Identification of bolted-joint interface models -...

Identification of bolted-joint interface models

H. Ahmadian and M. EbrahimiIran University of Science and Technology,Narmak, Teheran 16844, Irane-mail: [email protected]

J. E. MottersheadUniversity of Liverpool,Liverpool L69 3GH, U.K.e-mail: [email protected]

M. I. FriswellUniversity of Wales Swansea,Swansea SA2 8PP, U.K.e-mail: [email protected]

AbstractA linear dynamic model is developed for bolted joints and interfaces that can be incorporated into existingcommercial finite element codes. The joint interface is modelled using a thin layer of solid elements withisotropic material properties. The material properties of the layer are functions of normal and tangentialstiffness of the joint interface and are identified using experimentally measured data. It is critical to determineonly the simplest model possible which also captures the dominant physics of the joint. The objective is todevelop guidelines in designing models for bolted joints under different pre-stress conditions of the interfacezone. This work provides insight on how the model should be designed in the joint region.

1. IntroductionThe behaviour of joints significantly affects thedynamic response of many mechanical structures. Astructure consists primarily of linear substructuresconnected by joints of various sorts. These jointsintroduce considerable uncertainty in modellingbecause of the complicated and nonlinear physicsinvolved. Determining the relevant physics of eachjoint is critical to a validated full body model of thestructure. Modelling of joints has been investigatedby many researchers [1-9]. The three most commonmechanisms of joint mechanics are frictional slip atbolted or face-to-face compression joints, slapping atgaps, and reflection/transmission at locations ofmismatch of mechanical impedance. All threepostulated mechanisms have their own characteristicfeatures. For instance frictional slip becomessaturated at very high amplitudes and slapping pushesenergy from low to high frequencies. We intend to develop linear dynamic models forbolted joints that can be incorporated into existingcommercial finite element codes. It is critical to

determine only the simplest model possible whichalso captures the dominant physics of the joint. Evenwith fast modern computers the micro-mechanics ofjoint interfaces is not amenable to full systemmodelling. Face-to-face bolted flanges are widelyused in many structural assemblies. We investigatethe behaviour of a face-to-face bolted flangeconnection between two pipes. This makes an idealstarting place for a study of mechanical joints.

In this work, we determine the physics thatcharacterises the joint within a frequency range.While there are many active mechanisms in the joint,this work will try to identify the dominant ones andrepresent them using simplified models. The goal isto develop guidelines for designing of face-to-facecontact bolted-joint models. This provides insight onhow the model should be designed in the joint region.

The finite element model of each section of thestructure and its assembly is reported in Section 2where the predications of the finite element modelare compared with the modal test data. In Section 3 alinear dynamic model is presented. The joint

1741

Figure 1. Flanged Joint

parameters are then tuned so that a good correlationbetween the model predictions and the experimentalobservations is achieved. The updating procedure andits outcomes are reported in Section 4.

2. The FE modelThe test structure consists of two similar steel pipeseach welded at one end to a circular flange. Eachpipe has a length of 1 m, a diameter of 150 mm and athickness of 5 mm. The flanges are bolted to eachother using eight M22 bolts and nuts. Figure 1 showshow the two substructures are assembled to form aface-to-face compression joint.

A finite element model for each substructure isdeveloped using the finite element softwareMSC/NASTRAN. The models are formed using acombination of low order 4 noded plate elements(CQUAD4) and 8 noded solid elements (CHEXA). A good agreement was achieved between thepredictions of each separate substructure model andthe corresponding experimental observations. Thefinite element model of the assembled structure isshown in Figure 2.

In the initial model, nodes of the substructureslocated on the contact interfaces were mergedtogether. This provided much higher joint stiffnessthan the actual value. The first five bending modes

of the assembled model are compared with theexperimentally measured modes in Table 1. Theexperimental results were obtained by exciting theassembled structure using a roving hammer usingfree-free boundary conditions. Comparing theseresults, one notices higher values for the analyticallypredicted modes. However, the models of theindividual substructures were found to represent thephysical systems, therefore one should look for aprocedure to tune the uncertain joint parameters.

3. Joint modellingIn the finite element model, the joints betweensubstructures can be represented by interfaceelements. Interface elements have been developed tomodel the behaviour of joints with different loadingconditions. Two groups of interface elements arecommonly used: zero-thickness and thin-layerinterface elements. In the zero-thickness interfaceelements [1-3] it is assumed that the interface has azero thickness and a constitutive law, usuallyconsisting of constant values for both the shearstiffness and the normal stiffness, is defined. Thebehaviour of the thin- layer interface [4-6] isassumed to be controlled by a narrow band or zoneadjacent to the interface with different propertiesfrom those of the surrounding materials. The thin-

1742 PROCEEDINGS OFISMA2002 - VOLUME IV

Figure 2. Finite Element Model

layer element is treated as any other element of thefinite element mesh and is assigned specialconstitutive relations. When an interface element is used to model ajoint, a suitable constitutive relationship must beadopted. A number of interface constitutive modelshave been developed. Depending on the type ofanalysis performed, the interface physics may berepresented by quasi-linear or nonlinear models.Quasi-linear models [6-7] consider a constant valueof stiffness over a range of interface displacements,until yield is reached at the friction capacity of thejoint. In nonlinear models [8-9], the interface shearstress-displacement relationship is represented by amathematical function of higher order. Theinterface shear stiffness changes during shear,depending on the magnitude of the displacementand any other factors included in the model. Thecoupling between normal and shear deformations isoften ignored and is included in most of theconstitutive formulations found in the literature.__________________________________________

For the joint that behaves elastically in the closedstate the implemented constitutive relation can beexpressed as follows,

22

11

∆∆∆∆∆∆

ukτukτvkσ

s

s

n

===

(1)

where 1∆,∆ τσ and 2∆τ are respectively the elasticpart of the incremental normal and tangential stress,

1∆,∆ uv and 2∆u are the incremental relative normaland tangential displacements across the joint. Subscripts �1� and �2� denote two orthogonaltangential displacement directions in the plane of thejoint. Parameters kn and ks are penalty parameterswhich, respectively, simulate the no penetrationcondition of the joint face, and the stick or noslippage condition in the joint plane. Once the jointexceeds its elastic limit, further sliding will follow anelastic-plastic behaviour. When the interface elementis in a state of sliding, a quasi-linear constitutiveequation at each sliding increment can be defined as,

( )( )

( ) ��

��

�

��

��

�

��

��

�

−−

−−−=

��

��

�

��

��

�

2

122

2122

1

212

2

1

∆∆∆

/1//1

///1

∆∆∆

uuv

HkβkSymHββkHkβk

HµβkkHµβkkHkµk

ττσ

ss

sss

snsnnn

(2)

φµττ

τβ

ττ

τβkµkH sn tan,,

22

21

222

221

11,

2 =+

=+

=+=

where φ is the friction angle. In this work the joints are considered to behave(quasi-)elastically and in the closed state. This meansthat either there is no joint slippage and forcesapplied to the joints are less than the limit defined byfriction and cohesion, or if there is any slippage it isstable and the constitutive equation remains linear.In modelling the joints, we use thin layer interfaceelements as shown in Figure 3. It is easy to

___________________________________________incorporate an interface layer into the model byallowing the elements neighbouring the interface tohave a different constitutive relation from the rest. All the elements used in modelling of the structurehave isotropic material properties, corresponding tosteel, except for the interface elements. One shouldassign the interface elements to have a materialproperty that allows appropriate normal stiffness andshear stiffness in the joint. The general form of theconstitutive equation for the interface layer is,

SUBSTRUCTURING AND COUPLING 1743

��

�

��

�

�

��

�

��

�

�

��

��

�

=

��

�

��

�

�

��

�

��

�

�

zx

yz

xy

zz

yy

xx

zx

yz

xy

zz

yy

xx

εεεεεε

ccccccccccccccccccccc

σσσσσσ

66

5655

464544

36353433

2625242322

161514131211

(3)

Assuming that at each joint the properties of theinterface layer remain constant from one element tothe neighbouring one, there are 21 parameters foreach joint, i.e., cij, i,j = 1, �, 6, to be identified. Bothstates defined in equations (1) and (2) are members ofthe family defined in equation (3). Therefore byselecting the entries cij of the constitutive matrix asupdating parameters the joint physics in sliding ornon-sliding can be identified.

The selected parameters can be identified if theyhave a significant effect on the modal response of themodel. Otherwise the process of identifying theparameters will be ill-conditioned and physicalrealisable results would not be achieved. One mayintroduce relationships, or constraints, into theparameter identification procedures to avoid ill-conditioning [10-11].

In face-to-face contact the behaviour of the joint isgoverned mainly by normal stiffness and shearstiffness, whilst coupling terms have mainlysecondary effects. Therefore to avoid ill-conditioning problems in the updating procedure wechoose an isotropic constitutive law for the interfacelayer. By restricting the stress-strain relationship tobe isotropic, we are still able to define stiffnesses innormal and tangential directions independently.

The linear constitutive equations for isotropicCHEXA elements, which form the interface layer is,

Figure 3. Interface Layer

where E and G are the elastic and shear modulirespectively. Assigning these two as the updatingparameters, we allow appropriate shear and normalstiffness for the interface layer during updating.

4. Parameter identificationThe first five bending modes of the assembledstructures within the frequency range of 0-2kHz are___________________________________________

( )GEGEGλ

εεεεεε

GSymG

GGλ

λGλλλGλ

σσσσσσ

zx

yz

xy

zz

yy

xx

zx

yz

xy

zz

yy

xx

32,

22

2

−−=

��

�

��

�

�

��

�

��

�

�

��

��

�

++

+

=

��

�

��

�

�

��

�

��

�

�

(4)

_________________________________________________________________________________________

measured with free boundary conditions. Theexperimental natural frequencies are shown in Table1. These modes will be used to identify the unknown

parameters of the model by tuning the parameters sothat a good agreement between the predictions of themodel and the measured data is achieved.


The modes within the frequency range of interestcorrespond to different combinations ofdeformation, mostly oval shapes of the pipes�cross-sections, with no deformations at the jointinterface zone. In this frequency range onlybending like modes for the structure wereconsidered for updating. The behaviour of thestructure at the bending modes is dominated by thenormal stiffness nk and the shear stiffness sk at theinterface layer.

The identification procedure was performedusing the Design Sensitivity Module available inMSC/NASTRAN 2001. The modulus of elasticityand shear modulus of the thin-interface elementswere selected as the design variables. The objectivefunction for the optimisation procedure was definedas,

( )25

11/min � −

=i

ei

aii ωωW (5)

where e

iω and aiω

are, respectively, experimentallymeasured and analytically determined naturalfrequencies, and Wi is a real positive weightingfactor. The design sensitivity procedure inMSC/NASTRAN is based on an iterative linearisedeigenvalue sensitivity determined using the followingexpression:

( ) ( )( ) ( ) ( )

( ) ( )ai

Tai

ai

ai

Tai

ai

ei

φMφ

φMωKφωω

��

�� −

=−∆∆

2

22 (6)

As there was no preference between the modes, theweighing factors were set to unity, Wi = 1, �, 5 inequation (5). The initial values of the updatingparameters were selected as those corresponding tothe material properties of steel. A permissible rangeof variation for each parameter was also defined. The upper bound for the variation of the updatingparameters was set to 1.01 of the initial value and thelower bound was set to 0.0001 of the initial value. The selection of bounds were based on the fact thatthe interface layer introduces a local softening effectand sharply reduces the normal stiffness kn

represented by E and the shear stiffness ks

represented by G at the joint compared to the otherareas of the structure.Figure 4 shows changes in the objective function

Figure 4. Objective Function (E and G)

Figure 5. Parameter Convergence

when both E and G are varied independently.Convergence occurs at around 40 iterations. Changesin the design variables during the optimisationprocedure are shown in Figure 5. The stiffness of the interface layer is decreased by 4 orders ofmagnitude from its initial value.

The predictions of the updated model using onlyE are close to the test values when the normal stressin the joint is high, such as in the 1st, 3rd and the 5thmodes. But the prediction of the 2nd and 4th modesare not accurate. In the even modes, as shown inFigure 6, the joint acts as a node and is under highshear stress with small normal stress. By allowingboth E and G to vary in updating, we observeexcellent agreement between the model predictionsand the test results as shown in Table 1. The


Figure 6. Finite Element Mode Shapes

ModeNo.

Measured( )Hzωi

Initial Model( )Hzωi % error

Updating E( )Hzωi % error

Updating E and G( )Hzωi % error

12345

162.6603.8864.915851932

217.2615

999.716572111

33.51.815.64.59.2

163.4592.7865.314271931

0.4-1.8

0-9.9-0.2

163.7605.4865.815701932

0.60.20.1-0.9

0

Table 1. Table of Natural Frequencies


accuracy in predicting all modes with only twoparameters ensures that the identified interfaceparameters have physical merit and can be used in themodel for simulations of the dynamical behaviour ofthe structure in service.

5. ConclusionsA linear dynamic model for bolted joints using thethin layer interface theory is developed. Theparameters of the model are determined by an inverseapproach. The model can be easily incorporated intoexisting commercial finite element codes with theability to define the dominant physics of the joint. The method is demonstrated by identifying theinterface parameters of a bolted structure. Theresults show that at the interface layer the stiffness ofthe element is reduced significantly. This researchprovides insight on how the model should bedesigned in the joint region.References[1] R. E. Goodman, R. L. Taylor and T. L. Brekke,

A model for the mechanics of jointed rock,Journal of the Solid Mechanics and FoundationsDivision, ASCE, Vol. 94, pp. 637-559 (1968).

[2] F. E. Heuze and T. G. Barbour, New models forrock joints and interfaces, ASCE Journal of theGeotechnical Engineering Division, Vol. 108,pp. 757-766 (1982).

[3] G. Beer, An isoparametric joint interfaceelement for finite element analysis, InternationalJournal for Numerical Methods in Engineering,Vol. 21, pp. 585-600, (1985).

[4] C. S. Desai, M. M. Zaman, J. G. Lightner and H.J. Siriwardane, Thin-layer elements forinterfaces and joints, International Journal forNumerical and Analytical Methods ofGeomechanics, Vol. 8, No. 1, pp. 19-43 (1984).

[5] C. S. Desai, A. Muqtadir and F. Scheele,Interaction analysis of anchor-soil systems,Journal of Geotechnical Engineering, ASCE,Vol. 112, No. 5, pp. 537-553, (1986).

[6] P. C. Wong, F. H. Kulhawy and A. R. Ingraffea,Numerical modelling of interface behaviour fordrilled shaft foundations under generalizedloading, Foundation Engineering: CurrentPrinciples and Practice, ASCE GeotechnicalSpecial Publication, Vol. 22, pp. 565-579(1989).

[7] T. Matsui and K. C. San, An elastoplastic jointelement with its application to reinforce slopecutting, Soils and Foundations, Vol. 29, No. 3,

pp. 95-104 (1989).[8] T. D. Lau, B. Noruziaan and A. G. Razaqpour,

Modelling of construction joints and shearsliding effects on earthquake response of archdams, Earthquake Engineering and StructuralDynamics, Vol. 27, pp. 1013-1029 (1998).

[9] J. S. Lee and G. N. Pande, A new joint elementfor the analysis of media having discretediscontinuities, Mechanics Cohesive-FrictionalMaterials, Vol. 4, pp. 487-504 (1999).

[10] H. Ahmadian, J. E. Mottershead and M. I.Friswell, Regularisation methods for finiteelement model updating, Mechanical Systemsand Signal Processing, Vol. 12, No. 1, pp. 47-64(1998).

[11] M. I. Friswell, J. E. Mottershead and H.Ahmadian, Finite element modal updating usingexperimental test data: parameterization andregularization, Philosophical Transactions of theRoyal Society of London, Series A, Vol. 359,pp. 169-186 (2001).


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