modelling the behaviour of interface surfaces using …

APPLICATION OF THE TANGENT-STIFFNESS METHOD IN THEFINITE ELEMENT MODELLING OF THE BEHAVIOUR OF

INTERFACE SURFACES

S.O. EdelugoDepartment Of Mechanical Engineering, University Of Nigeria,

Nsukka, Nigeria, West AfricaPHONE: 2348037463298; e-mail: [email protected]

ABSTRACT:To simulate the behaviour of an interface formed between two contacting bodies using thefinite element method, a review of previous approaches employed by other authors whoworked with the same finite element method is carried out; their models and experimentaldata evaluated. A third-degree polynomial approximation approach is first undertaken tomodel the behaviour of the interface during loading, unloading and reloading. Anincremental method and an iterative procedure are both employed. Finally, the tangent-stiffness method in which, for every iterative step, a portion of the external load is applied andthe shear and normal stiffnesses are calculated is introduced. An example of a prismatic beamon a rigid base is analyzed and the predicted results are compared with those obtained fromexperiments.

Keywords: Modeling; Interface-surfaces; Finite-element-method; Loading-reloading-unloading; Third-degree- polynomials; Tangent-stiffness-method.

Symbols / Notations

ia area of contact of node I

intA apparent area of contact at the interface

1 2 3a , a , a coefficients of the τ-δ curve (loading branch)

1 2 3 slope and intercept of the straight lines obtained when a , a , a are plottedagainst pressure

0b distance of the unloading branch of the τ-δ curve from the original

1 2 3b , b , b coefficients of the τ-δ curve (loading branch)

1 2 3 slope and intercept of the straight lines obtained when b , b , b are plottedagainst pressure

c constant determined by the characteristic of the interface an defined byequation (1)

0c distance of the reloading branch of the τ-δ curve from the origin

1 2 3c , c , c coefficients of the τ-δ curve (reloading branches)

11 12c , c slope of and intercept of the straight lines obtained when

21

22 S.O. Edelugo

21 22 1 2 3c , c c , c , c are plotted against pressure

21 22c , c

1d deflections of the interface element ie number of elements to which node i is common

1e number of elements in the contacting bodies

2e number of interface elementsE equivalent Young’s modulus of the contacting bodies

zF normal load acting on the structure

z j∆ F incremental load acting on the structure during the jth iteration in the z-direction

z j∆ F normal force acting on beam i after the jthi

xb yb sb{F , F ,..., F } forces acting on beam i

nk normal stiffness of the interface

nj k normal stiffness of beam i for the jth iteration

sxk shear stiffness in the x-direction

syk shear stiffness in the y-direction

intk stiffness matrix of an interface element

bk stiffness matrix of an element representing the contacting bodies

sk structural stiffness matrixm constant determined by the characteristic of the surfaceP pressure at the interface

avP average interface pressure resulting from a uniform distribution of the first

z1incremental load F

jP i pressure in beam i after the jth iteration

1 1 2u , v ,….w deflections of the nodes of an interface beam in the x-,y-, and z-directions

λ normal deflection at the interface

z j1λ normal deflection in beam i due to force ∆Fi

τ shear stress at the interface

jτ limiting value of shear stress due to friction

1τ shear stress in the beam after the iteration in which the limiting value of shearstress is exceeded

τ shear stress in the beam after the first re-analysis

1. INTRODUCTIONWhen structures are analyzed by numericalmethods, it is sometimes acceptable to ignorethe influence of joints formed by theconstitutive elements of a structure. In theanalysis of interface surfaces in machinetools, however, they cannot be ignoredbecause their flexibility can account for afairly large percentage of the deflection

occurring at the cutting edge [1].In the case of bolted joints the stiffness

depends upon the bolt preload, the geometryof the flanges and bolt, the area over whichthe bolt load is applied and the characteristics(i.e. the surface roughness, hardness,machining methods employed, etc.) of themating surface. Several research workers [2-7] have considered the geometry of the joint

NIGERIAN JOURNAL OF TECHNOLOGY, VOL. 27 NO.2, SEPTEMBER 2008

23APPLICATION OF THE TANGENT-STIFFNESS METHOD IN THE FINITE ELEMENT MODELLING OF ...

and developed empirical relationships for thestiffness of the assembly and for the pressuredistribution along the interface. Some ofthese relationships are based upon thepressure cone theory.

The advent of large-scale computersand the rapid growth of the finite elementmethod have made it possible to obtainmore accurate models. Cullimore andEckart [8] extended an earlier empiricalrelationship [9] to cater for a three-boltthree-flange assembly. Fernlund [10]assuming that a single plate can replace twoplates estimated the contact pressure andarea but Gould and Mikic [11] using aniterative finite element technique haveshown that the assumptions made in [10]could lead to serious errors. Thompson et al[12] using an experimental/finite elementapproach found that the contact area isindependent of the bolt preload andbolthole diameter. In another paper, Jofiertet al [13] developed from finite elementstudies empirical relationships for theinterface pressure and contact area for ring-loaded flanges. Schafer [14], Mahtab andGoodman [15] extended the finite elementmodel by introducing bond elementsbetween the contacting bodies and modeledthe behaviour of the joint in shear.

The above models do not take intoaccount the characteristics of the contactingsurfaces.

Back et al [16] represented the normalstiffness of the interface by using beam andplate-like elements. However, their modelcaters only for the situation when theinterface or part of it is being continuouslyloaded. The model developed hereincalculates the stiffness of the interfacewhen it is being loaded, unloaded orreloaded. The model also takes into account

the shear stiffness of the interface. Thetangent-stiffness method [17] is used tocalculate these stiffness values and it isshown that this method gives results thatare more accurate than those obtained bythe method employed in [16].

2. NORMAL AND SHEARBEHAVIOUR OF AN INTERFACE

Several research workers have studieddeflections experimentally [1,18,19]. Briefly,when two specimens are brought into contactand a load normal to the interface is applied,the deflections produced lie on the curve OA(Fig. 1a). If the loading is stopped, for

1example at pressure p , and then graduallydecreased, the unloading curve, AB, is almostlinear and corresponds to the elasticbehaviour of the specimens. If the load isincreased again, the behaviour is elastic until

2the pressure, p is reached after which thedeflection follows the curve OAC again untilthe next unloading cycle. It has been found[1] that the curve OAC can be adequatelydescribed by the following equation:

λ = cp ...(1)m

Where λ is the deflection at the interface, p isthe pressure and c and m are constantsdetermined by the characteristics of theinterface.

In addition to normal loads, shearloads may act upon the joint. The behaviourof an interface loaded in shear has beenstudied,, amongst others ,by Kirsanova[20], Masuko [21] and Sanad [22]. Atypical shear deflection curve for aninterface is shown in Fig. 1b. This curve,which is obtained with a constant normalpressure and varying shear load acting onthe interface, can be divided into threeparts:


24 S.O. Edelugo

(a) The loading curve formed by thesegments OA, AB, BC, etc

(b) The unload branches A1, B2, C3, etc.;(c) The reload branches 1A, 2B, etc.

Fig . 1.Normal and shear behaviour

of an interface

Unlike the pressure-deflection curve, there isno known simple equation to represent theshear stress-deflection curves. It was decidedto use polynomials to represent each part ofthe curve. Several tests were conducted andit was found [23] that a third-degreepolynomial of the type:

1 2 3τ = a δ + a δ + a δ ... (2)2 3

represented the loading curve accurately. The

oterm a is absent because the curve passesthrough the origin.

The unload branches are disjointedbecause they start from different points onthe main curve. They have, however, thesame shape. This was verified by translating

the different branches so that they passedthrough the same point (i.e. the origin).Notice the resulting interloop. This importantproperty made it possible to group themtogether and represent them by a single third-degree polynomial:

1 2 3τ = b δ + b δ + b δ ...(3a)2 3

Each branch has to be uniquely identified andthis is done by expanding the above equation

oand including the term b ; whose value ismade equal to the amount by which thebranch has to be translated so that it passesthrough the origin.Therefore, for unloading:

o 1 2 3τ = b + b δ + b δ + b δ …(3b)2 3

Fig. 2 Approximation of the pressure-deflection curve

It was found that an identical procedure couldbe used for the reloading branches and thepolynomial to represent them is:

o 1 2 3τ = c + c δ + c δ + c δ …(4)2 3

The behaviour of the interface shown in fig.1b is for a specific value of normal pressure.If the normal pressure is changed, another buta similar τ - δ curve is obtained. For example,from the experimental results obtained, bySanad [22] for ground mild steel, thereloading branches at different normalpressures, when translated to pass through theoriginal, were approximated at follows:



p = 24 kgf/cm ; τ = 337.99δ - 116.34δ + 26.76δ2 2 3

p = 50 kgf/cm ; τ = 427.52δ - 115.45δ + 90.50δ2 2 3

p = 75 kgf/cm ; τ = 609.50δ - 264.80δ + 166.34δ2 2 3

p = 103 kgf/cm ; τ = 647.83δ - 280.89δ + 265.61δ 2 2 3

p = 128 kgf/cm ; τ = 724.83δ - 276.43δ + 361.66δ2 2 3

Figure 2 shows the values of the coefficientsof δ, δ , δ plotted against the normal2 3

pressure. The maximum value of pconsidered is 128kgf/mm because this was2

the maximum pressure, which theexperimental rig, designed and built by Sanad[22], could exert on the test specimens.Assuming that at higher normal pressure a

1 2 similar trend is exhibited, the values of c , c

3 and c (in equation 4) can be approximated tovary linearly with p. Thus, in equation form:

1 11 1 2 21 22 3 31 32c = c + c p, c = c + c p and c = c + c p

If these values are substituted into equation

o(4) and the term c is neglected because itsvalue is associated with a particular branch,a general equation to represent the reloadingbranches is obtained:

11 12 21 22 31 32τ = (c +c p)δ +(c +c p)δ +(c +c p) δ ... (5)2 3

For the load and unload branches, a similarlinear variation of the coefficients withnormal pressure was observed. Therefore,equation (2) and (3a) can be re-written as:

11 12 21 22 31 32τ = (a + a p)δ +(a + a p)δ +(a + a p) δ ... (6)2 3

and

11 12 21 22 31 32τ = (b +b p)δ +(b +b p)δ +(b +b p) δ ... (7)2 3

3. STIFFNESS MATRIX OF ANINTERFACE ELEMENT

To model the above non-linear behaviour,beam elements are introduced between thetwo contacting bodies. For the three-dimensional case, each interface node hasthree degrees of freedom: two (u,v) in theplane of the joint and one normal to it (w),The force-deflection relationship for aninterface element, I, is given below:

int {∆F }I = (k ) {d}i ... (8)

1 2Note that λ is given by (w – w ) and the

1 2 1 2shear deflection, δ, by (u – u ) or (v – v ).Although the shear and normal stiffnessesappear to be uncoupled, it was shown earlier[equations (5-7)] that the shear stiffness isdependent upon the normal pressure. In thecalculations below, the normal and shear

z1 z2 x1 x2forces ∆F (or - ∆F ), ∆F (or - ∆F ) and

y1 y2 zF (or - ∆F ) are designated by ∆F , ∆andi

y∆F respectively. The suffix, hereafter, refersi

to the iteration number.

4. D E T E R M I NA T I ON OF T H ENORMAL AND SHEAR STIFFNESS

The most commonly used methods to modelnon-linear behaviour are the step-wise,iterative and mixed methods. In the iterativemethod, the structure is fully loaded in eachiteration and the portion of the load that isnot equilibrated is used in the next step tocompute an additional increment ofdeflection.

This process is repeated until thedeviation from equilibrium becomesnegligible. Back et al [16] used a modifiedversion of this technique to model thepressure-deflection curve. The convergentcurve obtained by them was oscillatory innature; the oscillations were damped out byaveraging two consecutive sets of results.

In the step-wise method, which hasbeen used herein, the total load applied tothe system is subdivided into several partial


26 S.O. Edelugo

loads which need not be of equalmagnitude. During each iteration therelationship between the forces anddisplacements is assumed to be linear. Thisrelationship, i.e. the stiffness matrix asdefined by equation (8), can be evaluatedusing either the ‘secant modulus’ or the‘tangent-stiffness’ method. A description ofthese methods is given in [17]. In the firstiteration, the former is used; in the secondand subsequent iterations the latter is used.The accumulated values of the incrementsof displacement corresponding to the forceincrements, gives the total displacement forthe intended load. Given below is a detaileddescription of how these methods havebeen adapted to calculate the normal andshear stiffnesses of the interface when it isbeing loaded, unloaded or reloaded.

4.1 Normal stiffness during loadingThe process starts by subdividing the total

zexternal load F into several load increments.For the first iteration, it is assumed that theactual area of contact is equal to the apparent

intarea of contact, A . Therefore, for the

z1iteration, the incremental load, ∆F , isdistributed over the entire interface.

To calculate the force acting at aninterface node, it is necessary to know (i) the

1pressure at the node and (ii) the area, a , overwhich it exerts an influence.

For the first iteration, it is assumed that

z1 av∆F will cause a uniform pressure, p , acrossthe interface therefore its magnitude is given

z1/ int iby ∆F A . The area of influence, a ,ascribed to an interface node i can be shown

i x yequal to Σ ∫∫ N d d where e is the numbere

of elements the node is common to. Thisderivation is based upon the assumption thatthe pressure remains constant over theelement. It was found that the variation in thepressure had to be one degree lower than thatassumed for the displacement functions. The

stiffness matrices of the elements in thecontacting bodies were derived using linearshape functions. When a linear variation ofpressure was tried with these elementsincorrect pressure distribution were obtained.For example, in an axisymmetric problem,instead of the pressure maintaining a constantvalue around the circumference of a givencircle, it oscillated. Note that the sum of all

intthe areas of influence is equal to A and thatthe above expression merely helps to

intsubdivide A into region. Therefore, for thefirst iteration the load acting at node i is

z av i∆F = p a .i .

This concentrated force acting at the nodewill cause the asperities within its area ofinfluence to deform. This deformation can becaluculated from equation (1) and is

0represented by the deflection, λ , occurring ini

=i0the beam which is introduced at node i: λ

a . Hence the normal stiffness at thismvc p.

point on the interface is:

Fig.3:Variati

on of the polynomial coefficients with normalload



...(9)

This value corresponds to the slope of

0the line OA in Fig.3 For this first iteration;one cannot use the slope of the curve becauseits value is zero at the origin.

The value of the normal stiffness asgiven by equation (9) is now used in equation(8) to calculate the stiffness matrix for theinterface element. The above procedure isrepeated for all the interface elements. Thesematrices are now combined with the stiffnessmatrices of the elements representing thecontacting bodies to form the global stiffnessmatrix of the structure:

A finite element analysis of the structure isnow performed with the first incremental

z1load ∆F acting on it. From the resultingdisplacements, the relative deflectionbetween the two ends of beam i can be

1extracted. Let this be represented by λ .i

This pressure acting at the same node afterthe first iteration can now be calculated:

1 p = element force area of influencei

= Stiffness x deflection

= ... (10)

This value of pressure is now intersected with

1the curve to locate point A (fig.3). The slopeof the curve at this point is required tocalculate the stiffness of the beam element ifor the second iteration. Differentiatingequation (1) gives the slope of the curve at apoint:

... (11)

n iSince, k = (dp/dλ)a the normal stiffness of

the interface node i for the second iteration

is: ...(12)

The interface element stiffness matrices are

nrecalculated with the new values of k .

bMatrices [k ] are not recalculated because thedeflections occurring in the structure aresmall and its behaviiour is elastic. Therefore:

z2The second incremental load, ∆F , is appliedand another finite element analysis is

2performed. This causes deflection, λ , toi

2occur in element i and from it the pressure pi

in the beam at the end of the second iterationcan be calculated [equation (10)]:

... (13)

The normal stiffness for the third iteration is:

... (14)

This process is repeated until all theincremental loads have been applied. The final deflection of the interface at node i is

given by where k is the

number of iteration performed.

The pressure calculated from equation(13) is assumed to act over the part of theinterface, which is assigned to node i. Overthis region it is assumed to be constant. Thereis, however, with this model a discontinuityin the pressures occurring in adjacentregions. This discontinuity will decrease asthe number of nodes in the finite elementmodel is increased.

The magnitude of the incremental


28 S.O. Edelugo

zkforces, DF , are arranged to be either in ageometric or an arithmetic progression. Thisseems justifiable considering the exponential

nnature of equation (1). Since the value of kfor the first iteration is calculated using the

0, z1secant OA the first iteration load is F ischosen to be small (about 5 per cent of thetotal load). The greater the initial load, the

1 1greater is the magnitude of A B (fig. 3)which is the error incurred in the firstiteration.

4.2 Normal stiffness during unloading andreloading

As the total load is increased, the pressure inthe regions around the load also increases. Atthe same time, in the other regions thepressure decreases thus initiating theunloading process. As mentioned earlier,during unloading the joint behaves as if thetwo bodies were rigidly connected. Thisresults in curve AB (fig. 1a) that can beapproximated to a straight line; its slope isgiven by the equivalent Young’s modulus ofthe contacting bodies.

If, during the iterations, a beam is foundto go into tension, it indicates that unloadinghas begun. For the next iteration, its stiffnessis:

... (15)

This value is used as long as the beamcontinues to be unloading. When the sum ofthe incremental forces acting on it becomeszero, the beam is removed from the finiteelement model. This indicates that a certainamount of area at the interface has lostcontact. The deflection occurring in the beamwhen it is disconnected represents a

Bpermanent deflection, λ (fig. 1a) which hasoccurred at the interface.

In subsequent iteration, due to a changein loading conditions, the same regions of theinterface may begin to approach each other.When the relative deflection between a

Bcontact nodepair reaches λ (fig. 1a) thebeam is reintroduced. This indicates that thispart of the interface has been brought intocontact and reloading has begun. Thestiffness of the beam is calculated from theslope of the reloading curve BA. When thedeflection in the element reaches the valuecorresponding to point A. an equation similarto (14) is used once again to calculate thenormal stiffness of the interface element.

4.3 Calculation of the shear stiffness

sx syThe shear stiffness, k and k , are calculatedusing the stepwise method. The total force, asbefore is applied in increments. For all theiterations, including the first, the shearstiffness of a beam is calculated using thetangent-stiffness method

The value of τ/δ is evaluated from the slope,

dτ/dδ, of the loading, unloading or reloadingcurve. If equation (6) is used, the shearstiffness of beam i is:

1 2 3 i = (a + 2a δ + 3a δ ) a.2

1 11 12 1where a = (a + a p ), etc. This equation isused for all the iterations. Note that for thefirst iteration, the slope of the curve at theorigin is used and therefore



(a) Forconstantnormal

load

(b) For varying normal load

Fig. 4 Approximation of the τ-δ curve

For the second and subsequent iterations, thecumulative of the incremental sheardeflections is used to locate the point on thecurve at which the tangent is drawn. Therefore, for example, for the third iteration

where and are the

shear deflections occurring in the beam in the

first and second iterations.

Figure 4a shows how the loading curve isapproximated by lines OA, AB, etc.: theirslopes are the same as the tangent to thecurves. This approximation is valid only ifthe normal pressure remains constant. If anincrease in the shear force results in acorresponding increase in the normalpressure, as would happen if the plane of thejoint is not normal to the direction ofapplication of the load, a different curve hasto be used for each iteration. The shearbehaviour of the interface is nowapproximated by OA, AB, BC, etc. (fig. 4b)

1 2 3whose slopes, α , α , α , etc., are obtained

0 1 2 3from the curves, P , P , P , P , etc. Thesecurves represent the shear behaviour of theinterface at different normal pressures.

4.4 Coefficient of frictionThe model also takes into consideration thecoefficient of friction. By limiting the maximum shear force that can be transmittedthrough the joint, the coefficient of frictiondefines the position (point D, Fig. 5a) atwhich the loading curve stops and slip beginsto occur. The curve OPQ should berepresented by the tangents OA, AB, BC, CDand DE. DE represents the small amount ofslip, which is permitted to occur in localizedregions in the vicinity of the bolt load. Withthe tangent stiffness method, it is difficult tomodel the transition from CD to DE.Consider the situation at the interface whereone of the beams after k iterations has

kdeflected δ (fig. 5a), for the next iteration,the shear stiffness of the beam is calculatedfrom the slope of the curve at point P.


30 S.O. Edelugo

Fig. 5. Modeling the τ-δ near the limit of friction

At the end of this iteration, the shear acting at

1this beam is τ which exceeds the limiting

1 kvalue τ but the resulting deflection, δ . Thisis also illustrated in Fig. 5b. AB representsthe initial position of the beam. The beamshould deflect to A ‘B’ with B’B”representing the slip. However, the finiteelement model moves the beam to position

1 1A B . The sole reason for considering thecoefficient of friction is to reduce the excess

1 fstress (τ - τ ) and to correct the position of

1 1A B . It is not intended to model thebehaviour of the joint when gross-slip occurs;for this the reader is referred to [24,25]. Toreduce the excess stress, an approachsuggested in [26] has been adapted herein.The excess force, whose magnitude is given

1 f iby (τ - τ ). a , is applied at the interface nodesafter which the structure is re-analysed. The

beam AB under the action of the correcting

sforce ∆F (fig. 5b) now deflects to positions

1 1A B , which is nearer to the required1 1

position A’B”. After the first re-analysis, the

f1excess shear stress is reduced to (τ – τ ). It is1

suggested in [26] that the model is re-analysed several times until the excess

Fig. 6a Flow diagram for solving contact

problems



Fig. 6b Flow diagram for subroutine SHEAR

shear stress become zero. This was found tobe very expensive with respect to computingtime because several re-analysis are requiredfor every node where slip has just occurred.Because of this and the fact that the shearbehaviour of the interface plays a secondary

part in the case of fixed joints, it was decidedto perform only one re-analysis. It was foundthat this reduced the excess stress by 60 percent.5. COMPUTATIONAL PROCEDUREThe overall operation is described by the


32 S.O. Edelugo

flowchart shown in Fig. 6a. In addition to thedata required to describe the finite elementmodel such as nodal coordinates, elementnode numbers, the user must supply thevalues of c and m, the number of iterationsrequired, the coefficients which describe theτ-δ curve, and the manner in which theexternal load is to be divided intoincremental loads.To start with, the program calculates thestiffness matrices of the elementsrepresenting the contacting bodies. For the

n sinterface elements, the values of k and k arecalculated by subroutines NORM andSHEAR. The flowchart for NORM is shownin Fig. 6b. For the first iteration, the

ncalculation of k is straightforward. For thesecond and subsequent iterations, the valuesof BRANCH, SUMF and SIGMAF areexamined to decide which part of thepressure-deflection curve should be used to

ncalculate k . Note that SUMF and SIGMAF,the current and previous sums of theincremental forces sustained by the beam aregiven by the following expressions:

where k is the iteration which has just beencompleted.

A similar but more complex procedure is

sused in SHEAR to calculate the value of k .In a three- dimensional problem, two callsare made to this subroutine because the shearstiffness in the plane of the joint may not beequal.

After each iteration is completed, theshear force sustained by a beam is comparedwith the limiting friction force; if it exceedsthe latter, correction forces are calculated andincluded in the overall load vector. A re-analysis is performed before passing on tothe next iteration.

6. RESULTS AND DISCUSSION

6.1 Prismatic beam on a rigid baseThe example to be analysed is a simplerectangular prism (Fig. 7 inset) which lieswith one of its longitudinal narrow facesagainst a plate of infinite stiffness. Anexternal load is applied at mid-length andcompresses the prism against the rigid base.The prism is represented by in-planerectangular elements and the interface bybeam elements (not shown in the inset, Fig.7). The rigid base is simulated by fixing oneend of the beams. Because the problem issysmmetrical, suitable constraints areimposed on nodes which lie on the center-line and only one-half of the structure isanalysed.

This example was selected for tworeasons: experimental values of the interfacedeflections are available from Levina [28]and it has also been analysed by Back et al[16] using a ‘standard spring’ method. Thesetwo sets of results serve as a basis ofcomparison. Levina and Back et al did notconsider the shear stiffness of the interface,and therefore the finite element results shownin Fig. 7 take into account only the normalstiffness of the interface. In general, theinterface deflection is given by thecompression occurring in a beam element.Since, in this example, one end of each beamis constrained, the interface deflection isgiven directly by the downward deflection ofthe nodes lying on the interface i.e. nodeslabeled 1,6,11,16, etc. in Fig 7a. Figure 7bshows the pressure at each of the interfacenodes.

With the incremental method, theaccuracy obtained depends upon the numberof steps chosen by the user. This problem hasbeen analysed with three, five, ten and fifteensteps.



Increasing the steps from three to fiveresults in a relatively large increase in theaccuracy of the predicted results; the increasein the accuracy from ten to fifteen steps isonly marginal and does not justify the extracomputing effort. For this problem, Levinaquotes a value of c = 0.8 µm and m = 0.5 andmodulus of elasticity of 9500kgf/mm . The2

results obtained by the incremental methodconverge to a value slightly higher than theexperimental results. The results obtained bythe method suggested in [16] are even furtherremoved from the experimental values.

Fig. 7 Interface pressure and deflections for aplate on a rigid base

Another important observation is that if

the convergence curve were to be plotted forthe interface deflection at node 51, it wouldbe monotonic in nature and would converge

from ‘above’. However, in the case ofpressure (Fig.7b), the nature of theconvergent curve depends upon the positionof the node along the interface. For example,the pressure for nodes 36 to 51 convergesfrom ‘below’, whereas for nodes 6, 11, 16, 21and 26 it converges from ‘above’.

7. CONCLUSIONS

The tangent-stiffness method has beenadapted to model the normal and shearbehaviour of an interface. It was found thatthe loading, unloading and reloading

branches of the τ-δ curve could beapproximated by third-degree polynomials.The results are fairly accurate if the load issubdivided into five or more partial loads.The model has yielded results which agreewith the experimental values.

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34 S.O. Edelugo

9. Cullimore, M. S. G. and Upton K. A. Thedistribution of pressure between two flatplates bolted together. Int. J. Mech. Sci.,1964, vol. 6, pp 13-25.

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