Stochastic finite element method study on temperature sensitivity of the steel telecommunications towers

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1 CMM-11 Computer Methods i Mechaics 9 1 May 11, Warsaw, Polad Stochastic fiite elemet method study o temperature sesitivity of the steel telecommuicatios towers Marci M. Kamińsi 1 ad Jace Szafra 1 Professor, Departmet of Steel Structures, Faculty of Civil Egieerig, Architecture ad Evirometal Egieerig, Techical Uiversity of Łódź, Al. Politechii 6, 9-9 Łódź, POLAND Marci.Kamisi@p.lodz.pl, webpage: Assistat, Faculty of Civil Egieerig, Architecture ad Evirometal Egieerig, Techical Uiversity of Łódź, Al. Politechii 6, 9-9 Łódź, POLAND aceszafra@aol.pl Abstract This paper addresses a importat questio i structural aalysis how to efficietly model the ucertai temperature ifluece o the iteral forces i particular structural elemets of the steel telecommuicatio towers. The computatioal methodology proposed here is based o the traditioal Fiite Elemet Method eriched with the geeralized stochastic perturbatio techique ad the computatioal implemetatio is performed usig the FEM commercial system ROBOT oitly with the symbolic algebra computer system MAPLE. Cotrary to the previous straightforward differetiatio techiques, ow the local respose fuctio method is employed to compute ay order probabilistic momets of the structural state fuctios. The respose fuctio is assumed i the polyomial form, whose coefficiets are computed from the several solutios of the determiistic problem aroud the mea value of the radom temperature. This method is illustrated with the example of telecommuicatios tower with a height equal to 5 meters modelled as the 3D truss structure uder Gaussia radom temperature load, which leads to determiatio of probabilistic momets of the iteral forces ito this system. Keywords: stochastic fiite elemet method, steel structures, telecommuicatio towers, reliability aalysis 1. Itroductio The aalysis of structures with radom parameters plays a importat role i structural desig, optimizatio ad reliability modellig. There are several well established both mathematical ad umerical models eablig for iclusio of radomess i desig parameters ito the structural statics ad dyamics problems solutios [1,6]. Startig from the aalytical approaches based o the respose spectrum aalysis, through the crude Mote-Carlo stochastic simulatio util the computatioal spectral methods based o the Karhue-Loeve or polyomial chaos expasios of the iput radom fields. O the other had, there are some lower order stochastic perturbatio methods, however they have fudametal limitatios o the iput radom dispersio level, so that their applicatio to the real egieerig problems seems to be rather limited. This ew method is based o the Taylor expasio of ay desired order with radom coefficiets of all ucertai parameters ad state fuctios aroud their expected values. The secod ew idea here is a applicatio of the respose fuctio method i couctio with this geeralized stochastic perturbatio techique. We suppose that the structural respose may be represeted via some polyomial form of the iput radom parameter, whose order strogly depeds o the fial approximatio error. The coefficiets for this respose polyomium are computed from the several solutios of the origial problem obtaied for this parameter values tae aroud its mea value. The polyomial form of the respose fuctio leads to aalytical determiatio of its partial derivatives with respect to the iput radom parameter, which ca be fially employed for a symbolic determiatio of the probabilistic momets; it essetially differs from the previous straightforward solutio to the equatios of the icreasig order. The etire procedure is applied to aalyze the steel telecommuicatio tower i 3D stress state with radom temperature load applied uiformly o all the structural members. It shows that the geeralized stochastic perturbatio method coverges relatively fast (eight ad teth order approaches retur almost the same results) ad eables for fial determiatio of the reliability idices.. The goverig equatios of the problem We cosider the boudary value problem of the liear elastostatics cosistig of the followig equatios i the presece of radom temperature T: equilibrium equatios D T fˆ T, x,, x, y,z, (1) costitutive equatios T C T, x,, x, y,z, () geometrical relatios T D u T, x,, x,y,z, (3) Dirichlet boudary coditios u û, x, () vo Neuma boudary coditios T tˆ T, x. u N (5) u ``

2 CMM-11 Computer Methods i Mechaics 9 1 May 11, Warsaw, Polad The basic idea of the stochastic perturbatio approach is to expad all iput variables ad all the state fuctios of the give problem via Taylor series about their spatial expectatios usig some small parameter >, so that the followig expressio is employed for a radom temperature T ad the state variable q: 1 q q q T (6)! 1 T where T T T (7) T T T (8) deote the first ad the secod variatio of T about T. A symbol (.) represets the cosidered fuctio value tae at the expectatio of radom iput; all partial derivatives are evaluated at the expectatios. The, the expected value of ay state fuctio f(t) may be derived from the defiitio usig the expasio (6) i the followig way [3,]: 1 T q T pt E q q dt (9)! 1 T where E q T deotes the expected value of the fuctio qt ad where p T deotes the probability desity fuctio of exteral temperature. Quite similarly oe ca fid the higher order probabilistic momets of the structural respose. It is importat to metio that the itegratio provided i above equatio is ever carried out with ifiite upper ad lower bouds. Usually it is doe for some real values reflectig the physical meaig of the give radom iput variable. Aalogous issue is a expasio i the same relatio, which most frequetly is provided up to the teth order oly ad is a subect of the additioal umerical verificatio. Fially, we apply the perturbatio parameter ε=1, although iclusio of the aalytic expasio with respect to this parameter is also atural as far as the computer algebra system assists i all the computatioal procedures. We apply the formula () with the symmetry assumptio ad usig the itegral defiitios of all probabilistic momets of the iput radom variable to calculate the expectatio as q q 1 1 EqT q ( T) T! ( T) T T (1) q 6! 6 ( T)... 6 T A classical defiitio of the variace leads to the coefficiet of variatio illustratig a radom dispersio of the temperature; there holds qt T)] pt T) (11) q 1 q T! pt dt 1 T Higher order momets ad characteristics, lie the third ad the fourth oe are determied to recogize the type of probabilistic distributio of the structural state fuctios. Oe calculates the sewess ad urtosis from the followig relatios, where also stochastic perturbatio together with itegral defiitios of the momets eable for the discrete formulas. We compute i tur the sewess coefficiets as the ratio of the third cetral dt probabilistic momet ad the third power of the stadard deviatio i the followig way: S 3 qt T )] pt qt T )] pt ad urtosis as K qt T )] pt qt T)] pt dt dt dt 3 dt 3 (1) (13) As far as the perturbatio-based approach is implemeted, a reliable ad efficiet determiatio of higher order probabilistic momets may demad relatively larger expasios tha the expectatios but we apply oly lowest order terms to recogize the mai tedecies. It should be metioed that at this stage the proposed procedure is idepedet from a choice of the iitial probability distributio fuctio (oe ca employ the logormal oe, for istace [5]), however a satisfactory probabilistic covergece of the fial results may demad various legths of the expasios for radom variables with differet distributios. 3. Computatioal implemetatio As it is well-ow, the elastostatic aalysis i the Geeralized perturbatio-based Stochastic Fiite Elemet Method implemeted with the use of the respose fuctio approach proceed usig the followig determiistic algebraic equatios series: T q T R, K (1) where α=1,, idexes all iteratios ecessary for the respose fuctio approximatio, K ad R deote stiffess matrix ad the vector of odal forces. Let us remid that this is applied cotrary to the classical Direct Differetiatio Method, where oly a sigle zeroth order equatio, also of the determiistic ature, is solved T q T Q, K (15) ad this process is cotiued for higher order equatios havig the recursive form give below 1 K T T q T T Q T (16) The respose fuctios betwee the displacemet vector compoets ad radom temperature are assumed i the followig polyomial form [3,]: T ( ) 1 ( ) ( ) 1 q A T A T... A T, 1,..., N (17) where N is the total umber of degrees of freedom. These respose fuctios are foud usig the few traditioal FEM tests with varyig value of the radomized parameter ad, at the same time, some classical approximatio procedures implemeted i ay computer algebra system. Fially, we calculate the partial derivatives ecessary for probabilistic momets evaluatio from the recursive formula:

5 S - 9 6 S - 8 6 S - 7 6 6 5 S - 5 S - 5 6 S - 1 CMM-11 Computer Methods i Mechaics 9 1 May 11, Warsaw, Polad q ) i1 ( ) ( i A

1 (18) b It should be uderlied that radomess i the evirometal temperature is much more difficult i such a aalysis because practically temperature as the physical parameter may exhibit eormously large

3 5 S S S S - 5 S S - 1 CMM-11 Computer Methods i Mechaics 9 1 May 11, Warsaw, Polad q ) i1 ( ) ( i A b... A. 1 (18) b It should be uderlied that radomess i the evirometal temperature is much more difficult i such a aalysis because practically temperature as the physical parameter may exhibit eormously large radom fluctuatios; they are for sure sigificatly larger tha for traditioal structural parameters. Therefore, a usage of the stochastic perturbatio-based techique must proceed with as may orders as it is ustified by the probabilistic covergece aalysis. O the other had, such a algorithm eables for ay commercial or the academic FEM codes applicatio [].. Numerical experimets Computer aalysis was performed usig 3D model of the telecommuicatio tower with 5, meters height ad this tower was subected to the followig loadigs: self-weight, costat ad techology loads all adopted accordig to the Eurocodes. This model cosists of 7 two-oded beam elemets with liear shape fuctios ad 138 two-oded elastic 3D truss elemets oied i 77 odal poits (see Fig. 1), which was a subect of the eigefrequecy aalysis with radom parameters before [5]. Radom parameter i this aalysis is the temperature load of all the structural elemets varyig i the rage [-8 C, 5 C]. The temperature is treated as the Gaussia radom variable with the expected value T] equal to -15 C, while its stadard deviatio (through the coefficiet of variatio) is treated as the additioal desig parameter i this study. The FEM aalysis is performed usig the egieerig system ROBOT ad is utilized usig the geeralized stochastic perturbatio techique thas to the computer algebra system MAPLE, v. 13. First four probabilistic momets ad the additioal coefficiets are computed for this tower as the fuctios of the coefficiet of variatio of the temperature varyig i the rage α(t)=(.,.3). Temperature deformatio of the bars (Δl) i this system have bee tae recalculated i each temperature usig the well-ow formula statig that the elogatio of each member T T l, (19) l T where α T is thermal deformatio of the steel ad l stads for the iitial legth of the cosidered structural elemet. Sice the referece temperature is, the the explicit evirometal temperature is icluded ito this equatio. Therefore, taig this as the iitial Gaussia radom variable of the problem with the give first two probabilistic momets, oe may easily calculate the probabilistic characteristics of the deformatio of all bars. These are: the expected values lt l ET, E T () the variaces l T l VarT, Var T (1) th order cetral probabilistic momet lt l T, () T l T T, the coefficiet of variatio, asymmetry ad urtosis S l T ST, lt KT. K (3) A cycle of thirtee computatioal problems has bee carried out to determie the extremum values of iteral forces ad displacemets ito the tower. The structure give schematically i Fig. 1 is loaded with the dead load oly (excludig the wid effects) to show trasparetly the effects of the evirometal temperature, whose uiform value is applied o all structural members; some deformed cofiguratios are attached i Fig.. A A 6 6 S - 6 S - 5 S - 3 Fig. 1. FEM model of the telecommuicatio tower Fig.. Deformed cofiguratios of the tower for T=-8, 5 C A-A

4 CMM-11 Computer Methods i Mechaics 9 1 May 11, Warsaw, Polad Further aalysis of the probabilistic momets is performed with respect to two most loaded bars from segmets 5 (aroud the half of this tower) ad 9 (the lowest segmet). From the structural poit of view all the members are made of roud cross-sectios, Ø95 mm ad Ø8 mm ad we postpoe at all local bedig effects i this truss resultig from the bolted ad welded oits. The exemplary ormal forces variatios for those two bars are collected i Table o 1, where positive values are relevat to the compressio state, whereas egative to the tesio members. It is quite clear from this table that the compressio state i the lowest segmet of the tower icreases together with the exteral temperature the few times i the give temperature rage (the highest temperature the larger compressio). It should be uderlied that some selective symbolic aalysis is ecessary for all large scale probabilistic computatios cosiderig the few parameters or fuctios to be to determied ad draw for each degree of freedom, so that full, pure computatioal reliability aalysis without ay pre-selectio may be very time ad space cosumig, whereas fial visualizatio somewhat misleadig. Table 1. Normal forces variatios versus temperature chages Temperature F x i sectio 9 [N] F x i sectio 5 [N] -5 7,93 1,38-31,7 5,3-3 3, 9, - 37,3 13,15-1,7 17,7 3,6,99 1 6,76,9 9,87 8,8 3 53, 3,76 56,1 36, ,7,61 Next, we otice the respose fuctios relevat to the verified bars i two segmets with respect to the iput temperature withi the etire temperature rage. As oe may expect this iterrelatio is totally almost liear, without ay umerical discrepacies, local oscillatios, especially at the edges of the computatioal domai. It guaratees very good probabilistic covergece of the basic characteristics for those forces at least. The discrete results comig from the FEM computatios are mared i Figs. 3 ad with the blac dots, while the MAPLE symbolic processig result is a cotiuous red lie. Fig.. The respose fuctio of the ormal force, segmet o 5 Usig those fuctios we compute the expected values of the give forces for the mea value of the exteral temperature equal to -15 C (Figs. 5 ad 6 with respect to segmet o 9 ad o 5). Up to 1 th order expasios are compared to show a optimal legth of Taylor expasio ecessary i this particular case. Cosiderig the fact that the overall differeces are oticed i the total umerical error rage ad are ot quite clear from those graphs, we have attached the additioal Table o equivalet to the iput coefficiet of variatio α=,3. So that eglectig those margial fluctuatios oe ca coclude that the secod order is quite sufficiet i the view of a accurate determiatio of these particular variables. Nevertheless, they somewhat decrease together with a icrease of the iput coefficiet of variatio mared almost everywhere o the horizotal axis. The differeces i-betwee particular orders solutios are practically ivisible i Figs. 5 ad 6 startig from the 6 th order, while cosistetly with iitial coclusios, the secod order method is efficiet eough for. 1. Table. The expected values of ormal forces computer for various temperatures Perturbatio method order Expected values i the segmet 9 th Expected values i the segmet 5 th 38, , , , , , , , , ,11131 Fig. 3. The respose fuctio of the ormal force, segmet o 9 Fig. 5. The expected values of the ormal forces, segmet o 9

5 CMM-11 Computer Methods i Mechaics 9 1 May 11, Warsaw, Polad Fig. 6. The expected values of the ormal forces, segmet o 5 represeted by the covex curve. It starts from the begiig of the coordiates system, which perfectly reflects the fact that for zero iput radom dispersio the pure determiistic problem is obtaied. The perturbatio parameter ε is of rather smaller importace ad eve for maximum value of the parameter α the fial iterrelatio with the computed variace is almost liear, as it is illustrated i Fig. 8 above. It is worth to uderlie that the variability iterval for the perturbatio parameter is ot so small, because it obeys aroud % fluctuatios i plus ad i mius. A simple cosequece of the variaces ad the expected values computatios is the output coefficiet of variatio give i Fig. 9, which is directly proportioal to its iput couterpart. Aalyzig the particular values it is apparet that this system shows sigificat probabilistic dampig havig output coefficiets almost te times smaller tha the additioal iput parameters. There is practically o differece i-betwee various orders of the stochastic perturbatio-based aalysis i this case (see Fig. 9). Fig. 7. The variace of the ormal forces i the segmet o 9 Fig. 9. Coefficiet of variatio of ormal force, segmet o 9 Fially, the urtosis ad sewesses of the ormal forces i the segmets o 5 ad 9 are preseted computed accordig to the secod, fourth ad sixth order perturbatio theories, also as.,. 3. a fuctio of the iput coefficiet of variatio, for Fig. 8. The variace of the ormal forces i the segmet o 9 as the fuctio of the perturbatio parameter ε Figures 7 ad 8 show the variability of the most loaded elemet i segmet o 9 with respect to both iput coefficiet of variatio as well as the perturbatio parameter. Figure 6 shows that practically there is o differece i-betwee d, th ad 6 th order approximatios of this variace for iput coefficiet of variatio taig the values.,. 3. Similarly to previous aalyses i this area, a icrease together with a additioal icrease of the iput coefficiet is oliear ad Fig. 1. Kurtosis of the ormal forces i the segmet o 9

6 CMM-11 Computer Methods i Mechaics 9 1 May 11, Warsaw, Polad those two parameters are idepedet. Determiatio of urtosis although eeds higher order approximatios, sice the secod order results i small positive values, while higher orders fluctuate ito the clearly egative oes. Each time all those coefficiets (at least their absolute values) icrease together with the iput coefficiet of variatio, which eeds separate aalysis ad covergece computatioal studies. 5. Cocludig remars Fig. 11. Kurtosis of the ormal forces i the segmet o 5 Fig. 1. Sewess of the ormal forces i the segmet o 9 Computatioal aalysis provided i this paper documets that the temperature variatios typical for our climate may have a sigificat ifluece o the overall iteral forces ito the high ad tall spatial trusses lie the telecommuicatio towers studied here. These forces fluctuatios may result for higher segmets i the sig chage for the ormal forces, although a radomess i temperature, accordig to the proposed variability rage, has smaller importace i this cotext. The aalyzed elastic system exhibits probabilistic dampig sice a ratio of output to the iput coefficiets of variatio is decisively smaller tha the uity. Nevertheless, the output state variables for the tower appear to be Gaussia provided that the radom temperature is also Gaussia, but the method proposed is ot restricted to this type of radomess. This coclusio is draw o the basis of the sewess ad urtosis equal to with relatively small umerical error. This study may be exteded ext towards a fire simulatio ad its results o the reliability ad itegrity of the telecommuicatio towers, where temperature variatios will itermediately affect also some material properties of the structural steel, so that the output radomess will have apparetly oliear character. The iterestig issues i this cotext would be also elastoplastic aalysis of the tower structure, a applicatio of the alumiium istead of the stailess steel as the basic material as well as forced dyamics of the etire spatial truss i the presece of some radom parameters aalyzed separately or as a sigle iput vector with the partially correlated compoets. Acowledgmet The fiacial support of the Research Grat from the Polish Miistry of Sciece ad Higher Educatio uder the Research Grat No NN is highly appreciated. Refereces Fig. 13. Sewess of the ormal forces i the segmet o 5 The very apparet result i Figs. 1 ad 13 is that both coefficiets oscillate aroud, so that the output distributio may be treated with good accuracy as the Gaussia oe. Determiatio of the sewess is so stable, that secod, fourth ad sixth order quatities are exactly the same ad depeds almost liearly upo the iput coefficiet of variatio. This is a basic differece with respect to the Mote-Carlo aalysis, where [1] Bearoya H., Radom eigevalues, algebraic methods ad structural dyamic models, Applied Mathematics ad Computatio, 5(1), pp , 199. [] Hughes T.J.R., The Fiite Elemet Method - Liear Static ad Dyamic Fiite Elemet Aalysis. Dover Publicatios Ic., New Yor,. [3] Kamińsi M., Potetial problems with radom parameters by the geeralized perturbatio-based stochastic fiite elemet method. Computers& Structures, 88, pp. 37-5, 1. [] Kamińsi M., A itroductio to the plasticity problems usig the geeralized stochastic perturbatio techique. Computatioal Mechaics, 5(), pp , 1. [5] Kamińsi M., Szafra J., Eigevalue aalysis for high telecommuicatio towers with logormal stiffess by the respose fuctio method ad SFEM. Computer Assisted Mechaics ad Egieerig Sciece, 16, pp. 79-9, 9. [6] Kleiber M., Hie T.D., The Stochastic Fiite Elemet Method, Wiley, Chichester, 199.

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