Kinematically Admissible Failure Mechanisms for Plane Trusses

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1 OP Conference Series: Mterils Science nd Engineering PAPE OPEN ACCESS Kinemticlly Admissible Filure Mechnisms for Plne Trusses To cite this rticle: Jonn Kosowsk et l 0 OP Conf. Ser.: Mter. Sci. Eng. 00 View the rticle online for updtes nd enhncements. elted content - The relibility model of the fult-tolernt computing system with triple-modulr redundncy bsed on the independent nodes P A hmn nd E Yu Bobkov - Using specilized Mrkov chin in the relibility model of disk rrys AD-0 with dt mirroring nd striping P A hmn - The relibility model of the fult-tolernt border routing with two nternet services providers in the enterprise computer network P A hmn nd E Yu Bobkov This content ws downloded from P ddress..0. on //0 t 0:0

2 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 Kinemticlly Admissible Filure Mechnisms for Plne Trusses Jonn Kłosowsk, Pulin Obr, Jn Turnt Kielce University of Technology, Fculty of Civil Engineering nd Architecture, l. Tysiącleci PP, - Kielce, Polnd Lodz University of Technology, Deprtment of Technicl Mechnics nd nformtics, ul. Stefn Żeromskiego, 0- Łódź, Polnd jklosowsk@tu.kielce.pl Abstrct. The relibility models of stticlly indeterminte steel trusses re nlysed in this pper. The plne trusses re considered. For nlysed structures types of relibility model re determined. dentifiction of system relibility is bsed on studies on the trnsformtion from the sfe structurl system into the geometriclly vrible system (mechnism). These reserches intended to determine the kinemticlly dmissible filure mechnisms which contin miniml criticl sets of elements. For nlysed truss the formul setting out the number of mechnisms for ny number of repetble sections is determined. To identify the mechnism, spectrl nlysis of the liner stiffness mtrix is used. This stge of the study, in conjunction with considertion of sttic lods, llows to determine the minimum criticl set of rods corresponding to the most probble scenrio of the structurl dmge. For these nlysis the computer progrm bsed on the finite element method hs been creted within the Mthemtic environment.. ntroduction The sfety nd relibility of structures re very importnt in the design process nd on the opertion stge during the lifetime of structure. Steel trusses mong the most importnt nd the simplest structures. The most dngerous sitution for trusses is dmge of rods converting truss into mechnism cusing globl dmge of the system. The subject of relibility nlysis is to determine the probbility of filure, which is understood s exceeding ultimte limit stte [-]. System nlysis in ultimte limit stte of rod structures, nd consequently identifiction of relibility system, is bsed on studies on the trnsformtion the sfe structurl system into the mechnism. These reserch re intended to determine kinemticlly dmissible filure mechnisms (KAFM) which contin miniml criticl sets of elements (MCSE) [,,- 8]. MCSE is collection which is chrcterized by the property tht if only one element is operble, the entire system is operble the structure is ble to trnsfer cting lods. Exhustion of the lod of ll the elements included in the custive MCSE mkes the structure is converted into the geometriclly vrible system. elibility nlysis of structurl system consists of identifiction of relibility models nd estimtion of filure probbilities of individul modes nd the overll system. n this pper the relibility models of stticlly indeterminte plne trusses re constructed. This theme is very importnt becuse the Content from this work my be used under the terms of the Cretive Commons Attribution.0 licence. Any further distribution of this work must mintin ttribution to the uthor(s) nd the title of the work, journl cittion nd DO. Published under licence by Ltd

3 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 probbility of filure depends on the relibility model of structures. For these nlysis the computer progrm bsed on the finite element method hs been creted within the Mthemtic environment. This stge of the study, in conjunction with considertion of sttic lods, llows to determine the minimum criticl set of rods corresponding to the most probble scenrio of the structurl dmge.. Methods nd Theory.. Kinemticlly dmissible filure mechnisms To evlute the relibility of the system one of three types of relibility model (Figure ): seril, prllel nd mixed (prllel-seril or seril-prllel) system is used [,]. A system tht is functioning if nd only if ll of its components re functioning is clled seril system (Figure ). For such structures the higher number of members, the lower lod bering cpcity nd relibility. The relibility of the seril system is clculted s follows: S i k i () where i is relibility of single element nd k is the number of elements. The seril system is pproprite for structures tht re sttisticlly determinte. n this cse number of miniml criticl sets of elements is k (k MCSE) nd the number of custive elements MCSE is l=. ) i k i k MCSE (l=) k b) c) d) j m i i k k MCSE (l=k) j m i k i k j ji jk m mi mk i k i k j ji jk m mi mk m MCSE (l=k) MCSE (l=k) Figure. Types of relibility model: ) seril system, b) prllel system, c) prllel-seril system, d) seril-prllel system A system tht is functioning if t lest one of its components is functioning is clled prllel (Figure b). This system is pproprite for some structures tht re stticlly indeterminte nd the relibility is clculted s follows: k ( ). () S i i For prllel system there is one miniml criticl set of elements ( MCSE) nd number of custive elements MCSE is l=k. n the cse of complex structures, there is usully need to identify mixed systems, which re combintion of prllel nd seril systems [,8]. There re two bsic systems s prllel-seril (Figure c) nd seril-prllel (Figure d). The relibility for the first system is clculted s: m k S ( ji ), () j i where m is the number of columns nd k is the number of elements in columns. The relibility for the second system is clculted s:

4 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 k m S ji. () i j where k is the number of rows nd m is the number of elements in rows. These systems re chrcterized by multiple MCSE ssocited with KAFM. However in rel structures, mixed systems re usully more complicted [,]. Some exmples of the mixed systems re presented further in the pper... Spectrl nlysis of liner stiffness mtrix The spectrl nlysis is widely used in mny res of humn investigtions. Between others it is one of the most populr techniques in structurl helth monitoring procedures [0]. To identify mechnism of truss the spectrl nlysis of the liner stiffness mtrix is used: ( K L - ) q = 0 () where K L is the liner stiffness mtrix nd q is displcement vector. The eigenvlues of liner stiffness mtrix describe the energy sttes of the model, while the eigenvectors describe the form of its own deformtion. n the cse when ll the eigenvlues re greter thn zero there re no movements. Zero eigenvlues re relted to the finite or infinitesiml mechnisms, but in generl the informtion from the null-spce nlysis lone does not suffice to estblish the difference between them. The mechnism cn be considered s n eigenvector relted to zero eigenvlue. To estblish if the mechnism is infinitesiml it is necessry to pply the nonliner nlysis with the use of geometric stiffness mtrix []. The progrm bsed on the finite element method ppliction ws creted in Mthemtic environment to identify possible mechnisms in the nlyzed structure. The min functionlity of the progrm ws crry out spectrl nlysis for ll possible trusses hving removed given number of rods. To do tht ll possible combintions of given number of rods were generted nd liner stiffness mtrix ws ggregted. Next the eigenvlues of creted mtrix were serched nd zero eigenvlues ws identified. f zero eigenvlue ws found the structure ws clssified s the mechnism. As result of clcultions ll possible not repetble combintions of removed rods trnsforming truss to mechnism were generted.. Exmples The subject of further qulittive nlysis is supported, stticlly indeterminte plne truss which is composed of n repetble sections (Figure ). For this type of truss, the relibility nd dmge nlysis re considered in some ppers [,-]. n the pper five types of trusses, of vrying number of: repetble sections n, nodes ln nd rods le, re considered: n=, ln=, le=, n=, ln=, le=, n=, ln=8, le=, n=, ln=0, le=, n=, ln=, le=. Kinemticlly dmissible filure mechnisms (KAFM) which contin miniml criticl sets of elements (MCSE) re determined for ech cse. The relibility S of studied systems is clculted ssuming the sme relibility of ll the rods i =.

5 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 n= n= n= n= n= n ln- le- ln b 0 0 le- le- le le- ln- le- ln- Figure. Plne truss composed of n repetble sections with ln nodes nd le elements For truss which is composed of one repetble section (n=) two (, ) kinemticlly dmissible filure mechnisms re determined (Figure ) []. The KAFM consists of seven MCSE with two custive elements (l=) nd it is divided into two KAFM: A nd B. For A KAFM the filure of ny pirs of elements - results in the filure of the whole structure nd for B KAFM the structure filure occurs s the result of the filure of both cross-brces (,). The relibility A B for this mechnism is clculted s for the prllel-seril system (). The KAFM consists of one MCSE with two custive elements like KAFM but it is mixed system. The structurl filure is cused by the filure of one of cross-brces (,) nd one of the elements -. elibility is clculted s for the serilprllel system (). All KAFMs re connected in seril wy. Therefore, the relibility of the whole truss is clculted s follows: T ; For this truss number of mechnisms is nd ll mechnisms consist of two custive elements. A KAFM B KAFM KAFM MCSE (l=) MCSE (l=) MCSE (l=) Figure. Kinemticlly dmissible filure mechnisms for n= For truss which is composed of two repetble sections (n=) three kinemticlly dmissible filure mechnisms re determined (Figure ). Miniml criticl sets of elements (MCSE) re presented in the Figure. n this cse there re mechnisms: 0 which consist of two custive elements (l=) (- KAFMs) nd contining three custive elements (l=) ( KAFM). The relibility is clculted in the sme wy s t the first exmple: 8 T ; ;.

6 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 B KAFM 0 MCSE (l=) A KAFM 8 MCSE (l=) 8 KAFM 0 8 MCSE (l=) KAFM 8 0 MCSE (l=) Figure. Kinemticlly dmissible filure mechnisms for n= For truss which is composed of three repetble sections (n=) four kinemticlly dmissible filure mechnisms re determined (Figure ). Miniml criticl sets of elements (MCSE) re shown in the Figure. n this cse there re mechnisms: which consist of two custive elements (l=) (- KAFMs), 0 consisting of three custive elements (l=) ( KAFM) nd which consist of four custive elements (l=) (V KAFM). The relibility of the whole truss is clculted s follows: 0 T ; ; V ; V. A KAFM B KAFM 0 8 MCSE (l=) MCSE (l=) KAFM 0 8 MCSE (l=) KAFM 8 0 V KAFM MCSE (l=) 8 0 MCSE (l=) Figure. Kinemticlly dmissible filure mechnisms for n= For truss which is composed of four repetble sections (n=) five kinemticlly dmissible filure mechnisms re determined (Figure ). Miniml criticl sets of elements (MCSE) re shown in the Figure. n this cse there re mechnisms: which consist of two custive elements (l=) (- KAFMs), consisting of three custive elements (l=) ( KAFM), 0 which consist of four custive elements (l=) (V KAFM) nd consisting of five custive elements (l=) (V KAFM). The relibility of the whole truss is clculted s follows: T V V

7 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 ; ; ; V V. ; A KAFM 8 B KAFM MCSE (l=) 8 MCSE (l=) 0 KAFM 0 8 MCSE (l=) 8 KAFM 8 0 Figure. Kinemticlly dmissible filure mechnisms for n= For truss which is composed of five repetble sections (n=) six kinemticlly dmissible filure mechnisms re determined (Figure ). Miniml criticl sets of elements (MCSE) re shown in the Figure. n this cse there re mechnisms: 8 which consist of two custive elements (l=) (- KAFMs), consisting of three custive elements (l=) ( KAFM), which consist of four custive elements (l=) (V KAFM), 0 consisting of five custive elements (l=) (V KAFM) nd which consist of six custive elements (l=) (V KAFM). The relibility of the whole truss is clculted s follows: 8 0 T V ; ; V KAFM MCSE (l=) V KAFM MCSE (l=) MCSE (l=) V V ;

8 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 V ; V. V ; A KAFM 0 B KAFM 0 8 MCSE 8 (l=) 0 KAFM 0 MCSE (l=) 8 MCSE (l=) KAFM MCSE (l=) 8 0 V KAFM MCSE (l=) Figure. Kinemticlly dmissible filure mechnisms for n=. esults nd discussions The considertions performed in the pper llows to estblish reltions for every truss composed of n repetble sections for n. The number of nodes ln nd rods le is clculted s: ln n, () le n. The minimum number of filure rods converting structure into the geometriclly vrible system is l= nd mximl l=n+. The number of kinemticlly dmissible filure mechnisms is n+. The number V KAFM MCSE (l=) MCSE (l=) V KAFM

9 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 of miniml criticl sets of elements for KAFM is n, for n+, for V n+ etc., but the number of MCSE for KAFM is clculted s: l le n. () KAFM MCSE The formul setting out the number of mechnisms is computed s: lm lm lm n lm (8) n where lm is the number of mechnisms consisting of two custive elements (l=), lm n is the number of mechnisms which consist of n custive elements (l=n) nd lmn is the number of mechnisms consisting of n+ custive elements (l=n+): l = l =,,..., n l n lm lm n lm. 0 n n n l, 0 n l t is noticed tht the eqution () is not tken into ccount in the cse of truss which is composed of two repetble sections (n=). The first () nd the second () KAFM consist of mechnisms determined by (). KAFM is lwys the prllel-seril system of two custive elements. KAFM is the seril-prllel system in which the filure of one of cross-brces n repetble sections nd one of the nother elements cuses the filure of the whole structure. The n-th KAFM consists of mechnisms determined by () nd (n+)-th KAFM by (). To illustrte proposed pproch truss which is composed of eight repetble sections (n=8) is considered. The number of rods nd nodes is respectively le= nd ln=8. Nine kinemticlly dmissible filure mechnisms which consist of mechnisms re determined for this structure (Tble ). The relibility of this truss my be clculted using the nlysis presented for previous trusses. There is certin repetbility which llows to determine relibility s: V T 8 0 V ; ; ; V ; V ; V ; ; V. X V V V V, X ; () 8

10 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 Tble. Kinemticlly dmissible filure mechnisms for n=8 KAFM Number of MCSE Number of custive elements Number of mechnisms V 0 V 88 V V V 8 0 X. Conclusions n the pper the relibility of plne trusses is considered. New strtegy is presented to determine of the dominnt filure pths nd the structurl system filure probbility. The method llows to clculte the relibility for the stticlly indeterminte truss composed of n repetble sections. Presented method consists of three steps: identifiction of relibility models nd estimtion of filure probbilities of individul modes nd the overll system. To identifiction of relibility models kinemticlly dmissible filure mechnisms which contin miniml criticl sets of elements re determined. The spectrl nlysis of the liner stiffness mtrix is used to identify mechnism. The problem is solved using the finite element method implemented in Mthemtic environment. Bsing on obtined results the universl reltions on possible number of mechnism is determined for considered type of truss. eferences [] A. Biegus, Probbilistic nlysis of steel structures, (in Polish) PWN, Wrsw-Wroclw,. [] J. Murzewski, elibility of engineering structures, (in Polish) Arkdy, Wrsw 8. [] M. Gwóźdź, nd A. Mchowski, Selected studies nd clcultions for building structures using probbilistic methods, (in Polish) Crcow University of Technology Press, Crcow 0. [] K. Kubick, nd U. doń, Proposl for the ssessment of steel truss relibility under fire conditions, Archives of Civil Engineering, vol. (), pp. -, 0. [] A. Dudzik, nd U. doń, The relibility ssessment for steel industril building, Advnces n Mechnıcs: Theoreticl, Computtionl And nterdisciplinry ssues, pp. -, 0. [] Z. Kowl, nstruments of probbilistıc optimistion of lod bering cpcity nd relibility of stticlly indeterminte complex structures, Archıves of Civil Engineering, vol. 0 (), pp. -0, 0. [] Z. Kowl, The formtion of spce br structures supported by the system relibility theory, Archives of Civil nd Mechnicl Engineering, vol., pp. -, 0. [8] W. Dziubdziel, B. Kopociński, nd Z. Kowl, Ultimte bering cpcity of structurl systems with miniml criticl sets hving connection elements in pirs, (in Polish) Archives of Mechnics, vol., pp. -,. [] H. J. Li, nd Z. Z. Wng, "eserch on system relibility of plne steel truss," Applied Mechnics nd Mterils, Vols. -, pp. -, 0. [0] K. Dems, J. Turnt, Structurl dmge identifiction using frequency nd modl chnges, Bulletin of the Polish Acdemy of Sciences: Technicl Sciences, vol. (), pp. -, 0. [] W. Gilewski, J. Kłosowsk, nd P. Obr, Form finding of tensegrity structures vi Singulr Vlue Decomposition of compbility mtrix, Advnces n Mechnıcs: Theoreticl, Computtionl And nterdisciplinry ssues, pp. -, 0. [] V. Kltjri, A. Kveh, nd P. Mnsoorin, System relibility ssessment of redundnt trusses using improved lgebric force method nd rtificil intelligence, Asin Journl of Civil

11 OP Conf. Series: Mterils Science nd Engineering 80 (0) 00 doi:0.088/-8x///00 Engineering (Building nd Housing), vol. (), pp. -0, 0. [] V.S. Sundr, nd C.S. Mnohr, Updting relibility models of stticlly loded instrumented structures, Structurl Sfety, vol. 0, pp. 0, 0. [] A. Hi, A. Weigung, Y. Duohe, nd W. Binsheng, Sttic force relibility nlysis of truss structure with piezoelectric ptches ffixed to its surfce, Chinese Journl of Aeronutics, vol., pp. -, 00. [] A. Bgheri, H.. zeghi, nd G. Ghodrti Amiri, Detection nd estimtion of dmge in structures using imperilist competitive lgorithm, Shock nd Vibrtion, vol., pp. 0, 0. 0