper and lower bounds are large de-spite that fact that a large number of elements was used. For thisexample, the computed upper-bound was found to be 6.289 whichis lower than the best solution obtained previously in [29].

6. Conclusions

The performance of the two novel numerical limit analysis pro-cedures using finite element method in conjunction with second-order cone programming has been investigated. It has been shownthat when limit analysis problems are cast in the form of a SOCP,the resulting optimization problems can be solved rapidly by sucha efficient interior point algorithm, even though for cases when avery large number of variables involves. The proposed proceduresare enable to provide relatively good bounds on the actual collapseload multiplier since most solutions in existing references wereimproved. Moreover, the proposed procedures can handle effi-ciently problems of arbitrary geometry. The only drawback is thatthe solutions are highly sensitive to the geometry of the finite ele-ment mesh, particularly in the region of stress or displacement sin-gularities. An automatically adaptive mesh refinement scheme canbe performed to increase the accuracy of solutions. A well-knownbenefit from dual structure of limit analysis is that both the stressand velocity fields of the upper and lower bound problem can bedetermined. It is, therefore, relevant to investigate the performanceof an adaptive scheme based on a posteriori error estimate usingelemental and edge contributions to the bound gap [22,23].

References

[1] Johansen KW. Yield-line theory. London: Cement and Concrete Association;1962.

[2] Wood RH. Plastic and elastic design of slabs and plates. London: Thames andHudson; 1961.

[3] Hodge PGJ, Belytschko T. Numerical methods for the limit analysis of plates.Trans ASME, J Appl Mech 1968;35:796–802.

[4] Christiansen E, Larsen S. Computations in limit analysis for plastic plates. Int JNumer Methods Eng 1983;19:169–84.

[5] Andersen KD, Christiansen E, Overton ML. Computing limit loads byminimizing a sum of norms. SIAM J Sci Comput 1998;19:1046–62.

[6] Capsoni A, Corradi L. Limit analysis of plates – a finite element formulation.Struct Eng Mech 1999;8:325–41.

[7] Krabbenhoft K, Damkilde L. Lower bound limit analysis of slabs with nonlinearyield criteria. Comput Struct 2002;80:2043–57.

[8] Yan AM, Nguyen-Dang H. Limit analysis of cracked structures by mathematicalprogramming and finite element technique. Comput Mech 1999;24:319–33.

[9] Yan AM, Jospin RJ, Nguyen-Dang H. An enhance pipe elbow element –application in plastic limit analysis of pipe structures. Int J Numer Methods Eng1999;46:409–31.

[10] Phan-Hong Q, Nguyen-Dang H. Limit analysis of 2D structures using glidingline mechanism generated by rigid finite elements, Collection of papers fromProf. Nguyen-Dang Hungs former students. Vietnam National University, HoChi Minh City Publishing House; 2006. p. 447–60.

[11] Clough R, Tocher J. Finite element stiffness matrices for analysis of plates inbending. In: Proceedings of the conference on matrix methods in structuralmechanics, Ohio,Wright Patterson A.F.B.; 1965.

[12] Morley LSD. The triangular equilibrium problem in the solution of platebending problems. Aero Quart 1968;19:149–69.

[13] Gaudrat VF. A Newton type algorithm for plastic limit analysis. ComputMethods Appl Mech Eng 1991;88:207–24.


[14] Zouain N, Herskovits J, Borges LA, Feijoo RA. An iterative algorithm for limitanalysis with nonlinear yield functions. Int J Solids Struct 1993;30:1397–417.

[15] Yan AM, Nguyen-Dang H. Kinematical shakedown analysis with temperature-dependent yield stress. Int J Numer Methods Eng 2001;50:1145–68.

[16] Nguyen-Dang H, Yan AM, Vu DK. Duality in kinematical approaches of limitand shakedown analysis of structures, complementary, duality and symmetryin nonlinear mechanics. In: Gao David, editor. Shanghai IUTAM symposium;2004. p. 128–48.

[17] Andersen KD, Christiansen E, Overton ML. An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms. SIAM J Sci Comput2001;22:243–62.

[18] Andersen ED, Roos C, Terlaky T. On implementing a primal-dual interior-pointmethod for conic quadratic programming. Math Program 2003;95:249–77.

[19] Mosek, The MOSEK optimization toolbox for MATLAB manual, Mosek ApS;2008. <http://www.mosek.com>.

[20] Krabbenhoft K, Lyamin AV, Sloan SW. Formulation and solution of someplasticity problems as conic programs. Int J Solids Struct 2007;44:1533–49.

[21] Makrodimopoulos A, Martin CM. Upper bound limit analysis using simplexstrain elements and second-order cone programming. Int J Numer AnalMethods Geomech 2006;31:835–65.

[22] Ciria H, Peraire J, Bonet J. Mesh adaptive computation of upper and lowerbounds in limit analysis. Int J Numer Methods Eng 2008;75:899–944.

[23] Munoz J, Bonet J, Huerta A, Peraire J. Upper and lower bounds in limit analysis:adaptive meshing strategies and discontinuous loading. Int J Numer MethodsEng 2009;77:471–501.

[24] Debongnie JF, Applying pressures on plate equilibrium elements. Technicalreport, University of Liege, Belgium.

[25] Nguyen-Xuan H, Debongnie JF, The equilibrium finite element model and errorestimation for plate bending. International Congress Engineering MechanicsToday 2004, Ho Chi Minh City, Vietnam; August 16–20, 2004.

[26] Debongnie JF, Nguyen-Xuan H, Nguyen-Huy C, Dual analysis for finite elementsolutions of plate bending. In: Montero G, Montenegro R, editors. Proceedingsof the eighth international conference on computational structurestechnology, B.H.V. Topping, Stirlingshire, Scotland: Civil-Comp Press; 2006.

[27] Nguyen-Dang H. Direct limit analysis via rigid-plastic finite elements. ComputMethods Appl Mech Eng 1976;8:81–116.

[28] Christiansen E, Limit analysis of collapse states, Handbook of numericalanalysis, vol. IV, Amsterdam: North-Holland; 1996. p. 193312 [chapter II].

[29] Le CV, Gilbert M, Askes H. Limit analysis of plates using the EFG method andsecond-order cone programming. Int J Numer Methods Eng 2009;78:1532–52.

[30] Nguyen-Dang H, Konig JA. A finite element formulation for shakedownproblems using a yield criterion of the mean. Comput Methods Appl MechEng 1976;8:179–92.

[31] Le CV, Nguyen-Xuan H, Nguyen-Dang H. Dual limit analysis of bending plates.In: Proceeding of third international conference on advanced computationalmethods in engineering, Ghent – Belgium; 2005.

[32] Hopkins H, Wang A. Load-carrying capacities for circular plates of perfectly-plastic material with arbitrary yield condition. J Mech Phys Solids1954;3:117–29.

http://www.mosek.com





Uncertain linear structural systems in dynamics: Efficient stochasticreliability assessment

H.J. Pradlwarter, G.I. Schuëller *

Institute of Engineering Mechanics, University of Innsbruck, A-6020 Innsbruck, Austria


Article history:Received 2 April 2009Accepted 14 June 2009Available online 25 July 2009

Keywords:Finite element modelsLinear structuresDynamicsGaussian excitationUncertain structural parametersReliability


* Corresponding author.E-mail address: [email protected] (G.I. Schuëlle

A numerical procedure for the reliability assessment of uncertain linear structures subjected to generalGaussian loading is presented. In this work, restricted to linear FE systems and Gaussian excitation,the loading is described quite generally by the Karhunen–Loève expansion, which allows to model gen-eral types of non-stationarities with respect to intensity and frequency content. The structural uncertain-ties are represented by a stochastic approach where all uncertain quantities are described by probabilitydistributions. First, the critical domain within the parameter space of the uncertain structural quantitiesis identified, which is defined as the region which contributes most to the excursion probability. Eachpoint in the space of uncertain structural parameters is associated with a certain excursion probabilitycaused by the Gaussian excitation.

In order to determine the first excursion probability of uncertain linear structures, an integration overthe space of uncertain structural parameters is required. An extended procedure of standard Line sam-pling [P.S. Koutsourelakis, H.J. Pradlwarter, G.I. Schuëller, Reliability of structures in high dimensions, partI: algorithms and applications, Probabilistic Engineering Mechanics 19(4) (2004) 409–417; G.I. Schuëller,H.J. Pradlwarter, P.S. Koutsourelakis, A critical appraisal of reliability estimation procedures for highdimensions, Probabilistic Engineering Mechanics 19(4) (2004) 463–474] is used to perform the condi-tional integration over the space of uncertain parameters. The suggested approach is applicable to generaluncertain linear systems modeled by finite elements of arbitrary size by using modal analysis as exem-plified in the numerical example. Special attention is devoted to the efficiency of the proposed approachwhen dealing with realistic FE models, characterized by a large number of degrees of freedom and also alarge number of uncertain parameters.


1. Introduction

The potentials and abilities of modeling the behavior of fluids,solids and complex materials subjected to various forces andboundary conditions by finite element analyses (FEA) provides akey technology for further developments in the industrializedworld. Supported by the available relatively inexpensive and con-tinuously growing computer power, the expectations on FEA areshifting towards reliable and robust computational simulationsand predictions of physical events. For achieving this goal, answersto many open issues in computational mechanics need to be pro-vided as discussed e.g. in [10]. The need to design, given uncertainmaterial properties, production processes, operating conditionsand fidelity of mathematical computational (FE) models to repre-sent reality, leads to the concern of reliability and robustness ofcomputer-generated predictions. Without some confidence in thevalidity of simulations, their value is obviously diminished. Since

ll rights reserved.

r).

uncertainty is always present in non-trivial realistic applications,uncertainty propagation through the FEA is one of the importantissues which must be addressed for further developments.

In this paper, a computational efficient reliability estimationprocedure for uncertain linear systems subjected to dynamic sto-chastic loading is presented. The approach is designed to cope withlarge FE-models in terms of degrees of freedom, a large number ofuncertain input quantities and high variabilities in terms of coeffi-cient of variation (e.g. P10%). Uncertainties with respect to dy-namic loading and of the parameters describing the mechanicalproperties of the FE-model are propagated by a stochastic ap-proach: Inherent irreducible (aleatory) uncertainties as well asreducible (epistemic) uncertainties due to insufficient knowledgeor modeling capabilities are translated into a probability distribu-tion defining the input of the stochastic analysis. The reliability willbe assessed in terms of the first excursion probability of critical re-sponses. For this purpose, Line sampling [7,15] is further developedand extended. Efficient solutions of the first excursion probabilityfor deterministic systems are exploited to compute failureprobabilities conditioned on discrete realizations of the uncertain





H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86 75

structural properties. Computationally efficient procedures todetermine the stochastic response in terms of the Karhunen–Loèverepresentation of the conditional critical response are suggested byusing modal analysis, impulse response functions in combinationwith Fast Fourier Transforms (FFT). The associated conditional firstexcursion probabilities are evaluated by Line sampling. The shownprocedure is applicable for any Gaussian distributed excitation rep-resented by a Karhunen–Loève expansion and for an arbitrarilylarge FE-model. In a further step, the domain of uncertain struc-tural parameters which contributes most to the failure probabilityis identified by a recently developed estimation procedure whichallows to identify the most influential uncertain parameters amonga large number. Finally, the unconditional total failure probabilityis estimated by a novel Line sampling procedure, covering theimportant failure domain and efficiently integrating over thewhole parameter uncertainty space.

To demonstrate the applicability of the proposed approach forgeneral FE-models, a realistic FE-model for a twelve story buildingwith more than 24,000 DOF’s and 200 uncertain quantities isanalyzed.

2. Methods of analysis

2.1. General outline

This section, containing the theoretical developments, is subdi-vided into six subsections. The first subsection addresses the Karh-unen–Loève representation of the stochastic excitation. The seconddescribes the treatment of uncertain structural properties by a sto-chastic approach. The third considers impulse response functionsand its use to evaluate the critical response for linear systems con-ditioned on specific realizations of the uncertain structural proper-ties. The fourth subsection shows how Line sampling can beapplied to estimate the conditional first excursion probability. Inthe fifth subsection, the domain within the parameter space ofuncertainties is identified which contributes most to the firstexcursion probability. In the last and sixth subsection, the integra-tion over the entire high dimensional parameter space is carriedout by a novel Line sampling procedure.

2.2. Representation of uncertain excitation

Dynamic excitations acting on the structure are in many casesuncertain. However, although it might be impossible to describethese excitations in a deterministic sense, information on the ex-pected range and its variability is usually available. Such infor-mation can be described by the mean value, the standarddeviation of the fluctuation, and also the correlation in spaceand time.

The dynamic excitation f ðx; a;AðtÞÞ is a function of the spatialcoordinates x, the direction a and the amplitude AðtÞ as a functionof time t. This complex dependency on eight scalar quantitiesfx1; x2; x3;a1;a2;a3;A; tg is considerably simplified by the use of fi-nite element analysis (FEA). In FEA, the spatial coordinates x andthe direction of actions a are specified by the degrees of freedomand all forces are reduced to nodal forces. The excitation f cantherefore be interpreted as vector with m components, where thestructure is assumed to be specified by m degrees of freedom.Hence it suffices in FEA, and as implied here, to specify the dy-namic excitation by the m-dimensional vector f ðtÞ as a functionof time t.

The uncertainties of the excitation f ðtÞ are described mathemat-ically by a stochastic process [8,1,16] characterized as a function ofindependent random variables N ¼ ðN1;N2; . . . ;NnÞ and by deter-ministic (vector) functions f ðiÞðtÞ of time t. Most conveniently, each

of the independent random variables is assumed to follow a stan-dard normal distribution

qNiðniÞ ¼

1ffiffiffiffiffiffiffi2pp exp � n2

i

2

!ð1Þ

P½Ni 6 ni� ¼Z ni

�1qNiðxÞdx ¼ UðniÞ; ð2Þ

in which Uð�Þ denotes the cumulative standard normal distribution.Any Gaussian distributed process, is most conveniently describedby the so called Karhunen–Loève presentation (see e.g. [9,6,5,14]).

f ðt; nÞ ¼ f ð0ÞðtÞ þXn

i¼1

nif ðiÞðtÞ: ð3Þ

In the above, all vectors on the right hand side do have determinis-tic properties and the independent random variables assume thefollowing relations,

E½ni� ¼ 0; E½n2i � ¼ 1; E½ninj� ¼ 0 for i–j; ð4Þ

where E½�� denotes the mean or expectation. The representation (3)specifies uniquely the mean lf ðtÞ and the variance r2

fkðtÞ or standard

deviation rfk ðtÞ for each degree of freedom k, and the correlation ofthe uncertain excitation.

lf ðtÞ ¼ f ð0ÞðtÞ ð5Þ

r2fkðtÞ ¼

Xn

i¼1

f ðiÞk ðtÞh i2

ð6Þ

E½fjðt1Þfkðt2Þ� ¼Xn

i¼1

f ðiÞj ðt1Þf ðiÞk ðt2Þ: ð7Þ

However, the deterministic vector valued functions f ðiÞðtÞ are usu-ally not quantified a priori. They need to be determined from thesymmetric covariance matrix Cf ðtj; tkÞ. Assuming that the excitationis discretized at equidistant instants 0 P tk ¼ kDt P T , the covari-ance matrix is defined as given below,

Cf ðtj; tkÞ ¼ E½ðf ðtjÞ � lf ðtjÞÞðf ðtkÞ � lf ðtkÞÞ0�; 0 6 t1; t2 6 T; ð8Þ

where a prime ‘‘0‘‘ denotes the transposed vector and T is the consid-

ered duration.After solving the eigenvalue problem,

Cf U ¼ UK; ð9Þ

where U contains the eigenvector and K is a diagonal matrix ofeigenvalues, the eigen-solution for the n highest eigenvaluesk1 P k2 P � � �P kn are used. The deterministic vector values func-tions f ðiÞðtÞ are then determined by

f ðjÞðtkÞ ¼ffiffiffiffikj

p/½k�j; ð10Þ

where ½k� denotes the rows associated with tk and j the column ofthe eigenvector matrix. In practice it would be extremely difficult,if not unfeasible, to establish the covariance matrix Cf for all m de-grees of freedom simultaneously and all K time steps tk; k ¼ 1; . . . ;K .This would lead to a quadratic matrix of the size m� K , whichmight be beyond a manageable size. For a tractable solution, inde-pendent excitation processes are considered separately. A typicalexample for such a separation could be an earthquake excitationin a specified direction,

f ðtÞ ¼ �MI � aðtÞ ð11Þ

in which M is the mass matrix and I is vector with 1’s for all de-grees of freedom in the considered direction and zeros elsewhere.Hence the nodal forces f ðtÞ are for any time t fully correlated, andonly the correlations at different times tj – tk need to be described.Hence, after a discretization of the time t by tk ¼ kDt; k ¼ 1; . . . ;K,

76 H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86

the covariance matrix of the scalar acceleration aðtkÞ has only thesize K, for which the eigen-solution poses no difficulty, leading tothe representation

aðtkÞ ¼ að0ÞðtkÞ þXn

i¼1

niaðiÞðtkÞ ð12Þ

aðiÞðtkÞ ¼ffiffiffiffiki

pUki ð13Þ

f ðtkÞ ¼ �MI � að0ÞðtkÞ þXn

i¼1

niaðiÞðtkÞ

" #: ð14Þ

In practical applications, each of the deterministic Karhunen–Loèveexcitation terms f ðiÞðtÞ can be described by a constant vector FðiÞ de-fined over the range of degrees of freedom and a scalar functionhðiÞðtÞ as function of time t,

f ðiÞðtÞ ¼ FðiÞ � hðiÞðtÞ: ð15Þ

As shown above, such a representation applies for earthquake exci-tation, with F ¼ �MI.

2.3. Uncertain structural systems

Linear structural systems are described by a constant symmet-ric mass matrix M, damping matrix D and stiffness matrix K . Thesematrices are realistically considered to be associated with uncer-tainties. In the proposed stochastic parametric approach [1], theuncertainties of these matrices are also conveniently modeled asfunctions of independent random variables

H ¼ fH1;H2; . . . ;HSg ð16Þ

such that for any fixed set H ¼ h, these structural matrices are un-iquely specified. Hence, for any fixed set (vector) h, the structuralmatrices MðhÞ;DðhÞ and KðhÞ can be treated deterministically. It iscommon practice, to assume the random variables Hi to be stan-dard normally distributed, i.e. with zero mean E½Hi� ¼ 0 and unitvariance E½H2

i � ¼ 1. This assumption, however, does not imply thatthe structural components follow a Gaussian distribution or thatcorrelations among structural parts do not exist. Non-Gaussian dis-tributions are realized by non-linear relations of these basic stan-dard normally distributed random variables. Correlations amongdifferent structural parts are established by representing differentparts as a functions of one or several random variables.

2.4. Impulse response functions

Since the structural system is assumed to be linear, the law ofsuperposition is valid. This law implies, that the response uðt; hÞfor fixed structural properties h and dynamic excitation f ðtÞ has,analogous to (3), also a Karhunen–Loève representation,

uðt; h; nÞ ¼ uðhÞð0ÞðtÞ þXn

i¼1

niuðiÞðt; hÞ; ð17Þ

where each deterministic term uðiÞðt; hÞ; i ¼ 0;1; . . . ; n, is the solutionof a deterministic dynamic analysis, involving the constant sym-metric mass matrix MðhÞ, damping matrix DðhÞ and stiffness matrixKðhÞ in the equation of motion.

MðhÞ€uðiÞðt; hÞ þ DðhÞ _uðiÞðt; hÞ þ KðhÞuðiÞðt; hÞ ¼ f ðiÞðtÞ;80 6 i 6 n: ð18Þ

Hence, to specify the variance of the displacement response, nþ 1deterministic analyses are required. However, the dynamic re-sponse might depend considerably on the structural parametersspecified by the vector h.

One of the efficient ways to compute the associated dynamicdisplacement response uðiÞðtÞ is the use of the impulse response

function v ðiÞðtÞ and the Fast Fourier transform for computing theconvolution. The impulse response function is computed as freemotion with initial zero displacement and the initial velocity:

_v ðiÞð0Þ ¼M�1 � FðiÞ: ð19Þ

For illustration, Fig. 1 shows an example of the impulse responsefunction for the velocity and displacement response of a 2-DOFsystem.

Since the system is linear, also efficient modal analysis applies.After solving the eigenvalue problem,

KðhÞUðhÞ ¼MðhÞUðhÞKðhÞ ð20Þ

the impulse response vðt; hÞ is represented in modal coordinateszðt; hÞ

v ðiÞðt; hÞ ¼ UðhÞzðiÞðt; hÞ: ð21Þ

Then, the modal coordinates have initial values zðiÞj ð0; hÞ ¼ 0, and_zðiÞj ð0; hÞ is specified by the vector

_zðiÞð0; hÞ ¼ UTðhÞFðiÞ: ð22Þ

The free motion in modal coordinates has the explicit solution [8]

zðiÞj ðt;hÞ ¼_zðiÞj ð0;hÞ

x0jefjxj t sinðx0jtÞ ð23Þ

_zðiÞj ðt;hÞ ¼ _zðiÞj ð0;hÞefjxj t cosðx0jtÞ �

fjxj

x0jsin x0jt� �" #

ð24Þ

€zðiÞj ðt;hÞ ¼ � _zðiÞj ð0;hÞefjxj t

x2j

x0jð1� 2fjÞ sin x0jt

� �þ

2fjx0jxj

cos x0jt� ��

;

ð25Þ

where all the quantities fj;xj and x0j depend actually on the param-eter h:

xj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffikðiÞj ðhÞ

q; ð26Þ

x0j ¼ xj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� f2

j ðhÞq

: ð27Þ

The damping ratios fjðhÞ are, most advantageously, not specified bythe an explicit damping matrix DðhÞ, but by modal damping ratioswith a moderate increase with frequency xj. The damping ratiosmight be defined directly as function of uncertain quantities, e.g.fjðhÞ ¼ cj � expð�ajhkÞ with given value for cj and aj.

Suppose, the critical responses of interest, e.g. stresses, strains,accelerations, displacements, etc., are comprised in the vectoryðt; hÞ. Each component yiðt; hÞ of the vector must fulfill certainconditions in order to be regarded as reliable. It will be assumedthat these critical types of structural response can be representedby a linear combination of the displacement or acceleration re-sponse uðt; hÞ or €uðt; hÞ,

yðt; hÞ ¼ QðhÞuðt; hÞ; ð28Þ

where Q is a constant matrix, independent of time t, and a functionof the structural parameters which might depend on h. Similarly, asthe displacement response, the variability due to the random exci-tation can be cast into a Karhunen–Loève representation.

yðt; h; nÞ ¼ yð0Þðt; hÞ þXn

i¼1

niyðiÞðt; hÞ: ð29Þ

The impulse response function associated with FðiÞ is then:

wðiÞðt; hÞ ¼ QðhÞUðhÞzðiÞðt; hÞ: ð30Þ

The critical response yðiÞk ðt; hÞ due to the excitation term f ðiÞðtÞ istherefore the result of the convolution

mass 1

mass 2

0 2 4 6 8 10

−1

−0.5

0

0.5

1

Velocity impulse response

time t

velo

city

[m/s

]

mass 1mass 2

0 2 4 6 8 10−0.2

−0.1

0

0.1

0.2Displacements impulse response

time [s]

disp

lace

men

t [m

]

mass 1mass 2

Fig. 1. Impulse response function.


yðiÞk ðt; hÞ ¼Z t

0hðiÞðsÞwðiÞk ðt � s; hÞ ds: ð31Þ

Numerically, this integral is most efficiently computed by using FastFourier Transforms,

hðiÞðtÞ ! FFT ! hðiÞðxÞ; ð32ÞwðiÞk ðtÞ ! FFT ! wðiÞk ðxÞ; ð33ÞyðiÞk ðxÞ ¼ hðiÞðxÞwðiÞk ðxÞ; ð34ÞyðiÞk ðxÞ ! IFFT ! yðiÞk ðtÞ; ð35Þ

where FFT denotes the discrete Fast Fourier Transform and IFFT itsinverse.

For the efficiency of the presented approach, it should bestressed that a single modal FE analysis is sufficient to computethe Gaussian distributed response. The associated Karhunen–Loèverepresentation (29) provides the basis for the reliability assess-ment conditional on the structural parameters h.

2.5. Conditional reliability

2.5.1. First excursion probabilityIn this section, the conditional first excursion problem pf ðN; hÞ is

discussed for random Gaussian excitation and for the case where arandom realization of H assumes a certain set of deterministicparameters h. The first excursion probability is then defined asthe probability that the critical response exceeds at least oncewithin the considered time period ½0; T� the threshold bi of theith component of the critical response yðt; hÞ [8]. The threshold bi

might consist of a lower limit bi and an upper limit �bi as shownin Fig. 2 for a single critical response.

Fig. 2. First Excursions o

pi;f ðhÞ ¼ P½max06t6T

fyiðt; hÞgP �bi

[ min06t6T

fyiðt; hÞg 6 bi� ð36Þ

¼Z

giðh;nÞ60qðnÞdn ð37Þ

giðh; nÞ ¼minf�bi �max06t6T

fyiðt; h; nÞg; ð38Þ

min06t6T

fyiðt; h; nÞg � big:

This conditional first excursion probability corresponds to the caseof stochastic excitation, specified by the random vector N and itsprobability density function qðnÞ, subjected to a deterministic struc-tural system. In the next section, the integration pf ¼

Rpf ðhÞqðhÞ dh

over the whole domain of the probability density function qðhÞ willbe discussed in detail, leading to the unconditional first excursionproblem pf . To simplify the notations and the reliability problem,each critical response will be considered separately. Hence, the sub-script ‘‘i” is dropped in the following developments.

For this case, where the critical response yðtÞ can be representedby a Karhunen–Loève representation, as given in (29), several effi-cient numerical procedures have been developed recently to calcu-late the first excursion probability (e.g. [2,13]).

2.5.2. Line sampling in dynamicsIn this section, a procedure, based on Line Sampling (LS), is pre-

sented. Line sampling [7,15] is particularly efficient if a so calledimportant direction can be computed, which points towards thefailure domain nearest to the origin (see Fig. 3). Most importantly,it is not required that the vector a (see Fig. 3) points exactly to-wards the center of this domain, nor are any assumptions made

f thresholds b and �b.

Fig. 3. Line sampling in high dimensions.


regarding the shape of the limit state function gðh; nÞ ¼ 0 [7]. LS isrobust, since unbiased reliability estimates are obtained, irrespec-tively of the important direction a, and any deviation from the opti-mal direction merely increases the variance of the estimate. Thebasic procedure of LS is for completeness shown in the Appendix A.

For structures with the properties specified uniquely by h andwhich are excited by a Gaussian distributed loading process, theimportant direction is well known to be proportional to the re-sponse yðjÞðt; hÞ. Let cðt; hÞ be the standard deviation of the responseyðt; hÞ

c2ðt; hÞ ¼ Var½yðt; hÞ� ¼Xn

j¼1

½yðjÞðt; hÞ�2: ð39Þ

Then, the unit vector aðt; hÞ pointing towards the important direc-tion is specified by

ajðt; hÞ ¼yðjÞðt; hÞcðt; hÞ ð40Þ

It should be noted that the important direction is a continuous func-tion of time t. Moreover, linearity implies that any excitation de-fined by n and orthogonal to aðt; hÞ, will have a zero response attime t.

Since the important direction aðt; hÞ varies within the timeinterval 0 6 t 6 T , and the reliability index bðt; hÞ might be of sim-ilar size over a considerable portion of time, the efficiency and reli-ability of Line sampling can be increased by sampling in severalimportant directions a½i�

� �L

i¼1. The algorithm to determine theseimportant directions a½i�

� �L

i¼1 is presented in Appendix B. The pro-cedure involves a transformation of the KL vectors yðjÞðt; hÞ suchthat a½i� ¼ ei

� �L

i¼1, where ei are unit vectors of which the ith compo-nent contains the value one and zeros elsewhere.

In LS, each point n in the standard normal space is decomposedinto the one dimensional space ca½i� and the ðn� 1Þ dimensionalsubspace n?ða½i�Þ orthogonal to the direction a½i�; i ¼ 1;2; . . . ; L,

n ¼ ca½i�ðhÞ þ n?ða½i�Þ: ð41Þ

In the following it is assumed that ~nðjÞ� �N

j¼1 are independent samplepoints of the subspace n?ða½i�Þ drawn by direct Monte Carlo simula-tion. The random sample ~nðjÞ is generated by simulating first n inde-pendent standard normally distributed components of a vector �nðjÞ

and then subtracting the component which points towards thedirection of a½i�,

~nðjÞ½i� ¼ �nðjÞ � ða½i�T � �nðjÞÞa½i�: ð42Þ

For each independent random realization ~nðjÞ½i� , the probability of fail-

ure conditioned on the value ~nðjÞ½i� and h can be determined by

pf~nðjÞ½i� ; h

� �¼Z þ1

�1If n

ðjÞ½i� ðcÞ

� � 1ffiffiffiffiffiffiffi2pp exp � c2

2

dc

¼ U bðjÞ½i�

� �þU ��bðjÞ½i�

� �; ð43Þ

where Uð�Þ denotes the cumulative standard normal distribution,If ð�Þ is an indicator function of failure, and pf

~nðjÞ½i� ; h

� �denotes the

failure probability conditioned on the jth randomly selected linein the ith important direction n

ðjÞ½i� ðcÞ ¼ ~n

ðjÞ½i� þ ca½i� and on the struc-

tural parameter set h. The safe domain lies within the boundsbðjÞ½i� ;

�bðjÞ½i�

h i. Since the superposition law is valid, the response for

nðjÞ½i� ¼ ca½i� þ n

?ðjÞ½i� ðcÞ can be specified by the following function:

y t; c; nðjÞ½i��

¼ cy t; a½i��

þ y t; n?ðjÞ½i��

ð44Þ

y t; a½i��

¼Xn

k¼1

a½i�k yðkÞ½i� ðt; hÞ ð45Þ

y t; n?ðjÞ½i��

¼ yð0Þðt; hÞ þXn

k¼1

n?ðjÞk½i� yðkÞ½i� ðt; hÞ: ð46Þ

Then, the safe domain bðjÞ½i� ;�bðjÞ½i�

h ialong the jth random line in the ith

important direction is defined as

bðjÞ½i� ¼max�b� y t; n?ðjÞ½i�

� �y t; a½i�ð Þ � H �y t; a½i�

� � �8<:

þb� y t; n?ðjÞ½i�

� �y t; a½i�ð Þ � H y t; a½i�

� � �;0 6 t 6 T

9=; ð47Þ

�bðjÞ½i� ¼min�b� y t; n?ðjÞ½i�

� �y t; a½i�ð Þ � H y t; a½i�

� � �8<:

þb� y t; n?ðjÞ½i�

� �y t; a½i�ð Þ � H �y t; a½i�

� � �;0 6 t 6 T

9=; ð48Þ

in which HðyÞ ¼ 1 for y P 0 and HðyÞ ¼ 0 for y < 0. To each of theimportant directions fa½i�gL

i¼1, a disjunct failure domain is associatedby imposing the condition

bðjÞ½i� < nðjÞk ða½i�Þ < �bðjÞ½i� ; k – i and 0 < k 6 L: ð49Þ

For the very unlikely cases in which the above condition is not sat-isfied, the line will be ignored and another line is generated instead.

The independent estimates pf ðhðjÞÞ for the failure probability al-low to compute an unbiased estimate for the conditional failureprobability

pðjÞf ðhÞ ¼XL

i¼1

pðjÞf ½i�ðhÞ ð50Þ

pðjÞf ½i�ðhÞ ¼1Ni

XNi

k¼1

pf ð~nðkÞ½i� ; hÞ ð51Þ

pf ðhÞ ¼1N

XN

j¼1

pðjÞf ðhÞ ð52Þ

and the variance

Varðpf ðhÞ ¼1

NðN � 1ÞXN

j¼1

pðjÞf ðhÞ � pf ðhÞ� �2

; ð53Þ

which is the basis for deriving further confidence intervals.

2.6. Design point for stochastic structural systems

In this section, the domain which contributes most to theunconditional total failure probability pf will be determined. The-


oretically, the failure probability pf jh, conditioned on specific real-izations h, need to be integrated over the whole domain of pHðhÞ

pf ¼Z

pf ðhÞpHðhÞ dh: ð54Þ

Since the number of uncertain structural parameters, characterizedby the random quantities fH1;H2; . . . ;HSg, is in general large,numerical integration is not feasible. Basically the above integralcan be estimated by Monte Carlo sampling

pf ¼1N

XN

j¼1

pf ðhðjÞÞ; ð55Þ

where N independent realizations hðjÞ are drawn from the distribu-tion pHðhÞ and the associated conditional failure probability pf ðhðjÞÞis determined as described in the previous section. Such an ap-proach is only efficient in case the variability of pf ðhðjÞÞ is small,i.e. might be within one order of magnitude. However, in case thestructural variability is high, the conditional first excursion proba-bilities will vary over many orders of magnitude and direct MonteCarlo approaches will be very inefficient. Considering Eq. (54), itis not difficult to recognize that sampling in the neighborhood ofthe structural design point h�, characterized by

pf ðh�ÞpHðh�ÞP pf ðhÞpHðhÞ; ð56Þ

contributes most to the total (unconditional) failure probability. Inorder to identify this domain, composed of the standard normalvariables fn; hg, the point fn�; h�g with the minimal distance b tothe failure surface is determined at the most critical time t�1 (see(B.8)). Since the uncertainty in the excitation and of the structuralparameters, specified by n� and h�, respectively, are independentand therefore orthogonal in the standard normal space, this dis-tance is specified according to Pythagoras by

b2ðn; hÞ ¼ knk2 þ khk2: ð57Þ

Fig. 4 shows a sketch of the failure domain for this case, wherebðn; hÞ can only be determined in the complete space ðn; hÞ. Thethreshold b is reached for

knk2 ¼ b2

c2ðt�; hÞ ð58Þ

b ¼ min½�b� yð0Þðt�; hÞ; yð0Þðt�; hÞ � b�: ð59Þ

The failure point ðh�; n�Þ, with the highest probability density anddefined as the nearest failure point to the origin in standard normalspace, is derived by imposing the necessary condition

Fig. 4. Structural design point h� .

@b2ðn; hÞ@hk

¼ 0; k ¼ 1;2; . . . ;K; ð60Þ

which leads to the solution

h�k ¼b2

c3ðt�; h�Þ �@cðt�; h�Þ@hk

: ð61Þ

An accurate solution of the above relation can only be obtained inan iterative manner, where convergence is reached for s > S with�ðSÞ smaller than the tolerance,

h�ðsþ1Þk ¼ ð1�wÞh�ðsÞk þw

b2

c3ðt�; h�ðsÞÞ� @cðt

�; h�ðsÞÞ@hk

ð62Þ

�ðsþ1Þ ¼ kh�ðsþ1Þ � h�ðsÞk=kh�ðsþ1Þk; ð63Þ

where w ¼ 0:5 leads to a stable and fast convergence. However,there is not much gain in evaluating h� very accurately. A first orsecond step ðs ¼ 0;1Þ is usually sufficiently accurate for the integra-tion procedures as developed in the next section.

The above iterative evaluation of the structural design point h�

requires a gradient computation which might hamper the effi-ciency of the approach in case the number of uncertain structuralparameters is large. For such cases, a gradient estimation proce-dure – as shown in the Appendix C – might be used to improvethe computational efficiency.

2.7. First excursion probability for stochastic systems

In this section, the procedure to estimate the total uncondi-tional first excursion probability by integrating over the completespace of uncertain structural parameters is shown (see Eq. (54)).The proposed approach is to limit the application of LS to the sub-space of the structural parameters h. This in fact leads to robust re-sults, even for quite large uncertainties of the structuralparameters, since the important directions a½i�ðhÞ are then alwayscomputed in the optimal directions. The procedure is outlined asfollows:

1. Determine the design point h� with acceptable accuracy andcompute

bS ¼ kh�k ð64Þ

aS ¼h�

bSð65Þ

where the index ‘‘S” denotes the subspace of the structuraluncertainties.

2. Generate samples fh?ðjÞgNj¼1 using direct MCS in the subspace of

h, which are perpendicular to the vector h�.3. Compute for each parallel line for, say, five discrete points

hðjÞðckÞ; k ¼ 1; . . . ;5, as shown in Fig. 5

hðjÞðckÞ ¼ h?ðjÞ þ ckaS; ð66Þck ¼ bS þ ðk� 3ÞD; k ¼ 1;2; . . . ;5 ð67ÞD � 0:6 ð68Þ

the associated conditional failure probability pðjÞf ðckÞ.4. Estimate the conditional failure probability pðjÞf along each line,

Z
pðjÞf ¼
1ffiffiffiffiffiffiffi2pp

1

�1e�c2=2pðjÞf ðcÞ dc: ð69Þ

with quadratic interpolation of the function lnðpðjÞf ðcÞÞ by usingthe available discrete values. This estimation will be discussedsubsequently in more detail.

5. Estimate the mean and variance of the failure probability pf ðtÞaccording to Eqs. (52) and (53).

Fig. 5. Line sampling in the uncertain structural paramameter space.


It should be noted that pðiÞf ðcÞ > 0 for �1 < c <1, since the po-sition along each line specifies uniquely the properties of a struc-ture subjected to random excitation. Since pðiÞf ðcÞ is always apositive quantity, it is proposed to represent pðiÞf ðcÞ as the exponen-tial of a linear or quadratic polynomial,

pðiÞf ðcÞ ¼ exp½a0 þ a1c þ a2c2=2�; ð70Þ

where the coefficients are obtained by solving a least squareproblem

a0 þ a1cl þ a2c2l =2 ¼ ln pðiÞf ðclÞ

h i; l ¼ 1;2; . . . ;5 ð71Þ

The advantage of this approximation is that the infinite integral inEq. (69) can be represented in closed form for a2 < 1

pðiÞf ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� a2p exp a0 þ

a21

2ð1� a2Þ

: ð72Þ

A linear approximation of ln pðiÞf ðcÞh i

in the least square solutionshould be applied ða2 ¼ 0Þ, whenever a2 > 0:4, which might be

Fig. 6. First three mode shapes

due to random fluctuation of the estimates pðiÞf ðclÞ. Experienceshowed that the results using either a linear or quadratic approxi-mation are quite similar. However, a linear approximation is lesssensitive to random fluctuations of the estimate pðiÞf ðclÞ. Hence theintegration over all points along the line is efficiently approximated,requiring only three to five finite element analyses per line.

3. Numerical example

3.1. General remarks

The method, as developed in Section 2, is now exemplifiedwithin this section. Usually, proposed methods have been demon-strated by applying them to exceptionally small academic type ofexamples, where the feasibility of the approach to real world prob-lems remains an open issue, because computational difficulties andmany issues of computational efficiency are avoided. Since reliabil-ity evaluations in structural dynamics are computationally verydemanding, the computational efficiency of the proposed methodis naturally in the focus of interest. However, to be of practical va-lue, the efficiency of procedures should be demonstrated by apply-ing them to models which reflect the complexity of relevantengineering structures. By choosing a realistic FE-model, modelinga 12-story building made of reinforced concrete, the requirementof practical applicability is shown.

3.2. Structural system

3.2.1. GeometryA twelve story building with an additional cellar floor made of

reinforced concrete is considered. The FE-model consists of 4046nodes and 5972 elements using shell and 3-D beam elements formodeling the girders, resulting in 24,276 degrees of freedom.Fig. 6 provides a view of the building showing the displacementfield according to the first three mode shapes. The building isaxis-symmetric and consists of 13 floors of 24:0� 24:0 m. Thefoundation plate is 0.5 m thick and rests on soil modeled by elasticsprings. Fig. 7 shows a plan view for all floors, with the exceptionthat the cellar floor is surrounded by cellar walls of 0.3 m thick-ness. The height of each story is 4.0 m. The weight of the structureis carried by four groups of concrete walls forming a cross. The fourgroups of supporting concrete walls are connected by four girdersof type g-1 the height of which is 1.2 m and a width identical to the

of twelve story building.

Fig. 7. Floor plan of twelve story building supported by croncrete walls (w) andgirders of type g-1 and type g-2. Critical strains are considered at positions p-1 andp-2.


walls. The thickness of the supporting walls varies with respect tothe respective story. In the cellar floor and first and second floorthe wall thickness is 0.6 m, the next two floors (third and fourth)0.5 m, and 0.4 for the fifth until the twelfth floor. The concreteplate of 0.2 m thickness is supported also by girders of type g-2of height 0.5 m and constant width of 0.3 m. Above the concreteplate, a floor construction with the weight of 150 kg/m2 have beenassumed for all floors.

3.2.2. Uncertain structural propertiesConcrete and reinforced concrete are widely used in building

constructions. Its material properties, however, are hard to specifyby just a few characteristic quantities. The stiffness of reinforcedconcrete construction parts depends on many factors which arestill difficult to predict and to control. Due to inhomogeneitiesand possible cracking, the stiffness under dynamic loading revealsuncertainties. Moreover, it might depend also on the loading his-tory (see e.g. [3]). At the present state-of-the-art, quantitativemodels for reinforced concrete structures, which are both feasibleand realistic (including bond-slip, cracking, etc.) are not used yetfor ordinary structural analysis as they are still topics of active re-search. Uncertainties of the stiffness is partially the result of inher-ent random factors and to a large part by lack of knowledge of theactual mechanisms. A practical way to cover all these uncertaintiesis to take a single quantity, such as the Young’s modulus, to repre-sent the uncertainties in the stiffness, with the understanding thatthis quantity should cover the uncertainty of the stiffness and notonly of the physical elastic constant.

All structural uncertainties are assumed to be represented as afunction of 244 independent standard normal variables. The stiff-ness of the soil bedding is represented by 9 random variablesand the modal damping ratio by a single random variable. Themean value of the stiffness of the soil is assumed to be4.44e+08 N/m3 in the horizontal and 8.88e+08 N/m3 in the verticaldirection, with a coefficient of variation of 0.224. 169 random vari-ables do represent the Young’s modulus of various parts of thereinforced concrete structure.

The mean value for the Young’s modulus for the foundationplate and cellar walls are assumed to be 2.8e+10 N/m2 and its var-iability is modeled by a coefficient of variation of 0.215 and corre-

lation coefficients in the order of 0.86. The Young’s moduli of theremaining primary reinforced concrete parts are assumed some-what higher to be 3.2e+10 N/m2 with a coefficient of variation of0.144. This relatively large value reflects, to a considerable extent,the lack of knowledge on the dynamic stiffness. It implies also alarge correlation of the stiffness, which is reflected by a correlationcoefficient in the order of 0.7. The live loads (masses) are repre-sented by a function of five random variables per floor, which areassumed to be independent between the floors and correlatedwithin the floors. The mean value is assumed to be 100 kg/m2

and the coefficient of variation of 0.224 with correlation coeffi-cients of 0.8.

3.3. Dynamic excitation

In this numerical example, the dynamic excitation is due toearthquake ground motion in terms of ground accelerations. Fu-ture ground motions are highly uncertain, regarding its ampli-tudes, frequency content and durations, respectively. For physicalreasons, the average velocity and acceleration must vanish, andthe frequency content depends on the unknown source mechanismand is influenced substantially by local soil conditions. The inten-sity (acceleration amplitudes) depends on the distance to theearthquake source and the duration on the magnitude of the en-ergy release.

It is difficult to cover all these uncertainties by a credible exci-tation model. In engineering practice, the excitation model is se-lected such that the model covers uncertainties conditioned onthe acceptable hazard.

There are several options to derive a suitable realistic excitation.In the ideal case, many records from different events are available– which after a suitable scaling – are used further to establish thecovariance matrix (see (8)) from which the Karhunen–Loève termscan be deduced in a straightforward manner. A further option is toestablish the covariance matrix such that it is compatible to someresponse spectra requirements. In this work, an approach based onfiltered white noise is used. To cover the unknown frequency con-tent and random amplitudes of the acceleration, the model pro-posed in [4] is applied, where the acceleration is represented asthe output of the response of a linear filter excited by white noise

aðtÞ ¼ X2gv1ðtÞ þ 2fgXgv2ðtÞ �X2

f v3ðtÞ � 2ff Xf v4ðtÞ; ð73Þ

in which Xg represents the dominant frequency of the ground andXf ensure that the spectrum of the acceleration tends to zero for fre-quencies approaching zero. The linear filter is described by the dif-ferential equation [4]

ddt

v1

v2

v3

v4

8>>><>>>:

9>>>=>>>; ¼

0 1 0 0�X2

g �2fgXg 0 00 0 0 1X2

g 2fgXg �X2f �2ff Xf

26664

37775 �

v1

v2

v3

v4

8>>><>>>:

9>>>=>>>;þ

0wðtÞ

00

8>>><>>>:

9>>>=>>>;

ð74Þ

where wðtÞ denotes white noise. The intensity of wðtÞ can either becharacterized by its intensity IðtÞ or by the constant (over frequencyx, i.e. Sðx; tÞ ¼ S0ðtÞ) spectral density S0ðtÞ ¼ IðtÞ=ð2pÞ, which sat-isfy the following relation [8]

E½wðtÞwðt þ sÞ� ¼ IðtÞdðsÞ ¼ 2pS0ðtÞdðsÞ ð75Þ

where dð�Þ denotes Dirac’s delta function with the property dðsÞ ¼ 0for s – 0 and

R �� dðsÞds ¼ 1; � > 0. Since white noise is uncorrelated

for any s > 0, white noise excitation can be discretized by a sequenceof independent impulses with zero mean and standard deviationrIðkDtÞ or with linearly interpolated independent force amplitudeswith standard deviation rFðkDtÞ at subsequent time steps kDt

0 2 4 6 8 10 12 14 16−0.4

−0.2

0

0.2

0.4

[m/s

2 ]Various KHL−terms for horiz. acceleration

10−th30−th60−th80−th

0 2 4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

time in [s]

[m2 /s

4 ]

Variance of horizontal acceleration

Fig. 9. Variance of horizontal acceleration over time.


rIðkDtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIðkDtÞDt

qð76Þ

rFðkDtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiIðkDtÞ

Dt

r: ð77Þ

The non-stationary properties of earthquake ground motion is de-scribed by a time varying white noise intensity.

IðtÞ ¼ I0

0 for t 6 0t=2 for 0 < t < 21 for 2 6 t 6 8expð�0:16ðt � 8ÞÞ for t P 8:

8>>><>>>: ð78Þ

The following constants are used in this work: I0 ¼0:16 m2=s3; Xg ¼ 2p rad=s; fg ¼ ff ¼ 0:6, and xf ¼ 0:1Xg .

In a next step, the covariance matrix C ¼ ½Cij� of the accelerationis established with Dt ¼ 0:01 s, and 0 < i; j 6 1600 andCij ¼ E½aðiDtÞaðjDtÞ]. For this purpose, the impulse response func-tion of the linear filter is determined by the free motion of the filterwith zero initial displacement, and _v1ð0Þ ¼ _v3ð0Þ ¼ _v4ð0Þ ¼ 0 and_v2ð0Þ ¼ 1. To obtain the impulse response function of the acceler-ation aIRFðtÞ, Eq. (73) is used. The result is shown in Fig. 8. To estab-lish the covariance matrix, a single impulse at time kDt isconsidered first. The associated covariance is

CijðkÞ ¼Dt � IðkDtÞaIRFðði�kÞDtÞaIRFððj�kÞDtÞ for i P k and j P k

0 for i< k or j< k

�ð79Þ

Since the impulses at instants kDt are all independent, the coeffi-cients Cij are given by the sum

Cij ¼X1600

k¼0

CijðkÞ: ð80Þ

The diagonal terms Cii ¼ E½a2ðtÞ� specify the variance over the timeti ¼ iDt and are shown in Fig. 9. Applying further the procedure asdescribed in Section 2.1, the Karhunen–Loève functions aðjÞðtÞ (seeEq. (12)) are computed. For illustration, some of these functionsare shown in Fig. 10. Finally, it should be stated that also the verticalearthquake acceleration has been modeled. It turned out that forthe structure under consideration this vertical acceleration has onlya negligible effect on the critical response and hence it is not de-scribed further.

time in [s]

Fig. 10. Karhunen–Loève functions aðjÞðtÞ; j ¼ 10;30;60;80 of the horizontalacceleration.

0 1 2 3 4 5 6 7 8−2

−1

0

1

2

3

4

5

6

7

8

time in [s]

[m/s

2 ]

Impulse acceleration response of horizontal filter

Fig. 8. Acceleration impulse response function of filter.

3.4. Critical response

Under dynamic earthquake loading, two reinforced structuralcomponents are likely to exceed the critical strain. The strain invertical direction of the walls in the basement, forming a cross,might be critical. These walls support the weight of the entirestructure and are, in addition, highly strained by bending due tothe horizontal earthquake accelerations. Since the structure issymmetric, and the horizontal accelerations are in both directionsindependent and identically distributed, it suffices to consider asingle position, indicated by p-1 in Fig. 7.

Girders form the remaining critical parts, denoted as g-1 inFig. 7, which connect separated walls. These girders transmit largeshear forces from one group of walls to the counterpart, and con-tribute essentially to the bending stiffness of the twelve storybuilding. The critical part is the curvature of the girder at the con-nection to the walls. The position is indicated by p-2 in Fig. 7.

Table 1 provides an overview of the strains in these compo-nents, i.e. for all floors. Accepting as limit state function a vertical

Table 1Critical strain due to static weights and maximum standard deviation due to horizontal earthquake motion.

Floor Width Wall at pos. p-1 Girder g-1 at p-2

v. Strain static v. Strain std. dev. Curvature static Curvature std. dev.(–) (m) �10�6 �10�6 �10�6 �10�6

0 0.6 �143 266 �9 2811 0.6 �110 228 �27 4662 0.6 �100 181 �27 5483 0.5 �108 170 �32 6434 0.5 �97 134 �32 6545 0.4 �107 125 �41 7046 0.4 �93 80 �41 6757 0.4 �80 73 �42 6208 0.4 �66 60 �42 5449 0.4 �53 46 �42 45710 0.4 �39 33 �42 36711 0.4 �25 20 �42 28912 0.4 �18 12 �34 187


strain of 0.0016 for the walls without having to expect seriousdamage and a maximum acceptable curvature of 0.004 in the gird-ers g-1, the curvature in the girder are more likely to be exceeded.Hence, the reliability analysis will focus on the reliability of thegirders for not exceeding the limit state function the curvature of0.004. Taking into account the static solution of floor five,the limitstate function has the lower and upper bounds are b ¼ �0:004þ000041 ¼ �0:003959 and �b ¼ 0:004þ 000041 ¼ 0:004041, respec-tively.

3.5. Reliability of critical component

3.5.1. Estimation by direct monte carlo simulationBefore evaluating the first excursion probability, the Karhunen–

Loève terms (see Eqs. (29) and (35)) of the critical response, i.e. thecurvature at girder g-1, are determined. These terms yðjÞðt; hÞ de-pend only on the set of uncertain structural parameters h and theirevaluation requires a single FE analysis for each distinct set hðjÞ.Hence, the investigation of the variability of the response due touncertain structural properties is the computationally expensivepart. The efficiency of the procedure will therefore be governedby the required number of FE analyses; the larger the FE-model,the more important this number will be.

Fig. 11 shows some of these critical response functions for thenominal solution h ¼ 0. The associated first excursion problem isdetermined by Line sampling as outlined in Section 2.5, where

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0 2 4 6 8 10 12 14 16

time t [s]

KHL-terms of curvature of girder at p-2

1104080

Fig. 11. Karhunen–Loève terms yðjÞðtÞ; j1;10;40;80 of the curvature of girder g-1 atposition p-2 in floor five for h ¼ 0.

N ¼ 4000 lines have been used. Note that the number N is selectedsuch that the computation time is approximately one fifth of thetime for the FE analysis to compute the Karhunen–Loève terms.The failure probability assumes for the nominal system ðh ¼ 0Þthe value pf ¼ 4:67� 10�7 and a standard deviation of this esti-mate of rpf

¼ 1:46� 10�7. However, it is well known that the reli-ability is quite sensitive with respect to the variations of structuralproperties.

This sensitivity is shown by the results of Direct Monte Carlosimulation (see Eq. (55)) by randomly sampling over the uncertainstructural parameters fhðjÞg1600

j¼1 . Direct Monte Carlo sampling leadsto the estimate pf ¼ 2:32� 10�4 and the standard deviation ofrpf¼ 3:43� 10�5. Fig. 12 shows the histogram of all independent

estimates. It is observed, that the estimate for the failure probabil-ity varies over many orders of magnitude. Hence, the procedure ofDirect Monte Carlo Simulation cannot be used in an efficientmanner.

3.5.2. Critical domain of uncertain structural parametersThe domain of the structural parameters which contributes

most to the total failure probability has been derived in Section2.6. Eq. (61) provides a quantitative guidance to determine this do-main. Its solution, shown in Table 2, requires the gradient of thestandard deviation due to the dynamic excitation with respect tothe uncertain structural parameters. To avoid direct differentiationfor all 244 uncertain structural parameters, the gradient estimation

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16

%

k

Histogram of 1600 DMC estimates p_f=10^{-k}

Fig. 12. 1600 estimates by direct Monte Carlo of the stochastic structure.

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

2 4 6 8 10 12 14 16 18 20

Twenty inpendent estimates of pf by LS

Fig. 14. Twenty independent estimates of the total first excursion probability of theuncertain structure lead to the mean pf ¼ 0:00046 with standard deviationrpf¼ 0:000040.

Table 2The first three most important components (k) of the structural design point h�

computed by three iterations s ¼ 1;2;3.

s 0 1 2 3c�ðsÞ 0.00070 0.00190 0.00110 0.00111

k h�ð0Þk h�ð1Þk h�ð2Þk h�ð3Þk26 0 �1.92 �2.07 �2.05244 0 �1.52 �1.18 �1.17114 0 �0.69 �0.76 �0.75


procedure is employed [12,11] to compute these componentswhich significantly influence the standard deviation of the criticalresponse. It was observed that the variability of cðt�; hÞ is governedby only three random variables, namely the numbers 26, 244 and114. Number 26 controls the stiffness of the girder of all floors,number 244 controls the log-normally distributed damping ratio,and the number 114 the local stiffness deviation at the consideredgirder at floor 5. These three random variables cause 97% of the to-tal variability of cðt�; hÞ, i.e. Eq. (C.4) lead to � ¼ 0:03. The gradientestimation procedure has been used only in the first step, i.e. toidentify the most important parameters and to compute the asso-ciated accuracy in terms of �. This procedure required 44 FE anal-yses to arrive at the solution as shown. The iteration for s ¼ 1;2 inEq. (62) are then performed by finite differences of the three iden-tified random variables. Table 2 summarizes the result of the fail-ure point nearest to the origin in standard normal parameter space.

3.5.3. Line sampling within the uncertain structural parameter spaceThe procedure outlined in Section 2.7 is applied to compute effi-

ciently the total first excursion probability of the uncertain struc-tural system. Fig. 13 shows the five discrete values pðjÞf ðckÞ of theconditional failure probability at five points along 20 random linesby using bS ¼ 2:48 and D ¼ 0:6. The significant increase of the con-ditional failure probabilities in the direction of the design pointdemonstrates the high sensitivity of the failure probability with re-spect to the particular uncertain structural parameters.

Fig. 14 shows twenty independent estimates of the total firstexcursion probability. To obtain these results 100 FE analyses (20� 5) have been carried out. The estimated mean is pf ¼ 0:00046with a standard deviation of rpf

¼ 0:000040. When compared withresults of Direct Monte Carlo sampling in Fig. 12 in the uncertainparameter space, one observes that these results are approxi-mately by a factor of two larger. LS, however, explores the impor-

1e-05

0.0001

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18 20

Fig. 13. Five discrete conditional failure probabilities along twenty lines in thedirection of the structural design point using bS ¼ 2:48 and D ¼ 0:6.

tant domain systematically and is therefore more reliable. This canbe also seen by the substantial smaller coefficient of variation ofapproximately 8% compared with 16% and a 16 times larger num-ber of FE analyses when using Direct MCS.

4. Conclusions

The developments as shown here allow to draw the followingconclusions:

1. The reliability evaluation of linear structural systems (generalFE-models) with uncertain structural parameters subjected togeneral Gaussian dynamic excitation is feasible. To the authorsknowledge, the feasibility of a reliability evaluation in dynamicsfor a large FE-model and a large number of structural uncertain-ties is demonstrated the first time in a numerical example.

2. A novel procedure to identify the critical uncertain parametershas been introduced.

3. Efficiency is gained by performing modal analysis, impulseresponse functions combined with Fast Fourier Transforms(FFT) and a new Line Sampling approach which focuses on theimportant failure domain within the uncertain parameterspace.

4. The presented approach is accurate and robust, also for caseswhere the uncertainties of the structural parameters are large.

5. The reliability of linear systems is quite sensitive to structuraluncertainties – as demonstrated in the numerical example –and therefore must not be ignored.

Acknowledgement

This work has been supported by the Austrian Science Founda-tion (FWF) under contract number P19781-N13 (Simulation Strat-egies for FE Systems under Uncertainties), which is gratefullyacknowledged by the authors.

Appendix A. Line sampling in a single direction

Line sampling (LS) [7,15] has been developed for estimating lowprobabilities of failure in high dimensional reliability problems. LSis particularly efficient if a so called important direction can becomputed, which points towards the failure domain nearest to


the origin (see Fig. 3). LS is robust, since unbiased reliability esti-mates are obtained, irrespectively of the important direction a,and any deviation from the optimal direction merely increasesthe variance of the estimate.

Suppose the important direction a has been estimated by anappropriate procedure, e.g. gradient estimation, stochastic searchalgorithm, design point, etc. Let’s assume further that the reliabil-ity is computed in standard normal space of dimension n. Then,any point n in the n- dimensional standard normal space is repre-sented by the projection on a

c ¼ aTn ðA:1Þ

and the ðn� 1Þ�dimensional space n?ðaÞ:

n ¼ caþ n?ðaÞ ðA:2Þ

LS is performed by selecting random points f~n?ðjÞðaÞgNj¼1 in the space

n?ðaÞ by applying direct Monte Carlo Simulation. Random lines arespecified by the variable c and ~n?ðjÞðaÞ

nðjÞðcÞ ¼ caþ ~n?ðjÞðaÞ: ðA:3Þ

For each random line, the safe domain bj < c < �bj is determined,leading to an estimate for the failure probability

pðjÞf ¼ UðbjÞ þUð��bjÞ; ðA:4Þ

where Uð�Þ denotes the cumulative standard normal distribution.Finally, an unbiased estimate for the failure probability is obtainedfrom the average

pf ¼1N

XN

j¼1

pðjÞf : ðA:5Þ

Appendix B. Evaluation of important directions in lineardynamics

To obtain these specific important directions, aðtÞ at discrete in-stances ft�i g

Li¼1 are selected.

a½i�ðhÞ ¼ aðt�i ; hÞ ðB:1Þ

The instance t�i is determined in the ðnþ 1� iÞ-dimensional sub-space n½i� � n, defined as the subspace orthogonal to all previouslyselected important directions fa½j�gi�1

j¼1 and defining the initial condi-tion n½1� ¼ n

ha½j�; n½i�i ¼ 0 8 j ¼ 1; . . . ; i� 1; ðB:2Þ

where h�; �i denotes the inner product. The ðnþ 1� iÞ-dimensionalsubspace n½i� is computed by the linear transformation

n½i� ¼ n½i�1� � R½i�; i ¼ 2;3; . . . ; L ðB:3Þ

where R½i� is a ðnþ 1� iÞ � ðn� iÞ dimensional orthonormal matrixorthogonal to a½i�. This matrix is established by populating first allentries of R½i� by independent random numbers and applying subse-quently the well known Gram-Schmidt orthogonalization algo-rithm. The deterministic KL functions yðjÞðt; hÞ need also to betransformed because of the change of orientation. Define

Y½i� ¼ ½yð1Þ½i� ðt; hÞ; yð2Þ½i� ðt; hÞ; . . . ; yðnþ1�iÞ

½i� ðt; hÞ� ðB:4Þ

with the initial condition yðjÞ½1�ðt; hÞ ¼ yðjÞðt; hÞ, these terms are alsotransformed by

Y½i�ðt; hÞ ¼ Y½i�1�ðt; hÞ � R½i�; i ¼ 2;3; . . . ; L ðB:5Þ

The variance c2½i�ðt; hÞ of the critical response yðt; hÞ in these subspace

n½i� is determined analogous as in Eq. (39)

c2½i�ðt; hÞ ¼

Xnþ1�i

j¼1

yðjÞ½i� ðt; hÞh i2

ðB:6Þ

The important direction a½i� for LS is selected at time t�i ; a½i� ¼ aðt�i Þ,

when the excursion probability assumes its highest value, or equiv-alently when the reliability index biðtÞ assumes it smallest value:

a½i�ðhÞ ¼ aðt�i ; hÞ ðB:7Þ

min�b� yð0Þðt�i Þ

c½i�ðt�i Þ;yð0Þðt�i Þ � b

c½i�ðt�i Þ

" #6 min

�b� yð0Þðt; hÞc½i�ðt; hÞ

;yð0Þðt; hÞ � b

c½i�ðt; hÞ

" #;

0 6 t 6 T ðB:8Þ

Appendix C. Gradient estimation

In the following, the actual evaluation of the derivatives@cðt�; h�ðsÞÞ=@h�ðsÞk is discussed, since it might considerably affectthe efficiency of the procedure. For the case where only very fewuncertain structural quantities are specified, differentiation for allparameters fhkgK

k¼1 by finite differences can be carried out in astraight forward manner, requiring additional K finite elementanalyses (FEA). In general, however, the number K of uncertainstructural quantities might be quite large, where K FEA would seri-ously reduce the efficiency of the procedure. Although, the numberof uncertain structural properties might be large, only a limitednumber of uncertain parameters have usually a significant effecton the standard deviation cðt�; hÞ. These quantities are sometimesknown a priori and often they cannot be specified with sufficientconfidence. Suppose an estimate where only I < K uncertain quan-tities significantly influence the standard deviation cðt�; hðsÞÞ. It isthen proposed to compute first these derivatives @cðt�; hðsÞÞ=@hðsÞi2I

by finite difference. These directly evaluated derivatives are de-noted as

ji2fIg ¼ @cðt�; hðsÞÞ=@hðsÞi2fIg ðC:1Þ

Since the set fIg is just an estimate, a check whether indeed noimportant quantities hjRI are missing is necessary. For this purpose,the standard deviation cðt�; hÞ has to be computed for the setfcðt�; hðjÞ þ DðkÞgI

k¼0, with the following properties:

Dð0Þi ¼ 0; i ¼ 1;2; . . . ;K ðC:2Þ

Dðj>0Þi ¼ d

�1 for 0 6 ui;j < 0:5þ1 for 0:5 6 ui;j 6 1

�ðC:3Þ

with 0:01 6 d 6 0:1 and uniformly distributed independent randomnumbers 0 < ui;j < 1 drawn by a random number generator. Theaccuracy achieved by this procedure can be estimated by comparingthe Euclidean norm of the vector b and the vector c,

� ¼ kckkbk ðC:4Þ

where the components of the vectors are defined as follow:

bk ¼ cðt�; hðsÞ þ DðkÞÞ � cðt�; hðsÞÞ; k ¼ 1;2; . . . ; J ðC:5Þck ¼ bk �

Xi2fIg

jiDðkÞi ðC:6Þ

In case � < 0:2, the estimate might be regarded as sufficiently accu-rate, otherwise the estimate requires improvements. For an efficientprocedure for improvements it is referred to [12,11]. An abbreviatedversion is shown in the following:

Components hj, which are likely to have a great effect, can berecognized by the correlation of DðkÞj and the component ck. It issuggested to determine for all components fj R fIgg the estimate


jj ¼PJ

k¼1DðkÞj ckPJ

k¼1 DðkÞj

� �2 ¼PJ

k¼1DðkÞj ck

Jd2 ðC:7Þ

In a further step, the component m with the absolutely largest esti-mate jjmjP jjjRfIgj is selected and the derivative jm is computed byfinite differences. The improvement is measured by � after updatingthe vector c

ck#ck � jmDðkÞm : ðC:8Þ

This procedure is repeated until a satisfactory accuracy is obtained.If the improvements are insignificant, the sample size J needs to beenlarged as shown in [11].

References

[1] Adomian G. Stochastic Systems. New York: Academic Press; 1983.[2] Au SK, Beck JL. First excursion probabilities for linear systems by very efficient

importance sampling. Probabilistic Engineering Mechanics 2001;16:193–207.[3] Chryssanthopoulos MK, Dymiotis C, Kappos AJ. Probabilistic evaluation of

behaviour factors in ec8-designed r/c frames. Engineering Structures 2000;22:10281041.

[4] Clough RW, Penzien J. Dynamics of Structures. Intern. Studented. Auckland: McGraw-Hill; 1975.

[5] Ghanem R, Spanos P. Stochastic Finite Elements: A SpectralApproach. Springer-Verlag; 1991.

[6] Karhunen K. Über lineare Methoden in der Wahrscheinlichkeitsrechnung.American Academy of Science Fennicade Series A 1947;37:3–79.

[7] Koutsourelakis PS, Pradlwarter HJ, Schuëller GI. Reliability of structures in highdimensions, part I: algorithms and applications. Probabilistic EngineeringMechanics 2004;19(4):409–17.

[8] Lin YK. Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill,Inc., McGraw-Hill Company; 1967.

[9] M. Loève, Fonctions aleatoires du second ordre, supplemeent to P. Levy. InProcessus Stochastic et Mouvement Brownien. Gauthier Villars, Paris, 1948.

[10] W.L. Oberkampf, S.M. DeLand, B.M. Rutherford, K.V. Diegert, K.F. Alvin,Estimation of total uncertainty in computational simulation. TechnicalReport, Sandia National Laboratories, Albuquerque, NM SAND2000-0824,USA, 2000.

[11] Pellissetti MF, Pradlwarter HJ, Schuëller GI. Relative importance of uncertainstructural parameters, part II: applications. Computational Mechanics2007;40(4):637–49.

[12] Pradlwarter HJ. Relative importance of uncertain structural parameters part I:algorithm. Computational Mechanics 2007;40(4):627–35.

[13] Pradlwarter HJ, Schuëller GI. Excursion probability of non-linear systems.International Journal of Non-Linear Mechanics 2004;39(9):1447–52.

[14] Schenk CA, Schuëller GI. Uncertainty Assessment of Large Finite ElementsSystems. Berlin/Heidelberg/New York: Springer-Verlag; 2005.

[15] Schuëller GI, Pradlwarter HJ, Koutsourelakis PS. A critical appraisal ofreliability estimation procedures for high dimensions. ProbabilisticEngineering Mechanics 2004;19(4):463–74.

[16] Soong TT, Grigoriu M. Random Vibration of Mechanical and StructuralSystems. New Jersey: Prentice-Hall; 1993.





A finite element procedure for multiscale wave equationswith application to plasma waves

Haruhiko Kohno a,*, Klaus-Jürgen Bathe b, John C. Wright a

a Plasma Science and Fusion Center, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USAb Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA


Article history:Received 8 April 2009Accepted 1 May 2009Available online 2 June 2009

Keywords:WavesMulti-scaleSpectral methodFinite elementsPlasmaRadio frequencies


* Corresponding author.E-mail address: [email protected] (H. Kohno).

a b s t r a c t

A finite element wave-packet procedure is presented to solve problems of wave propagation in multi-scale behavior. The proposed scheme combines the advantages of the finite element and spectral meth-ods. The basic formulation is presented, and the capabilities of the procedure are demonstrated throughthe solution of some illustrative problems, including a problem that characterizes the mode-conversionbehavior in plasmas.


1. Introduction

Although much research effort has been spent on solvingwave propagation problems, the accurate solution of many suchproblems is frequently still difficult, in particular when multi-scale behavior is involved. Of course, in general, a numericalsolution needs to be employed [1,2]. However, when wave num-bers vary in magnitude over the domain, the wave numbersmay be very large in certain regions, in particular where thereis resonance, requiring a fine mesh to capture this fine-scalevariation accurately. In this case, if conventional numericalmethods are used, even though most of the domain shouldnot require a fine mesh, we still have to provide the fine dis-cretization for the entire domain. The reason is that the waveswill travel throughout the domain and we cannot predict pre-cisely prior to the analysis where resonance will occur. Indeed,frequently, it is the objective of the analysis to detect the re-gions of resonance.

A specific example in mind is the solution of waves in plasmas.Applying radio-frequency waves in order to raise plasma tempera-tures is an important subject of research for nuclear fusion. Mucheffort has been directed to uncover the mechanisms of electromag-netic wave propagations in plasmas (see Refs. [3,4] and the refer-ences therein) and computer programs to solve wavepropagations in plasmas have been developed [5–7]. Since in these

ll rights reserved.

numerical solutions also the phenomenon of mode conversionneeds to be addressed, the usual numerical techniques to solvewave propagation problems are not efficient.

To solve wave propagation problems accurately, the spectralmethod [8] or spectral finite element method have been used[9–11] and good results have been obtained in certain analyses.However, these methods can be computationally expensive, andmore importantly, the methods show intrinsic difficulties in satis-fying the boundary conditions for arbitrary-shaped domains. Sincein many wave propagation analyses, the domain considered is geo-metrically complex, the available spectral techniques may not beeffective.

Another possibly more efficient approach is to utilize basicinterpolation functions that are enriched with waves. This meansin essence to construct special interpolation functions that aremore amenable to capture the desired response. This approach israther natural to increase the effectiveness of the finite elementmethod for the solution of specific problems, and has been pursuedfor a long time, like for example (that is, not giving an exhaustivelist of references) in the analysis of wave propagations [12–14],global local solutions [15,16], piping analyses [17], the develop-ment of beam elements [18], and in fluid flow analyses [19,20].Such methods have lately also been referred to as partition of unitymethods or extended finite element methods, see for example [21–24]. In addition, recently, discontinuous Galerkin methods [25] andrelated techniques have been researched for the solution of wavepropagation problems, but these techniques are computationallyvery expensive to use.




88 H. Kohno et al. / Computers and Structures 88 (2010) 87–94

Whenever such a problem-specific method is proposed, thegenerality for a specific class of problems and effectiveness are cru-cial. For plasma wave problem solutions, Pletzer et al. proposed awave-packet approach using the Gabor functions as envelopes[26]. Although this method has several good features, five param-eters need to be selected, where it is difficult to find near optimalchoices. Also, since the values of the Gabor functions are nonzero inthe entire calculation domain, a cutoff value has to be defined. Fur-thermore, it is difficult to incorporate general boundary conditions.

Our objective in this paper is to present a finite element schemein which basic finite element interpolations are used enriched withwave packets. The method is quite simple and is based on the stan-dard finite element method [1] and spectral method [8], but doesnot have the above-mentioned disadvantages. It turns out thatthe resulting interpolation functions have the same structure asthose proposed in References [12,13] but can be applied to a muchbroader range of problems. Specifically, the procedure can also beused to solve a range of plasma wave propagation problems, forexample in which mode conversion occurs. In these cases, waveswith dramatically different wavelengths can exist in localized re-gions, which are determined by sophisticated plasma models con-sidering kinetic effects. An important point is that the governingequations corresponding to the kinetic models include integrals,since the dielectric tensor is evaluated by integrating over thewhole of velocity space and past particle trajectory time. For thatreason, the methods referenced above [12–14,21,22] cannot di-rectly be used to solve such plasma wave problems, because theyuse solutions of some specific differential equations. Our approachutilizes classical finite element interpolations with spectral enrich-ments, and can be applied to the equations including integrals aswell as general differential equations. The combined interpolationtechnique can be used to easily satisfy Dirichlet boundary condi-tions and solve for many different wave numbers in one solution.

We first present the numerical procedure in detail and then givethe solutions of some test problems, including a problem model-ling wave behavior in plasmas. We show that the proposed finiteelement method gives more accurate results than the conventionalfinite element method for wave propagation problems. While weonly consider one-dimensional linear problems, there is consider-able intrinsic potential of the method to be effective for multi-dimensional and even nonlinear solutions.

Piecewise-linearenvelope

Waves1

Piecewise-linearenvelope

Waves

Fig. 1. Schematic diagram of a linear wave-packet interpolation function.

2. Finite element wave-packet approach

The method proposed here is based on three important fea-tures: the technique can be thought of as using the interpolationsof the traditional finite element method enriched by waves, theresultant global coefficient matrix is sparse as in finite elementmethods, and the boundary conditions are easily incorporated.The purpose of this section is to describe each feature in detail.

2.1. Foundation of the numerical method

The basis of the proposed scheme is a weak form of theweighted residual method [1]. Consider a general one-dimensionalordinary differential equation written as L[u] + f(x) = 0, where L isan ordinary differential operator. Let u be an approximate numer-ical solution. The numerical solution u is determined such that thefollowing integral equation is satisfied:Z

XhðxÞðL½u� þ f ðxÞÞdXþ

ZC

hðxÞðB½u� � B½u�ÞdC ¼ 0; ð1Þ

where h(x) is a weight function, B is an operator for the boundaryterm, X and C denote the calculation domain and its boundary,respectively. Using the standard Galerkin approach, the numerical

solution and the weight function are given by the same type ofinterpolation functions, which are formulated next.

2.2. Linear, quadratic and Hermitian wave-packet interpolationfunctions

The interpolation functions are constructed by multiplyingsinusoidal functions by well-known finite element interpolationfunctions. First, the numerical solution u and the weight functionh are expressed using the linear or quadratic wave-packet interpo-lation functions g(i,j) as follows:

uðxÞ ¼ gði;jÞðxÞuði;jÞ; hðxÞ ¼ g�ði0 ;j0 ÞðxÞhði0 ;j0Þ; ð2Þ

where the superscript * denotes the complex conjugate; u(i,j), hði0 ;j0 Þare nodal complex variables in the coordinate-frequency spaceidentified by the global node number i (i

0) and the harmonic number

j (j0). Here the summation convention applies to the subscripts i and

j. Since our methods utilize a finite element interpolation function asan envelope function, the value of the envelope function is one atsome nodal point xk and zero at every xj (j – k). This allows the func-tions g(i,j) to be defined locally as follows.For the linear case:

gða;jÞðnÞ ¼12ð1þ nanÞ exp i2pmj xe þ

Dx2

n

� �� : ð3Þ

For the quadratic case:

gða;jÞðnÞ ¼nan2ð1þ nanÞ þ ð1� n2

aÞð1� n2Þ� �

exp i2pmj xe þDx2

n

� �� ;

ð4Þwhere i, xe, Dx and n are the imaginary unit, the x-coordinate at thecenter of an element, the length of an element and the coordinatevariable in the calculation space (�1 6 n 6 1), respectively; thephysical space is then related to the calculation space byx = xe + (Dx/2)n. The subscript a denotes the local node number,and the values of na are n1,2 = �1, 1 for the linear case andn1,2,3 = �1, 1, 0 for the quadratic case, respectively. The wave num-bers 2pmj are determined by mj = jm, where m is the fundamental fre-quency and j is an integer in the range �(NF � 1)/2 6 j 6 (NF � 1)/2with the cutoff number of harmonics NF. Here NF P 1 is an odd inte-ger. The schematic profile of a linear wave-packet interpolationfunction is shown in Fig. 1. As we will see in numerical examplesin Section 4, the quadratic wave-packet interpolation is actuallymore effective.

Another possibly more efficient wave-packet approach can beestablished by employing Hermitian cubic beam functions [1]where then the nodal values and also the derivative values at thenodes are used. This makes the expressions for the numerical solu-tion and the weight function slightly different from Eq. (2):

uðxÞ ¼ gði;jÞðxÞ~uði;jÞ; hðxÞ ¼ g�ði0 ;j0 ÞðxÞ~hði0 ;j0Þ: ð5Þ

Here the Hermitian wave-packet interpolation functions comprisetwo different expressions:

gði;jÞ ¼ g1ði;jÞ ~uði;jÞ ¼ uði;jÞ for 1 6 i 6 Nx;

¼ g2ðk;jÞ; ¼ u0ðk;jÞ for Nx þ 1 6 i 6 2Nx

ð6Þ

H. Kohno et al. / Computers and Structures 88 (2010) 87–94 89

(same applies to ~hði;jÞ), where Nx is the total number of nodes, andthe value of the subscript k is related to the value of i by k = i � Nx.In a similar way to the linear and quadratic wave-packet interpola-tions, these functions in Eq. (6) can be written locally as follows:

g1ða;jÞðnÞ ¼

14ðnþ naÞ2ð�nanþ 2Þ exp i2pmj xe þ

Dx2

n

� �� ;

g2ða;jÞðnÞ ¼

Dx8ðnþ naÞ2ðn� naÞ exp i2pmj xe þ

Dx2

n

� �� ;

ð7Þ

where n1,2 = �1, 1. The real-valued profiles of the Hermitian wave-packet interpolation functions are shown in Fig. 2.

For a real-valued solution, we can easily derive the followingrestrictions from Eqs. (2) and (5):

uða;jÞ ¼ u�ða;�jÞ;u0ða;jÞ ¼ u0�ða;�jÞ; ð8Þ

where the equation involving derivatives is of course only consid-ered for the Hermitian wave-packet interpolation functions. Theserelations reduce the number of unknowns to half and consequently,the size of the global matrix to a quarter. Using Eq. (8), for example,we can modify the linear wave-packet interpolation functions asfollows:

uðxÞ ¼ gaða;0Þuða;0Þ þ

XðNF�1Þ=2

j¼1

gbða;jÞu

ðRÞða;jÞ þ gc

ða;jÞuðIÞða;jÞ

h i¼ gða;mÞ~uða;mÞ;

ð9Þ

with

gaða;0Þ ¼

12ð1þ nanÞ;

gbða;jÞ ¼ ð1þ nanÞ cos 2pmj xe þ

Dx2

n

� �� ;

gcða;jÞ ¼ �ð1þ nanÞ sin 2pmj xe þ

Dx2

n

� �� ð10Þ

gða;mÞ ¼ gaða;0Þ ~uða;mÞ ¼ uða;0Þ for m ¼ 0;

¼ gbða;jÞ; ¼ uðRÞða;jÞ for 1 6 m 6 ðNF � 1Þ=2;

¼ gcða;kÞ ¼ uðIÞða;kÞ for ðNF � 1Þ=2þ 1 6 m 6 NF � 1;

ð11Þ

where uðRÞða;jÞ, uðIÞða;jÞ are the real and imaginary parts of u(a,j), respec-tively, and the subscripts j, k and m in Eq. (11) are related to one an-

(a)

-1.0 -0.5 0.0 0.5 1.0

Re(

g)

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

g1(1,j) g1

(2,j)

Fig. 2. Profiles of the Hermitian wave-packet interpolation functions together with theirRe½g2

ða;jÞ � vs. n.

other by j = m, k = m � (NF � 1)/2. Of course, if we consider a generalplasma wave, the numerical solution is always complex, and henceEqs. (8)–(11) are not applicable.

An interesting observation is that for j = 0 all the wave-packetinterpolation functions given in Eqs. (3), (4), and (7) reduce tothe usual finite element interpolation functions as a result ofmj = jm = 0. Thus, for NF = 1, the present interpolation scheme con-sists only of the conventional finite element interpolation func-tions, and indeed the present wave-packet approach is identicalto the conventional finite element method when NF = 1 (see Sec-tion 2.3). We will see that this property leads to the straightfor-ward treatment of the boundary conditions.

The present scheme results in a relatively low computationalcost since the global matrix is sparse. This sparsity is due to the lo-cal interpolation of wave packets. As an example, we show the dis-tribution of the global matrix elements for the case of using theHermitian functions in Fig. 3, where the nonzero regions areblock-diagonalized with a regular bandwidth of 3NF.

As an illustration, consider a one-dimensional sine-wave prob-lem described by u00 + a2u = 0 in the range 0 6 x 6 1 subject to theboundary conditions u(0) = 0, u0(1) = a. Here a is a constant withcos a = 1. The exact solution for this problem is then given byu = sin (ax). Fig. 4a shows a numerical solution obtained by the lin-ear finite element wave-packet approach for a = 4p, m = 0.5, Nx = 2and NF = 9. As seen, with only one element used, we obtain virtu-ally the exact analytical results. This is the desired result sincethe method is based on the Fourier decomposition technique sothat any smooth function should be reproduced by the combina-tion of sinusoidal waves with different wave numbers regardlessof the value of Nx. Fig. 4b is a semi-log plot of the error norm, whichis defined by k � k � ½

Rðu� uÞ2dx=

Ru2dx�1=2, as a function of NF. We

notice that the error decreases logarithmically with the number ofharmonics for NF P 5. Due to this feature, the present wave-packetapproach can yield more accurate results compared to the conven-tional finite element method by orders of magnitude.

2.3. Imposing the boundary conditions

An important feature of the present method is the ease ofimposing the boundary conditions. Consider a one-dimensionalproblem governed by a second-order differential equation. Whenimposing the Dirichlet boundary condition, we choose a weightfunction whose value is forced to be zero at the boundary in the

(b)

-1.0 -0.5 0.0 0.5 1.0

Re(

g)

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

g2(1,j)

g2(2,j)

envelope functions for Dx = 0.1 and mj = 100: (a) plot of Re½g1ða;jÞ� vs. n; and (b) plot of

Fig. 3. An example of the structure of the global matrix for the analysis using theHermitian finite element wave-packet method.

(a)

(b)

NF

0 2 4 6 8 10 12 14 16 18 20

Nor

m o

f er

ror

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e+1

x0.0 0.2 0.4 0.6 0.8 1.0

u (N

umer

ical

sol

utio

n)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Fig. 4. The numerical results obtained by the linear finite element wave-packetmethod for m = 0.5, Nx = 2: (a) the calculated wave for NF = 9; (b) the norm of error asa function of NF.


same way as in the conventional Galerkin finite element methods.But an important point to notice is that the interpolated nodal val-ues u(i,j) (or ~uði;jÞ) are not identical to the nodal values of the numer-

ical solution uðxÞ. Thus, for example, if we intend to exactly satisfythe Dirichlet boundary condition at the boundary x = xb (the right-hand side boundary), the following equality must be satisfied:

uðxbÞ ¼ gði;jÞðxbÞuði;jÞ ¼ ub ðfor the linear=quadratic caseÞ;uðxbÞ ¼ gði;jÞðxbÞ~uði;jÞ ¼ ub ðfor the Hermitian caseÞ

ð12Þ

For ub being real, Eq. (12) leads to

PðNF�1Þ=2

j¼�ðNF�1Þ=2cosð2pmjxbÞuðRÞðNx ;jÞ � sinð2pmjxbÞuðIÞðNx ;jÞ

h i¼ ub;

PðNF�1Þ=2

j¼�ðNF�1Þ=2sinð2pmjxbÞuðRÞðNx ;jÞ þ cosð2pmjxbÞuðIÞðNx ;jÞ

h i¼ 0;

8>>>><>>>>:

ð13Þ

where we note that Eq. (13) does not lead to a unique solution forNF > 1. However, the following choice always satisfies the boundarycondition for any m and xb:

uðRÞðNx ;jÞ ¼ ub for j ¼ 0;

¼ 0 for j – 0;

uðIÞðNx ;jÞ ¼ 0 for any j:

ð14Þ

This corresponds to the concept of imposing the exact boundary va-lue in the conventional finite element component (j = 0). Note thatEq. (14) is consistent with the statement in Section 2.2; the presentscheme reduces to the conventional finite element method forNF = 1.

On the other hand, the proposed method only approximatelysatisfies the Neumann boundary conditions, again as in the conven-tional finite element method. For the linear or quadratic wave-packet approach, the value of the weight function at the boundarycan be arbitrary. The boundary term in the discretized equation iscalculated in the same way as in standard finite element methods.For the Hermitian wave-packet approach, we specify h0ði;jÞ ¼ 0 at theNeumann boundary and choose the boundary nodal values in asimilar way to the Dirichlet boundary condition as follows:

u0ðRÞðNx ;jÞ ¼ u0b for j ¼ 0;

¼ 0 for j – 0;

u0ðIÞðNx ;jÞ ¼ 0 for any j:

ð15Þ

Here we assume that the Neumann boundary condition is imposedat x = xb. In general, the above choice does not exactly satisfy theNeumann boundary condition because

dudx

��x¼xb

¼ 2Dx

dg1ða¼2;jÞðnÞ

dnuða¼2;jÞ þ

dg2ða¼2;jÞðnÞ

dnu0ða¼2;jÞ

!��n¼1;x¼xb

¼ 2Dx

dg1ð2;jÞðnÞdn

uð2;jÞ

��n¼1;x¼xb

þ u0b

ð16Þ

In general, the first term on the right-hand side is nonzero so thatdu=dxjx¼xb

–u0b. For NF = 1, the scheme reduces to the conventionalHermitian finite element method, and then the Neumann boundarycondition is exactly satisfied.

3. A required condition in the fundamental frequency

In the present scheme, we need to specify three numericalparameters: Nx, NF and m. Here we derive one required conditionfor a proper choice of m related to the value of Nx.

First of all, an important point is that every integral in the lo-cally discretized equations can be written in the following form:

I ¼Z 1

�1

Xn¼0

Cnnn

!expðanþ bÞdn; ð17Þ

(a)

(b)

x0.0 0.5 1.0 1.5 2.0

u (N

umer

ical

sol

utio

n)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

x

0.0 0.5 1.0 1.5 2.0

u (N

umer

ical

sol

utio

n)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Fig. 5. Numerical solutions of the wave propagation problem through differentmedia: (a) u = 0.5 sin (8px) in 0 6 x < 1 and u = sin (4px) in 1 < x 6 2 (case 1); (b)u = 0.125 sin (64px) in 0 6 x < 1 and u = sin (8px) in 1 < x 6 2 (case 2).


where

a ¼ ipðmj � mj0 ÞDx; b ¼ i2pðmj � mj0 Þxe: ð18Þ

Here n P 0 takes integer values, and Cn are the coefficients deter-mined depending on the differential equations considered. Now let

FðnÞ ¼Z 1

�1nn expðanþ bÞdn: ð19Þ

Then Eq. (17) is simply expressed by I ¼P

n¼0CnFðnÞ. Consider firstthe case of mj – mj0 (i.e., j – j0). For n P 1 one can rewrite Eq. (19) asfollows:

FðnÞ ¼ nn

aexpðanþ bÞ

� �1

�1� n

aFðn� 1Þ: ð20Þ

For n = 0 we have:

Fð0Þ ¼Z 1

�1eanþbdn ¼ 1

aðeaþb � e�aþbÞ: ð21Þ

Thus, using Eqs. (20) and (21) we obtain F(n) for any value of nthrough successive calculations. For mj ¼ mj0 ðj ¼ j0Þ, the integral inEq. (19) is easily solved as follows:

FðnÞ ¼Z 1

�1nndn ¼ 1

nþ 1½1� ð�1Þnþ1�: ð22Þ

These analytical expressions are desirable since we do not need toapply any numerical integration to the integral shown in Eq. (17);consequently, the computation of each term is fast without anumerical error due to numerical integration.

Now, using Eqs. (20)–(22), consider the following two impor-tant limits: |a| ?1 and |a| ? 0. Assume that a given differentialequation is discretized by properly choosing finite element wave-packet interpolation functions. For |a| ?1, we find thatjIj¼j0 j=jIj–j0 j ! 1 and jIj¼j0 j ! 1 for j – 0 in a non-sparse block(i, i0), where jIj¼j0 j and jIj – j0 j are the integrals obtained by addingup all the discretized derivative terms for j = j0 and j – j0, respec-tively, expressed in the form of Eq. (17). On the other hand, for|a| ? 0, we find that jIj – j0 j=jIj¼j0 j ! 1 and jIj – j0 j ! 1 in a non-sparse block (i, i0). Of course, the numerical solutions for thesecases do not make any sense. Therefore, a required conditionshould be |a| � 1, i.e., mDx � 1, for which the magnitude of everyterm in Eq. (17) is about like in the conventional finite element dis-cretization (j = j0 = 0). The physical interpretation of this constraintis that the waves in the wave packet should have at least onewavelength in an element (see Fig. 1).

4. Numerical examples

In this section, we illustrate the performance of the finite ele-ment wave-packet approach using three test problems. First, wesolve a wave propagation through different media, then we solvethe problem described by the Airy-type equation, whose exactsolutions are available for comparison with the numerical results.Finally we solve a more difficult problem which models themode-conversion behavior of the radio-frequency waves in plas-mas described by the Wasow equation. We take the last two exam-ples from Ref. [26]. In all solutions we use uniform meshes andwhen we compare solution accuracies with the accuracy obtainedusing the conventional finite element method we employ the factthat the solutions are real and use the same number of unknowns(see Section 2.2).

4.1. Wave propagation through different media

Consider the wave propagation problem through different med-ia described by the following equation:

d2u

dx2 þ a2u ¼ 0; 0 6 x 6 2; ð23Þ

where a2 ¼ a2I for 0 6 x < 1 and a2 ¼ a2

II for 1 < x 6 2. We assumethat sin aI = sin aII = 0 and cos aI = cos aII subject to the boundaryconditions u(0) = 0 and u0(2) = aII. The exact solution is thenuI = (aII/aI)sin (aIx) in the range 0 6 x < 1 and uII = sin (aIIx) in1 < x 6 2. Here we consider two cases: aI = 8p, aII = 4p in case 1and aI = 64p, aII = 8p in case 2.

The discretized equation for Eq. (23) is:ZXe

dg�ða;j0 Þdx

dgðb;jÞdx�a2g�ða;j0 Þgðb;jÞ

!dx �uðb;jÞ �g�ða;j0 Þ

dudx

��Neumann boundary

¼ 0:

ð24Þ

As for the parameters used in the numerical scheme, we set thenumber of envelope positions (i.e., nodes), the cutoff number of har-monics and the fundamental frequency to Nx = 9, NF = 5, m = 1.8(Nx = 21, NF = 11, m = 6.0) for the linear, quadratic wave-packetmethods and Nx = 5, NF = 5, m = 1.5 (Nx = 11, NF = 11, m = 6.0) for theHermitian wave-packet method in case 1 (case 2).

The profiles of the numerical solutions obtained by the Hermi-tian wave-packet method are shown in Fig. 5. Fig. 6 shows the

(a)

(b)x

u -

erro

r

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Hermitian FEM (0.1 x error)Hermitian wave-packet (1000 x error)

x

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

u -

erro

r

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Hermitian FEMHermitian wave-packet (100 x error)

Fig. 7. Comparison of the numerical error for the wave propagation problemthrough different media between the finite element wave-packet method and theconventional finite element method: (a) case 1; and (b) case 2.

x

0.0 0.2 0.4 0.6 0.8 1.0

u

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Fig. 8. Exact solution of the Airy-type equation for a = 21p/2.

(a)

(b)x

u -

erro

r

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Linear wave-packet (10 x error)Quadratic wave-packet (1000 x error)Hermitian wave-packet (1000 x error)

x

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

u -

erro

r

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08


Fig. 6. Comparison of the numerical error for the wave propagation problemthrough different media among the three different wave-packet methods: (a) case1; and (b) case 2.


comparison of the numerical error (u� u) for the linear, quadraticand Hermitian wave-packet approaches. As seen, the error is con-siderably smaller if we use higher-order envelope functions,although the difference between the quadratic and Hermitianwave packets is small. Fig. 7 shows the comparison of the numer-ical error between the present wave-packet method and the con-ventional finite element method with Nx = 25 in case 1 andNx = 121 in case 2, both of which utilize the Hermitian interpola-tion functions. We see that the numerical results obtained usingthe Hermitian wave-packet method are several orders of magni-tude more accurate than the results obtained using the standard fi-nite element method. Especially, the result in Fig. 7b demonstratesthat a sufficient number of harmonics yields rapid convergence fora smooth function as for the standard Fourier series (see Fig. 4b).

4.2. Airy-type equation

Second, the methods are applied to the following second-orderdifferential equation:

d2u

dx2 þ a2ð1� 2xÞu ¼ 0; 0 6 x 6 1; ð25Þ

whose exact solution is described by the Airy function: u =Ai[(a/2)2/3(2x � 1)]. Here the coefficient a is fixed at 21p/2 (the same

value as in Ref. [26]), and the corresponding boundary conditions aregiven by u0(0) = �8.3239 and u0(1) =�9.8696� 10�5. Fig. 8 shows theprofile of the corresponding exact solution. The fundamentalfrequency, the numbers of envelope positions and Fourier modes

x0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

u -

erro

r

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08


Fig. 9. Comparison of the numerical error for the Airy-type equation among thethree different wave-packet methods.

x0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

u -

erro

r

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Hermitian FEM (100 x error)Hermitian wave-packet (10000 x error)

Fig. 10. Comparison of the numerical error for the Airy-type equation between thefinite element wave-packet method and the conventional finite element method.

x

(a)

0.0 0.2 0.4 0.6 0.8 1.0

u

-10

-5

0

5

10

15

20

25

30

Fig. 11. Numerical solution of the Wasow equation: (a) m


are m = 2.0, Nx = 9 for the linear, quadratic cases, Nx = 5 for the Hermi-tian case, and NF = 5.

Fig. 9 gives the numerical error for the three different wave-packet approaches. As before, the errors obtained using the high-er-order wave-packet interpolations are much smaller than theerror obtained using the linear interpolation. Also, it is observedthat the higher-order finite element wave-packet methods arecomparable in accuracy with the Gabor element method developedby Pletzer et al. [26]. Fig. 10 shows the comparison of the numer-ical error between the present wave-packet method and the con-ventional finite element method (with Nx = 25), both of whichutilize the Hermitian interpolation functions. Again, it is observedthat the numerical result using the Hermitian wave-packet methodis much more accurate; note that the error-scale differs by two or-ders of magnitude.

4.3. Wasow equation

Lastly, we consider the numerical solution of the Wasow equa-tion, which models the mode conversion effects of radio-frequencywaves in plasmas. The equation considered here is given by

d2

dx2þk2½1�0:5ðx�0:5Þ�( )

d2

dx2þk2½1�160ðx�0:5Þ�( )

uþau¼ 0;

06 x6 1; ð26Þ

where k2 = 2 � 103 and a = 8 � 106 subject to the boundary condi-tions u(0) = 0, u(1) = 1 and u0(0) = u0(1) = 0 (the same boundary con-ditions as in Ref. [26]). Eq. (26) implies the formation of multiscalewaves with different wave numbers by a factor of 320. Here a com-parison is made between the finite element wave-packet methodand the conventional finite element method, both utilizing the Her-mitian interpolation functions which can be straightforwardly ap-plied to this fourth-order differential equation. As numericalparameters, we choose m = 10.5, Nx = 10 and NF = 11.

Since an analytical solution to this problem is not available, wefirst calculate the problem with a very fine mesh using the Hermi-tian interpolation functions, and utilize the obtained result as a‘‘quasi-exact” solution. Fig. 11 shows the numerical solution ob-tained with 1000 elements. We see that the fast and slow wavesare coupled on the left half of the domain (see Fig. 11b), while onlythe slow wave having a shorter wavelength is evanescent on the

x

u

-3

-2

-1

0

1

2

3

(b)

0.0 0.1 0.2 0.3 0.4 0.5

acroscopic oscillation; and (b) fine scale oscillation.

x0.0 0.2 0.4 0.6 0.8 1.0

u -

erro

r

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Hermitian FEM (0.1 x error)Hermitian wave-packet

Fig. 12. Comparison of the numerical error for the Wasow equation between thefinite element wave-packet method and the conventional finite element method.


right half. This is also confirmed in Eq. (26); although the sign ofr1 = k2[1 � 0.5(x � 0.5)] is always positive in the entire domain,the sign of r2 = k2[1 � 160(x � 0.5)] changes from positive to nega-tive at x = 0.5. The former corresponds to propagation of the fastwave at every point, whereas the latter corresponds to evanes-cence of the slow wave on the right half of the domain. The mixingof these very different waves makes it more difficult to accuratelysolve the Wasow equation compared to the equations in the previ-ous problems.

A comparison of the numerical error (u� uquasi�exact) betweenthe finite element wave-packet method and the conventional finiteelement method is shown in Fig. 12. Again, the present wave-pack-et approach gives a more accurate numerical result compared tothe conventional finite element solution.

5. Conclusions

We presented in this paper a finite element wave-packet meth-od for the analysis of waves through media, and solved some illus-trative problems. The method is in particular directed to solvewaves in plasmas accurately with a reasonable computational cost.The key idea is to enrich the usual finite element interpolationswith wave packets. We see that this approach results into somefavorable features drawing from both, conventional finite elementand spectral methods. First, the interpolation functions are locallydefined in the same way as in the conventional finite elementmethods, which is effective for programming. Second, this localdefinition results in the formation of a sparse global matrix. Third,all the integrals in the discretized equation are analytically solved,yielding simple expressions (of course, numerical integration couldbe used and probably has to be used for wave equations of higherdimensions). Fourth, the boundary conditions are easily incorpo-rated in the discretized equation. In fact, the Dirichlet and Neu-mann boundary conditions are treated in a similar way as in theconventional finite element methods. Fifth, using the wave packetscan give more accurate results than using the corresponding con-ventional finite element methods under the same computationalcosts.

Plasma wave equations can be far more complex than the one-dimensional equations we solved here, but the one-dimensionalequations/solutions exhibit some of the fundamental characteris-

tics of these more complex waves. In further research the methodshould be applied to and tested in two- and three-dimensionalsolutions with nonuniform meshes. Also, a mathematicalconvergence analysis should be pursued to identify the rate andorder of convergence, and the optimal value of fundamentalfrequency.

Acknowledgements

We would like to thank Prof. Jeffrey Freidberg and Dr. Paul Bon-oli of M.I.T. for their valuable comments on this work for the appli-cation to plasmas. This work was supported in part by DoEContract No. DE-FG02-99ER54525.

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A constraint Jacobian based approach for static analysis of pantograph masts

B.P. Nagaraj a,1, R. Pandiyan a,2, Ashitava Ghosal b,*

a ISRO Satellite Centre, Bangalore 560 017, Indiab Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India


Article history:Received 10 April 2009Accepted 21 September 2009

Keywords:Pantograph mastsDeployable structuresNull-spaceJacobianStiffness matrix


* Corresponding author. Tel.: +91 80 2293 2956; faE-mail address: [email protected] (A.

1 Spacecraft Mechanisms Group.2 Flight Dynamics Group.

This paper presents a constraint Jacobian matrix based approach to obtain the stiffness matrix of widelyused deployable pantograph masts with scissor-like elements (SLE). The stiffness matrix is obtained insymbolic form and the results obtained agree with those obtained with the force and displacement meth-ods available in literature. Additional advantages of this approach are that the mobility of a mast can beevaluated, redundant links and joints in the mast can be identified and practical masts with revolutejoints can be analysed. Simulations for a hexagonal mast and an assembly with four hexagonal mastsis presented as illustrations.


1. Introduction

Deployable structures can be stored in a compact configurationand are designed to expand into stable structures capable of carry-ing loads after deployment. In their general form, they are made upof a large number of straight bars (links) connected by revolutejoints and with one or more cables used for deployment or increas-ing the stiffness of the deployed structure (see, for example, [1,2]).Initially, the whole assembly of bars can be stowed in a compactmanner and, when required, can be unfolded into a predefinedlarge-span, load bearing structural form by simple actuation ofone or more cables. This characteristic feature makes them emi-nently suitable for a wide spectrum of applications, ranging fromtemporary structures that can be used for various purpose inground to the large structures in aerospace industry. Deployable/collapsible mast are often used for space applications since in theircollapsed form they can be easily carried as a spacecraft payloadand expanded in orbit to a desired size. Many deployable systemsuse the pantograph mechanism or scissor-like elements (SLE’s).Typically, an SLE has a pair of equal length bars connected to eachother at an intermediate point with a revolute joint. The joint al-lows the bars to rotate freely about an axis perpendicular to theircommon plane. Several SLE’s are connected to each other in orderto form units which in plan view appear as regular polygons withtheir sides and radii being the SLE’s. Several such polygons, in turn,

ll rights reserved.

x: +91 80 2360 0648.Ghosal).

are linked in appropriate arrangements leading to deployablestructures that are either flat or curved in their final deployed con-figurations. The assembly is a mechanism with one degree of free-dom from the stowed/folded configuration till the end ofdeployment. The deployment is through an active cable and afterdeployment the assembly is a pre-tensioned structure. Activecables control the deployment and pre-stress the pantograph andpassive cables are pre-tensioned in the fully deployed configura-tion. These cables have the function of increasing the stiffness inthe fully deployed configuration [3].

1.1. Kinematics and mobility

The kinematics of multi-body mechanical systems can be stud-ied by use of relative coordinates [4], reference point coordinatesas used in the commercial software ADAMS [5] or Cartesian coor-dinates (also called natural/basic coordinates) [6]. In Refs. [7,8],Garcia and co-workers have used Cartesian coordinates to obtainthe constraints equations for different types of joints and for kine-matic analysis of mechanisms. Typical pantograph masts are over-constrained mechanisms according to Grübler–Kutzbach criteria,and in Ref. [9], Cartesian coordinates have been used to study thekinematics and mobility of deployable pantograph masts – theauthors use the derivative of the constraint equations and developan algorithm to obtain redundant link and joints in over-con-strained deployable masts, perform kinematic analysis and obtainglobal degrees of freedom. The key advantage of Cartesian coordi-nates is that the constraint equations are quadratic (as opposed totranscendental equations for relative coordinates), and, hence theirderivatives are linear. As shown in [9], these features allows easier





96 B.P. Nagaraj et al. / Computers and Structures 88 (2010) 95–104

manipulation and simplification of expressions in a computeralgebra system to obtain symbolic expressions and closed-formsolutions for the kinematics of pantograph masts. A disadvantageof Cartesian coordinates is that the number of variables is typicallylarger and tends to be (on average) in between relative coordinatesand reference point coordinates. However, for analysis of panto-graph masts, the number is not too large and could be handledwithout much difficulty in the computer algebra system, Mathem-atica, used in this work.

The masts in their deployed configuration become pre-ten-sioned structures. For pre-stressed structures with pin jointed bars,the necessary condition for the structure to be statically and kine-matically determinate is given by the Maxwell’s rule

3j� b� c ¼ 0 ð1Þ

where, j is the number of joints, b is the number of bars or links andc is the number of kinematic constraints. Calladine [10] generalizedthe Maxwell’s rule as

s ¼ b� r

m ¼ 3j� c � r

3j� b� c ¼ m� s ð2Þ

where, m is the number of internal mechanisms, s is the number ofstates of self-stress, and r is the rank of the equilibrium matrix. Thisequation is referred to as the extended Maxwell’s rule. The values mand s depends on the number of bars and joints, topology of theconnection and on the geometry of the frame work. The numericalvalues of the vectors describing s and m, for a given system, can bedetermined from the singular value decomposition (SVD) of theequilibrium matrix. The concept of using a Jacobian matrix to eval-uate the mobility was first presented by Freudenstein [11] for anover-constrained mechanism. Later, the first and higher order deriv-atives of constraint equations has been used for under constrainedstructural systems to evaluate mobility and state of self-stress byKuznetsov [12,13]

1.2. Structural matrix

The mechanism at the end of deployment becomes a pre-ten-sioned structure and the structural matrices are useful for evaluat-ing the stiffness/displacement of the SLE masts in the deployedconfiguration. In literature, researchers have used various methodsfor formulating the structural matrix for an SLE. These are termedas force method [14], displacement method [15] and equivalent con-tinuum model [16]. We describe each of these methods in briefbelow.

1.2.1. Force methodIn the force method, as used by Kwan and Pellegrino [14], the

SLE is discretised into four beam elements. The equilibrium, comp-atability and flexibility matrices are derived for a typical beam ele-ment in a local coordinate system using shear force and bendingmoment relationships. These equations are transformed to the glo-bal coordinate system by using the rotation matrices and areassembled for the four beam elements, which make up the SLE.The equilibrium matrix is reduced in size by matrix partitioningand by setting the end moments to zero [18]. In this approachone can evaluate the number of self-stress states and the numberof infinitesimal mechanisms of the given system by using singularvalue decomposition (SVD) of the equilibrium matrix [19].

1.2.2. Displacement methodThe displacement method is used by Shan [15] to formulate

stiffness matrix for the SLE. In his approach, each link of the SLEis called an uniplet. One uniplet of the SLE is modeled as an assem-

bly of two beam elements with mid node at the pivot point of SLE.The stiffness matrix was partitioned to have the translation termsand rotational terms in order. The final reduced stiffness matrix isobtained by condensing and removing the rotational degrees offreedom of all the three nodes. In Ref. [20], the authors have formu-lated the stiffness matrix for two uniplets, called as a duplet, byusing the stiffness matrix of the uniplet developed above. Matrixpartitioning is used to get the reduced stiffness matrix which con-denses the translational degrees of freedom of the pivot node.

1.2.3. Equivalent continuum modelThis approach was used to predict the stiffness characteristics

of deployable flat slabs when they are subjected to normal loads[16,17]. In this method, the SLE is considered as an equivalent uni-form beams that deflects identically to the given loading as that ofan SLE. The flat large deployable structure is substituted with anequivalent grid of uniform beams running in particular directionsThe beams are rigidly connected to each other. This arrangementis reduced to an equivalent orthotropic plate of constant thicknessand stiffness matrix is obtained. The results predicted by thismethod are approximate unlike above methods and hence can onlybe used for initial design phase which reduces the computationaltime. In an exact finite element modeling the storage spacerequirements are large for large number of SLE units due to thecomplicated pivotal connections and hinged connections that re-quire more than one nodal point to be described accurately. Theequivalent approach can significantly reduce the computationaleffort during preliminary design stage.

1.2.4. Comparison of existing methodsThe force method gives the additional information about the

states of self-stress and infinitesimal mechanisms. The displace-ment method or equivalent continuum model does not give thisinformation. The force method uses two matrix reductions whichreduces the matrix of dimension 18� 14 to 12� 8 in the first step.Further in the second step the matrix dimension is reduced from12� 8 to 10� 6, to obtain the final reduced equilibrium matrix.The displacement method has a stiffness matrix of dimension18� 18 for the two assembled beam elements with six degreesof freedom at each node. By condensing the rotational degrees offreedom at all the nodes the matrix dimension reduces to 9� 9.The reduced matrix has only translational degrees of freedom ateach node. The equivalent continuum approach is useful for verylarge repetitive structures. However, this method does not givethe accurate results when compared to other two methods and,hence, can be used only for initial design phase to reduce compu-tational time.

As mentioned earlier, at the end of deployment we get a struc-ture capable of bearing loads, and in this paper, we extend the ap-proach in [9] to the static analysis of deployable pantograph masts.We present a new approach to formulate the structural matricesfor a typical SLE using Cartesian coordinates, the kinematic equa-tions of the SLE/pantograph element, and the constraint Jacobianmatrix. These matrices are derived by using the symbolic compu-tation software Mathematica [21]. The results of formulations ob-tained by this approach matches exactly with the results of forceand displacement based methods. Our approach has the advanta-ges of the force method in evaluating the states of self-stress andinfinitesimal mechanisms. However, in our approach, the final re-duced equilibrium matrix can be obtained in a single step unlikein the force and displacement methods. In addition, the constraintequations of the links and joints are useful in studying the kine-matics behavior of pantograph masts during deployment, in evalu-ating the redundancy in the links/joints of these over-constrainedsystems, and in obtaining the final degrees of freedom of thedeployable masts. In literature the successive SLE joint connection

Fig. 2. Stacked planar SLE mast – (a) fully deployed and (b) partially deployed.

B.P. Nagaraj et al. / Computers and Structures 88 (2010) 95–104 97

are assumed to be spherical joints. In a practical pantograph mast,two revolute joints with intersecting axis are used. In this work, wehave used the revolute joint constraints for the SLE connected tothe successive SLEs by revolute joints.

This paper is organized as follows: In Section 2, we present abrief description of the deployable masts considered for the analy-sis. and present the constraint equations for the links, joints andthe SLE with the Jacobian matrix. In Section 3, we present themathematical approach for the evaluation of stiffness matrix forthe SLE and the detailed equations are presented in an AppendixA. In Section 4, we present the stiffness matrix for the cables usedin pantograph masts. In Section 5, we present the additional con-straints and the stiffness matrix due to revolute joints. In Section6, we illustrate our approach by using a planar stacked mast andthree-dimensional SLE based masts. Finally, in Section 7, we pres-ent the conclusions.

2. Kinematic description of the SLE masts

In this section a brief description of the SLE masts and formula-tion of the constraint equations are presented for the sake of com-pleteness (see, [9], for details). In the next section, we use theseequations to derive stiffness matrices.

The simplest planar SLE is shown in Fig. 1. The revolute joint inthe middle connects the two links of equal length. The assemblyhas one degree of freedom. Fig. 2 shows a two-dimensional stackedSLE mast [3]. This consists of four SLEs stacked one above the other.The deployment angle b can vary continuously from b ¼ 0�, whenthe assembly is fully folded, i.e. lying flat on its base and all linksare collinear, to b ¼ 45� which corresponds to the fully deployedconfiguration. This has eight passive cables connecting the adja-cent joints of the SLEs. The cables are taut in the fully deployedconfiguration and slack at all other configurations. One active cablewhich is firmly connected to the joint 3 of SLE mast, runs over apulley at joint 4, zig-zags down the SLE following the route shownin the figure (it runs over a pulley at each kink) and, after passingover a pulley at joint 1, is connected to the motorised drum locatedbelow the base. This mast remains stress free during folding. It canbe deployed simply by turning the drum below the base and thuswinding in the active cable. When the passive cable is taut thedeployment is complete. At this stage the active cable is woundin little more to set up a state of self-stress in the system. Usuallyit is desirable that all the passive cables be in a state of pre-tensionwhile the structure is operational to avoid the possibility of someof them might going slack when the mast is subjected to the actionof external loads; it is easiest to aim for uniform state of pre-stress

Fig. 1. Basic planar module of SLE.

in all cables. The uniform pre-stress can be obtained by introducingthe second active cable [3].

The triangular SLE mast can be created with three SLE’s. Thestacked triangular SLE mast [2] is shown in Fig. 3. This has twelvepassive cables and an active cable. The active cable is firmly at-tached to joint 5. The double loops are connected at the intermedi-ate joints as shown in Fig. 3. This pre-tensions uniformly all thecables in the mast. A drum is used to wind the active cable.

The function of passive cables are (a) for termination of deploy-ment, (b) increasing the stiffness of fully deployed structure, and(c) setup a state of pre-stress in the fully deployed structure result-ing in pre-tensioning of all passive cables. An active cable is suchthat its length reduces monotonically as the structure deploys.The functions of active cable is to (a) control the deployment pro-cess, (b) setup a state of pre-stress in a fully deployed structureresulting in pre-tensioning the whole system, and (c) eliminationof backlash at all joints. More than one active cable is often intro-duced in some structure. In practice it is advisable to have no lessthan two active cables to ensure minimum level of redundancyshould an active cable fail. However it is impractical to introducemany active cables in the structure because different cables mayrequire independent winding mechanisms and control units. Astructure with passive and active cables remains essentially stressfree in folded/partially folded configurations and is pre-stressed inthe fully deployed state. These structures have high stiffness whenfully deployed.

2.1. Formulation of constraints

In this section we derive the kinematic constraint equations forthe SLE. We will use the Cartesian/natural coordinates [6] to modelthe SLE. The natural or Cartesian coordinates are defined at thelocations of the joints and unit vectors along the joint axis to definethe motion of the link completely. In the natural coordinate systemthe constraint equations originate in the form of rigid constraintsof links and joint constraints.

Consider a SLE shown in Fig. 1. This is considered as an assem-bly of two links 1–2 and 3–4 with a pivot p. The link 1–2 with pivotp is considered as an assembly of two link segments 1–p and p–2

Fig. 3. Stacked triangular SLE mast – (a) full mast, (b) passive cables, and (c) active cable.


with lengths l1 and l2, respectively. Likewise, the link 3–4 with pi-vot p is considered as an assembly of two link segments 3–p andp–4 with lengths l3 and l4, respectively.

2.1.1. Rigid link constraintA rigid link is characterized by a constant distance between two

natural coordinates i and j. This is given by

rij � rij ¼ L2ij ð3Þ

where rij ¼ ½ðXi � XjÞ; ðYi � YjÞ; ðZi � ZjÞ�T with ðXi;Yi; ZiÞ; ðXj;Yj; ZjÞare the natural coordinates of i, j, respectively, and Lij is the distancebetween i and j. Using this equation for the SLE of Fig. 1 we get thefollowing systems of equations for the four segments:

ðXp � X1Þ2 þ ðYp � Y1Þ2 þ ðZp � Z1Þ2 � l21 ¼ 0

ðX2 � XpÞ2 þ ðY2 � YpÞ2 þ ðZ2 � ZpÞ2 � l22 ¼ 0

ðX3 � XpÞ2 þ ðY3 � YpÞ2 þ ðZ3 � ZpÞ2 � l23 ¼ 0

ðXp � X4Þ2 þ ðYp � Y4Þ2 þ ðZp � Z4Þ2 � l24 ¼ 0 ð4Þ

2.1.2. Constraint for SLEReferring to Fig. 1, the node p is a pivot, the link segments 1�p

and p�2 of link 1�2 are aligned at pivot. Hence, the cross-productof the two adjacent link segments 1�p and p�2 is given by

r1p � rp2 � l1l2 sin /1 ¼ 0 ð5Þ

here, /1 is the angle between the two link segments (equal to 0 de-grees for a pantograph mast). Similarly the constraint equation forthe link 34 is given by

r3p � rp4 � l3l4 sin /2 ¼ 0 ð6Þ

where, /2 the angle between the two link segments (0 degree inour case). Eqs. (5) and (6) along with rigid link constraint equations(Eq. (4)) of the four link segments 1p, p2, 3p and p4 form the com-plete set of equations for a single SLE.

2.1.3. System constraint equationsThe rigid constraint equations and joint constraints of SLE can

be written together as

fjðX1; Y1; Z1;X2; . . . ; ZnÞ ¼ 0 for j ¼ 1; . . . ; nc ð7Þ

where, nc represents the total number of constraint equationsincluding rigid link and joint constraints of SLE, and nt ¼ 3n is thetotal number of Cartesian coordinates of the system. The derivativeof the constraint equations give the Jacobian matrix and can besymbolically written as

½J�dX ¼ 0 ð8Þ

Since, Eq. (8) is homogeneous, one can obtain a non-null dX if thedimension of the null-space of ½J�nc�nt

is at least one. The existenceof the null-space implies that the mechanism possess a degree offreedom along the corresponding dX [6]. The null-space of [J] canbe obtained numerically.

The dimension of the null-space is the degree of freedom/mobility of the deployable system. The deployable systems willhave large number of links and joints arranged in a repetitive pat-tern. Using the above equations one can evaluate the possiblechange in degree of freedom of the deployable system with addi-tion of each link/joints and also can identify the redundant links/joints in the system [9].

The deployable systems at the end of deployment lock and thecables attached to the successive joints get pre-stressed there byreducing the mechanism to a structure. Using the null-spacedimension of the Jacobian matrix one can evaluate the minimumnumber of cables required to reduce the mechanism to structure.

3. Stiffness matrix for the SLE

In this section we present a method to evaluate the stiffnessmatrix for SLE from the constraint Jacobian matrix discussed inthe previous section. The SLE is considered to have constant crosssectional area and uniform material properties. The cross section ofthe SLE remains plane and perpendicular to the longitudinal axisduring deformation. The longitudinal axis which lies within theneutral surface does not experience any change in length. TheSLE beam is long and slender and the transverse shear and rotaryinertia effects are negligible. These assumptions allows the use ofEuler–Bernoulli beam theory.

3.1. Stiffness matrix from length constraints

From the length constraint equations, the elongation in thestructural members, dL, can be related to the system displace-ments, dX, as


½Jm�dX ¼ dL ð9Þ

where the Jacobian matrix, ½Jm�, can be obtained from Eq. (4) (seeAppendix) and dX; dL are given by

dX ¼ ½dX1; dY1; dZ1; dX2; dY2; dZ2; dX3; dY3; dZ3; dX4;

dY4; dZ4; dXp; dYp; dZp�T

dL ¼ ½dl1; dl2; dl3; dl4�T

If the elongation dL are elastic, the member forces, dT, can be ex-pressed with a diagonal matrix of member stiffnesses as givenbelow:

½Sm�dL ¼ dT ð10Þ

where, the member stiffness matrix ½Sm� for the length segments ofSLE is given by

½Sm� ¼

A1E1l1

0 0 0

0 A2E2l2

0 0

0 0 A3E3l3

0

0 0 0 A4E4l4

266666664

377777775

In the above equation E is the Young’s modulus, A the cross sec-tional area the diagonal elements correspond to the axial stiffnessdue to elongation of the link segments, and dT ¼ ½dT1; dT2; dT3; dT4�T

are the forces in the link segments.The equilibrium matrix for the reference configuration can be

written in terms of the transpose of the Jacobian matrix and wecan write

½Jm�TdT ¼ dF ð11Þ

where dF is given by

dF ¼ ½dF1x; dF1y; dF1z; dF2x; dF2y; dF2z; dF3x; dF3y; dF3z; dF4x; dF4y; dF4z;

dFpx; dFpy; dFpz�T

with the right-hand side denoting the load components at nodes.To be statically determinate, the load must be in the column

space of the equilibrium matrix, in which case it is the equilibriumload. Substituting the Eqs. (9) and (10) in Eq. (11), we get

½Jm�T ½Sm�½Jm�dX ¼ dF ð12Þ

The above equation can be written as

½Km�dX ¼ dF ð13Þ

where, ½Km� ¼ ½Jm�T ½Sm�½Jm� is the elastic stiffness matrix of four

length segments of the SLE.

3.2. Stiffness matrix due to bending

The rotations d/ in the structural members can be obtainedfrom the cross-product Eqs. (5) and (6) of SLE. The rotations arethe constraint variations related to the system displacements dXand in terms of the Jacobian matrix [J] are given as

½J12�dX12 ¼ d/1

½J34�dX34 ¼ d/2 ð14Þ

The detailed matrix is given in Appendix.In the above equation dX12 is the vector ½dX1; dY1; dZ1; dX2; dY2;

dZ2; dXp; dYp; dZp�T and dX34 is the vector ½dX3; dY3; dZ3; dX4; dY4;

dZ4; dXp; dYp; dZp�T . These are the displacements of the link 1–p–2and 3–p–4, respectively. Finally, d/1 ¼ ½d/1x; d/1y; d/1z�

T and

d/2 ¼ ½d/2x; d/2y; d/2z�T are the rotations in the global coordinate

system.The transformation matrix relating the global and local coordi-

nate system is given by

d/01 ¼ ½R�d/1

d/02 ¼ ½R�d/2 ð15Þ

where d/01 and d/02 are the rotations in the local coordinate systemand [R] is the transformation matrix relating the local and globalcoordinate systems [22]. The transformation matrix [R] is given by

½R� ¼ ½RH�½RU�½RW� ð16Þ

where,

½RW� ¼

CxffiffiffiffiffiffiffiffiffiffiC2

xþC2z

p 0 CzffiffiffiffiffiffiffiffiffiffiC2

xþC2z

p

0 1 0�CzffiffiffiffiffiffiffiffiffiffiC2

xþC2z

p 0 CxffiffiffiffiffiffiffiffiffiffiC2

xþC2z

p

266664

377775

½RU� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

x þ C2z

qCy 0

�Cy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

x þ C2z

q0

0 0 1

266664

377775

½RH� ¼

1 0 0

0 cos H sin H

0 sin H cos H

2664

3775

where, for the nodes i and j with length L, Cx ¼Xi�Xj

L ; Cy ¼Yi�Yj

L andCz ¼

Zi�Zj

L , and H is the angle from one of the principal axis of crosssection of the SLE beam. Using Eqs. (14) and (15) we get

d/01 ¼ ½R�½J12�dX12

d/02 ¼ ½R�½J34�dX34 ð17Þ

Considering the bending deformation of the links and neglect-ing torsion the above equations can be written as

d/001 ¼ ½J1�dX12

d/002 ¼ ½J2�dX34 ð18Þ

where, d/001 ¼ ½d/01y; d/01z�T and d/002 ¼ ½d/02y; d/02z�

T .Combining the above equations, we can write

d/00 ¼ ½Jn�dX ð19Þ

where, d/00 ¼ ½d/001; d/002�T . The relation between the forces and mo-

ments is given by

dF ¼ ½JTn�dM00 ð20Þ

where, dM00 ¼ ½dM012y; dM0

12z; dM034y; dM0

34z�T .

If the rotations d/00 are elastic, the member moments dM00 canbe expressed with a diagonal matrix of member stiffnesses as givenbelow:

½Sn�d/00 ¼ dM00 ð21Þwhere, the member stiffness matrix ½Sn� for the SLE is given by

½Sn� ¼

3E1 Izl1þl2

0 0 0

0 3E1Iy

l1þl20 0

0 0 3E2 Izl3þl4

0

0 0 0 3E2Iy

l3þl4

26666664

37777775

In the above equation E is the Young’s modulus, Iz and Iy are thesecond moment of area of cross section about Z and Y axes, respec-tively, and the diagonal elements correspond to the bending stiff-ness of the links 1–p–2 and 3–p–4 about Z and Y axis.

Fig. 4. Typical truss element with coordinates.


By using a similar procedure as given in previous section, the fi-nal equations can be obtained by substituting the Eqs. (19) and(21) in (20). We get

½Jn�T ½Sn�½Jn�dX ¼ dF ð22Þ

and this equation can be written as

½Kn�dX ¼ dF ð23Þ

where, ½Kn� ¼ ½Jn�T ½Sn�½Jn� is the elastic stiffness matrix for the SLE. By

combining the stiffness matrix due to length constraint Eq. (13) andSLE constraint Eq. (23), we get

½Ks�dX ¼ dF ð24Þ

where, ½Ks� ¼ ½Km� þ ½Kn� is the elastic stiffness matrix due to lengthand SLE constraints

3.3. Rank of stiffness matrix

The stiffness matrix is given by

½Ks� ¼ ½Js�T ½Ss�½Js� ¼ ½Js�

Tð½S�s �T ½S�s �Þ½Js� ¼ ð½Js�½S

�s Þ�

Tð½Js�½S�s �Þ ð25Þ

where ½S�s � is a diagonal matrix whose diagonal elements are squareroot of the diagonal elements of [Ss] and [Js] is the Jacobian matrixof a single SLE. Since for a regular bar frame works all elements from½Ss� are positive, the rank of ½Js�½S

�s � takes it value from the rank of ½Js�.

Furthermore

rankð½Ks�Þ ¼ rankðð½Js�½Ss�ÞTð½Js�½Ss�ÞÞ ¼ rankð½Js�½Ss�Þ¼ rankð½Js�Þ ð26Þ

3.4. Comparison with other methods

In Ref. [15], the displacement method was used to derivethe stiffness matrix. The link 1–2 is also called as an uniplet inthe reference paper. By using the first two length constraint equa-tions in (4) of the link 1–p–2 and the SLE constraint Eq. (5) wecan formulate the Jacobian matrix. The stiffness matrix for theuniplet can be computed by using Eq. (24). It can be observedthat the stiffness matrix, obtained by our method, matchesexactly with the matrix formulated in Ref. [15] by the displace-ment method.

In Ref. [14], the authors have used force method to arrive at thestiffness matrix. Using the length constraint Eq. (4) and SLE con-straint Eqs. (5) and (6) we can formulate the Jacobian matrix as de-scribed in the previous section. By using the coordinate system ofRef. [14] and making the substitutions in Jacobian matrix Eq. (8),we can observe that matrix obtained by our method is same asthe matrix shown in Eq. (37) of Ref. [14] obtained by the forcemethod. It can be observed that the transpose of this matrix relatesthe forces and the moments of SLE.

Fig. 5. A spherical joint replaced by two revolute joints.

4. Equation for the cable

As already described earlier, cables are added in the maststo enhance their stiffness. These cables are slack in the stowedconfiguration and are taut at the end of deployment. A cablecan be assumed to be bar in the taut configuration. The stiff-ness matrix for the bar may be found in many textbooks andis described below for completeness. For the bar shown inFig. 4 connecting the joints i and j, the stiffness matrix is givenby

½Kc� ¼AcEc

lc

r2 rs rt �r2 �rs �rt

rs s2 st �rs �s2 �st

rt st t2 �rt �st �t2

�r2 �rs �rt r2 rs rt

�rs �s2 �st rs s2 st

�rt �st �t2 rt st t2

26666666666664

37777777777775

where, r ¼ Xi�Xj

lc; s ¼ Yi�Yj

lcand t ¼ Zi�Zj

lc. In the above equation cross

sectional area is denoted by Ac , Young’s modulus is denoted by Ec

and the bar/cable length is denoted by lc .By combining the stiffness matrix of SLE elements and the cable

we can write

½K�dX ¼ dF ð27Þ

where the total stiffness matrix [K] is given by ½Ks� þ ½Kc�.

5. Revolute joint constraints

The two adjacent SLE’s are connected by revolute joints asshown in Fig. 5. This enforces additional constraints of the form

Table 1Input data: coordinates of joints for hexagonal mast.

Joints X coordinate (mm) Y coordinate (mm) Z coordinate (mm)

Joint 1 0.0 0.0 0.0Joint 2 500.0 �866.0254 0.0Joint 3 1500.0 �866.0254 0.0Joint 4 2000.0 0.0 0.0Joint 5 1500.0 866.0254 0.0Joint 6 500.0 866.0254 0.0Joint 7 0.0 0.0 700.0Joint 8 500.0 �866.0254 700.0Joint 9 1500.0 �866.0254 700.0Joint 10 2000.0 0.0 700.0Joint 11 1500.0 866.0254 700.0Joint 12 500.0 866.0254 700.0

Fig. 6. Hexagonal SLE mast in the deployed configuration.

Table 2[J] matrix details for hexagonal mast.

Contents Size of[J]

Null-space

Remarks

+SLE 1 (20,39) 21+SLE 2 (28,42) 18+SLE 3 (36,45) 15+SLE 4 (44,48) 12+SLE 5 (52,51) 10+SLE 6 (60,54) 10 SLE – 6 is

redundant

+FACE 1 (62,54) 8+FACE 2 (64,54) 6+FACE 3 (66,54) 5+FACE 4 (68,54) 4+FACE 5 (70,54) 4 Revolute joints are

redundant+FACE 6 (72,54) 4 Revolute joints are

redundant

+Boundary conditionsðX1 ¼ Y1 ¼ Z1 ¼ 0Þ

(75,54) 1 Mechanism

+Cable 1–2 (76,54) 0 Structure


rip � Um � Lip cosða1Þ ¼ 0rjq � Un � Ljq cosða2Þ ¼ 0 ð28Þ

where the unit vectors Um and Un are along the revolute joint axis asshown in Fig. 5. The angles a1 and a2 are the angles between theunit vectors and r. In our approach, Lagrange multiplier is used toenforce these constraints on the stiffness matrix equations pre-sented in the previous section.

5.1. Lagrange multiplier method

For a steady state discrete linear system with potential energyfunctional P� expressed by

P� ¼ 12

UT ½K�U� UT F ð29Þ

where, U is the displacement of the structural system, [K] is thestiffness matrix and F is the external load, the equilibrium equa-tions can be found for the condition which make the variation ofP� stationary. We get

dP� ¼ dUTð½K�U� FÞ ¼ 0 ð30Þ

with respect to the admissible virtual displacements dU. Since dU isarbitrary the above equation yields

½K�U ¼ F ð31Þ

The constraint equations due to revolute joints can be written in thegeneral form as

/ðUÞ ¼ ½C�U� D ¼ 0 ð32Þ

where ½C�p�q is the constraint matrix, p is the number of constraintequations, q is the number of variables and D is a vector ofconstants.

In the Lagrange multipliers method the potential function isappended with the revolute joint constraints and we get

P ¼ P� þ kT/ðUÞ ð33Þ

where, k ¼ ½k1; . . . ; kp� are the Lagrange multipliers. The stationary ofthis functional P is

dP ¼ dP� þ dUTðCTkÞ þ dkTðCU� DÞ ¼ 0 ð34Þ

For arbitrary dk and dU the above equation gives

K CT

C 0

" #Uk

� �¼

FD

� �ð35Þ

The advantage of Lagrange multiplier method is that the con-straints are satisfied exactly but this is at the expense of largerset of equations. This method also gives the magnitudes of con-straint forces since the Lagrange multipliers can be obtained bysolving Eq. (35).

6. Results and discussion

In this section, the degree of freedom and the redundancy of thejoints/links are first obtained for a hexagonal mast. The stiffness ofthe mast is then evaluated and the variation in stiffness with addi-tion of cables is presented. We also present the stiffness evaluationfor an assembly of four hexagonal masts.

6.1. Degree of freedom and redundancy evaluation

The hexagonal mast built out of SLE’s, is presented in Fig. 6. Themast has six SLEs. Each SLE has 4 rigid link constraints and 6 SLE

constraint equations at the pivot point. Fixed boundary conditionsare used at joint 1. The coordinates of joints of the mast arepresented in Table 1. The results of null-space analysis of the con-straint Jacobian matrix are presented in Table 2. It is observed fromthe table that the dimension of null-space reduces on adding eachSLE and the null-space does not change for the last SLE. Hence, thelast SLE is redundant. The above analysis assumes spherical joints

Fig. 7. Stacked SLE units of Ref. [14].

0 5 10 15 20 25 30 35 40 45−10

0

10

20

30

40

50

60

70

Angle of deployment (deg)

Stiff

ness

(N/m

m)

Transverse to mastAlong the mast

Fig. 8. Axial and lateral stiffness during deployment.

Fig. 9. Hexagonal SLE mast in the deployed configuration with cables.

Table 3Variation of stiffness with addition of cables for hexagonal mast.

Stiffness inX direction(N/mm)

Stiffness inY direction(N/mm)

Stiffness inZ direction(N/mm)

Mast with top or bottom cables 24.27 53.24 11.32Mast with only vertical cables 27.88 30.48 7.39Mast with top and bottom cables 28.61 98.36 18.33Mast with all cables 31.73 138.72 23.85


for the points connected by the adjacent SLEs. The revolute jointconstraints are added further for each face and null-space is eval-uated. It can be observed from the table that the null-space reducesfor addition of revolute joints on each face. The null-space does notchange for the revolute joints added for the face 5 and face 6.Hence these joints are redundant.

By adding the boundary condition the mast will be a single de-gree of freedom system. It can be observed that the last SLE and therevolute joints on the face 5 and face 6 are redundant from thekinematic point of view. In order to reduce this mast to a structurea cable can be added at any two successive joints. The simulation isfurther continued by adding a cable between the joint 1 and joint2. The null-space dimension of the Jacobian matrix is zero indicat-ing that the mast a structure.

6.2. Stiffness evaluation

In this section the stiffness matrix (see Eq. (24)) developed inprevious section for the SLE is used along with the Lagrange mul-tiplier (see Eq. (35)) to evaluate the stiffness of the mast.

Fig. 7 shows the two-dimensional straight deployable structureconsisting of four pantograph units presented in Ref. [14] and anactive cable zig-zagging across the pantograph. A constant tensionspring keeps the active cable pre-tensioned in all configurations.The structure is deployed from nearly flat b ¼ 1:0� to the configu-ration shown in Fig. 7, b ¼ 45� , by shortening gradually the activecable. The cables have AE ¼ 1:5� 105 N and the pantograph unitshave AE ¼ 3:5� 106 N and EIz ¼ 9:6� 107 N mm2. The length ofarm is 1000 mm. The tip stiffness of the assembly as it deploys isevaluated by applying two forces of 0.5 N to top joints in X and Ydirections. Fig. 8 presents the axial and lateral stiffness of thesystem. These results matches with those presented in literature[14].

Fig. 9 shows a hexagonal mast in the deployed configuration.This mast has six SLEs and the six cables in the top, six cables in

the bottom and six vertical cables are connected as shown in thefigure. These cables are slack during deployment and becomes tautat the end of deployment. The SLE and cables have the cross sec-tional area, A, of 138:23 mm2 and 1:0 mm2, respectively. theYoung’s Modulus E of the SLE and cables are 70000.0 N/mm2 and63000.0 N/mm2, respectively. The second moment of inertia Iz, ofSLE is 8432.0 mm4. An unit load is applied at joint 10. The stiffnessof the mast due to these loads were found to be 31.73 N/mm,138.72 N/mm and 23.85 N/mm in X, Y and Z direction,respectively.

In order to study the sensitivity of the mast stiffness withcables the simulations were carried out by adding top, bottomand vertical cables individually and in combinations. The results

Fig. 10. Assembly of four hexagonal masts in the deployed configuration with cables.

Table 5Variation of stiffness with addition of cables for assembled hexagonal mast.

Stiffness inX direction(N/mm)

Stiffness inY direction

(N/mm)

Stiffness inZ direction(N/mm)

Mast with top or bottom cables 32.01 104.31 17.56Mast with only vertical cables 40.46 81.17 10.28Mast with top and bottom cables 65.44 175.42 27.25Mast with all cables 114.23 326.64 39.26

Table 4Input data: coordinates of joints for assembled hexagonal mast.

Joints X coordinate (mm) Y coordinate (mm) Z coordinate (mm)

Joint 1 0.0 0.0 0.0Joint 2 500.0 �866.0254 0.0Joint 3 1500.0 �866.0254 0.0Joint 4 2000.0 0.0 0.0Joint 13 2000.0 �1732.10 0.0Joint 14 3000.0 �1732.10 0.0Joint 15 3500.0 �866.0254 0.0Joint 16 3000.0 0.0 0.0Joint 27 4500.0 �866.0254 0.0Joint 28 5000.0 0.0 0.0


are presented in Table 3. It can be observed that the stiffness doesnot increase significantly when either top, bottom or verticalcables are used individually. The stiffness increases by more than50% when top and bottom cables are used together. The stiffnessfurther increases by additional 30% or more when all threecables, namely top, bottom and vertical, are used together. Thestiffness in Y direction is found to be higher than in the othertwo directions.

Fig. 10 shows a deployed mast consisting of four hexagonalmasts. This mast has nineteen SLEs and the nineteen cables inthe top, nineteen cables in the bottom and sixteen vertical cablesare connected as shown in the figure. Fixed boundary conditionsare used at joint 1. The coordinates of the bottom joints of the mastare presented in Table 4. The other coordinates are symmetrical.The top coordinates are located at 700 mm along Z axis. The cablesare slack during deployment and becomes taut at the end ofdeployment. The geometrical and material properties of the SLE

and cables are same as in the example of the single hexagonalmast presented earlier. An unit load is applied at joint 31. The stiff-ness of the mast due to these loads were found to be 114.23 N/mm,326.64 N/mm and 39.26 N/mm in X, Y and Z direction,respectively.

In order to study the sensitivity of the mast stiffness with cablesthe simulations were carried out by adding top, bottom andvertical cables individually and in combinations. The results arepresented in Table 5. It can be observed that the stiffness doesnot increase significantly when either top, bottom or verticalcables are used individually. The stiffness increases by morethan 55% when top and bottom cables are used together. Thestiffness further increases by additional 45% or more when all threecables, namely top, bottom and vertical, are used together. Thestiffness in Y direction is found to be higher than in the othertwo directions.

7. Conclusions

In this paper, Cartesian coordinates and symbolic computa-tions have been used for kinematic and static analysis of three-dimensional deployable SLE masts. The mobility of the mastswere evaluated from the dimension of null-space of the Jacobianmatrix formed by the derivative of the constraint equations. Thestiffness matrix for the SLE was obtained from the constraintJacobian. The stiffness matrix obtained by our approach is sameas those obtained with the force and displacement methods ofliterature. The main advantage of the constraint Jacobian basedapproach are (a) ease of obtaining the stiffness matrices, (b) deter-mination of mobility and the redundant joints/links of the mast,and (c) ease of incorporating revolute joint constraints by usingLagrange multipliers. The stiffness due to cables, an integral partof deployable masts, are also considered. The constraint Jacobianapproach was used for the analysis of a hexagonal mast and aassembled hexagonal mast, and the stiffness of the masts in dif-ferent directions were obtained. The approach presented in thispaper can be extended to masts of different shapes and to stackedmasts.

Appendix A. Stiffness matrix for the SLE

The matrices ½Jm� and ½Jmn� associated with the stiffness matrixfor SLE are given by

½Jm� ¼

X1�Xp

l1

Y1�Yp

l1

Z1�Zp

l10 0 0 0 0 0 0 0 0 Xp�X1

l1

Yp�Y1l1

Zp�Z1l1

0 0 X2�Xp

l2

Y2�Yp

l2

Z2�Zp

l20 0 0 0 0 0 Xp�X2

l2

Yp�Y2l2

Zp�Z2l2

0 0 0 0 0 0 X3�Xp

l3

Y3�Yp

l3

Z3�Zp

l30 0 0 Xp�X3

l3

Yp�Y3l3

Zp�Z3l3

0 0 0 0 0 0 0 0 0 X4�Xp

l4

Y4�Yp

l4

Z4�Zp

l4

Xp�X4l4

Yp�Y4l4

Zp�Z4l4

0BBBBBB@

1CCCCCCA

½Jmn� ¼

0 Zp�Zn

lmlnYn�Yp

lmln0 Zm�Zp

lmlnYp�Ym

lmln0 Zn�Zm

lnlmYm�Yn

lmlnZn�Zp

lmln0 Xp�Xn

lmlnZp�Zm

lmln0 Xm�Xp

lmlnZm�Zn

lmln0 Xn�Xm

lmlnYp�Yn

lmlnXn�Xp

lmln0 Ym�Yp

lmlnXp�Xm

lmln0 Yn�Ym

lmlnXm�Xn

lmln0

2664

3775


References

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[2] You Z, Pellegrino S. Cable stiffened pantographic deployable structures, Part 1:triangular mast. AIAA J 1996;34(4):813–20.

[3] Kwan AKS, Pellegrino S. Active and passive elements in deployable/retractablemasts. Int J Space Struct 1993;8(1–2):29–40.

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[8] Garcia De Jalon J, Unda J, Avello A. Natural coordinates of computer analysis ofmulti body systems. Comput Methods Appl Mech Eng 1986;56:309–27.

[9] Nagaraj BP, Pandiyan R, Ghosal A. Kinematics of pantograph masts. Mech MachTheory 2009;44(4):822–34.

[10] Calladine CR. Buckminister Fuller’s Tensigrity structure and Clerk Maxwell’srule for the construction of stiff frames. Int J Solids Struct 1978;14:161–72.

[11] Freudenstein F. On the verity of motions generated by mechanism. J Eng Ind,Trans ASME 1962;84:156–60.

[12] Kuznetsov EN. Under constrained structural systems. Int J Solids Struct1988;24(2):153–63.

[13] Kuznetsov EN. Orthogonal load resolution and statical-kinematic stiffnessmatrix. Int J Solids Struct 1997;34(28):3657–72.

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Vibration modes and natural frequencies of saddle form cable nets

Isabella Vassilopoulou a,*, Charis J. Gantes b,1

a Laboratory of Metal Structures, School of Civil Engineering, National Technical University of Athens, 12, Irinis Avenue, 15121 Pefki, Greeceb Laboratory of Metal Structures, School of Civil Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, GR-15780 Zografou, Athens, Greece


Article history:Received 10 April 2009Accepted 9 July 2009Available online 7 August 2009

Keywords:Cable netVibration modesModal transitionDeformable edge ring


* Corresponding author. Tel.: +30 210 6141055; faxE-mail addresses: [email protected] (I. Va

tral.ntua.gr (C.J. Gantes).1 Tel.: +30 210 7723440; fax: +30 210 7723442.

The objective of this paper is to investigate the dynamic behaviour of cable networks, in terms of theirnatural frequencies and the corresponding vibration modes. A multi-degree-of-freedom cable net modelis assumed, having circular plan view and the shape of a hyperbolic paraboloid surface. The cable sup-ports are considered either rigid or flexible, thus accounting for the deformability of the edge ring. Onthe basis of numerical analyses, empirical formulae are proposed for the estimation of the linear naturalfrequencies, taking into account the mechanical and geometrical characteristics of the cable net and thering, expressed in the form of appropriate non-dimensional parameters. The sequence of the symmetricand antisymmetric modes of the network and the occurrence of modal transition can be predicted in rela-tion to one of these parameters, in analogy to single cables. The differences between a network with rigidcable supports and one with boundary ring, concerning the eigenmodes and the corresponding eigenfre-quencies are identified.


1. Introduction

Many cable net roofs have been designed and constructed sincethe completion of the first saddle form cable network, being theroof of the North Carolina State Fair Arena at Raleigh, in 1953,which opened a new chapter in architecture and engineering. Thistype of structures constitutes the most competitive solution forcovering large spans, not only from aesthetic, but also from struc-tural and economical point of view; such tensile structures provideappealing architectural shapes, while the loads are carried to thesupports developing only tension in the members, leading thusto the best exploitation of the material.

Saddle form cable networks usually have a rectangular, rhom-boid, circular or elliptical plan and the shape of their surface is thatof a hyperbolic paraboloid, where the two opposite boundaryedges are higher than the other two. In this way, convex andconcave curvatures develop in two perpendicular directions,respectively. The cables are anchored to a contour ring, usuallymade of prestressed concrete, having a closed box cross-section.Cable nets belong to the family of tension structures characterizedby geometrically nonlinear behaviour. The system becomes stifferas the deformation of the cables increases, as long as they remainin tension. The stiffness of these structures is obtained by thepretension of the cables and by the opposite curvatures of

ll rights reserved.

: +30 210 7723442.ssilopoulou), chgantes@cen-

the net. The pretension must be high enough to ensure the avoid-ance of cable slackening under any loading combination, becausein that case the net becomes soft and may undergo large deforma-tions. On the other hand, opposite curvatures are necessary to en-able pretension, and to provide bearing capacity for loads directedboth downwards, such as snow and upwards, such as wind suction.Moreover, nets of low curvatures are not very stiff and thus,unsuitable for large spans.

The continuously increasing interest of engineers in cable struc-tures has led to a growing demand to understand and, if possible,to predict their behaviour. The response of cables under staticloads is quite well known, but their complicated dynamic responseis still under investigation. Because of their lightness, they are vul-nerable to dynamic excitation, especially due to wind action, whichmay lead to large amplitude oscillations, overstressing of cablesand causing fatigue problems at the cable anchorages. In addition,because of their geometrical nonlinearity, cables are vulnerable tomany kinds of resonance during their wind-induced vibrations. Inundamped or lightly damped linear or nonlinear systems, whenthe loading frequency equals the eigenfrequency, even a weakexcitation may lead to unbounded vibrations, with a continuouslyincreasing amplitude, in which case the system is said to be in thewell-known state of fundamental or primary resonance. In nonlin-ear systems, secondary resonances may also emerge: when theloading frequency X may be expressed as X = (1/n) �x, where nis an integer, whereas x the eigenfrequency, superharmonic reso-nance may occur and when the loading frequency is larger than thefrequency of the system, related to it with the expression X = n �x,where n is again an integer, phenomena of subharmonic resonance







Nomenclature

1A first antisymmetric mode about x or y-axis1S first symmetric mode of the cable net (cable net with ri-

gid or flexible supports)1SS first symmetric mode of the system (cable net with flex-

ible boundary ring)2A first antisymmetric mode about x and y-axesA cable cross-sectional areaAr edge ring cross-sectional areab ring cross-section widthD cable diameterE elastic modulus of cable materialEr elastic modulus of ring materialf maximum initial sag of the cablesg gravitational constant

Ir ring moment of inertiaL diameter of circular cable net plan viewN number of cables in each directionT0 cable initial pretensionx x-coordinate of the cable nety y-coordinate of the cable netz z-coordinate of the cable netb non-dimensional frequency of the cable netc non-dimensional parameter for the ringk2 non-dimensional parameter for the cable netq cable unit weightqr ring unit weightx natural frequencyX loading frequency

106 I. Vassilopoulou, C.J. Gantes / Computers and Structures 88 (2010) 105–119

are possible. In both cases, the system responds in such a way thatthe free oscillation term does not decay to zero, in spite of presenceof damping and in contrast to the linear solution. Especially sub-harmonic oscillations, that occur for excitations with frequenciesmuch larger than the natural frequency of a system, can havepotentially catastrophic effects. Moreover, when two or more line-arized natural frequencies, which depend on the geometrical andmechanical characteristics of the structure, are related with theexpression n1 x1 þ n2 x2 þ � � � þ nn xn � 0; where n1, . . ., nn posi-tive or negative integers, the corresponding modes may be stronglynonlinearly coupled, leading to internal resonances. If an internalresonance occurs in a system, energy imparted initially to one ofthe modes, will be continuously exchanged among all the modesinvolved in that resonance [1]. All these phenomena, related tothe linear natural frequencies as well as the loading frequency, ren-der unpredictable the dynamic response of a nonlinear system.

The vibration modes and natural frequencies of individualcables exhibit interesting phenomena that have been studied bymany researchers. First, Pugsley [2] gave some semi-empiricalexpressions for the first three in-plane modes of a sagged sus-pended chain, which could represent a hanging inextensible cablewithout pretension. Ahmadi-Kashani [3] used numerical methodsto calculate the frequencies of an inclined hanging inextensiblecable and compared the above semi-empirical formulae of Pugsleywith the numerical results. New approximate formulae were given,for the first four in-plane and out of plane natural frequenciesapplicable to a wide range of sag/span ratios and inclination angles.Irvine and Caughey [4] focused their attention on the in-plane andout-of-plane vibrations of a sagged suspended cable, and derivedspecific formulae for the corresponding elastic natural frequenciesof the symmetric and antisymmetric modes, comparing them withexperimental results. In their study they considered the cable asextensible, with a sag-to-span ratio up to 1/8, and introduced theparameter k2, which involves the cable’s geometry and elasticityand governs the symmetric in-plane modes. When this parameteris very large, the cable may be considered inextensible, and when itis very small, the cable profile approaches that of a taut string. Theyalso verified the existence of crossover points, at which modaltransition occurs, for specific values of this parameter, whichmeans that for the first crossover point the frequencies of the firstsymmetric and the first antisymmetric in-plane modes are equal,and the system is characterized by 1:1 internal resonance betweenthese two modes. Rega and Luongo [5] noticed crossover points forthe inextensible suspended cable with movable supports, for awide range of values of the sag-to-span ratio. Rega et al. [6] studiedthe influence of quadratic and cubic nonlinearities of a simplesagged cable in the modes and the modal crossovers, using a sim-

pler expression for the parameter k2, concluding that an infinitenumber of crossover points exists, instead of the single one ofthe linear theory. In [7,8] inclined cables are explored; it is notedthat their mode shapes are totally different from those of horizon-tal cables, and the phenomenon of frequency avoidance is identi-fied, meaning that, while in frequency crossover two naturalfrequencies become close, in frequency avoidance they always re-main apart and never coincide. Hybrid modes are also detected, i.e.a mixture of symmetric and antisymmetric shapes, which are asimportant as the symmetric ones, regarding the amplification ofthe cable tension.

Gambhir and Batchelor [9] investigated the influence of variousparameters, such as the cable cross-sectional area, the initial pre-tension, the sag-to-span ratio, as well as the surface curvature,on the natural frequencies of 3D cable nets. Talvik [10] noticedthat, in a cable network with an elliptical flexible contour ring,the first vibration mode involves mostly the contour ring, whilethe next four modes are determined only by cable netdeformations.

In [11] an attempt to calculate the natural frequencies of a cablecross with analytical methods is presented. The system consists oftwo crossing cables with equal spans and sag-to-span ratio. Eachsegment is modelled as one or two elements. The mass is consid-ered either lumped or consistent. The formulae of the first threeeigenfrequencies are derived, considering all three components ofdisplacement unconstrained. The horizontal vibrations have equalfrequencies, which, for sag-to-span ratios less than 1/8, are largerthan the frequency referring to the vertical vibration, taking thusthe place of the second and third eigenfrequencies of the system.A crossover point is noted for sag-to-span ratios larger than 1/8and the frequency of the vertical vibration becomes the thirdeigenfrequency of the system. Seeley et al. [12] examined the nat-ural frequencies and mode shapes of a circular concave cable net-work consisting of circular and radial cables. The sag f of the netwas obtained by the static loading and the range of sag-to-span ra-tio was between 1/9 and 1/15. They derived an approximate for-mula of the fundamental circular frequency of the net, involvingonly the sag and the sag-to-span ratio, similar to the one givenby Pugsley for the catenary, mentioned above. They concluded thatthe first natural frequency of the concave cable net is close to theaverage of the uncoupled in-plane and out-plane fundamental fre-quencies of an individual cable with the same sag/span ratio. Theyalso noticed that, only the higher order frequencies depend on theextensibility of the network, expressed by a parameter in terms ofthe elastic modulus of the cable material, the cable cross-sectionalarea, the number of the radial cables, the diameter of the networkand the uniformly distributed dead load. Buchholdt [13] suggested

I. Vassilopoulou, C.J. Gantes / Computers and Structures 88 (2010) 105–119 107

to calculate the eigenfrequencies of the deformed structure underthe permanent loads and the wind load produced by the meanwind velocity, and then add the response due to the fluctuatingcomponent. He reported measured frequencies of a saddle-shapednet roof, with a circular plan of 125 m diameter, for the first sevenmodes having frequencies from 0.74 Hz to 1.12 Hz. Several com-puterized methods of analysis and other numerical techniqueswere developed to calculate the linear natural frequencies andnonlinear static or dynamic response of cable networks and mem-branes, by solving the governing equations of motion [14–19].

This paper deals with saddle form cable networks, with a circu-lar plan view. The cable ends are considered either fixed or an-chored to a boundary ring. Their behaviour under static loadsdepends mainly on the geometry of their boundaries, on the curva-tures, on the stiffness of the system, which is determined by thecable axial stiffness and the ring flexural stiffness, and on the levelof cable pretension. Furthermore, their dynamic response dependsalso on the mass of the system, consisting of the cable mass, thering mass and the mass of the roof’s cladding, on the structuraldamping, as well as on the amplitude and the frequency of theexcitation. A first attempt to provide insight to some very interest-ing aspects of the dynamic response of saddle form suspendedroofs with fixed cable ends, was presented by the authors in [20].Conducting parametric modal analyses of the cable nets with fixedcable ends, several similarities with the dynamic behaviour of asimple sagged suspended cable were revealed, with respect to theireigenmodes and eigenfrequencies. The introduction of a parameterk2 for saddle form cable nets, similar to the one for simple cables,makes possible the prediction of the modes sequence and of thecrossover points at which modal transition occurs. Empirical for-mulae were proposed for the natural frequencies’ estimation ofthe first symmetric and antisymmetric modes. In [21] some simi-larities and differences between a network with rigid cable sup-ports and one with the cables anchored to a flexible edge ringwere highlighted.

The dynamic behaviour of these two systems is further exploredand compared in the present work, regarding their natural fre-quencies and vibration modes. Crossover points at which modaltransitions occur are also detected for both systems. Semi-empiri-cal formulae for the estimation the frequencies of the first vibra-tion modes are provided and proposed to be used at apreliminary design stage. Solving analytically three dimensionalcable structures turned out to be practically impossible, due to

Fig. 1. Geometry of the ne

their complex nonlinearity; thus numerical analyses have beenconducted.

If the main loading frequencies are known, e.g. the frequencyspectrum of the wind action, the natural frequencies of a systemprovide important information on the nature of the response todynamic loads and on the potential danger of nonlinear dynamicresponse and resonance phenomena. For preliminary structuraldesign and initial selection of form as well as cable cross-sectionsand pretensions, instead of setting up multi-degree of freedomnumerical models in order to calculate the natural frequencies ofa cable net system, an empirical formula is preferable. Knowingthe frequencies of the system and the parameters that influencethem, it is possible to improve the design of such structure, avoid-ing internal or secondary resonances, protecting the cable net fromoscillations of large amplitude and the cables from fatigue symp-toms at their anchorage points.

2. Cable net model and assumptions

The model adopted for this work is a three-dimensional cablenet, with the geometry of a hyperbolic paraboloid surface and a cir-cular plan view of diameter L (Fig. 1). The net consists of N cables ineach direction, arranged in a grid of equal distances. The sag of thelongest main and secondary cables is equal to f which is also con-sidered as the sag of the roof. All cables have a circular cross-sec-tion with diameter D and area A and their material is assumedinfinitely linearly elastic with Young-modulus E; they are modelledby truss elements, that can sustain only tension, with initial strainequal to T0/EA, where T0 is the cable pretension. The cable unitweight is equal to q. A lumped mass matrix is used for the analysis.Each part of a cable between two adjacent net intersection points ismodelled with one truss element. All three translational degrees offreedom are considered free for all internal nodes of the net.

For the cable net with fixed cable ends, the supports at all cableends are considered as pinned. Without loss of generality, the edgering, if considered, has a square box cross-section of width b, wallthickness b/10, with cross-sectional area Ar, moment of inertia Ir,unit weight qr and elastic modulus Er (Fig. 2). The z-displacementof the ring’s nodes is restrained. The displacement in the x-direc-tion is not permitted for the two nodes of the ring with coordinatex = 0, and respectively, the y-displacement is not permitted for thetwo ring nodes with coordinate y = 0, in order to avoid rigid body

t with rigid supports.

Fig. 2. Geometry of the net with flexible edge ring.


motion. Thus, the radial deformation of the ring is allowed, but notthe overall rotation about the z global axis.

The net is uniformly prestressed and no structural damping istaken into account in the present work. Linear modal analysesare performed to calculate the eigenmodes and eigenfrequencies.For the calculation of the natural frequencies of the system, thegeometry and stiffness of the equilibrium state under prestressingare considered. All analyses have been carried out with the finiteelement software ADINA [22,23], which was validated for analysesof this type by comparisons with analytical results of the equationof motion for several cases of single cables as well as for a single-degree-of-freedom cable net, having the shape of a cross. For allthese cases complete agreement of analytical and numerical re-sults was observed.

3. Cable net with rigid supports

3.1. The vibration modes

In order to investigate the vibration modes of a cable net withrigid supports, parametric analyses were performed for a largenumber of cable net models with different geometrical andmechanical characteristics, eight of which are presented here, withthe characteristics listed in Table 1 and the cross-sectional diame-ter D of all cables varying between 10 mm and 60 mm.

The cable net’s eigenmodes can be distinguished in symmetricand antisymmetric ones. The former ones consist of symmetricvertical components and antisymmetric horizontal componentsabout both horizontal axes x and y, while the latter ones consistof antisymmetric vertical components and symmetric horizontalcomponents with respect to one, or to both horizontal axes. In case

Table 1Characteristics of the cable nets with rigid supports.

Cases

1 2 3 4 5 6 7 8

N 25 25 25 25 25 25 25 35L (m) 100 100 100 100 100 100 50 50f/L 1/20 1/20 1/35 1/35 1/20 1/20 1/20 1/20T0 (kN) 400 600 400 400 400 400 400 400E (GPa) 165 165 165 165 165 148.5 165 165q (kN/m3) 100 100 80 100 80 90 100 100

two modes are antisymmetric with reference to x or y axis, theyhave similar shapes and equal eigenfrequencies. In this work, thefirst four modes are thoroughly examined, which are the first sym-metric mode of the net, denoted as 1S, the first similar antisym-metric modes about x or y axis, respectively, which are treated asone mode, denoted, both of them, as 1A and the first antisymmetricmode about both horizontal axes, which is denoted as 2A (Fig. 3).

Examining the natural modes of the cable nets, frequency cross-overs and modal transitions, similar to those described for the sim-ple suspended cable in Section 1, were also observed. To describethese phenomena, a parameter k2 is herein introduced for cablenets, similar to that for a simple cable, proposed by Rega et al.[6], which is given by the following expression:

k2 ¼ EAT0

fL

� �2

ð1Þ

The modes of the cable net are found to depend on the parameterk2, in a similar manner as for the simple cable. This has been verifiedby means of numerical analyses for all cable nets examined. Morespecifically:

(a) for k26 0.80 the first eigenmode of the system is the 1S

mode, while the second and third eigenmodes are the 1Amodes. The fourth eigenmode is the 2A mode. The 1S modehas a natural frequency smaller than the one of the 1Amodes, which in turn, is smaller than the frequency of the2A mode, that is x1S < x1A < x2A. For k2 = 0.80 the first threeeigenmodes have equal natural frequencies, which meansx1S = x1A, accounting for the first crossover point. Thesequence of the first four eigenmodes is shown in Fig. 4.

(b) for 0.80 < k26 1.00 the natural frequencies of the first two

eigenmodes, which are the 1A modes, are equal and smallerthan that of the 1S mode, which is the third cable neteigenmode, followed by the 2A mode. This means x1A <x1S < x2A. For k2 = 1.00 the natural frequencies of the 3rdeigenmode – which is the 1S mode – and the fourth eigen-mode – which is the 2A mode – are equal, that is x1S = x2A

(second crossover point). The first four eigenmodes havethe sequence shown in Fig. 5.

(c) for 1.00 < k26 1.17 (Fig. 6) a transition between the 3rd

and 4th eigenmodes occurs. Thus, the 1S mode becomesthe 4th cable net eigenmode, while the 1A modes remain

Fig. 3. The first four vibration modes of a cable net with rigid supports.

Fig. 4. The first four eigenmodes of a cable net with rigid supports for k26 0.80.

Fig. 5. The first four eigenmodes of a cable net with rigid supports for 0.80 < k26 1.00.

Fig. 6. The first four eigenmodes of a cable net with rigid supports for 1.00 < k26 1.17.

Fig. 7. The first four eigenmodes of a cable net with rigid supports for 1.17 < k2.


the first two eigenmodes. This means x1A < x2A < x1S. Fork2 = 1.17 the natural frequencies of the 1st, 2nd and 3rdeigenmodes are equal, that is x1A = x2A (third crossoverpoint).

(d) for 1.17 < k2 (Fig. 7) a transition between the 3rd and thefirst two eigenmodes occurs. The 2A mode becomes the firstmode of the system. The 1A modes become second and third,while the 1S remains the fourth mode. This means x2A <x1A < x1S.

The above limits of k2 refer to the first four eigenmodes of acable net with rigid supports. Transitions among higher modes alsooccur for different values of k2.

3.2. The natural frequencies

In this section parametric analyses are presented for all cases ofTable 1, in order to examine the relation between the natural fre-quencies and the characteristics of the cable net, and especiallythe parameter k2. Different values are given to this parameter bychanging the cable cross-sectional area between 10 mm and60 mm, keeping the characteristics of Table 1 constant. The 1S,1A and 2A modes are examined again. The analysis results areshown in the charts of Fig. 8, where the parameter k2 is plottedon the horizontal axis and the normalised frequency x=

ffiffiffiffiffiffiffiffiffiffiffiffiðg=LÞ

pon the vertical axis for all eight cable nets of Table 1, where g isthe gravitational constant considered equal to 10 m/s2.

Case 1

0

30

60

90

0 1 2 3 4 5 6

λ2

ω/(

g/L

)0.5

Case 2

0

30

60

90

0 1 2 3 4 5 6

λ2

ω/(

g/L

)0.5

Case 3

0

30

60

90

0 1 2 3 4 5 6λ2

ω/(

g/L

)0.5

Case 4

0

30

60

90

0 1 2 3 4 5 6λ2

ω/(

g/L

)0.5

Case 5

0

30

60

90

0 1 2 3 4 5 6λ2

ω/(

g/L

)0.5

Case 6

0

30

60

90

0 1 2 3 4 5 6λ2

ω/(

g/L

)0.5

Case 7

0

30

60

90

0 1 2 3 4 5 6λ2

ω/(

g/L

)0.5

Case 8

0

30

60

90

0 1 2 3 4 5 6λ2

ω/(

g/L

)0.5

1S1A2A

Fig. 8. Normalised natural frequencies of a cable net with rigid supports vs. k2.


From the charts some important remarks can be elicited:

� as the parameter k2 increases, indicating that the net becomesstiffer, the natural frequencies decrease.

� the natural frequencies and eigenmodes do not depend on thenumber of the cables in each direction (cases 7, 8).

� if k2, q, E, L and f/L are kept constant, changing the level of pre-tension does not affect the natural frequencies (cases 1, 2).

� keeping k2, q, E, T0 and L constant, the natural frequenciesdecrease as the sag-to-span ratio f/L decreases (cases 1, 4and 3, 5).

� keeping k2, q, E, T0 and f/L constant, the natural frequenciesincrease as L decreases (cases 1, 7).

� on the other hand, if k2, T0 and f/L remain constant, thenatural frequencies increase with respect to 1=

ffiffiffiffiqp (cases1, 5).

� moreover, if k2, T0 and f/L remain constant, the naturalfrequencies do not change if the ratio

ffiffiffiffiffiffiffiffiffiffiffiffiðE=qÞ

premains the same

(cases 1, 6).� finally, the frequency crossovers occur at the same values of the

parameter k2 for all cable nets, as already pointed out in Section3.1.

(1S)

0

5

10

15

20

0 1 2 3 4 5 6

λ2

β

case 1case 2case 3case 4case 5case 6case 7case 8

(1A)

0

5

10

15

20

0 1 2 3 4 5 6

λ2

β


(2A)

0

5

10

15

20

0 1 2 3 4 5 6

λ2

β


Fig. 9. Parameter b for the 1S, 1A and 2A modes of a cable net with rigid supports vs. k2.

Table 2Error mean value (MV) and standard deviation (SD) of the empirical formulae of cablenets’ natural frequencies xe.

Cases Error (xn �xe)/xn

1 (%) 2 (%) 3 (%) 4 (%) 5 (%) 6 (%) 7 (%) 8 (%)

1S mode MV �5.4 �4.5 �0.9 �0.9 �5.4 �5.4 �5.6 �5.7SD 3.4 4.9 7.9 7.9 3.4 4.3 3.6 3.6

1A mode MV �3.1 �2.4 �1.1 �1.1 �3.1 �3.4 �3.4 �3.4SD 3.5 3.6 2.9 2.9 3.5 3.9 3.5 3.5

2A mode MV �3.0 �3.5 �4.0 �4.0 �3.5 �3.4 �3.5 �3.7SD 1.6 0.5 0 0 0 0 0 0


3.3. Empirical formulae

A new non-dimensional parameter b is introduced, in order toinclude all the above information in one chart. This parameter rep-resents the non-dimensional cable net frequencies

b ¼ Lxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q=ðEgÞ�

pf=L

ð2Þ

In the charts of Fig. 9, plotting on the horizontal axis the parameterk2 and on the vertical one the parameter b for each one of thefrequencies of the net, it is noted that each natural frequency fol-lows the same curve for all cable nets. Based on the results of theprevious modal analyses, it is possible to produce approximatemathematical formulae estimating the natural frequencies of thecable nets. According to the above charts, there is a relation be-tween the two non-dimensional parameters b and k2, for each ofthe three modes, which can be expressed as follows:

k2 � bn ¼ 7n )

EAT0

fL

� �2

� Lxe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½q=ðEgÞ�

pðf=LÞ

" #n

¼ 7n )xe ¼ 7 � fL�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEg

qL2

!vuut ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT0

EAðLfÞ2

� �n

s

ð3Þwhere the subscript e denotes that this is an empirical expression ofthe eigenfrequencies, n = n1S = 3 for the 1S mode, n = n1A = 2.5 forthe 1A modes and n = n2A = 2 for the 2A mode. Thus, Eq. (3) be-comes, for the three modes respectively:

x1S;e ¼ 7 � fL�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEg

qL2

!vuutffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

T0

EALf

� �2" #

3

vuut ð4Þ

x1A;e ¼ 7 � fL�ffiffiffiffiffiffiffiffiffiffiffiffið Eg

qL2Þs ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

T0

EALf

� �2" #

2:5

vuut ð5Þ

x2A;e ¼7L�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigT0

qA

� �sð6Þ

The error of the above formulae is calculated for all the results ob-tained from the parametric modal analyses performed for each case,as the ratio (xn �xe)/xn where xn and xe are the net’s frequenciescalculated by numerical methods and by the empirical formulae,respectively. The mean value and the standard deviation of the er-ror are calculated and tabulated in Table 2 for each case. The accu-racy of the empirical formulae is also illustrated in Fig. 10, and it isconsidered as very satisfactory for preliminary design purposes.

4. Boundary ring

In order to extend the above investigation into the natural fre-quencies of a cable net anchored to a deformable edge ring, at first,the ring itself is examined, without the cables. Conducting a para-metric linear modal analysis, the first natural frequency of such astructure is calculated, considering four different geometries ofthe boundary, given in Table 3. The ring modulus of elasticity Er

varies between 30 GPa, 34 GPa, 37 GPa and 39 GPa, accountingfor the concrete categories B25, B35, B45 and B55, respectively,according to the DIN codes. The ring’s cross-section has the shapeof a square box, as given in Fig. 2, with width b taking the valuesb = 5.00 m, b = 6.50 m and b = 8.00 m for the models with diameterL = 100 m, while for the model with diameter L = 50 m the width bvaries between b = 2.00 m, b = 3.50 m and b = 5.00 m.

Case 1-8 (1S)

0

5

10

15

20

0 5 10 15 20ω1S,n

0 5 10 15 20ω1A,n

ω1S

,e

Case 1-8 (1A)

0

5

10

15

20

ω1A

,e

Case 1-8 (2A)

0

5

10

15

20

ω2A

,e

0 5 10 15 20ω2A,n

Fig. 10. Empirical formulae for natural frequencies fits numerical data.

Fig. 11. Ring’s first vibration mode (in-plane mode).

Table 3Characteristics of the edge ring.

Cases

1 2 3 4

L (m) 100 100 100 50f/L 1/20 1/20 1/35 1/20qr (kN/m3) 25 35 25 25


The first vibration mode is characterized by an in-planebreathing motion of the ring (Fig. 11). The results of the naturalfrequencies are plotted in Fig. 12 where on the horizontal axisthe non-dimensional parameter c is represented, defined as:

c ¼ ErIr

qrArL3 ð7Þ

and on the vertical axis the non-dimensional eigenfrequencyxr=

ffiffiffiffiffiffiffiffiffiffiffiffiðg=LÞ

p: From the chart, one can conclude that the relation be-

tween the non-dimensional natural frequency of the edge ringwithout cables and the parameter c is the same for all cases.

In [24] the natural frequency of any mode of vibration is given,concerning the flexural vibration of a plane circular ring:

xr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiErIrg

qrArR4

i2ð1� i2Þ2

i2 þ 1

s¼ 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiErIrg

qrArL4

i2ð1� i2Þ2

i2 þ 1

sð8aÞ

When i = 1, xr = 0 and the ring moves as a rigid body. For i = 2, thering performs the fundamental mode of flexural vibration and Eq.(8a) becomes:

xr ¼ 10:73

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiErIrg

qrArL4

sð8bÞ

Although, the above formula refers to a plane ring, it can also beused for the boundary ring of a hyperbolic paraboloid roof. The errorof the above formula is calculated for all the results obtained fromthe parametric modal analyses performed for each case, as the ratio(xrn �xre)/xrn, where xrn and xre are the ring’s frequencycalculated by numerical methods and by Eq. (8b), respectively.The mean value and the standard deviation of the error are calcu-lated and tabulated in Table 4 for each case. The accuracy of the for-mula is illustrated in Fig. 13, and is considered as sufficient for allpractical purposes.

When i = 3, the calculated frequency corresponds to the ring’ssecond mode, which is the first antisymmetric vibration mode,being 2.8 times larger than the first eigenfrequency of the ring. This

mode is not taken into account, because it cannot affect the vibra-tion of the net, as will be shown next.

5. Cable net with boundary ring

5.1. The natural modes

A large number of models are also examined, considering bothcomponents of the suspended roof, the cable net and the deform-able edge ring. It is concluded that in case the flexibility of theboundary ring is taken into account, among the eigenmodes of

Cases 1-4

0

20

40

60

80

0 5 10 15 20 25 30 35 40 45

γ

ωr/

(g/L

)0.5

case 1

case 2

case 3

case 4

Fig. 12. Normalised first natural frequency of the ring without cables vs. non-dimensional parameter c.

Cases 1-4

30

40

re


the system there is also the in-plane mode of the ring, described inthe previous section, which produces a symmetric verticalvibration of the cable net (Fig. 14). For common values of the ring’sstiffness and cable net stiffness in terms of pretension and cablecross-sectional area, the ring’s in-plane mode is the firsteigenmode of the system and the corresponding frequency canbe estimated by Eq. (8b), with negligible influence of the cablenet. In this case, the following four eigenmodes are the same vibra-tion modes of the cable net examined in Section 3.1 with negligibleinfluence of the ring, and their frequencies can be expressed byEqs. (4)–(6) with small errors. For very high values of the ring’sstiffness, its in-plane mode becomes of higher order, while the cor-responding eigenfrequency still follows the law given by Eq. (8b).In this case the first four modes of the system are the vibrationmodes of the net examined previously, with frequencies that stillfollow Eqs. (4)–(6). Between these first four modes and the ring’sone, other vibration modes appear, most of them higher order net’smodes, but also hybrid ones, involving the ring and the net into thevibration.

For illustration purposes, a cable net is considered with diame-ter L = 50 m, f/L = 1/20, T0 = 100 kN, D = 30 mm, q = 100 kN/m3,E = 165 GPa, while the ring’s characteristics are Er = 39 GPa,qr = 25 kN/m3, and the width b of the square box takes the valuesb = 2.00 m and b = 5.00 m. For the first structure the first mode isthe ring in-plane mode and the vibration modes of the net follow,while for the second one, for which the ring is much stiffer than thecable net, the ring in-plane mode is the eighth mode of the systemand the net modes are first. In Figs. 15 and 16 the first eight vibra-tion modes and the deformed ring are shown for both structures,where one can distinguish the in-plane mode of the ring.

For intermediate values of the ring’s stiffness, the symmetricvibration of the net and the in-plane one of the ring are not dis-tinct; it is not possible to distinguish which mode represents a purevibration of the net affecting also the ring and which one is mainlya vibration of the ring that produces a symmetric oscillation to thenet. Consequently, in what follows, the 1st symmetric mode of thesystem is examined, whether this is produced mainly due to a netsymmetric vibration or a ring in-plane one. The corresponding fre-quency will be named as x1SS.

Table 4Error mean value (MV) and standard deviation (SD) of the formula of ring’seigenfrequency xre.

Cases Error (xrn �xre)/xrn

1 (%) 2 (%) 3 (%) 4 (%)

MV 0.2 0.2 1.3 0.1SD 0.1 0.1 0.1 0.3

5.2. The natural frequencies

A sample of the above investigation is presented in this section,for the cable nets with characteristics given in Table 5. Parametricanalyses are performed in order to evaluate the influence of theedge ring deformability to the net’s vibration modes, by varyingthe ring stiffness Er Ir and the ring mass qr Ar. The frequencies ofthis system are compared with those of the cable net with rigidsupports.

In Table 6 the parameter k2 and the frequencies are given for eachnet with rigid supports. Keeping the ring unit mass constant andequal to 25 kN/m3, the ring’s influence on the system’s frequencyof the first symmetric mode (x1SS) is examined, by varying theelastic modulus Er and the ring cross-section width b as in Section4, accounting for realistic values of the ring’s stiffness and cross-sectional area. The variation of the non-dimensional first five naturalfrequencies of the system (including the double 1A frequency) withrespect to the ring stiffness is given in the charts of Fig. 17. In thesecharts, x1S, x1A, x2A are the frequencies of 1S, 1A and 2A modes ofthe net, respectively, while x1SS is the frequency of the firstsymmetric mode of the system. The change of the combinedsystem’s non-dimensional frequency x1SS of the first symmetricmode with respect to the ring stiffness is shown in the charts ofFig. 18. In these charts, the frequencies of the three systems are com-pared, the ring without the cables (xr), as calculated from Eq. (8b),the cable net without the ring (x1S) and the cable net with the ring(x1SS). For the first case, keeping the elastic modulus Er and thecross-section width b constant and equal to 37 GPa and 5.00 mrespectively, the ring unit weight qr varies between 25 kN/m3 and50 kN/m3. The variation of the non-dimensional frequency x1SS ofthe system with respect to the mass of the ring, is shown in Fig. 19.

0

10

20

0 10 20 30 40ωrn

ω

Fig. 13. Formula for the first natural frequency of the ring without cables fitsnumerical data.


From the charts some important conclusions can be drawn:

� As the stiffness of the ring increases the frequency x1SS

increases.� If q, E, L, f/L and the stiffness of the ring are kept constant, an

increase of the level of pretension increases slightly the fre-quency x1SS (cases 1, 2).

� Keeping q, E, T0, L, f/L and the ring stiffness constant, the fre-quency x1SS increases as the cable diameter D increases (cases2, 3).

� Moreover, if k2, T0 and f/L remain constant, for the same levels ofthe ring’s stiffness, the frequency x1SS does not change, if theratio

ffiffiffiffiffiffiffiffiffiffiffiffiðE=qÞ

premains the same (cases 3, 6).

� Keeping q, E, T0, L and the ring stiffness constant, the frequencyx1SS decreases as the sag-to-span ratio f/L decreases (cases 1, 5).

Fig. 15. The first eight eigenmodes of the cable n

Fig. 14. The first five eigenmodes of the c

� For the same levels of the ring’s stiffness and keeping f/L, T0, E, qconstant, the frequency x1SS increases as L decreases (cases 1, 7).

� On the other hand, if the ring stiffness, E, T0, L, and f/L remainconstant, the frequency x1SS increases slightly as the cable unitmass q decreases (cases 3, 4).

� The frequency x1SS does not depend on the number of cables ineach direction (cases 7, 8).

� The frequency x1SS, for low levels of the ring’s stiffness, is the fre-quency of the in-plane mode of the ring and can be calculatedusing Eq. (8b), but as the stiffness increases the frequency divergesfrom the curve of the above equation and tends to become equal tothe x1S of the net with rigid supports (Fig. 18).

� The presence of the edge ring changes the frequency of the 1Smode of the net (x1S), up to 36%, with respect to the one ofthe net with rigid supports. This occurs in case 5, in which the

et with ring cross-section width b = 2.00 m.

able net with the flexible edge ring.

Table 5Characteristics of the cable nets with the flexible edge ring.

Cases

1 2 3 4 5 6 7 8

N 25 25 25 25 25 25 25 35L (m) 100 100 100 100 100 100 50 50f/L 1/20 1/20 1/20 1/20 1/35 1/20 1/20 1/20D (mm) 40 40 60 60 40 63.2 40 40T0 (kN) 400 600 600 600 400 600 400 400E (GPa) 165 165 165 165 165 148.5 165 165q (kN/m3) 100 100 100 80 100 90 100 100


first natural frequency x1SS of the combined system approachesthe frequency of the 1S mode (x1S) of the net with rigid supports(Fig. 18) causing this change to the net’s frequency when thetwo components are taken into account (Fig. 17). In the othercases, instead, the change of the frequency of the 1S mode ofthe net (x1S), arises at only 11%.

� The frequency of the 2A mode (x2A) remains unchanged in pres-ence of the edge ring (Fig. 17).

� The frequency of the 1A modes (x1A) does not change more than3.4% due to the deformability of the edge ring (Fig. 17). This isthe case because, for realistic values of ring flexural stiffnessand cable axial stiffness, the antisymmetric vibration mode ofthe boundary ring is always much larger than the first four fre-quencies of the cable net. Thus, it cannot influence significantlythe antisymmetric vibration mode of the net.

� If the stiffness of the ring is kept constant, and the mass of theedge ring increases, the frequency x1SS decreases (Fig. 19).

5.3. Empirical formulae

Based on the aforementioned results of our investigation, it isconcluded that the first symmetric mode of a cable net with a flex-ible boundary ring, depends on the ratio of the stiffness of the ring

Fig. 16. The first eight eigenmodes of the cable n

and that of the cable net. If the ring is flexible enough with respectto the cable net, the first symmetric mode is the in-plane mode ofthe ring. On the other hand, when the ring is much stiffer than thecable net, it behaves as a rigid support to the cables, and its vibra-tion mode is one of the higher order modes, while the first sym-metric mode of the system is the one of the cable net. Thestiffness ratio of the two components of such a system is expressedas the ratio of the natural frequencies of the two independent sys-tems and constitutes the criterion that indicates whether the firstsymmetric mode of the system will be the in-plane mode of thering or the first symmetric mode of the net.

et with ring cross-section width b = 5.00 m.

Case 1

0

10

20

30

40

50

60

0 5 10 15

γ

ω/(

g/L

)0.5

ω1Sω1Αω2Aω1SS

Case 2

0

10

20

30

40

50

60

0 5 10 15

γω

/(g/

L)0.

5

ω1Sω1Αω2Aω1SS

Case 3

0

10

20

30

40

50

60

0 5 10 15

γ

ω/(

g/L

)0.5

ω1Sω1Αω2Aω1SS

Case 4

0

10

20

30

40

50

60

0 5 10 15

γ

ω/(

g/L

)0.5

ω1Sω1Αω2Aω1SS

Case 5

0

10

20

30

40

50

60

0 5 10 15

γ

ω/(

g/L

)0.5

ω1Sω1Αω2Aω1SS

Case 6

0

10

20

30

40

50

60

0 5 10 15

γ

ω/(

g/L

)0.5

ω1Sω1Αω2Aω1SS

Case 7

0

20

40

60

80

0 10 20 30 40 50

γ

ω/(

g/L

)0.5

ω1Sω1Αω2Aω1SS

Case 8

0

20

40

60

80

0 10 20 30 40 50

γ

ω/(

g/L

)0.5

ω1Sω1Αω2Aω1SS

Fig. 17. Normalised natural frequencies of the cable net with ring vs. c.

Table 6Eigenfrequencies of the nets with rigid supports.

Cases

1 2 3 4 5 6 7 8

k2 1.30 0.86 1.94 1.94 0.42 1.94 1.30 1.30x1S (s�1) 14.011 15.848 12.544 14.024 9.984 12.544 28.021 28.054x1A (s�1) 13.205 15.015 11.614 12.895 11.157 11.614 26.410 26.424x2A (s�1) 12.940 15.221 10.566 11.813 13.014 10.566 25.880 25.874


Case 1

0

10

20

30

40

50

60

0 5 10 15γ

ω/(

g/L

)0.5

ω1S-fixedω1SSωr-ring

Case 2

0

10

20

30

40

50

60

0 5 10 15γ

ω/(

g/L

)0.5


Case 3

0

10

20

30

40

50

60

0 5 10 15γ

ω/(

g/L

)0.5


Case 4

0

10

20

30

40

50

60

0 5 10 15γ

ω/(

g/L

)0.5


Case 5

0

10

20

30

40

50

60

0 5 10 15γ

ω/(

g/L

)0.5


Case 6

0

10

20

30

40

50

60

0 5 10 15γ

ω/(

g/L

)0.5


Case 7

0

20

40

60

80

0 10 20 30 40 50γ

ω/(

g/L

)0.5


Case 8

0

20

40

60

80

0 10 20 30 40 50γ

ω/(

g/L

)0.5


Fig. 18. Normalised natural frequencies of the cable nets vs. c (x1S for the cable net with fixed cable ends, x1SS for the first symmetric mode of the cable net with the ring, xr

for the ring with no cables).

Case 1

0

10

20

200 300 400 500

30

ρrΑr [kN/m]

ω1S

S/(

g/L

)0.5

Fig. 19. Normalised natural frequency x1SS vs. qrAr.


From the charts of Fig. 18, it is evident that if the ring’s frequencyxr is less than, approximately, 65% of x1S, then the first natural fre-quency of the combined system is close to the ring’s frequency andthe first symmetric mode of the system is the in-plane mode of thering. As the ring becomes stiffer and its frequency increases, the com-bined system’s first natural frequency approaches asymptoticallythe frequency (x1S) of the cable net with rigid supports and the firstsymmetric mode of the system is the symmetric mode of the net.

Hence, if the ring’s frequency xr, according to Eq. (8b) is com-puted less than 65% of the x1S, then the frequency x1SS of the firstsymmetric mode of the cable net with edge ring can be evaluatedfrom this equation. If, on the other hand, it results to more than

Case 1

0

10

20

30

40

50

60

0 5 10 15

γ

ω1S

S/(

g/L

)0.5

n

e

Case 2

0

10

20

30

40

50

60

0 5 10 15

γ

ω1S

S/(

g/L

)0.5

n

e

Case 3

0

10

20

30

40

50

60

0 5 10 15

γ

ω1S

S/(

g/L

)0.5

n

e

Case 4

0

10

20

30

40

50

60

0 5 10 15

γ

ω1S

S/(

g/L

)0.5

n

e

Case 5

0

10

20

30

40

50

60

0 5 10 15

γ

ω1S

S/(

g/L

)0.5

n

e

Case 6

0

10

20

30

40

50

60

0 5 10 15γ

ω1S

S/(

g/L

)0.5

n

e

Case 7

0

20

40

60

80

0 10 20 30 40 50

γ

ω1S

S/(

g/L

)0.5

n

e

Case 8

0

20

40

60

80

0 10 20 30 40 50

γ

ω1S

S/(

g/L

)0.5

n

e

Fig. 20. Numerical data (n) and empirical formula (e) for the frequency x1SS of the system’s first symmetric mode vs. c.

Table 7Error mean value (MV) and standard deviation (SD) of the empirical formula of thesystem’s frequency x1SS.

Cases Error (x1SS,n � x1SS,e)/x1SS,n

1 (%) 2 (%) 3 (%) 4 (%) 5 (%) 6 (%) 7 (%) 8 (%)

MV �2.3% �1.0% �7.5% �6.0% �1.2% �7.5% �3.6% �5.6%SD 2.4% 1.7% 4.9% 2.8% 3.7% 4.9% 7.7% 9.0%


65%, the frequency x1SS depends on the value of x1S and it may beapproximated by:

If xr < 0:65x1S; then x1SS ¼ xr ¼ 10:73

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiErIrg

qrArL4

sð9aÞ

if xr P 0:65x1S; then x1SS ¼ x1S 1� 0:35 � 0:65 �x1S

xr

� �2" #

ð9bÞ

Case 1-8 (ω1SS)

0

10

20

30

0 10 20 30ω1SS,n

ω1S

S,e

Fig. 21. Empirical formula for the frequency x1SS fits numerical data.


In Fig. 20, the frequency x1SS of the system’s first symmetric modecalculated by numerical methods and by the empirical formula isplotted vs. the non-dimensional parameter c. The error of the aboveformulae is calculated for all results obtained from the parametricmodal analyses performed for each case, as the ratio (x1SS,n

� x1SS,e)/x1SS,n where x1SS,n and x1SS,e are the frequency x1SS

calculated by numerical methods and by the empirical formula,respectively. The mean value and the standard deviation of theerror are calculated and tabulated in Table 7 for each case. The accu-racy of the empirical formulae is also shown in Fig. 21 for all eightcases of Table 5, and is evaluated as satisfactory for preliminarydesign purposes. Since the presence of the edge ring does notinfluence significantly the frequencies of the net’s antisymmetricvibration modes, the empirical formulae, given by Eqs. (5) and (6)can also be used for the case of a cable net with cables anchoredto a deformable edge ring.

6. Conclusions

The dynamic behaviour of cable nets having rigid supports, hasbeen thoroughly investigated and many similarities with the caseof a simple suspended cable have been observed. The introductionof a parameter k2 for cable nets makes possible the prediction of themodes appearance sequence and of the crossover points, at whichmodal transition occurs. The semi-empirical formulae provided inthis work give satisfactory results and are suggested for predictingthe frequencies of the first four vibration modes of the net.

In case the deformability of the flexible contour ring is takeninto account, the dynamic behaviour of the system becomes morecomplicated. The existence of the ring negligibly influences theantisymmetric modes of the cable net, but an in-plane mode ofthe ring produces a symmetric vertical vibration of the net, influ-encing significantly the frequency of the first symmetric mode ofthe net, with respect to the one of the cable net with fixed ends.Another semi-empirical formula is proposed for estimating the fre-quency of the system’s 1st symmetric mode, either produced bythe symmetric vibration of the net, involving the ring, or by thein-plane mode of the ring, involving also the net.

Knowledge of the natural frequencies of a nonlinear system andthe relations between them provides us an important informationabout the existence of eventual internal resonances. It is possibleto use the proposed formulae to calculate the natural frequenciesof the cable net, with rigid or flexible supports, at a preliminary de-sign stage, in order to avoid 1:1 internal resonances between thefirst vibration modes, which may lead to oscillations of largeamplitude, with a continuous exchange of energy between themodes involved in the resonance.

References

[1] Vakakis A. Introduction in nonlinear dynamics. Lecture notes. Greece: NationalTechnical University of Athens; 2002.

[2] Pugsley AG. On the natural frequencies of suspension chains. Quart J MechAppl Math 1949;II(4):412–8.

[3] Ahmadi-Kashani K. Vibration of hanging cables. Comput Struct1989;31(5):699–715.

[4] Irvine HM, Caughey TK. The linear theory of free vibrations of a suspendedcable. Proc Roy Soc London, Ser A Math Phys Sci 1974;341(1626):299–315.

[5] Rega G, Luongo A. Natural vibrations of suspended cables with flexiblesupports. Comput Struct 1979;12:65–75.

[6] Rega G, Vestroni F, Benedettini F. Parametric analysis of large amplitude freevibrations of a suspended cable. Int J Solids Struct 1984;20(2):95–106.

[7] Burgess JJ, Triantafyllou MS. The elastic frequencies of cables. J Sound Vib1987;120(1):153–65.

[8] Triantafyllou MS, Grinfogel L. Natural frequencies and modes of inclinedcables. J Struct Eng 1986;112(1):139–48.

[9] Gambhir ML, deV Batchelor B. Finite element study of the free vibration of 3Dcable networks. Int J Solids Struct 1978;15:127–36.

[10] Talvik I. Finite element modelling of cable networks with flexible supports.Comput Struct 2001;79:2443–50.

[11] Leonard JW. Tension structures, behavior and analysis. New York: McGraw-Hill, Inc.; 1988.

[12] Seeley GR, Christiano P, Stefan H. Natural frequencies of circular cablenetworks. J Struct Div 1975;101(5):1171–7.

[13] Buchholdt HA. An introduction to cable roof structures. London: ThomasTelford; 1999.

[14] Porter Jr DS, Fowler DW. The analysis of nonlinear cable net systems and theirsupporting structures. Comput Struct 1973;3:1109–23.

[15] Morris NF. Dynamic response of cable networks. J Struct Div1974;100(10):2091–108.

[16] Morris NF. Modal analysis of cable networks. J Struct Div 1975;101(1):97–108.[17] Swaddiwudhipong S, Wang CM, Liew KM, Lee SL. Optimal pretensioned forces

for cable networks. Comput Struct 1989;33(6):1349–54.[18] Stefanou GD. Dynamic response of tension cable structures due to wind loads.

Comput Struct 1992;43(2):365–72.[19] Tabarrok B, Oin Z. Dynamic analysis of tension structures. Comput Struct

1997;62(3):467–74.[20] Vassilopoulou I, Gantes CJ. Modal transition and dynamic nonlinear response

of cable nets under fundamental resonance. In: Proceedings of the eighthHSTAM international congress on mechanics, vol. II, Patras, Greece; 2007, p.787–94.

[21] Vassilopoulou I, Gantes CJ. Vibration modes and dynamic response of saddleform cable nets under sinusoidal excitation. In: Proceedings of EuromechColloquium 483, geometrically non-linear vibrations of structures, FEUP,Porto, Portugal; 2007. p. 129–32.

[22] ADINA (Automatic Dynamic Incremental Nonlinear Analysis) v8.4. USA:ADINA R&D, Inc.; 2006.

[23] ADINA (Automatic Dynamic Incremental Nonlinear Analysis) v8.4. Theory andmodeling guide, ADINA solids and structures, vol. I. USA: ADINA R&D, Inc.;2006.

[24] Timoshenko S. Vibration problems in engineering. New York: D. Van NostrandCompany, Inc.; 1937.





Maximizing the fundamental eigenfrequency of geometrically nonlinearstructures by topology optimization based on element connectivityparameterization

Gil Ho YoonSchool of Mechanical Engineering, Kyungpook National University, Republic of Korea


Article history:Received 10 July 2009Accepted 20 July 2009Available online 13 August 2009

Keywords:Topology optimizationInternal element connectivityparameterization methodModal analysisNonlinear structure


E-mail addresses: [email protected], ghy@knu

This paper pertains to the use of topology optimization based on the internal element connectivityparameterization (I-ECP) method for nonlinear dynamic problems. When standard density-based topol-ogy optimization methods are used for nonlinear dynamic problems, they typically suffer from two mainnumerical difficulties, element instability and localized vibration modes. As an alterative approach, theI-ECP method is employed to avoid element instability and a new patch mass model in the I-ECP formu-lation is developed to control the problem of localized vibration modes. After the I-ECP based formulationis developed, the advantages of the proposed method are checked with several numerical examples.


1. Introduction

Topology optimization methods are used for design purposes incivil and mechanical engineering [1–4]. In particular, layouts of lin-ear dynamic structures, whose stiffness and mass matrices areindependent on load and displacements, have been topologicallyoptimized to reduce vibration or noise at target frequencies withinpre-assigned design limits (see [5–16] and references therein formore details). In other words, by controlling the first eigenfrequen-cy or some of the lowest eigenfrequencies of a linear structure, itsdynamic characteristics could be improved indirectly. However, areview of current topology optimization methods for nonlinear dy-namic problems highlights the difficulties related to unstable ele-ments in geometrical nonlinear static analysis [2,17–24] anddifficulties associated with the highly localized vibration modesin the modal analysis [12–16,25–29]. This study examines howthese difficulties can be resolved and investigates the use of topol-ogy optimization for maximizing the fundamental eigenfrequencyof geometrically nonlinear structures.

In order to calculate eigenfrequencies of a geometrically nonlin-ear structure, it is necessary to perform a two-step numerical anal-ysis; a modal analysis should be performed after a nonlinear staticanalysis. First, the deformation of a structure subject to a load atwhich nonlinear responses are detected in the structure shouldbe computed using a standard nonlinear static solver commonly

ll rights reserved.

.ac.kr

with Newton–Raphson iteration. Second, we perform a modalanalysis with the mass matrix and the tangent stiffness matrixcomputed at the last Newton–Raphson step of the first nonlinearstatic analysis [30,31]. In density-based topology optimization,flipped elements having negative Jacobian values – referred to asthe unstable elements – inevitably appear around void areas sim-ulated by elements with weak Young’s moduli during the Newton–Raphson iteration of the first nonlinear static analysis. Becausethese unstable elements make the Newton–Raphson iteration di-verge, it is difficult to carry out topology optimization with thenonlinear static analysis. These difficulties have been overcomeby developing numerical methods such as the modified Newton–Raphson iteration, the element removal and reintroduction meth-od, and the displacement loading method [17,20,21]. In additionto the first difficulty of the unstable elements, the modal analysisfor topology optimization problem is also suffering from highlylocalized vibrating modes in low-density regions having relativelylow eigenvalues. Considering one or some of the lowest modes intopology optimization, these highly localized vibrating modes aretroublesome. To resolve them, an optimization formulation maxi-mizing localized buckling modes with a mesh independent filterhave been proposed [12,26,27,29]. Alternatively several techniquesinvolving mass density interpolation have also been proposedwithout changing optimization formulation [14,25].

In this research, to overcome the above mentioned difficulties,an alternative topology optimization approach called the internalelement connectivity parameterization (I-ECP) method was



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G.H. Yoon / Computers and Structures 88 (2010) 120–133 121

employed. Unlike the standard density-based approach, this newmethod defines a structural layout by defining connectivity amongsolid elements using zero-length links, as shown in Fig. 1 [19,32–

(a)

SIMP approach

Design variable

Solidelement

Weakelement

SIMP approach

Design variable

Solidelement

Weakelement Stron

linksStronlinks

(b)

Fig. 1. Modeling by the density-based SIMP and the I-EC

Fig. 2. Solution procedure for th

34]; it may be possible to interpret the element connectivityparameterization (ECP) method as a numerical method imposingweak continuity constraints among elements using the penalty

I-ECP approach

Patch

Design variable

Weaklinks

g

I-ECP approach

Patch

Design variable

Weaklinks

g

(c)

P (internal-element connectivity parameterization).

e nonlinear modal problem.

122 G.H. Yoon / Computers and Structures 88 (2010) 120–133

or the Lagrange multiplier (see related discussions in [20]). Be-cause the material properties of solid elements do not change dur-ing optimization iterations, the unstable element of nonlinearstructures can be eliminated. Depending on how solid elements

Fig. 3. Effects of the applied load on eigenfrequencies for a slender beam. (a) Probleeigenvalues for nonzero force, and (d) curves of eigenfrequency versus applied load for

are connected with zero-length links, two ECP methods exist[32]. One is the external ECP (E-ECP) method in which solid ele-ments are directly connected, and the other is the internal ECP(I-ECP) method, where the solid elements are indirectly connected

m definition, (b) eigenmodes and eigenvalues for zero force, (c) eigenmodes andthe beam.

Fig. 4. e-th planar rectangular patch consisting of a plane finite element and four zero-length links.


through outer nodes. Because of the computational advantage inthe latter, this study employs the I-ECP method [32,35].

For successful analysis and optimization, this study also devel-ops a novel way of mass modeling formulation for the I-ECP meth-od. So far, because of the ambiguity in the definition of theformulation of a mass matrix, the I-ECP method has not been ap-plied to dynamic problems. It might have been obvious to definea mass matrix for a solid element inside a patch, but it was notapparent that a mass matrix could be defined for zero-length links;here ‘‘define” means calculating and assembling the mass matrix inthe framework of the finite element (FE) procedure. Moreover,from several empirical test results discussed in Section 2.3, it ap-pears that assembling the mass matrix of a solid element to the de-grees of freedom of the solid element complicates the optimizationprocess due to the presence of highly localized modes in patches

Fig. 5. Mass modelings using the I-ECP method. (a) A straightforward model(assigning the mass matrix at the solid element to the degrees of freedom of theinner nodes), and (b) the proposed patch mass matrix model (assigning the massmatrix to the degrees of freedom of the outer nodes).

[12–14,26–28]. To avoid the complication, an alternative approachin which the mass matrix of a solid element is assembled to the de-grees of freedom of the outer nodes is presented in this study.Adopting the new mass modeling method proposed here and themass interpolation functions proposed in [14,25], the difficultyarising from highly localized modes inside patches could be over-come and topology optimization could be carried out for nonlineardynamic system.

This paper is organized as follows. In Section 2, we provide anoverview of the basic notations and governing equations for non-linear modal analysis. We use the I-ECP method and study themass modeling of the method. The parameterization of design vari-ables and the sensitivity analysis of the I-ECP method are dealtwith in Section 3. In Section 4, two numerical examples in whichthe fundamental eigenfrequency of two-dimensional structures ismaximized are presented to show the potential of the proposedmethod in Section 4. In Section 5, some observations are maderegarding designs of nonlinear dynamics system and future workis discussed.

Fig. 6. Solid element: (a) a finite element model, and (b) four eigenvalues andeigenmodes.


2. Modal analysis of nonlinear structures using the I-ECPmethod

2.1. Formulation of nonlinear vibration problem

The vibration motion of a structure with practical dynamicloads is represented by sinusoidally varying eigenmodes as wellas static equilibrium displacements [30,31,36,37]. In the case of alinear structure, the static equilibrium displacements are zeroand only the sinusoidal eigenmodes remain. Thus, the linear eigen-value problems can be solved by using load-independent stiffnessand mass matrices can be solved. In contrast, to calculate eigen-values and their associated eigenmodes for a nonlinear structure,the modal analysis depicted in Fig. 2 should be used along withthe tangent stiffness matrix computed for the deformed domainfor given static loads and the mass matrix [30,31]. Consequently,eigenfrequencies of a geometrically nonlinear structure are depen-dent on current static displacements even for a structure made oflinear elastic material.

For nonlinear static analysis, the notations given in [30,31] werefollowed. By respectively denoting the updated displacements and

Fig. 7. Comparison of mass models obtained using the ECP method ðlmax=kstructurediagonal ¼ 104Þ

methods.

displacements at time t + Dt of a generic point of a body by DU andtþDtU, the following update rules can be obtained:

tþDtUðkÞ ¼ tþDtUðk�1Þ þ DUðkÞ; tþDtUð0Þ ¼ tU ð1ÞtKðk�1Þ

T DUðkÞ ¼ RðtþDtUðk�1ÞÞ ðsee ½19;20�Þ ð2Þ

where the superscript (k) denotes the kth iteration step in the New-ton–Raphson method. The incremental residual and tangent stiff-ness matrix are denoted by RðtþDtUðk�1ÞÞ and tKðk�1Þ

T , respectively.After solving the nonlinear static equation (Eq. (2)), the followingmodal analysis should be carried out, where the displacement-dependent tangent stiffness matrix tþDtKT and the displacement-independent mass matrices, M are used

ðtþDtKT �x2MÞU ¼ 0;UT MU ¼ I ð3Þ

Here, x and U are eigenfrequencies and associated eigenmodes,respectively. The identity matrix is I is of the same size as the asthe mass or stiffness matrix.

In order to investigate how eigenvalues of nonlinear structuresare affected significantly by applied forces in practice, we consid-ered the simple beam shown in Fig. 3a. The left side of the beam

. (a) A straight mass model, (b) a present patch mass model, and (c) modes of two


is clamped and a concentrated force is applied to the right side. Thedeformed shape of the beam determined by nonlinear static anal-ysis in Fig. 3c and the associated eigenmodes obtained from themodal analysis of Eq. (3) indicate that the eigenfrequencies varynonlinearly with the magnitude of the applied force; the first andsecond angular speeds at under the applied force of 0.003 N are in-creased approximately by 130% and 14%. However, the third angu-lar speed is decreased by 8%. As observed in the figure, theeigenfrequencies are influenced significantly by the applied loadsand associated deformations. Fig. 3c plots the eigenmodes of theundeformed structure; if the modes were plotted at the deformedstructure, it would be difficult to compare them with those of lin-ear analysis as shown in Fig. 2. Thus, later in the document, wepresent eigenmode plots of examples at undeformed structures.

2.2. Nonlinear static analysis using the ECP method

The I-ECP method enables the realization of a layout differentfrom that of the element density-based method. To underline thebasic concepts of these two methods and some fundamental differ-ences between them, consider the layout in Fig. 1a, which may beobserved during topology optimization iterations. When the stan-dard element density method is employed, design variables de-fined for each element and their corresponding Young’s moduliare varied to realize the layout of Fig. 1a in terms of the potentialenergy. Although this method is robust and simple, it suffers fromnumerical instability called the unstable element with negative

Fig. 8. Ratios of the condensed stiffness matrix to mass matrix with various mass interpobehavior, (b) a linear mass interpolation function of Eq. (25), (c) a power mass interpola(27) proposed in [14].

Jacobian values in topology optimization for geometrically nonlin-ear structures [2,17–21,23,24].

As opposed to the density-based methods, the I-ECP methoddoes not change the material properties of plane or cubic finite ele-ments discretizing a domain during topology optimization asshown in Fig. 1c. To define a layout, all elements are disconnectedfrom other elements and nodes at the same locations are con-nected using one-dimensional links. To reduce the computationtime, the static condensation scheme, which condenses out forthe degrees of freedom of inner nodes of the I-ECP patch fromthe global stiffness matrices, was proposed in [32]. A detail con-densation implementation of the I-ECP method can be found in[35].

To formulate the I-ECP method for nonlinear static analysis, theeth patch shown in Fig. 4 is considered along with the assumptionof geometrical nonlinearity. The nodes connecting elements arenamed the outer nodes and those defining plane elements arenamed the inner nodes. The displacements of the outer nodesand inner nodes are denoted by tþDtuðkÞe;out and tþDtuðkÞe;in in the kth iter-ation, respectively. The displacements are then updated as follows:

tþDtuðkÞe;out

tþDtuðkÞe;in

" #¼

tþDtuðk�1Þe;out

tþDtuðk�1Þe;in

" #þ

DuðkÞe;out

DuðkÞe;in

" #ð4Þ

where DuðkÞe;out and DuðkÞe;in denote the updated displacements for theouter nodes and the inner nodes, respectively, and are calculatedby the following equation:

lation functions for one element in Fig. 7 (SIMP penalty: 3). (a) The stiffness matrixtion function of Eq. (26), and (d) a C1 continuous mass interpolation function of Eq.


kI;e ¼ leðceÞI8�8 ðwhere I8�8 is a 8� 8identity matrixÞ ð5Þ

kI;e �kI;e

�kI;e kI;e

� �þ

0 00 tkstructure;ðk�1Þ

T;e

" #( )DuðkÞe;out

DuðkÞe;in

" #¼

Rðk�1Þe;out

Rðk�1Þe;in

" #

ð6Þ

The link stiffness of the eth patch, le, is a function of the designvariable ce. The stiffness matrix and the residual force terms of theouter and the inner nodes of the eth patch are denoted bytkstructure;ðk�1Þ

T;e , Rðk�1Þe;out and R

ðk�1Þe;in , respectively.

For the I-ECP, Rðk�1Þe;out and R

ðk�1Þe;in can be formulated as

Rðk�1Þe;out

Rðk�1Þe;in

" #¼

tþDtRe

0

� ��

0tþDt

0fstructure;ðk�1Þe

� ��

tþDt0f link;ðk�1Þ

e;out


e;in

24

35 ð7Þ


e;out

tþDt0 f link;ðk�1Þ

e;in

24

35 ¼ kI;e �kI;e

�kI;e kI;e

� � tþDtuðk�1Þe;out

tþDtuðk�1Þe;in

" #ð8Þ

In Eq. (7), the externally applied force on the outer nodes and theinternal force acting on the inner nodes are denoted by tþDtRe andtþDt

0fstructure;ðk�1Þe , respectively. Because the degrees of freedom of

the inner nodes are independent of those of the other nodes, thestatic condensation scheme can be applied.

tkðk�1ÞCon;e ¼ ðkI;e � kI;e kI;e þ tþDtkstructure;ðk�1Þ

T;e

� ��1kI;eÞ ð9Þ

tkðk�1ÞCon;e DuðkÞe;out ¼ R

ðk�1Þe;out þ kI;e kI;e þ tkstructure;ðk�1Þ

T;e

� ��1Rðk�1Þe;in ð10Þ

The global tangent matrix is assembled as

tKðk�1ÞCon ¼

XNp

e¼1

tkðk�1ÞCon;e ð11Þ

where Np is the total number of patches. Then the following systemof equations is solved iteratively for DUðkÞout, which is the global dis-placement vector for the outer nodes:

tKðk�1ÞCon DUðkÞout ¼ R

ðk�1ÞCon ð12Þ

where Rðk�1ÞCon ¼

XNp

e¼1

Rðk�1Þe;out þ kI;e kI;e þ tkstructure;ðk�1Þ

T;e

� ��1Rðk�1Þe;in

� �ð13Þ

Fig. 9. Cantilever problem in the case where the ends of the cantilever are clamped.(a) Problem definition, (b) a symmetric linear result with an evenly distributedinitial guess, and (c) an unsymmetrical result with an unsymmetrical initial guess.

2.3. Modeling of mass matrix in I-ECP method for modal analysis

Eigenfrequencies of the nonlinear system that can be deter-mined using the I-ECP method are also calculated by modal analy-sis (Eq. (3)). The tangent stiffness matrix can be easily computed byEq. (2). But it is not clear how the mass matrix in Eq. (3) is con-structed for the I-ECP method, which is one of the limitations ofthis method. The construction of the mass matrix is examined inthe following sections.

2.3.1. Formulation 1: straightforward method for mass matrixmodeling

For dynamic analysis, assigning mass matrices to each solid ele-ments is straightforward as shown in Fig. 5a. If the stiffness valuesof the links of a patch are sufficiently large, some eigenvalues of anI-ECP patch become equivalent to the eigenvalues of a solid ele-ment in principle. For example, consider the two-dimensional solidelement in Fig. 6a under the assumption of small displacements.Because the degrees of freedom of the bottom nodes are clamped,there are 4 real and 4 infinite eigenvalues in Fig. 6b. Here an I-ECP

patch assembling the mass matrix of the solid element to the innernodes is constructed in Fig. 7a. The modal analysis equations of thesolid element and the patch are formulated as follows

For the solid element of Fig. 6:

tþDtkstructureT;e �x2me

h i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

8�8

/½ �|{z}8�1

¼ 0|{z}8�1

ð14Þ

For the I-ECP patch of Fig. 7a:

kI;e �kI;e

�kI;e kI;e þ tþDtkstructureT;e

" #�x2 0 0

0 me

� �( )|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

16�16

ue;out

ue;in

" #|fflfflfflfflffl{zfflfflfflfflffl}

16�1

¼ 0|{z}16�1

ð15Þ

Here, u, ue,out, and ue,in are the eigenmodes of the solid element andthe sub-vectors of the eigenmodes for the outer and the inner nodesof the patch in Fig. 7a, respectively. The stiffness matrix and themass matrix of the solid element are denoted by tkstructure

T;e and me,respectively. In Eq. (15), note that the mass matrix of the solid ele-ment is assembled for the degrees of freedom of the inner nodes.Further, the tangent stiffness matrix tkstructure

T;e is assembled for thedegrees of freedom of the inner nodes. Because four new nodesare added when constructing a patch in the I-ECP method, thereare eight additional modes between the inner and the outer nodesin addition to the eight eigenmodes of the solid element. As seenin Fig. 7c, the first four eigenvalues and eigenmodes of the outernodes become almost identical with the high stiffness value of thelinks.

In contrast to the case of links with a high stiffness value, highlylocalized vibrating modes inside the patch of Fig. 7a have beenfound as side effects when the stiffness value of links becomessmall. In other words, with a small stiffness value, the fundamentaleigenmode of the patch will be the motion observed between thesolid element and links inside the patch. As this phenomenon


can be observed for any patch with a small stiffness value, manyhighly localized modes will appear. Furthermore, because thetopology optimization formulation for dynamic structures usuallyinvolves the consideration of the fundamental eigenfrequency orsome of the lowest eigenfrequencies, these localized modes ofpatches are not desirable. Consequently, although the proposedmethod, which assigns the mass matrices to solid elements is sim-ple, it is not applicable to topology optimization.

2.3.2. Formulation 2: present mass matrix approachTo resolve the above-mentioned problems related to the local-

ized modes in patches, this paper presents a new concept assem-bling the mass matrices of solid elements in patches into thedegrees of freedom of the outer nodes, as shown in Fig. 5b. Unlikethe straightforward method, the solid element of the patch is mod-eled as massless, which makes the eigenmodes between the innerand the outer nodes numerically infinite. To formulate this patchmass matrix approach, Eq. (15) is modified as below.

For the patch mass matrix :

kI;e �kI;e

�kI;e kI;e þ tþDtkstructureT;e

" #�x2 me 0

0 0

� �( )ue;out

ue;in

" #¼ 0 ð16Þ

Fig. 10. Eigenfrequency histories for (a) a design with an initial uniform density

This approach has the following features.

(1) Compared to Eq. (15), the mass matrix of the solid element issimply assembled into the degrees of freedom of the outernodes, which makes it simple to implement.

(2) This simple modification of the mass matrix resolves thelocal modes observed between the inner and the outer nodescompletely. For example, Fig. 7b and c shows the result ofthe reanalysis of the eigenmodes using this patch matrixapproach, where four real eigenvalues and eight infiniteeigenvalues are existing.

(3) It is important to note that the local modes observed amongpatches, which correspond to local modes among void ele-ments in element-density-based approaches, still exist.Therefore, links with very small stiffness values can producehighly localized vibrating modes which cause non-conver-gence in topology optimization.

For the efficient construction of the present patch mass model, thefollowing condensation scheme is used for the global stiffnessmatrix and mass matrix:

ðkI;eue;out � kI;eue;inÞ �x2meue;out ¼ 0 ð17Þ

, and (b) an optimized design with an unsymmetrical density distribution.


� kI;eue;out þ kI;e þ tþDtkstructureT;e

� �ue;in ¼ 0 ð18Þ

tkðk�1ÞCon;e ¼ kI;e � kI;e kI;e þ tþDtkstructure;ðk�1Þ

T;e

� ��1kI;e

� �ð19Þ

tKCon ¼XNp

e¼1

tkðk�1ÞCon;e ;MCon ¼

XNp

e¼1

me ð20Þ

Finally, the following modal equation is solved for obtaining theouter nodes, Uout

½tKCon �x2MCon�Uout ¼ 0 ð21Þ

Fig. 11. Optimized results considering the geometrical nonlinearity. (a) A modelwith an arbitrary force, (b) with F = 10 N, and (c) with F = 20 N.

3. Topology optimization formulation

3.1. Material Interpolation

For the topology optimization of dynamic problems, the valuesof two mechanical material properties, i.e., link stiffness and mate-rial density, are interpolated with respect to the design variable(c). In this paper, the following interpolation function is used forthe link stiffness in Eq. (5) (see [32,35] for more detail):

le ¼ acen

1þ ð1� cne Þsþ b s ¼ a� s

kstructurediagonal � k

!ð22Þ

a ¼ lmax � lmin; b ¼ lmin ð23Þcmin � ce � 1; cmin ¼ 0:001 ð24Þ

where k is the number of degrees of freedom per node and s and nare penalties. A diagonal term of the linear stiffness matrix is de-noted by kstructure

diagonal . The upper and lower bounds of the stiffness oflinks are denoted by lmax and lmin, respectively. Numerical examplesindicate that with sufficiently large and small values for lmax andlmin and appropriate penalties, similar results can be obtained (see[35] for the effects of these links in linear static structure casesand Section 4 for numerical examples in dynamic structure cases).Here, lmax and lmin are set to 104 � kstructure

diagonal and 10�4 � kstructurediagonal ,

respectively. In all numerical examples, we assumed n = 3, k = 2,and s = 10 assumed that because optimal layouts similar to thoseobtained in the solid isotropic material with penalization (SIMP)method could be obtained. Fig. 8a shows the ratio of the condensedstiffness matrix to the normal stiffness.

The mass matrix can be interpolated as follows:

me ¼m0ce ð25Þ

where m0 is the nominal mass stiffness matrix. Numerical testsshow that this interpolation function causes highly localized vibra-tion modes inside I-ECP patches having links with low stiffness val-ues [12,26–28].When the present patch mass method is used, it isobserved that highly localized vibrating modes do not appear be-tween the inner and outer nodes because the solid plane elementinside an I-ECP patch only has stiffness and is massless, as shownin [16]. However, the localized modes continue to appear at the out-er nodes, which are locally vibrating when the design variables havelow values. That is because the ratio of the condensed stiffness ma-trix of the Eq. (19) to the mass matrix for the I-ECP patch becomesalmost zero, similar to the case of the SIMP method; this results inthe presence of localized vibration modes in patches with low linkstiffness values (Fig. 8). (See [12,14,26–29] for localized vibrationmodes and sensitivity analysis of the SIMP method.) To suppressthese local modes, some mass interpolation functions that havebeen verified in [12,14,25] are also tested here. By employing thefollowing material interpolations, the locally vibration modes canbe suppressed as expected

Material model 1 : me ¼m0ce; ce > 0:1m0c6

e ; ce � 0:1

ð26Þ

Material model 2 :

me ¼m0ce; ce > 0:1

m0ðc1c6e þ c2c7

e Þ; ce � 0:1; c1 ¼ 6� 105; c2 ¼ �5� 106

(

ð27Þ

3.2. Topology optimization formulation

For the sake of simplicity, we consider topology optimization tomaximize only the fundamental eigenfrequency frequency of ageometrically nonlinear structure, which is defined as

Maxc

Minj¼1;...;j

fxjg

s:t:XNp

e¼1

qeðceÞve � V�

RðtþDtUÞ� ¼ 0ðtþDtKTðtþDtU�Þ �x2MÞU ¼ 0

ð28Þ

where qe, ve and V* are the element density, element volume, andprescribed volume limit, respectively. The converged displacementsobtained by solving the nonlinear static equation are denoted byt+DtU*, and the number of candidate eigenfrequencies is denotedby J. To determine optimal topologies using a gradient-based opti-mizer, the sensitivity analysis of the jth eigenfrequency, xj, shouldbe performed with respect to the design variable. In this study, thefollowing sensitivity equation can be derived for the patch massmethod (see [26–28,38]):

dxj

dce¼ 1

2xjðue;out � ue;inÞ

T dle

dceðue;out � ue;inÞ �x2

j uTe;out

dme

dceue;out

� �ð29Þ

Fig. 12. Eigenfrequency histories for (a) a design with F = 10 N, and (b) a design with F = 20 N.

1 Up the best of our knowledge, the benchmark problem has been solved withoutconsidering a fixed mass by an evenly distributed initial design only [10,14].Consequently, it appears that the first eigenfrequency, whose mode is the bendingmode, is maximized. Therefore, from our numerical examples, we could obtain theunsymmetrical design shown in Fig. 9c with a randomly distributed initial guess.These empirical numerical tests imply that the optimization problem referred to hasmany local optima. Furthermore, eigenfrequencies and associated eigenmodes of afew initial iterations influence the solutions of the optimization problem.


4. Topology optimization of nonlinear dynamic problems

In order to illustrate the potential of the developed method,two-dimensional optimization problems with different loads areconsidered. Unless stated otherwise, uniform initial guesses of csatisfying given mass constraints are used. The dimension, materialproperties and magnitude of the loads are arbitrarily chosen toshow the effectiveness of the present I-ECP method. For successfuloptimization, it is important to use a proper optimization algo-rithm such as sequential linear programming, sequential quadraticprogramming, and the method of moving asymptotes. In thisstudy, the method of moving asymptotes is used as an optimiza-tion algorithm; however other optimization algorithms can alsobe used [39]. To get rid of checkerboard patterns inside the designdomain, there are many regulation methods such as mesh-inde-pendent filtering, slope-constraint, density-filtering, and morphol-ogy filtering. In this paper, the mesh-independent sensitivityfiltering is used (see Ref. in [40] for more details).

4.1. Example 1: cantilever beam

A cantilever beam with clamped ends (Fig. 9a) is consideredfirst. A finite element model of the design domain is constructedby using 5000 (200 � 25) patches containing bilinear Q4 elements.The volume considered is constrained to be less than 50% of the do-main volume. Fig. 9b and c, and Fig. 10 show topological layouts

obtained by the proposed ECP method, and iteration histories.With a zero force (F = 0 N), the topological layout in Fig. 9b, whichis comparable to the layouts presented in [14,15] using the SIMPmethod, could be obtained using a constant initial guess. Resultsof empirical tests revealed that another local optimum designcould be obtained using an unsymmetrical initial guess in Fig. 9c.1

To consider the effect of large displacements on the optimal lay-outs, a concentrated point load is now applied at the center of thebottom surface of the beam in the downward direction (Fig. 11a).With sufficiently large loads (F = 10 N and 20 N) that elicit nonlin-ear responses in the beam, the unsymmetrical designs in Fig. 11band c, whose eigenfrequency histories are shown in Fig. 12, couldbe obtained. Table 1 and Fig. 13 compare the eigenfrequencies ofthe four solutions with respect to the applied loads.

From the analysis of the numerical results presented in Fig. 13,it is observed that the optimal layouts considering the geometricalnonlinearity in Fig. 11 are optimized for the given loads. AtF = 10 N, the fundamental eigenfrequency of the nonlinear design

Table 1Eigenfrequencies of optimized designs for the clamped beam.

Design Fig. 9b Fig. 9c Fig. 11b Fig. 11c

Linear 223.06 (rad/s) 176.26 (rad/s) 256.99 (rad/s) 241.72 (rad/s)

Loading 1 (F = 10 N) 225.86 (rad/s) 189.92 (rad/s) 304.33 (rad/s) 297.24 (rad/s)

Loading 2 (F = 20 N) 222.63 (rad/s) 196.57 (rad/s) 322.41 (rad/s) 343.16 (rad/s)

Fig. 13. Load and eigenfrequency curves for the optimized results for the clamped beam.

Fig. 14. Numerical tests for examining the effects of link stiffness values. (The values in parentheses angular speed.)

ig. 15. Beam structure with a point load. (a) Problem definition, and (b) result 1 (anear optimized result).


of Fig. 11b, optimized for F = 10 N, are better than that of the de-sign of Fig. 11c, optimized for F = 20 N, whereas the eigenfrequencyof Fig. 11c is better than that of Fig. 11b at F = 20 N. Furthermore, alarger force (F = 20 N) produces a straight beam whose length islonger than the length of a straight beam of Fig. 11b. This exampleshows that a physically sound structure can be obtained using thepresent ECP method.

Using Fig. 14 that shows numerical test results with differentupper and lower bounds for the link stiffness values, similar localoptimized results can be obtained in the case of both sufficientlylarge and sufficiently small link stiffness values (see [35]).

4.2. Example 2: beam structure with a point load or bending load

In the second example, a beam with clamped boundaryconditions is considered (Fig. 15a). The design domain is discret-ized by 200 � 30 patches. To avoid obtaining a trivial void struc-ture, the densities of two elements at the loading point are fixedas 20 kg/m3. The volume is also constrained to be less than 50%of the design domain. Fig. 15b shows an obtained linear designwith a uniform initial density distribution for F = 0 N. It is foundthat this result is similar to the results obtained using the SIMPmethod too [14]. Subsequently, optimization problems with differ-ent loads were considered (Fig. 16). The layouts of the results max-imizing the first eigenfrequency in Fig. 16 are, in a certain sense,

equivalent to those layouts obtained by solving the static compli-ance minimization problem (see Ref. [19] for more details). Thismight be explained by the fundamental structural eigenfrequencybeing proportional to the square root of the global stiffness.Fig. 16d plots the first frequencies against the applied loads foreach design. After exceeding near F = 0.5 N, the lower beam in

Fli

Fig. 16. Optimized results for point loading. (a) Problem definition, (b) result 2 (aresult with a point load of 1.5 N), (c) result 3 (a result with a point load of 5 N), and(d) the curves of eigenfrequencies with respect to the applied point load. Fig. 17. Optimized results for the bending loading case. (a) Problem definition, (b)

result 2 (result for a bending moment of 1.25 Nm), (c) result 3 (a result for abending moment of 2.5 Nm), and (d) the curves of eigenfrequencies with respect tothe applied bending moment.


Fig. 15b exhibits the buckling phenomenon. Consequently, the fun-damental structural eigenfrequency of Fig. 15b decreases. In con-trast, because the nonlinear designs of Fig. 16b and c can supportlarger forces without bucklings, their frequencies do not decreaseat the given loads. It was observed that small density perturbationsin the linear design of Fig. 15b can cause bucklings of the internal

Table 2Eigenfrequencies of optimized designs for a cantilever beam subject to a point load.

Design Fig. 15b

Linear 8.43 (rad/s)

Shear (F = 1.5 N) N/A (buckling of solid elements)

Shear (F = 5 N) N/A (buckling of solid elements)

bar structure. Therefore, investigations on methods to controlstructural behaviors after post-buckling and eigenfrequencyshould be carried out in future. Table 2 compares the eigenfre-quencies of the present designs.

Fig. 16b Fig. 16c

8.25 (rad/s) 7.74 (rad/s)

8.12 (rad/s) 7.77 (rad/s)

3.83 (rad/s) 7.87 (rad/s)

Table 3Eigenfrequencies of optimized designs for a cantilever beam subject to a bending moment.

Design Fig. 15b Fig. 17b Fig. 17c

Linear 8.43 (rad/s) 6.61 (rad/s) 6.81 (rad/s)

Bending (BM = 1.25 Nm) N/A (buckling of solid elements) 8.51 (rad/s) 8.28 (rad/s)

Bending (BM = 2.5 Nm) N/A (buckling of solid elements) 8.46 (rad/s) 8.99 (rad/s)


In addition to the point load, changes in the layout designs withthe bending moments were investigated (Fig. 17). Similar to thecase of the point loads, a large bending moment makes the left-end-lower beam prone to buckling as shown in Fig. 17d. Conse-quently, the first structural eigenfrequency of the linear design ofFig. 15b is decreasing with respect to the magnitude of bendingmoment. Considering the geometrical nonlinearity, different topo-logical layouts, which make the left-end parts stiffer and can sup-port given larger moments, can be obtained in Fig. 17. Theeigenfrequencies for these designs are given in Fig. 17d and Table3. An investigation of the deformed shapes indicates that thecurled left-end-upper beam of Fig. 17 becomes straight for the gi-ven bending moment, thereby increasing the overall stiffness. Thisexample also shows that the present I-ECP method can be used todesign for nonlinear dynamic structures.

5. Conclusions

This paper pertains to topology optimization using the internalelement connectivity parameterization (I-ECP) method in order tomaximize the first eigenfrequency of a geometrically nonlinearstructure. To calculate the eigenfrequencies of a nonlinear struc-ture, modal analysis should be conducted after nonlinear staticanalysis using the Newton–Raphson method. However, in theframework of the standard density-based method, the modal andstatic analysis of nonlinear structures are hindered due to theinstability of flipped elements among weak elements and the pres-ence of locally vibrating modes. To the best of our knowledge, ow-ing to these difficulties, there has not been any research ontopology optimization in the context of nonlinear dynamicstructures.

To overcome the shortcomings of density-based methods, thispaper proposes the use of the ECP method. This method was devel-oped for compliance minimization problems involving large dis-placements, yet it was not applied to dynamic problems becauseof the ambiguity in the mass model. To examine whether distrib-uted mass models can be constructed using the ECP method, thispaper investigates the construction of two computational massmodels. One mass model relates the mass matrix, formulated bythe finite element method, to the degrees of freedom of a solid ele-ment inside a patch, while the other relates the same mass matrixto the degrees of the freedom of outer nodes forming a patch.Showing good agreement with each other for a solid element witha high stiffness value for links, the first mass model with a smallstiffness value for links inherently has highly localized vibratingmodes inside patches which can be troublesome on topology opti-mization. Therefore, this paper employs the second mass model(the patch mass modeling), which is relatively free from localmodes, in preference to the direct method shown in an examplewith one element (Fig. 7).

To illustrate the potential of the proposed approach, twonumerical examples with sufficiently larger loads that elicit non-linear responses from the structure considered are presented. Forthe sake of simplicity in optimization and sensitivity analysis, onlythe first eigenfrequency is considered. The optimized results showthat it is possible to solve topology optimization problems by con-

sidering a geometrically nonlinear structure and using the I-ECPmethod. Moreover, we verified that results similar to those ob-tained in the case of the compliance minimization problem canbe obtained by maximizing the fundamental structuraleigenfrequency.

It is suggested that in future studies, the proposed approach canbe extended to layout designs by considering the post bucklingphenomena, the crashworthiness, and other dynamic aspects alongwith material nonlinearities. A study should also be performed tocompare the proposed ECP method and the discontinuous Galerkinmethod may be required.

Acknowledgement

This research was supported by the Grant of the Korean Minis-try of Education, Science and Technology – The Regional Core Re-search Program and by the Kyungpook National UniversityResearch Fund, 2008.

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