Stability Problems of Pyramidal von Mises Planar Trusses with Geometrical Imperfection

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1 Stabty Probems of Pyramda von Mses Panar Trusses wth Geometrca Imperfecton MARTIN KALINA 1a 1 Insttute of Structura Mechancs 1 Brno Unversty of Technoogy 1 Facuty of Cv Engneerng, Veveří Str. 95, 60 00, Brno 1 CZECH REPUBLIC 1a kana.m1@fce.vutbr.cz Abstract: - The present paper deas wth an anayss of pyramda von Mses panar trusses wth or wthout nta geometrc. Modes wth rea geometrca and matera parameters were apped. The modes had the cross secton of a hot-roed stee profe. The paper descrbes a dynamc reaxaton method whch montors the potenta energy of the von Mses panar truss oaded by dspacement of top hnge. The transformed potenta energy was depcted n fat and three-dmensona maps. Statc eqbrum paths and postons of chosen bfurcaton ponts were found. Key-Words: - Pyramda von Mses panar truss, Nonnear system, Dynamca reaxaton method, Potenta energy, Statc eqbrum paths, Bfurcaton pont 1 Introducton Stabty s the capacty of a structura eement or a system to resst to forces dsturbng ts equbrum poston [1]. The stabty oss means a structure coapse, whch can occur due to a very sma dspacement. The sma dspacement can be a random whch s, n the structure eement, due to producton or assemby [-4]. The stabty and nta s are key phenomena nfuencng the reabty of sender thn-waed pate grders [5-8] and bar structures [9], [10]. There exst numerous studes examnng the nfuence of s on the stabty of bar structures oaded by statc oadng, see, e.g., [11], [1]. Many of safety crtca objects are assessed by appyng the methods of reabty theory or probabstc rsk assessment [13-16]. The nstabty of statc equbrum of the structure n reaton to nta s s generay a negatve phenomenon whch can be studed by the senstvty anayss [17-0]. The majorty of reabty studes of common budng structures are amed at mt states, and form an mportant subset of decson-makng probems based on mutpe crtera [1-3]. Optmzaton probems of rea sender structures are usuay soved on behaf of geometrcay and matera nonnear soutons by the fnte eement method [4-6]. Theoretca possbtes of modern geometrcay nonnear agorthms are, however, much arger than those generay apped to engneerng approaches. The appcaton of usua geometrcay nonnear methods to anayse the deformatons of rea structures gves the deformatons whch are reatvey sma n comparson wth engths of bars of frame systems. The chaenge of basc research became the stabty probems of eastc pyramda von Mses panar trusses wth arge dspacements [7]. A arge range of stabty probems can be studed on behaf of agorthms of modern nonnear mechancs [8]. The von Mses panar truss s a bar structure consstng of two bars connected wth one another n the top hnge [9], [30]. The study of von Mses panar truss s mportant for understandng the stabty probems of these bar structures. The am of the presented study, whch foows work pubshed n [3], s an anayss of stabty oss of hgh von Mses panar trusses wth nta s. The stabty anayss s soved, appyng the geometrcay nonnear souton. The conons of statc equbrum of the nonnear system are cacuated by means of the so-caed dynamc method whch searches for the deformatons of von Mses panar truss n dependence on forced dspacement of top hnge n the coordnate grd. The montorng of potenta energy of the top hnge makes t possbe to fnd the stabe statc eqbrum state, the unstabe statc eqbrum state and ther mutua transton between each other (bfurcaton pont) of the von Mses ISSN: Voume 1, 016

2 panar truss. The maps of transformed potenta energes were apped to ceary ustrate the vaues of potenta energes. Mode and method used Fg.1 presents the mode of the von Mses panar truss dvded n hnges and assembed of segments transferrng the norma stffness of transatona sprng, and the bendng stffness of rotatona sprng, see [3]. where s orgna ength of the segment, s the ength of the segment after dspacement. To cacuate the anguar dspacement of rotatona sprngs, the equaton w be specfed: y 1 y d 1, tan, (4) x x 1 where φ s the anguar dspacement of segment (the segment between hnges and +1). The curvature of the eft bar represents the nta geometrca (bow ). Ths geometrca was consdered here n form of one haf wave of the functon snus. w x L x a sn, (5) Fg.1 Von Mses panar truss The nterna force F n the transaton sprng, and the nterna moment M φ n the rotatona sprng w be cacuated as: F k d, M k d, (1) where k s the stffness of transatona sprng, d s the eongaton of transatona sprng, k φ s the stffness of rotatona sprng, and d φ s the anguar dspacement of rotatona sprng. The potenta energy U, accumuated between the transatona and the rotatona sprngs, w be, for the von Mses panar truss, subsequenty wrtten as: 1 U k n 1 d k n 1 where n s the number of transatona sprngs, and n φ s the number of rotatona sprngs. The deformaton state of the descrbed von Mses panar truss s accuratey gven by the poston of each hnge. Ths poston of hnge s determned by the coordnate (x, y ), where denotes the hnge ndex. The foowng equaton w be used to cacuate the eongaton of transatona sprng: d, d, x x y y, 1 1 () (3) where a s the amptude of one haf-wave of functon snus, L s the bar ength. Incudng ths nto the engths of segments, and of anges of rotatona sprngs, the bar w be modfed (bent) st n the nta state, so that the fna shape of von Mses panar truss w have zero stress. The mode s formuated as a nonnear dynamc system. Statc eqbrum paths of the mode are found by dynamca reaxaton thanks to the near vscous dampng force actng on the hnges. The mass of the bar s concentrated n hnges. Due to ths assumpton, the dynamc equatons can be derved, appyng the Newton prncpe, n the foowng equaton: dx dy dvx vx, dvy vy, 1 Rx cvx, m 1 R y cv y, m where c s the dampng coeffcent, m s the mass of the hnges, v x and v y are the veocty vector components of the hnges, and R x and R y are the vector components of the resutant force R by whch the sprngs act on the hnge. Ths system s soved by the Sympectc Euer method [31], whch was generated by rearrangement of the expcte Euer method. 3 Mappng of potenta energy The mode of the von Mses panar truss s oaded by the controed dspacement n pont x n/, y n/ (further on, ony top hnge) [3]. At each dspacement of top hnge, the von Mses panar (6) ISSN: Voume 1, 016

3 truss reaxes for a certan tme. After ths tme, the potenta energy of the top hnge s recorded. The controed dspacement takes pace from coordnate x from the vaue of haf of span s/ n horzonta drecton aong the coordnate x, unt the coordnate x of top hnge reaches the vaue of the bar ength. Vertca dspacement of the hnge s mapped wthn the <s/; L>. In the step when x=l, the vertca dspacement of top hnge takes pace opposte to the drecton of coordnate y, and the top hnge starts dspacng back to the orgna coordnate x, recorded wthn the nterva <L; s/>. In the step, when x = s/, the vertca dspacement of the top hnge takes pace agan opposte to the drecton of coordnate y, and the top hnge starts dspacng n the same drecton as t was n the nta step. The dspacement of the top hnge n the drecton y can be wrtten as the nterva <h; 0>. The whoe course of the mappng s so repeated unt the top hnge reaches the ast step. Ths occurs, as soon as the coordnate y = 0 and, at the same tme, the coordnate x = L. In Fg., there s presented the dspacement of the top hnge. The step of controed dspacement s denoted p. 4 Anayss of von Mses panar trusses The profe of bars was hot-roed IPE400, wth cross-secton area A = 8446 mm, second moment of area about mnor axs s I = m 4, and Young s moduus s E = 10 GPa. The reaxaton tme t re = 10 s. Modes were generated both wth and wthout nta geometrca. The nta geometrca was ntroduced on the eft bar n form of a haf wave of the functon snus wth a = 0.1 m. The pyramdaty changed accordng to ange α, see Fg.1. The foowng von Mses panar trusses: α = 45, α = 55 and α = 60 were studed, on the bass of parameter α. In Fg.3 to Fg.8, there s presented the transformed potenta energy (vertca axs) of these modes. The transformed potenta energy s wrtten wthn the nterva. The grafc presentaton was reguated by the conon that, f there occurs the vaue hgher than 1. n cacuaton of transformed potenta energy, ths vaue w be set as 1.. The top hnge dspaces n the step p = 0.0 m n coordnate raster x 0;8 ~ 6 and y 0;4 ~ 5. Fg. Descrpton of shftng of top hnge The potenta energy of top hnge s recorded n such a way. To be possbe to fnd when the von Mses panar truss s n the stabe statc eqbrum state or n the unstabe statc eqbrum state, t s sutabe to draw the vaues of transformed potenta energy U n a cear ustraton. The vaue of transformed potenta energy s obtaned from the reaton of potenta energy of the top hnge U to the vaue of potenta energy of the top hnge beng on the coordnate x n the haf span of the von Mses panar truss U, for the approprate coordnate y. x n / For any coordnate y, ths equaton can be wrtten n the foowng way: U U (7) U xn / Fg.3 U for von Mses truss wth α = 45 wthout Fg.4 U for von Mses truss wth α = 45 wth ISSN: Voume 1, 016

4 Fg.5 U for von Mses truss wth α = 55 wthout Fg.6 U for von Mses truss wth α = 55 wth Fg.8 U for von Mses truss wth α = 60 wth Thanks to ntroducton of nto the modes, the dagrams of potenta energy of the top hnge dspacng wthn the rectanguar raster x, y are not symmetrca, see Fg.8, Fg.6 and Fg.4. Extremes of transformed potenta energes, and/or courses of statc eqbrum paths can be found by a more throughout study. These extremes are dvded nto mnmums of potenta energy, stabe statc eqbrum state, and maxmums of potenta energy, and nto unstabe statc eqbrum state. Bfurcaton ponts can be found n modes wth s. The ponts are concerned where the stabe statc equbrum state s n contact wth the unstabe statc equbrum state. Coordnates and vaues of the transformed potenta energy of bfurcaton ponts are gven n Tabe 1. Ange Symbo Coord. X Coord. Y U 45 B 1.98 m.39 m B 4.07 m 4.11 m B m.13 m B 4.46 m 1.86 m Fg.7 U for von Mses truss wth α = 60 wthout It can be seen n Fg.3 that the transformed potenta energy of top hnge does not decrease at ts dspacement. The von Mses panar truss s becomng stabe. Opposte to t, by ntroducton of or by ncreasng the pyramdaty, the transformed potenta energy decreases. 60 B m 4.71 m B 6.16 m 1.57 m Tabe 1 Coordnates and vaues of transformed potenta energes U of bfurcaton ponts ISSN: Voume 1, 016

5 5 Concuson The present paper studes stabty probems by transformed potenta energy of pyramda von Mses panar trusses. The mathematca procedure of the apped method of dynamc reaxaton whch montored the transformed potenta energy of top hnge was demonstrated. The process of generaton of fat and three-dmensona maps of transformed potenta energy was presented. The map has shown that, for the von Mses panar truss wth ower pyramdaty, the transformed potenta energy does not decrease. Further on, the maps have shown that ntroducton of or ncrease of pyramdaty nfuences the stabty of the von Mses panar truss. Statc eqbrum paths were determned and postons of bfurcaton ponts were dentfed, based on these facts. Acknowedgement The artce was eaborated wthn the framework of project GAČR S. References: [1] T. V. Gaambos, Gude to Stabty desgn Crtera for Meta Structures, John Wey & Sons, 1998, p [] Z. Kaa, J. Mecher, L. Pukcky, Matera and Geometrca Characterstcs of Structura Stees Based on Statstca Anayss of Metaurgca Products, Journa of Cv Engneerng and Management, Vo.15, No.3, 009, pp [3] Z. Kaa, Eastc Latera-torsona Buckng of Smpy Supported Hot-roed Stee I-beams wth Random Imperfectons, Proceda Engneerng, Vo.57, 013, pp [4] J. Vaeš, Effect of Random Axa Curvature on the Performance of Open and Cosed Secton Stee Coumns, AIP Conference Proceedngs, Vo.1558, 013, pp [5] Z. Kaa, J. Kaa, Resstance of Pate Grders Under Combned Bendng and Shear, In Proc. of the 3rd WSEAS Int. Conf. on Engneerng Mechancs, Structures, Engneerng Geoogy (EMESEG 10), Corfu Isand (Greece), 010, pp [6] J. Kaa, Z. Kaa, Large-defecton-theory Anayss of the Effect of Web Inta Curvature on the Utmate Strength of Stee Pate Grder, AIP Conference Proceedngs, Vo.1389, 011, pp [7] Z. Kaa, J. Kaa, M. Škaoud, B. Tepý, Senstvty Anayss of the Effect of Inta Imperfectons on the () Utmate Load and () Fatgue Behavour of Stee Pate Grders, Journa of Cv Engneerng and Management, Vo.11, No., 005, pp [8] Z. Kaa, J. Kaa, Resstance of Thn-waed Pate Grders under Combned Bendng and Shear, WSEAS Transactons on Apped and Theoretca Mechancs, Vo.5, No.4, 010, pp [9] A. Omshore, Verfcaton of Desgn Procedures of Structura Stabty usng Probabstc Methods of Reabty Anayss, AIP Conference Proceedngs, Vo.1479,, No.1, 01, pp [10] Z. Kaa, Reabty Anayss of the Latera Torsona Buckng Resstance and the Utmate Lmt State of Stee Beams, Journa of Cv Engneerng and Management, Vo.1, No.7, 015, pp [11] Z. Kaa, J. Vaeš, Modeng and Statstca Approaches to Latera-torsona Buckng, Advanced Materas Research, Vo.969, 014, pp [1] J. Vaeš, Anayss of Load-carryng Capacty of Thn-waed Cosed Beams wth Dfferent Senderness, Advanced Materas Research, Vo.969, 014, pp [13] Z. Kaa, Infuence of Parta Safety Factors on Desgn Reabty of Stee Structures - Probabty and Fuzzy Probabty Assessments, Journa of Cv Engneerng and Management, Vo.13, No.4, 007, pp [14] Z. Kaa, Senstvty and Reabty Anayses of Latera-torsona Buckng Resstance of Stee Beams, Archves of Cv and Mechanca Engneerng, Vo.15, No.4, 015, pp [15] A. Omshore, Uncertanty Anayss of the Cross-sectona Area of a Structura Member, In Proc. of the 4th WSEAS Int. Conf. on Engneerng Mechancs, Structures, Engneerng Geoogy (EMESEG 11), Corfu Isand (Greece), 011, pp [16] A. Omshore, Uncertanty Forecastng n Cv Engneerng, Probems and Appcatons, In Proc. of the 6th WSEAS Int. Conf. on Apped and Theoretca Mechancs (MECHANICS '10), Athens (Greece), 010, pp [17] Z. Kaa, Senstvty Anayss of Carryng Capacty of Stee Pane Frames to Imperfectons, AIP Conference Proceedngs, Vo.1048, 008, pp [18] Z. Kaa, J. Kaa, Senstvty Anayss of Stabty Probems of Stee Structures usng She Fnte Eements and Nonnear Computaton Methods, WSEAS Transactons ISSN: Voume 1, 016

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