International Journal of Solids and Structures 48 (2011) 775–784

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsols t r

Definition and updating of simplified models of joint stiffness

F. Gant a,⇑, Ph. Rouch b, F. Louf a, L. Champaney a

a LMT-Cachan, 61 Avenue du Président Wilson, F-94230 Cachan, Franceb Laboratoire de Biomécanique, 151 Boulevard de l’Hôpital, 75013 Paris, France

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

Article history:Received 15 July 2010Received in revised form 4 November 2010Available online 12 November 2010

Keywords:Model updatingConstitutive relation errorAeronautical joint modelingFastenerSimple model

0020-7683/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijsolstr.2010.11.011

⇑ Corresponding author. Tel.: +33 1 47405983; fax:E-mail addresses: gant@lmt.ens-cachan.fr, floren

Gant), philippe.rouch@ensam.eu (Ph. Rouch), louf@champaney@lmt.ens-cachan.fr (L. Champaney).

The objective of this work is to define a simple linear model of joints used in aeronautics and to updatethis model efficiently.

Industrial designers usually resort to semi-empirical linear joint models to represent the behavior ofthe joints of a large aeronautical structure. Here, we propose to develop a one-dimensional linear jointmodel which is capable of representing the behavior of every joint of a large structure globally whileenabling local nonlinear reanalysis of the most highly loaded joints. Work on nonlinear reanalysis isnot considered in this paper.

In order to solve the numerical difficulties encountered in some of modeling situations, an updatingstrategy based on the constitutive relation error is proposed. Since the updating efficiency is significantlyaffected by the ratios of the stiffnesses of the different parts of the model, the strategy consists in rigid-ifying some parts of the model in order to control the updating accuracy and the rate of convergence. Thenumerical results of a standard model and a rigidified model illustrate the updating improvementsallowed by the strategy.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Joints are often used in aeronautics because they make theassembly and maintenance tasks easier for the manufacturers.However, joint properties such as machining quality, friction orpreloading are hard to control during manufacturing and lead todifferences in behavior from one fastener to another. These irregu-larities can induce overloading and failure at joint locations duringthe global loading of the structure.

Therefore, the representation of the actual behavior of a singlejoint or a set of joints is a real issue in structural mechanics. Froma computational mechanics point of view, joints create a dilemma.When dealing with large structures such as aircraft, the joints aretoo small and too numerous for each to be modeled with a detailed3D geometry: an Airbus aircraft uses more than one million boltsand several million rivets. However, no realistic simulation canbe undertaken without taking them into account. Due to thesecomputational limits, industrial design is usually carried outemploying a two-scale method. First, simulations using a linearrepresentation of the global structural level are performed. The lin-ear modeling consists of shell or plate elements (representing thestructural parts of the aircraft) connected by various types of

ll rights reserved.

+33 1 47402240.t.gant@lmt.ens-cachan.fr (F.lmt.ens-cachan.fr (F. Louf),

springs (representing the joints). Most of the joint representationsare based on semi-empirical models (Huth, 1986; Tate and Rosen-feld, 1946). This first simulation provides an estimation of the dis-tribution of the joint loads in the structure. Finally, the most loadedjoints are identified and local reanalyses are performed with 3Dnonlinear modelings in order to verify damage criteria. Fig. 1 illus-trates the design method. Usually, uncertain representation andsensitivity analysis are also carried out. At the global levels, animportant effort on the quality of the estimated distribution ofthe joint loads must be made. In the present paper, only the firstpart of the design, i.e. the estimation of a reliable distribution ofloads employing a linear modeling, is considered.

The earliest works on joint modeling in aeronautics were devel-oped for static loading using semi-empirical models based onsprings (Huth, 1986; Tate and Rosenfeld, 1946). Simple analyticaljoint models were developed (McCarthy et al., 2006; Yen, 1978)along with one-dimensional FEM models (Baumann, 1982; Ekhand Schön, 2008) and multi-dimensional representations (Bortmanand Szabó, 1992; Champaney et al., 2008; Chen et al., 1995; IngvarEriksson, 1986; Izumi et al., 2005; Kelly, 2005; McCarthy et al.,2005). A comparison of 4 types of FEM joint models was presentedin (Kim et al., 2007). Techniques have also been developed fordynamics (Segalman et al., 2003), where joints play a crucial roleas dampening elements of the structure.

When many fasteners are used, industrial designers usuallyprefer 1D joint representations on the global structural level. Oneof the main objectives is to obtain the loading conditions (Wei-Xun

Fig. 1. Illustration of the design method.

776 F. Gant et al. / International Journal of Solids and Structures 48 (2011) 775–784

and Chun-Tu, 1993) of the most highly loaded joint in order to per-form a local nonlinear reanalysis of that fastener later. Designersusually consider the structure as completely known, and representthe joints with a connector tied to the structure with only twonodes. The simplicity of the 1D model has several advantages. Sucha model introduces no additional degree of freedom and the loadtransfer at the joints can be analyzed easily thanks to the one-dimensional representation. These techniques are very efficientin dealing with incompatible meshes for the different structuralparts because the nodes of the joint elements rarely match thoseof the structural elements. Many types of elements have alreadybeen studied in the automotive industry for spot-welded struc-tures in the context of car crash analysis (Xiang et al., 2006).Nonlinear spring and beam elements including elastic–plasticand damage behavior were developed in Combescure et al.(2003) and Langrand and Combescure (2004). Thanks to the repre-sentation of the complex behavior of a joint by a 1D model, theerror compared to reality is globalized, which simplifies theinterpretation of the analysis. Finally, these joint models can beupdated in order to reduce the error in the overall behavior ofthe structure.

With such modeling approximations, the identification of theparameters of a joint for an actual structure including nonlinearbehavior is a real challenge. The choices of the mathematical mod-el, of the updating method and of the experimental data are crucialand greatly affect the quality of the updated model. In order to im-prove the quality of the simulated distribution of joint loads com-pared to the real structure, an updating process can be performedthanks to experimental data achieved on some points of the realstructure. The modeling of joints with 1D linear springs generallyleads to high values of spring stiffness compared to the plate stiff-ness. Then, difficulties can be observed due to this stiffness ratiowhen dealing with updatings of spring stiffnesses.

The objective of model updating methods is to minimize thedistance between the behavior of the real structure and that of asimulated structure by modifying the numerical model. Recentworks on identification of joint parameters in dynamics were pre-sented in (Jalali et al., 2007). A review of model updating methodswas given in (Mottershead and Friswell, 1993). A first group ofmethods, called ‘‘direct methods’’, consists in modifying the struc-tural matrices directly with no physical rationale. Some examplesin this group are the methods based on minimum norm corrections(Baruch, 1982; Berman and Nagy, 1983) and control theory (Kaoukand Zimmerman, 1994; Zimmerman and Kaouk, 1992). The other

group of methods, called ‘‘indirect’’ or ‘‘parametric’’ methods, isbased on model updating through the correction of some physicalparameters using three possible approaches: input residuals (Far-hat and Hemez, 1993), output residuals (Lammens et al., 1995;Piranda et al., 1991), and the constitutive relation error (Ladevèzeand Chouaki, 1999; Mottershead and Friswell, 1993).

In the present study, a double-lap multiple joint related to atypical aeronautical joint is considered. The fasteners are modeledusing a 1D linear spring model in order to enable the numericalextension of this work to more complex structures. The objectiveof this work is to analyze and improve the updating process for thistype of structural model. The first section presents the updatingapproach based on the constitutive relation error method chosenfor this work. Then, the properties of this updating are analyzedfor a 1D model of the multiple joint. This first spring modeling isgenerally used in aeronautical pre-design. In some cases, difficul-ties occur and decrease the updating efficiency. Finally, an adjust-ment to the model which consists in rigidifying some parts of thestructure is considered in order to improve the updating proper-ties. Updating processes are performed using simulated experi-mental data of joint loads. The updating robustness of thedifferent joint modelings is tested with data simulated from differ-ent linear models.

2. Model updating

In this section, bolted or riveted joints are represented bynumerical models in which each fastener is a spring. This springrepresentation leads one to choose an updating method based ona parametric approach: the stiffnesses of the springs are unknowna priori and these are the parameters to be updated. Updatingbased on the constitutive relation error was developed for freeand forced vibration analysis (Ladevèze and Chouaki, 1999) and,by introducing the dissipation error into the updating technique,for material or contact nonlinearities (Chouaki et al., 2000).Recently, work has also been done on the localization and sizingof multiple cracks (Faverjon and Sinou, 2008) and on updatingbased on uncertain measurements (Faverjon et al., 2009). Here, thismethod is applied to the static problem of a structure with multi-ple fastened joints loaded in tension. Updating is performed usingexperimental or simulated data (displacements or loads) corre-sponding to sensors located at different points of the structure.Fig. 3 shows the geometry of the problem being considered.

F. Gant et al. / International Journal of Solids and Structures 48 (2011) 775–784 777

2.1. The reference problem

A standard finite element technique is used to solve the prob-lem and model updating is applied directly to the discretized rep-resentation. Since the updating procedure uses the constitutiverelation error, only the main principles will be discussed. Moredetails can be found in (Chouaki et al., 1998). Finite element fieldvectors are denoted {�} and operators are denoted [�]. Let us con-sider an approximate structure Xh subjected to prescribed bodyforces {fd} within Xh. At the boundary oXh of the structure, dis-placements {Ud} and forces {Fd} are prescribed over @1Xh and@2Xh, respectively. The reference problem consists in finding thedisplacement field {U} and the stress field [r] which verify a setof equations divided into two groups:

� The reliable equations:– the kinematic constraints;– the equilibrium equations.� The less reliable equations:

– the constitutive equations.

H is Hooke’s tensor, which is positive definite and symmetrical.The uniqueness of the solution is assumed to be guaranteed byDrucker’s stability conditions. In order to define the constitutiverelation error, we introduce the pair s({U}, [r]) composed of a‘‘static’’ quantity [r], denoted [rs], and a ‘‘kinematic’’ quantity{U}, denoted {Uc}. The problem becomes:

Find s 2 Sad

which minimizes g2ðs0Þ; where s0 2 Sad

���� ð1Þ

The constitutive relation error g2 applied on a linear model is de-fined as:

g2ðfUg; fVgÞ ¼ 12fU � VgT ½K�fU � Vg ð2Þ

where [K] is the stiffness matrix of the problem and {V} is definedby [rs] = H(e({V})). The admissibility of s means that s verifies thereliable equations. The minimization of g2 leads to the closest pos-sible verification of the less reliable equations.

2.2. Experimental data

This initial model is updated using data which are usuallyobtained from experiments on a real structure or virtual testingon complex modelings. Here, the data being considered are the dis-placements and loads transferred by the bolts. These experimentaldata are divided into two groups:

� The reliable data:– the locations of the sensors;– the directions of the sensors.� The less reliable data:

– the magnitudes feUg and feQ g of the measured displacementsand loads.

This categorization of the data will be used in this example, butcan vary depending on the problem being treated. The appliedforces are considered to be perfectly known and are not updated.

2.3. The modified constitutive relation error

In order to take into account the information given by theexperimental data, we define the modified constitutive relation er-ror as:

e2ðsÞ ¼ g2ðsÞ þ r1

1� r1PufUg � feUg��� ���2

þ r2

1� r2PqfQg � feQ g��� ���2

ð3Þ

where the added terms represent the square of a distance to the lessreliable measured quantities. The ri weighting coefficients are firstused to balance the weight of the error on measurements comparedto the constitutive relation error. Then, the coefficients are adjusteddepending on one’s degree of confidence in the measurements. Pu

and Pq are Boolean projection operators. Thus, the problem canbe reformulated as:

Find s 2 Sad

which minimizes e2ðs0Þ; where s0 2 Sad

���� ð4Þ

The admissibility of s means that s verifies the reliable equations.The minimization of e2 leads to the closest possible verification ofthe less reliable equations and measurements. The modified consti-tutive relation error becomes:

e2ðfUg; fVgÞ ¼ 12fU � VgT ½K�fU � Vg

þ r1

1� r1PuU � eUn oT

½Gu� PuU � eUn oþ r2

1� r2PqQ � eQn oT

½Gq� PqQ � eQn oð5Þ

Matrices [Gu] and [Gq] are used to represent the error in the mea-surements. These matrices are chosen such that their terms havethe same energy dimension as g2. Vector {Q} can be expressed as{Q} = [B]{U}, [B] being the extraction operator of the load vectors.The following expression is chosen:

½Gu� ¼ ½K� ð6Þ

The choice of [Gq] and ri will be explained in the next section con-cerning the updating application. Thanks to the stationarity of theLagrangian L, the minimization of the error can be reformulated as:

L ¼ e2ðfUg; fVgÞ þ fkgTð½K�fVg � fFgÞ ð7Þ

where {F} and {k} represent, respectively the vector of the nodal val-ues of the prescribed force fields and the vector of the Lagrangianmultipliers. The term {k}T([K]{V} � {F}) was added to make {V} stat-ically admissible. This formulation leads to the resolution of a crit-ical point problem. Expressing the stationarity of the Lagrangian L,the problem can be solved with the following simultaneousequations:

½K� þ r1

1� r1½Gu� þ

r2

1� r2½Gq�

� �fUg

¼ fFg þ r1

1� r1½Gu� eUn o

þ r2

1� r2½Gq� eQn o

ð8Þ

fkg ¼ fU � Vg ð9Þ

½K�fVg ¼ fFg ð10Þ

2.4. Minimization of the error

In order to solve the critical point problem, the gradient of theerror must be calculated. One can observe that L({U}, {V}) =e2({U}, {V}) when ({U}, {V}) is the solution of the minimization prob-lem. Hence:

de2

dci¼ dL

dcið11Þ

αk = x

e2 (x),

y ε(x) e2(x)

y1(x)

yε(x)

y0(x)

f(0)

Domain of validity of Armijo rule

0

Fig. 2. Illustration of Armijo rule.

778 F. Gant et al. / International Journal of Solids and Structures 48 (2011) 775–784

ci being the parameters of the model. This leads to the followingexpression:

dLdci¼ @L@fUg

@fUg@ci

þ @L@fVg

@fVg@ciþ @L@fkg

@fkg@ciþ @L@ci

ð12Þ

Since ({U}, {V}, {k}) is the solution of the minimization problem, itleads to the stationarity of the Lagrangian:

@L@fUg ¼

@L@fVg ¼

@L@fkg ¼ 0 ð13Þ

Finally, the gradient of the error can be calculated directly using thepartial derivative of the Lagrangian:

de2

dci¼ @L@ci

ð14Þ

Introducing (9), Expression (14) can be rewritten as:

de2

dciðfUg; fVgÞ ¼ 1

2fU � VgT @½K�

@cifU þ Vg

þ r1

1� r1PuU � eUn oT @½Gu�

@ciPuU � eUn o

þ r2

1� r2Pq½B�U � eQn oT @½Gq�

@ciPq½B�U � eQn o

þ 2r2

1� r2Pq

@½B�@ci

U� �T

½Gq� Pq½B�U � eQn oð15Þ

2.5. The minimization algorithm

Now, the gradient expression can be used to find a minimum ofthe constitutive relation error using a minimization algorithm. Agradient descent is performed and optimized with a line searchstrategy based on the Armijo rule (Armijo, 1966). At each step k

Fig. 3. Geometry of

of the gradient descent, the optimization consists in finding ak suchas {ck+1} minimize e2({ck+1}) with:

fckþ1g ¼ fckg � akfre2ðfckgÞg and fre2ðfckgÞg ¼

@e2

@c1ðfckgÞ

..

.

@e2

@cnðfckgÞ

8>>><>>>:

9>>>=>>>;ð16Þ

where {ck} is the list of parameters ci known at step k. By defining atstep k:

f : x # e2ðfckg � xfre2ðfckgÞgÞ ð17Þ

and the functions ye:

yeðxÞ ¼ f ð0Þ þ exf 0ð0Þ ð18Þ

the optimum ak is chosen as the first calculated x that verifies:

yeðxÞ > f ðxÞ ð19Þ

Fig. 2 illustrates the application of Armijo rule at a step k. A commonvalue of e = 0.5 is chosen in the further applications of the updatingmethod.

A stopping criterion is usually defined based on a relative errorchosen as:

e2r ¼

e2

12 fU0 þ V0gT ½K0�fU0 þ V0g

ð20Þ

where {U0}, {V0} and [K0] denote, respectively the vectors {U} and{V} and operator [K] at the first iteration of the algorithm.

3. Updating example

3.1. Geometry

The updating process described previously was applied to adouble-lap joint loaded in tension. This joint consisted of three alu-minum plates fastened by 4 titanium joints. It represents a typicalstructure of joint assemblies employed in aeronautical applications(Champaney et al., 2008). Fig. 3 describes the configuration of theproblem and Table 1 lists the parameters.

3.2. The one-dimensional model

This first model (Fig. 4) was based on the representation of eachelement by a 1D structure. The structure was clamped at one endand loaded in tension at the other end. Due to symmetry proper-ties, only half of the actual structure was represented. The modelhad 9 degrees of freedom (the displacements ui). Springs and barelements were used for the joints and the plates, respectively.The parameters of the model are given in Table 2. We will referto the model presented in Fig. 4 of this section as the ‘‘first model’’.A model which is more suitable for updating, designated as the‘‘rigidified model’’, will be considered in Section 4.

the structure.

Table 1The parameters of the problem.

Parameter Value

Young’s modulus 70 GPa (aluminum)Cross section of the outer plates 2.063 � 10�4 m2

Cross section of the middle plate 3.968 � 10�4 m2

Length L0 176.2 mmLength L 47.6 mmLoad F 5000 N

Table 2The parameters of the model.

Parameter Value

Young’s modulus 70 GPa (aluminum)Upper bar cross section 4.126 � 10�4 m2 (2 � 2.063 � 10�4 m2)Lower bar cross section 3.968 � 10�4 m2

Length L0 176.2 mmLength L 47.6 mmLoad F 5000 N

Table 3The measurements used for the updating.

Set 1 Set 2

Chosen values (N m�1) ki 1.932 � 108 1.932 � 1010

Load 1 1533 2414Measurements obtained (N and m) Load 2 937 36

Load 3 950 38Load 4 1579 2510Displacement 8.29 � 10�5 7.49 � 10�5

F. Gant et al. / International Journal of Solids and Structures 48 (2011) 775–784 779

3.3. Construction of the experimental data

A numerical model was used to simulate the virtual set ofexperimental data to be used to update the model. In order tostudy the robustness of the updating process with respect to therelative stiffnesses, two sets of experimental data were derivedfrom the model presented above using two sets of spring stiffness-es: a first set with the stiffnesses of the springs equivalent to thestiffnesses of the bars, and a second set with the stiffnesses ofthe springs about 100 times greater. The stiffness values of Set 1(of the same order of magnitude as those of the bar elements) wereobtained using Huth’s formula (Huth, 1986). The measured datawere the loads (Load i) transferred by each spring (i), plus the dis-placement u9 shown in Fig. 4. Table 3 summarizes the parameterschosen and the data obtained.

The results show that the greater the difference between thestiffnesses, the larger the difference between the loads transferredby the inner and the outer bolts. The stiffnesses of the bar elementswere approximately 1 � 108 N m�1.

3.4. Application of the updating process and results

Two initial states were chosen for the initialization of theupdating process: a first state with values of the spring stiffnessessmaller than the expected resulting values (target values of theupdating process), and a second state with values greater thanthe expected values. The parameters to update are the stiffnesseski. In Eq. (5), {Q} was expressed as:

fQg ¼

Q 1

Q 2

Q 3

Q 4

8>>><>>>:

9>>>=>>>; ð21Þ

where Qi denotes the load transferred by Fastener i. In order toachieve a homogeneous error, matrix [Gq] in Eq. (15) was chosen as:

½Gq� ¼

1=k1 0 0 00 1=k2 0 00 0 1=k3 00 0 0 1=k4

26664

37775 ð22Þ

In this case, the parameters ri affect the rate of convergence and theaccuracy of the results of the updating process. An optimum value

Fig. 4. Schematic vie

was chosen for this modeling by repeating the updating sequenceseveral times with different trial values of ri and comparing the con-vergence properties. Tested values of ri are ri = 1 � 10k withk = {�12,�11, . . . ,2,3}. Thanks to these trials, a compromise leadsto set the parameters ri to r1 = r2 = 1 � 10�8 for each updating ofthe problem treated in this section.

The results of Fasteners 1 and 4 were almost identical, as werethose of Fasteners 2 and 3. Therefore, only the results of Fasteners1 and 3 will be presented in the following figures.

3.4.1. Updating of the first model based on Set 1The updating was performed with the bolt stiffnesses initialized

at 1.932 � 104 N m�1 and 1.932 � 109 N m�1. Figs. 5–8 show, respec-tively the evolutions of the relative error, of the spring stiffnesses,of the displacement and of the bolt load during the updating. Fig. 5presents the error evolution of 3 different trials of ri values.

3.4.2. Updating of the first model based on Set 2The updating was performed with the bolt stiffnesses initialized

at 1.932 � 104 N m�1 and 1.932 � 1011 N m�1. Figs. 9 and 10 show,respectively the evolutions of the relative error and of the springstiffness during the updating.

3.4.3. Discussion of the first modelAfter each of these four updating steps, the updated quantities

of interest appear to have approached the ‘‘experimental data’’ cor-rectly. Contrary to Set 1, the updating based on Set 2 seems to havefailed to achieve the expected connector stiffnesses. A furtherstudy of the effects of parameters ri confirms that it only affectsthe rate of convergence of the updating. The tested values areri = 1 � 10k with k = {�12, . . . ,�4}. So the shape of the data

w of the model.

0 20 40 60 80 100 12010−12

10−10

10−8

10−6

10−4

10−2

Iteration

Rel

ativ

e er

ror

Lower initializationUpper initialization

Fig. 5. Evolutions of the relative error during the updating.

0 20 40 60 80 100 120104

105

106

107

108

109

1010

Iteration

Stiff

ness

(N/m

)

k1: lower initialization

k3: lower initialization

k1: upper initialization

k3: upper initialization

Stiffness: target

Fig. 6. Evolutions of the stiffness of Fasteners 1 and 3 during the updating.

0 20 40 60 80 100 12010−5

10−4

10−3

10−2

10−1

Iteration

Dis

plac

emen

t (m

)

Updated u; lower initializationUpdated u; upper initializationDisplacement: target

Fig. 7. Evolutions of the displacement during the updating.

0 20 40 60 80 100 120200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Iteration

Load

(N)

Load 1: lower initializationLoad 3: lower initializationLoad 1: upper initializationLoad 3: upper initializationLoad 1: targetLoad 3: target

Fig. 8. Evolutions of the fastener load during the updating.

0 200 400 600 800 1000 120010−12

10−10

10−8

10−6

10−4

10−2

Iteration

Rel

ativ

e er

ror

Lower initializationUpper initialization

Fig. 9. Evolutions of the relative error during the updating.

0 200 400 600 800 1000 1200104

106

108

1010

1012

Iteration

Stiff

ness

(N/m

)

k1: lower initialization

k3: lower initialization

Stiffness: targetk1: upper initialization

k3: upper initialization

Fig. 10. Evolutions of the stiffness of Fasteners 1 and 3 during the updating.

780 F. Gant et al. / International Journal of Solids and Structures 48 (2011) 775–784

evolutions remains the same: the connector stiffnesses are still notreached. In addition, for both sets, the convergence rate appears tobe higher when the updating is initialized with lower values. Table4 details the number of iterations for each case of updating. Thesefigures appear to be too large for such a quite simple structure.

One can assume that the weak sensibility of the connector stiff-nesses on the modified constitutive relation error leads to the lackof efficiency of the updating. In the following section, a parametricstudy is performed to illustrate the sensibility of the connectorstiffnesses with the first model, and then verify this previous

Table 4First model: total iterations of the updatings.

Updating case Set 1 Set 1 Set 2 Set 2

Lower Init. Upper Init. Lower Init. Upper Init.

Iterations 43 111 1107 93

Fig. 11. Color map of the modified constitutive relation error along with theconvergence paths using measurement Set 2 and the first model.

F. Gant et al. / International Journal of Solids and Structures 48 (2011) 775–784 781

assumption. Fig. 11 shows that k2�3 does not affect the error. Thisobservation is due to the high value of the target stiffness of k2 andk3 compared to the stiffness of the structure. The raised issue is ajoint modeling problem.

4. The proposed rigidified model

Some difficulties arose in the updating of the previous modelbased on Set 2 in terms of convergence toward accurate values ofthe quantities of interest and updated spring stiffnesses. In orderto analyze the updating behavior, bolt sets {1,4} and {2,3} were as-sumed to evolve identically. With this assumption, it was possibleto plot the constitutive relation error surface along with the con-vergence paths (Fig. 11). In Fig. 11, one can observe a line of globalminima with almost the same value. This error surface shape ex-plains why the algorithm experienced difficulties (large numberof iterations) in finding the actual global minimum. Therefore,we developed a second model in order to control the shape ofthe surface. The objective is to soften the structures of the updatedparameters and, thus, end up with a faster and more accurateupdating process. The updating difficulties leaded by the weak sen-sibility of some parameters are independent of the updating meth-od. Consequently, in the following work, improvements arepresented on the joint model, while the updating method doesnot change.

Fig. 12. Configuration of

4.1. The one-dimensional rigidified model

In modeling jointed structures, it is usually assumed that allparts of the structure, except the joints, are fully known. The firstmodeling uses quite local properties regarding the definition ofconnection between the joint and the structure. Therefore, the areaof interaction between the joint and the structure can be enlargedin order to include the effects of the geometry and the behavior ofthe connector (e.g. the bolt hole, the behavior of the contact areaaround the joint, . . . ) in the joint behavior. Finally, when dealingwith updatings on real structures, the stiffness of the joint modelto update is supposed to represent the behavior of the real fastenerincluding its surrounding structure. We assumed that with the firstmodel the efficiency of the updating algorithm is affected by theratio of the bar stiffnesses to the spring stiffnesses. We modifiedthe model in order to vary this ratio and evaluate its effect onthe updating. This second representation was based on the previ-ous model, except that the spring was used to represent not onlythe elasticity of the fastener, but also that of part of the surround-ing structure. The modified model is shown in Fig. 12, with hatchedzones representing the rigidified structure. Adding part of thestructure to the spring element decreases the overall stiffness ofthe set {fastener + part of the structure}. This enables the stiffness-es of the bar elements and of the springs to be adjusted usingparameter p in order to avoid excessive differences between them.This parameter corresponds to the proportion of the initial struc-ture which is not included in the spring representation. The newparameters to update are the stiffnesses k0i.

The application on the 1D model of multiple joints leads tomodify simply the length of the bar elements. An application ona more complex structure with much more degrees of freedomwould lead to rigidifying the elements located in the area repre-sented by hatched zones in Fig. 12.

A parametric analysis of the modified constitutive relation errorwas carried out for Set 2. Fig. 13 uses level sets to show the evolu-tion of the location of the global minimum of the error as a func-tion of p.

When p tends to 1 (no rigid part), the location of the global min-imum becomes less accurate. With p > 0.9 and high values of k02�3,the level sets are parallel and vertical. This means that in that zoneparameter k02�3 does not affect the error. In the same situation, butwith p = 1, updating difficulties were observed. Therefore, p = 0.2was chosen and used for all subsequent updates. Table 2 showsthe detailed values of the parameters used for that model. Theparameters to be updated were the spring stiffnesses k0i.

4.2. Application of the updating process

The same model as before was used to simulate the experimen-tal data. The detailed values used are shown in Table 3. Again,updating was carried out using both lower values and upper valuesof the spring stiffnesses. Parameters ri are chosen equal tor1 = r2 = 1 � 10�10. The objective of this modification of the firstmodel is to soften the representation of the joint in the structure.

the rigidified model.

108 109 1010 1011106

107

108

109

1010

1011

1012

Stiffness of bolts 1 and 4 (N/m)

Stiff

ness

of b

olts

2 a

nd 3

(N/m

)

1e−09

2e−09

5e−09

1e−08

2e−08

5e−08

1e−07

2e−07

5e−07

1e−06Error (J)

p = 0.9

p = 0.95p = 0.99

p = 1

p = 0.7

p = 0.1

Fig. 13. Evolution of the location of the global minimum of the constitutive relationerror.

0 10 20 30 40 50 60 70 8010−12

10−10

10−8

10−6

10−4

10−2

Iteration

Rel

ativ

e er

ror

Rigidified model: lower initializationRigidified model: upper initializationFirst model: lower initializationFirst model: upper initialization

Fig. 14. Updating of the rigidified model: evolution of the relative error during theupdating compared to the first model.

0 10 20 30 40 50 60 70 8010

4

105

106

107

108

109

1010

Iteration

Stif

fnes

s (N

/m)

k’1: rigidified model, lower initialization

k’3: rigidified model, lower initialization

k’1: rigidified model, upper initialization

k’3: rigidified model, upper initialization

k1: 1st model, lower initialization

k3: 1st model, lower initialization

k1: 1st model, upper initialization

k3: 1st model, upper initialization

Fig. 15. Updating of the rigidified model: evolution of the stiffness during theupdating compared to the first model.

0 5 10 15 20 25 3010−5

10−4

10−3

10−2

10−1

Iteration

Dis

plac

emen

t (m

)

Updated u: rigidified model, lower initializationUpdated u: rigidified model, upper initializationUpdated u: 1st model, lower initializationUpdated u: 1st model, upper initializationDisplacement: target

Fig. 16. Updating of the rigidified model: evolution of the displacement during theupdating compared to the first model.

0 10 20 30 40 50 60 70 80200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Iteration

Load

(N

)

Load 1: rigidified model, lower initializationLoad 3: rigidified model, lower initializationLoad 1: rigidified model, upper initializationLoad 3: rigidified model, upper initialization

Load 1: 1st model, lower initialization

Load 3: 1st model, lower initialization

Load 1: 1st model, upper initialization

Load 3: 1st model, upper initializationLoad 1: targetLoad 3: target

Fig. 17. Updating of the rigidified model: evolution of the bolt load during theupdating compared to the first model.

0 100 200 300 400 500 60010−12

10−10

10−8

10−6

10−4

10−2

Iteration

Rel

ativ

e er

ror

Rigidified model, lower initializationRigidified model, upper initialization1st model, lower initialization1st model, upper initialization

Fig. 18. Updating of the rigidified model: evolution of the relative error during theupdating compared to the first model.

782 F. Gant et al. / International Journal of Solids and Structures 48 (2011) 775–784

0 100 200 300 400 500 600 70010

4

106

108

1010

1012

Iteration

Stif

fnes

s (N

/m)

k’1: rig. model, lower initialization

k’3: rig. model, lower initialization

k’1: rig. model, upper initialization

k’3: rig. model, upper initialization

k1: 1st model, lower initialization

k3: 1st model, lower initialization

k1: 1st model, upper initialization

k3: 1st model, upper initialization

Fig. 19. Updating of the rigidified model: evolution of the stiffness during theupdating compared to the first model.

Table 5Rigidified model: total iterations of the updatings.

Updating case Set 1 Set 1 Set 2 Set 2

Lower Init. Upper Init. Lower Init. Upper Init.

Iterations 30 27 163 518

F. Gant et al. / International Journal of Solids and Structures 48 (2011) 775–784 783

Therefore, rather than match the parameters used in simulation ofSection 3.3, the identified parameters are expected to be lower.

4.2.1. Updating of the rigidified model based on Set 1The updating was performed with the bolt stiffnesses initialized

at 1.932 � 104 N m�1 and 1.932 � 109 N m�1. Figs. 14–17 show,respectively the evolutions of the relative error, of the spring stiff-nesses, of the displacements and of the bolt load during theupdating.

4.2.2. Updating of the rigidified model based on Set 2The updating was performed with the bolt stiffnesses initialized

at 1.932 � 104 N m�1 and 1.932 � 109 N m�1. Figs. 18 and 19 show,respectively the evolutions of the relative error and of the springstiffness during the updating.

Fig. 20. Color map of the modified constitutive relation error along with theconvergence paths using measurement Set 2 and the rigidified model.

4.2.3. Discussion of the rigidified modelAs expected, the updated spring stiffnesses (about

1 � 108 N m�1) were lower than those obtained with the previousmodel (1 � 1010 N m�1). These results lead one to think that updat-ing with the rigidified model can improve both the convergence(detailed in Table 5) of the algorithm and the converged value. In-deed, in Fig. 20, the error surface of the rigidified model shows thatthe algorithm produced a valid global minimum. Furthermore,most of the more complex structures used in aeronautics mainlyconsist in geometrical repetitions of simple assembly of multiplejoint like the one treated in this paper. One can assume that theparametrical study of the parameter p can be performed on a sim-ple structure only once, and then be reused on the entire structure.Then, Fig. 13 underlines the importance of rigidifying the structure.The choice of p value does not affect much the shape of the mini-mum when p < 0.9. Finally the parametrical study of the parameterp can be considered as an optional point.

5. Conclusions

In this paper, we studied a method of updating a model of anaeronautical multiple joint based on the constitutive relation error.The main objective was to define a relatively simple joint modeland develop an efficient updating algorithm for this model whichcould be easily extended to very complex structures. A complex is-sue of model updating of joints was considered and some answershave been found by proposing this updating algorithm.

The updating of the constitutive relation error applied to a firstone-dimensional model of the multiple joint with some specifictargeted data was found to be inefficient. An improved modelwas defined by rigidifying part of the structure surrounding thejoint in order to adjust the ratio of the stiffnesses. The updated val-ues showed that the efficiency of the updating process is greatly af-fected by the ratio of the structural stiffnesses to the springstiffnesses. The application of the updating process to the rigidifiedmodel led to faster convergence and a more accurate updatedmodel.

This updating technique can be extended to 3D structural mod-els obtained by using plates or shells to represent the structurewhile still representing the fasteners by springs. Moreover, prob-lem of mesh incompatibility between the connector nodes, whichoften occurs in industrial design, are generally solved by rigidifyingparts of the plates or shells near the spring nodes. Therefore, theproposed strategy appears to be convenient for aeronautical appli-cations considering its high compatibility with industrial methods.

The use of this method would enable one to consider the appli-cation of updating to larger structural problems. Also, the imple-mentation of this method in popular FE analysis codes couldallow designers to become more confident about their calculationson a global structural level compared to their current semi-empir-ical joint models. The locations of the most highly loaded jointswould be identified more effectively, which would make the localnonlinear reanalysis of these joints more relevant.

An extension of this study will develop an updating strategybased on this improved model. This strategy consists in a represen-tation of the scattering due to behavior uncertainties of quasi-iden-tical structures and experimental errors.

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