Verified Solution for a Statically Determinate Truss Structure with Uncertain Node Locations

Transcription

1 Nov. 00, Volue, No. (Seial No. 6 Jounal of Civil Engineeing and Achitectue, ISSN 9-79, USA Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations Andew P. Sith, Jügen Galoff and Host Wekle Faculty of Copute Science, Univesity of Applied Sciences/HTWG Konstanz, Konstanz D 780, Geany Faculty of Civil Engineeing, Univesity of Applied Sciences/HTWG Konstanz, Konstanz D 780, Geany Abstact: We conside a statically deteinate stuctual tuss poble whee all of the physical odel paaetes ae uncetain: not just the ateial values and applied loads, but also the positions of the nodes ae assued to be inexact but bounded and ae epesented by intevals. Such uncetainty ay typically aise fo ipecision duing the pocess of anufactuing o constuction, o ound-off eos. In this case the application of the finite eleent ethod esults in a syste of linea equations with nueous inteval paaetes which cannot be solved conventionally. Applying a suitable vaiable substitution, an iteation ethod fo the solution of a paaetic syste of linea equations is fistly eployed to obtain initial bounds on the node displaceents. Theeafte, an inteval tightening (puning technique is applied, fistly on the eleent foces and secondly on the node displaceents, in ode to obtain tight guaanteed enclosues fo the inteval solutions fo the foces and displaceents. Key wods: Tuss systes, finite eleent ethod, uncetain paaetes, inteval aithetic.. Intoduction Many souces of uncetainty exist in odels fo the analysis of stuctual echanics pobles. These include, e.g., easueent ipecision, anufactuing o fabication ipefections, and ound-off eos. An uncetain quantity is often assued to be unknown but bounded, i.e., lowe and uppe bounds fo this quantity can be povided (without assigning any pobability distibution. Theefoe, these quantities can be epesented by intevals. Inteval aithetic, e.g., Refs. [,, povides the eans to keep tack of such uncetainties thoughout the whole coputation. Consequently, the esult, which is again an inteval quantity, is guaanteed to contain the exact esult. The nueical ethod ost fequently used in stuctual echanics is the finite eleent ethod (FEM. Its accuacy is affected by discetisation and ounding eos and odel and data uncetainty. In Coesponding autho: Jügen Galoff, PhD, pofesso, eseach fields: nueical analysis, inteval coputations, atix analysis. E-ail: galoff@htwg-konstanz.de. this pape we focus on paaetic uncetainty and ounding eos. The souce of paaetic uncetainty (soeties also called data uncetainty is the lack of pecise data needed fo the analysis. In the FEM, paaetes descibing the geoety, ateial, and loads ay be uncetain. Paaetic uncetainty ay esult fo a lack of knowledge (episteic uncetainty o educible uncetainty, e.g., loads ae not exactly known, o an inheent vaiability (aleatoy uncetainty o ieducible uncetainty in the paaetes, e.g., ateial paaetes ae only known to vay within known bounds, cf. [9. Fo a decade o oe the inteval aithetic appoach has been used to handle paaete uncetainty in the application of the FEM to pobles in stuctual echanics, e.g., Refs. [,, 0,, -, to nae but a few. Most of these papes conside the case of affine paaetic dependency. Typically, oe advanced odels involve polynoial o ational paaete dependencies, in which case the coefficients of the systes of linea equations to be solved ae polynoial o ational functions of the paaetes. In Ref. [ we pesent an appoach to

2 Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations solve such systes. Theein we eploy a geneal-pupose fixed-point iteation using inteval aithetic and an efficient ethod fo bounding the ange of a ultivaiate polynoial ove a given box based on the expansion of this polynoial into Benstein polynoials [6, 7. As an exaple, we discuss a two-bay two-stoy fae involving and 7 paaetes. The poble that the lengths of the bas of a tuss syste ae uncetain, due to fabication eos, is consideed in Ref. [0. Howeve, in eal-life pobles, not only the lengths ae uncetain but also the positions of the nodes ae not exactly known. To the best of ou knowledge, this poble has not been consideed so fa in the liteatue. In this pape we pesent a siple odel with uncetain node locations, consisting of six linea tuss eleents joined at five nodes. As well as uncetain node coodinates, the ateial values (Young s odulus and coss-sectional aea and loading foces ae also inteval paaetes. As a consequence of the uncetain node locations, both the eleent lengths and angles in the poble ae also inteval values. With a suitable choice of vaiable substitution fo the angles appeaing in the syste atix, the esulting paaetic syste of linea equations is fistly solved by the afoeentioned geneal-pupose paaetic fixed-point iteation. Howeve, the tightness of the esulting displaceent intevals is not wholly satisfactoy. Theefoe, two inteval puning techniques ae applied to copute and contact the inteval enclosues fo the eleent foces and node displaceents. These intevals ae copaed to a tight inne estiation of the tue inteval solution obtained by a Monte Calo siulation. This pape is oganised as follows. The next section consists of a bief intoduction to inteval aithetic. The odel is then pesented in detail in Section, along with its paaete values. The collection of ethods used to solve the poble, including the iteation ethod fo paaetic systes and two inteval puning techniques ae descibed in Section. The nueical esults ay be found in Section and we conclude with soe suggestions fo continuation of this wok.. Inteval Aithetic Let IR denote the set of the copact, nonepty eal intevals. The aithetic opeation o { +,,, /} on IR is defined in the following way. If a [ a, a, b [ b IR, then a + b [ a + a + ( a b [ a a ( a b [in{ a a a ab},ax{ a a a ab}, ( b [in{ b}, ( ax{ b}, if 0 b. As a consequence of these definitions we obtain the inclusion isotonicity of the inteval aithetic opeations: If a,b IR with a a and b b then it holds that a o b a o b. ( Note that soe elations known to be tue in the set R, e.g., the distibutive law, ae not valid in IR. Hee we have the weake subdistibutive law a ( b + c ab + ac fo a, c IR (6 By IR n and IR nxn we denote the set of n -vectos and n -by- n atices with enties in IR, espectively. Futhe details on aithetic with intevals ay be found in Refs. [,.. The Model We conside the siple echanical tuss stuctue copising five nodes connected by six linea eleents as depicted in Fig. ; the eleents ae nubeed in cicles and the coodinates of the nodes ae also given. Two of the nodes, and, ae fixed; the othe thee ae fee-oving. A downwad loading foce of 0 kn is sepaately applied to both nodes and. Upon loading, we wish to copute the displaceents of nodes -, viz. u, v, u, v, u, v, and the esultant noal foces in all six eleents,

3 Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations S, K, S6. Each of these is an inteval quantity since the uncetainty in the input data causes uncetainty in the solution. We wish to copute intevals which tightly contain the tue anges of values fo each of these vaiables. The uncetain paaetes ae as follows (see also Table : The positions of the five nodes of the tuss (befoe loading ae subject to an uncetainty of ± 0.00 in both the x - and y -diections. With etes as the coodinate units, this coesponds to a vaiation of ±. Coespondingly, the eleents ae of uncetain length (depending upon configuation, they ay vay upto ± 0. The poduct of the eleents coss-sectional aea with the Young s odulus is subject to an uncetainty of ± %. The noinal value is taken as an IPE 60 steel eleent ( A 0.c, 8 E.*0 kn/. This esults in EA : [0099,0. Note that thee is a single, global EA paaete. The loading foces applied to all nodes ae subject to an uncetainty of ± kn in both the x -and y -diections. This applies even to nodes which do not have a loading foce applied (i.e., node.. Methodology Ou solution pocedue consists of the following stages: ( Application of a vaiable substitution to geneate the sybolic syste stiffness atix appeaing in the FEM in tes of the inteval paaetes; ( initial enclosues fo the node displaceents obtained by applying a paaetic solve to the inteval syste; ( initial enclosues fo the eleent foces coputed fo these node displaceents; ( an inteval tightening ethod applied to the eleent foces; ( an inteval tightening ethod applied to the node displaceents.. Finite Eleent Method Fig. Mechanical Tuss Model with Six Eleents. Table Inteval Paaetes fo the Tuss Model. Paaete Noinal Value Uncetainty Young s odulus * aea ± 0 kn 00 kn EA ( ± % x, (0, ± 0.00 (0,0 ± 0.00 Node (, ± 0.00 coodinates (,0 ± 0.00 Loading foces ( y ( x, y ( x, y ( x, y ( x, y x, y (,0 ± 0.00 F F 0kN, 0kN ± kn F F 0kN, 0kN ± kn x, y F F 0kN, 0kN ± kn x, y The usual FEM [, 9 poceeds by the asseblage of a single lage syste of linea equations. Fo each stuctual eleent in the poble (see Fig., an eleent stiffness atix is ceated, expessed in tes of cos θ, sin θ, EA, and L, the eleent length. Fig. Aangeent of a single eleent connecting left-hand and ight-hand nodes.

4 Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations Since the node locations ae uncetain, the angles of the vaious eleents ae also inteval quantities. Howeve, the angles and the eleent lengths ae only iplicit inteval paaetes. Theefoe, by eans of the following substitutions, we can eaange each eleent atix so that it is expessed only in tes of the explicit inteval paaetes, viz. EA and the node coodinates, ( x l, yl and ( x, y : x xl (7 L y (8 L L ( x xl + ( y (9 This yields the following eleent stiffness atix: EA k (( x x + ( y y ( x xl ( x xl ( y ( x xl ( x xl ( y ( x x ( y y ( y y ( x x ( y y ( y y l l l l l l l l ( x x ( x x ( y y ( x x ( y y l l l l l ( x xl ( x xl ( y ( y ( x xl ( y ( y yl (0 The global syste stiffness atix K is assebled in the usual way. If we wee to solve the esultant syste of equations in the conventional fashion, i.e. by substituting each of the vaiables by its liteal value (in this case, intevals instead of floating-point nubes, we would need to apply a linea syste solve (e.g., inteval Gaussian eliination [,, using inteval aithetic whee equied. Howeve, we will see that in the inteval case such an appoach is hopeless. Instead, we ust stoe the syste atix in sybolic fo. It is woth entioning that altenative eleent stiffness atices can be obtained by the use of the following altenative tansfoation: t t,, ( + t + t whee θ t tan. In this case the eleent stiffness atix is as follows: EA L( + t ( t( ( t( t( t( t ( ( t( t( t t( t t t( k ( We shall use the foe tansfoation; the elative eits of each ae biefly discussed in Section... Paaetic Syste Solution We now have a syste of linea equations fo the node displaceents u i, vi, i,, : K u F, ( whee K is the global syste stiffness atix assebled fo the eleent stiffness atices k i, i, K,6, u ( u v T u v u v is the vecto of node displaceents and ( Fx y F x F y F x F F T y is the vecto of loading foces. F We will now conside the geneal case of a syste of linea equations with inteval paaetes. Suppose we have a linea syste A( p x b( p, ( whee the coefficients of the atix A ( p and the vecto b ( p ae functions of n paaetes vaying within given intevals aij( p aij( p, K, pn, bi ( p bi ( p, K, pn, i, j, K,, ( T p p ([ p, K,[. (6 [ The set of solutions to the above syste, called the paaetic solution set, is Σ A ( p, b( p,[ p : { x R p n Σ( A( p x b( p fo soe p [ p}. (7 The set Σ is copact if A ( p is nonsingula fo evey p [ p. Fo a nonepty bounded set S R, define its inteval hull by S : {[ s IR S [ s}. It is geneally expensive to obtain Σ o Σ, so instead we seek an inteval vecto Ω fo which it is guaanteed that Ω Σ Σ. We apply a geneal-pupose self-veified ethod fo bounding the solution set of a paaetic linea syste, which does not assue any paticula stuctue aong the paaete dependencies. This ethod deives fo inclusion theoy fo nonpaaetic pobles, see Ref. [6 and the efeences theein. In Ref. [6, a staightfowad genealisation to linea systes with linea paaete dependencies is given. The coesponding theoes can be odified and applied to linea systes

5 Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations involving nonlinea paaete dependencies [,. The following is a geneal foulation of the enclosue ethod fo linea systes involving abitay paaetic dependencies. Theoe Conside a paaetic linea syste defined by Eqs. ( (6. Let R R, [ y IR, ~ x R be given and define [ z IR, [ C IR by [z : { R ( b( p A( p ~ x p [ p}, (8 [C : { I R A( p p [ p}, (9 whee I denotes the identity atix. Define [ v IR by eans of the following Gauss-Seidel iteation T [ vi : {[ z + [ C ([ v, K,[ vi,[ yi, K,[ y } i, i. (0 If [ v [ y with [ vi [ yi, i, then R and evey atix A ( p with p [ p ae egula, and fo evey p [ p the unique solution xˆ A ( p b( p of the syste defined by Eqs. ( -(6 satisfies x ˆ ~ x + [ v. In ou coputations we have chosen R A ( and ~ ( x R b( p, whee p ( is the idpoint of [ p and ( ( A A( p. The above theoe genealises [6, Theoe.8 by equiing a shap enclosue of C( p : I R A( p fo p [ p, instead of using the inteval extension C ([ p. Exaples deonstating the application of the genealised inclusion theoe can be found in [,. A detailed desciption of this algoith can be found in [.. Initial Node Displaceent Intevals Afte applying the above paaetic solve, we have peliinay inteval enclosues fo the node displaceents. By eans of the following foula fo the FEM [9, evaluated using inteval aithetic, EA Si ( ui, ( L whee S is the esulting noal foce (eithe tension i o copession in eleent i and u ( i ul v l u T v is the vecto of displaceents fo the i th eleent s left- and ight-hand nodes, peliinay inteval enclosues fo the eleent foces can also be obtained.. Inteval Tightening (Eleent Foces At each fee-oving node (nodes,, and in ou exaple, all foces (eleent foces and loading foces ust be in equilibiu, in both the x - and y -diections. Fo exaple, at node, the following ust hold: S + Fx S + S ( S + y + θ ( F S S sin These can be eaanged to give one o oe explicit foulae fo each eleent foce. Again at node, we have: S + S Fx S ( S S S S S S + S S S S S F S S y + F + F S S x y + F + F x y ( (6 (7 (8 (9 We apply the following inteval tightening (also soeties known as puning technique: Using the cuent values fo the eleent foces, evaluate each of the above foulae in tun, to obtain new inteval enclosues fo the foces. Fo each eleent foce, the cuent inteval value is intesected with the new coputed enclosue(s, yielding a naowe o identical inteval. This pocedue is iteated (fo all nodes as desied until the set of esulting set of intevals fo

6 6 Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations the eleent foces do not contact any futhe.. Inteval Tightening (Node Displaceents Using Eq. ( instead of the foce equilibiu equations, the above pocedue could siply be applied in a siila fashion in ode to contact the intevals fo the node displaceents. While this does indeed achieve a significant contaction of the displaceent intevals, thei widths ae still wide than one would like. This educed effectiveness is due to the geate nube of inteval quantities appeaing in Eq. (. We theefoe eploy a slightly oe sophisticated puning technique. Fistly note that L i, the length of eleent i, is elated to its noal foce S i by EA( Li Li 0 Si, (0 L whee i 0 i 0 L is the stating length of eleent i (i.e., befoe any loading foces ae applied. Fo this pocedue, we take the new, tight enclosues fo the eleent noal foces obtained above, and fo Eq. (0 we use inteval aithetic to calculate an inteval value fo L, i, K,6. These L i inteval values will stay fixed. Now conside eleent i. Assue fo the tie being that the displaceent of its left-hand node is a known point value, as is the angle θ i. Given that we know the length, L i, to within cetain bounds, what is the set of possible displaceents of the ight-hand node ( u, v which will satisfy this length equieent? As illustated in Fig., this set is bounded by two paallel lines which ae pependicula to the eleent. Now conside eleent j, whee i j, whee eleents i and j shae the sae ight-hand node. Making the sae assuptions about eleent j, and povided that the two eleents ae not paallel, the set of possible displaceents which will satisfy both length equieents is bounded by the intesection of i Fig. Bounds on the displaceents of a node due to two eleents of inteval length. two such pais of paallel lines, which descibes a paalleloga (see Fig.. By taking the sallest bounding box suounding this paalleloga, we obtain new bounds fo ( u, v. Howeve, the displaceents of the left-hand nodes and the angles of the eleents ae not point values, which coplicates the issue. We thus pusue a cobinatoial solution: Let each of these inteval values take eithe thei left -o ight -endpoint. Thee ae 6 6 possible peutations aong { ui, vi, θ i, u j, v j, θ j} l l l l. Fo each such peutation, we can copute the paalleloga intesection and its bounding box. We copute the sallest bounding box containing all these paallelogas as a new inteval enclosue fo ( u, v. Taking each node in tun, the ethod thus poceeds as follows: Take evey possible pai of non-paallel eleents which eet at the node, in tun. Fo exaple, at node we ay conside eleents and, which eet thee, followed by eleents and (but not eleents and, which ae paallel, i.e., the intesection of thei inteval angles θ and θ is non-epty.

7 Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations 7 Fo each such pai, copute an inteval enclosue fo the displaceent of thei coon node, in the x - and y -diections, as above. Take the intesections of these new inteval(s with the cuent values fo the displaceent of the node. Again, this pocedue is iteated (fo all nodes as desied until the set of esulting set of displaceent intevals do not contact any futhe.. Results In this section the afoeentioned ethods ae applied to the odel descibed in Section. We ai to copute intevals fo the eleent foces and node displaceents that ae guaanteed to contain the tue solution, which consists of the set of inteval anges fo these quantities when each inteval paaete is allowed to vay independently within its doain. Ou solution intevals should enclose the tue solution as tightly as possible, iniising the oveestiation associated with the well-known dependency poble in inteval aithetic, e.g., Ref. [. To obtain the initial values fo the displaceents u i, v i, i,,, an existing ipleentation of the paaetic syste solve fo the Matheatica envionent has been used [. The othe steps have been ipleented in C++; apat fo the Monte Calo ethod, which is un by way of copaison to estiate the tue inteval solution to the poble, these ae inteval ethods with inteval vaiables and paaetes, using inteval aithetic in place of floating-point aithetic. The coputational esults ae thus guaanteed, even accounting fo ounding eos. The C++ inteval libay filib++ [7 is eployed.. Monte Calo Siulation We wish to fistly copute a tight inne estiation to the tue inteval solution to the poble, fo which the well-known Monte Calo ethod is used. This is only done so as to obtain a close appoxiation to the tue esult, so as to be able to judge the quality of the guaanteed solution obtained by the othe ethods. All stating inteval paaetes ae eplaced by point values which ae andoly chosen within thei doains, and the point poble is solved, using the standad FEM. The esult intevals ae coputed as the inteval hulls (see Section. of the solutions to the point pobles. Sufficient (hee, 0 6 point pobles ae un in ode to povide a elatively tight inne estiation to the tue inteval solution. The inne estiations fo the eleent foces and node displaceents obtained by the Monte Calo siulation ae given in Table.. Finite Eleent Method With the vaiable substitution (7-(9, the FEM yields a syste of inteval equations. Howeve, the liteal intevals appeaing in the syste stiffness atix K ae of sufficiently lage width that it is not Table Copaison of esults. Inne Estiation (Monte Calo Oute Estiation (Paa. Sol. & Tightening S [ , [0.687,8.607 S [6.690,76.86 [6.7680,8.776 S [ , [ ,-.679 S [-7.760,.8 [-.6896,7.806 S [08.670, [07.097, S 6 [-0.089, [-0.60, u [ , [ , v [-0.000, [ , u [ , [ , v [-0.00, [ , u [ , [ , v [ , [ ,

8 8 Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations possible to solve the syste using inteval Gaussian eliination with patial pivoting, e.g., Ref. [8 and [,. A naive application of the FEM in the inteval case will alost always fail o delive esult intevals that ae hopelessly wide. We note that by using the altenative vaiable substitution ( the inteval enties of K actually becoe slightly naowe. Howeve this does not suffice fo the syste to becoe solvable. Also, this tansfoation is less suitable fo the paaetic solution, due to the pesence of iplicit inteval paaetes fo the eleent lengths (which depend on the explicit paaetes fo the node coodinates. This causes the esult intevals fo the diplaceents to be wide, since this dependency is not taken into account.. Paaetic Solution The paaetic solve fo Section. delives intevals fo the node displaceents which ae given as the stating values in Table. Applying Eq. (, these values ae used to geneate intevals fo the eleent foces, which ae given as the stating values in Table. By theselves, these esult intevals fo the displaceents ae athe wide and thus not copletely satisfactoy. The esulting intevals fo the eleent foces ae uch too wide and ae unsatisfactoy. The coputation tie was. seconds on a PC with an AMD Athlon-6 GHz pocesso unning the Matheatica envionent. It should be noted that such a paaetic solution apidly becoes vey tie-consuing fo lage systes.. Inteval Tightening (Eleent Foces The esults of applying the inteval tightening pocedue (Section. to the eleent foces obtained above ae given in Table. The intevals convege apidly in the fist couple of iteations; 0 iteations suffice to achieve convegence to 8 decial places, fo which the coputation tie is negligible. The final intevals fo the eleent foces ae given in Table. Copaed to the inne estiates obtained Table Results of inteval tightening on the displaceents. Node Displaceent Value Stating Values u [ , v [ , u [ , v [ , u [ , v [ , Iteation u [ , v [ , u [ , v [ , u [ , v [ , Iteation u [ , v [ , u [ , v [ , u [ 0.008, v [ , Iteation u [ , v [ , u [ , v [ , u [ , v [ , fo the Monte Calo ethod, we see that the intevals fo S, S, and S 6 ae tight. Those fo S, S, and S ae not quite so tight, but still acceptable.. Inteval Tightening (Node Displaceents The esults of applying the inteval tightening pocedue (Section. to the node displaceents obtained by the paaetic solution ae given in Table. Hee, iteations suffice to achieve convegence to 8 decial places. Again, the coputation tie is negligible. By copaing with the inne estiates fo the Monte Calo ethod (see Table, we see that the inteval enclosues fo the node displaceents ae all

9 Veified Solution fo a Statically Deteinate Tuss Stuctue with Uncetain Node Locations 9 Table Results of inteval tightening on the eleent foces. Eleent Foce Value Stating Values S [.96888, S [ 8.806,98.8 S [ 0.8, S [ 90.66, S [ 88.77, S [ , Iteation S [.96888, S [ 8.806,98.8 S [ 0.8, S [ 9.907, S [ ,9.896 S [ , Iteation S [0.80,.087 S [ 8.806,98.8 S [ 0.8, S [ 7.798, S [0.0678,8.777 S 6 [ , Iteation S [0.068, S [.9879, S [ , 0.98 S [ ,7.9 S [ ,6.677 S [ , Iteation 0 S [0.687,8.607 S [6.7680,8.776 S [ ,.679 S [.6896,7.806 S [07.097, S [ 0.60, of a siila quality, about twice the width of the tue solution. This is a noticable ipoveent on the values obtained fo the paaetic solve alone. 6. Conclusions We have consideed a statically deteinate tuss odel fo which the node locations, as well as all othe paaetes, ae uncetain. We have pefoed a suitable vaiable substitution in ode to apply a paaetic solve and have devised inteval tightening pocedues fo both the eleent foces and node displaceents, which delive a significant ipoveent to the esults. Though the use of inteval aithetic, the esult intevals ae guaanteed to contain the tue solution. The eaining oveestiation is due to soe lingeing occuences of the dependency poble, at least in the cuent foulation. Initial investigations have shown that it ay be possible to ipove the esults obtained by the paaetic solve, by augenting the syste of equations and the syste stiffness atix with additional equations and vaiables fo the eleent foces (, adding exta dependencies to the syste. Howeve, the esultant ipact on the tightening pocedue is inial. In futue, we wish to exploe how effectively the ethod ay be applied to tuss stuctues with a geate nube of eleents and nodes. As a fist attept, we conside in Ref. [8 the statically indeteinate tuss stuctue which is obtained fo the odel descibed in Section by the addition of a seventh eleent connecting nodes and. It is necessay to exploit onotonicity aguents to keep the oveestiation sall. Acknowledgeents We gatefully acknowledge suppot fo the State of Baden-Wüttebeg, Geany, and thank Pofesso E. Popova fo aking he Matheatica softwae available to us. Refeences [ G. Alefeld and J. Hezbege, Intoduction to Inteval Coputations, Acadeic Pess, New Yok, 98. [ K.-J. Bathe, Finite Eleent Pocedues, Pentice-Hall, Englewood Cliffs, 99. [ G. Coliss, C. Foley and R. B. Keafott, Foulation fo eliable analysis of stuctual faes, Reliable Coputing ( [ O. Dessobz, F. Thouveez, J. P. Laîné and L. Jézéquel, Analysis of echanical systes using inteval coputations applied to finite eleent ethods, Jounal of Sound and Vibation 9 ( (

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