Available online at ScienceDirect. Procedia Engineering 150 (2016 )

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1 Avalable onlne at ScenceDrect Proceda Engneerng 150 (2016 ) Internatonal Conference on Industral Engneerng, ICIE 2016 Analyss of Systems wth Unlateral Constrants through the Fnte Element Method n the Form of a Classcal Mxed Method A. V. Ignatyev a, *, V. A. Ignatyev a, E. V. Onschenko a a Volgograd State Unversty of Archtecture and Cvl Engneerng, 1 Akademcheskaya Street, Volgograd , Russa Abstract The paper gves an algorthm for the analyss of systems wth unlateral constrants through the fnte element method n the form of a classcal mxed method. Gven that both dsplacements and forces are varables n the system of governng equatons, the procedure for the ncremental parametrc loadng as a part of the analyss of structurally nonlnear systems allows varyng the respectve parameter, tendng from «hard» loadng based on the load parameter towards «soft» loadng based on the dsplacement (confguraton) parameter. The effcency of the algorthm s shown by the example of a beam wth unlateral constrants The Authors. Publshed by by Elsever Ltd. Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the organzng commttee of ICIE Peer-revew under responsblty of the organzng commttee of ICIE 2016 Keywords: fnte element method n form of classcal mxed method; systems wth unlateral constrants 1. Introducton Systems wth unlateral constrants are often encountered n engneerng practce and are consdered structurally nonlnear,.e. systems where the analytcal model vares n the process of loadng, erecton or operaton [1-]. The ssue of the analyss of systems wth unlateral constrants has been a focus of consderable nterest snce the 50s of the twenteth century. The smplest cases for whch exact solutons can be found were consdered as early as n the works of I.M. Rabnovch [4, 5]. Currently, the analyss of structurally nonlnear systems wth unlateral constrants s most frequently carred out applyng the lnear and nonlnear programmng methods. * Correspondng author. Tel.: E-mal address: algnat70@yandex.ru The Authors. Publshed by Elsever Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the organzng commttee of ICIE 2016 do: /j.proeng

2 A.V. Ignatyev et al. / Proceda Engneerng 150 ( 2016 ) The paper of A.V. Perelmuter [6] descrbes the analyss of a system wth unlateral constrants as reduced to the quadratc programmng problem. The paper [7] focused on the analyss of systems wth unlateral constrants suggests a drect method of successve approxmatons termed by the authors as the method of compensatng loads accordng to ts physcal meanng. The papers [7-9] by some specfc examples show the possble ways of the method applcaton for the analyss of systems wth unlateral lnearly elastc, nonlnearly hardenng and softenng constrants. Other approaches to the problem are proposed n [10-17]. The most comprehensve revew of works devoted to analyss of systems wth unlateral constrants s contaned n [7, 8]. Ths paper descrbes the algorthm for the analyss of systems wth unlateral constrants based on the classcal mxed fnte element method [10, 15]. Its basc dfference from the tradtonal dsplacement-based FEM s that the nodal varables of the fnte element mesh nclude both dsplacements and forces. Accordngly, whle constructng the system of governng equatons, nstead of the FE stffness matrx, we use the FE response matrx for the acton of each of the varables at unt values. Such system of governng equatons offers broad opportuntes for the analyss of the system's behavor both wth parameters varyng and acton type varyng. Ths means the ncremental addtonal loadng parameter can be vared from «hard» ncremental addtonal force loadng, where the system confguraton s determned for each step, toward «soft» ncremental addtonal stran loadng through settng the system confguraton and fndng the matchng loadng. The analyss of systems wth unlateral constrants as structurally nonlnear systems applyng the ncremental loadng procedure requres tracng at each step for gaps fllng between the structure and unlateral constrants, for changes of the reactons sgns n these constrants and for the detachment of the structure from the constrants. The most sutable method to trace and change the analytcal model at the stage followng any partcular step s the classcal mxed fnte element method. The algorthm for the analyss of bar systems wth blateral constrants applyng ths method s set out n the papers [10, 18, 19]. 2. Constructon of the system of governng equatons Consder the example of a contnuous beam wth unlateral constrants subject to a gven load (fg.1, ). For ths beam we determne ts stress-stran state after each loadng step (confguraton and forces) and the fnal stress-stran state upon reachng the specfed level of loadng. Fg. 1. (a) beam wth unlateral constrants; (b) prmary system of the mxed method for the rectangular fnte element; (c) th fnte element.

3 1756 A.V. Ignatyev et al. / Proceda Engneerng 150 ( 2016 ) The man system of mxed type as appled to the beam under consderaton s shown n fg.1,b. Fgure 1,c separately shows the th fnte element (FE) whch s assumed to correspond to the beam secton between supports (-1),. The process of the constructon of response matrx for such FE together wth the matrx structure and analytcal expressons for ts coeffcents are provded n [1, 7, 18]. The system of governng equatons for the gven beam can formally be presented as two groups of equatons equlbrum equatons and stran compatblty equatons. Equlbrum equatons are equaltes to zero responses n addtonal constrants ntroduced to the prmary system. Stran compatblty equatons are equaltes to zero mutual dsplacements of fnte elements' cross-sectons at the cuttng ponts (removal of FE mddle contrants). For the th fnte element between nodes (-1) and and for the th node. 1 R1 1. R 0 equlbrum equatons, 1 R, +1- numbe, 4 R2 0 rs of FEs 2. 5, 0 stran compatblty equato ns. 6, 0 (1) For lnearly deformable systems, the resolvng equatons system s composed of equatons (1) and n the matrx form s expressed as follows: A q P 0. (2) Where A s the global response matrx for the entre beam, and P s the load mpact vector. r r q r p 0. q p () r s the response matrx for the ntroduced constrants at ther unt dsplacements ( 1 r s the response matrx of the ntroduced constrants for force varables n the removed constrants of the man system, r T s the matrx of dsplacements along the removed constrants at unt dsplacements of the ntroduced constrants, s the matrx of dsplacements along the removed constrants at unt values of force varables n these constrants, r p s the subvector of responses at the set load n the ntroduced constrants, p s the subvector of dsplacements along the removed constrants at the set load mpact. In the expanded form the equatons (1) for the beam under consderaton (fg.1) are gven below: Where q ), 1 1, R R R c q q q P 1 l 1 l M R4 R2 q 5 q 5 q 6 q l l. 5, q 5 q q4 5, P 12 EI 2 l , q 6 q4 6, P EI (4) In these equatons:

4 A.V. Ignatyev et al. / Proceda Engneerng 150 ( 2016 ) , 1 P s the load n the prmary system reduced to node, c s stffness of the th elastc support, s the gap between the beam and the th 1 elastc support, q q 1 1 and q4 q 2 are varables pertanng to the same node. The problem of determnng the geometrc confguraton of the beam wth gven elastc unlateral constrants and preset load s solved applyng the followng algorthm. 1. The statcally determnate beam s analyzed wth the elastc one-sde supports removed (.e. c 0 ). The system of equatons (2) s solved, and the values of lnear dsplacements q along the support constrants are found. 2. The obtaned values q are compared wth the values of gaps. If some values comply wth q, then these nodes are assumed to have blateral constrants.. The beam analyss s carred out applyng the new analytcal model, and the values q are found at all the nodal ponts. 4. The obtaned values q are checked for consstency wth the condtons ) q, b) N c q If condton ) s met, these nodes are assumed to have blateral constrants, otherwse, the sprng stffness n ths node s assumed equal to zero (where ndcated n the analytcal model). If condton b) s met, ths node s assumed to have a blateral constrant, otherwse, the constrant s removed,.e. c The beam analyss s carred out applyng the new analytcal model. 7. The cycles are repeated untl the beam confguraton concdes at cycle (k) and cycle (k+1).. Test example The algorthm s verfed applyng the analyss of a cantlever bar on elastc supports (fg.2, ) for whch both the analytcal soluton [2] and numercal soluton obtaned usng LIRA-SAPR software package [20] are known. The related man system for the fxed method s shown n fg.2, b. Fg. 2. (a) beam wth unlateral constrants; (b) prmary system of the mxed method for the rectangular fnte element. Subject to the assumed drectons of the coordnate axes and the desgnatons of the varables, the system of resolvng equatons s expressed as follows:

5 1758 A.V. Ignatyev et al. / Proceda Engneerng 150 ( 2016 ) l l R2 P2 q7 q9 M2 q7 q8 q9q10 R q9 P q l l l M q9 q10 q11 q12 R4 P4 q11 M4 q11 q l l l 7 q7 q1 q2 8 q8 q2 12 EI 2 EI l l l l 9 q9 q2 q4 q1 q 0; 10 q10 q2 q 4 0, 12EI 2 2 EI l l l l 11 q q4 q6 q5 q11 12 q12 q4 q EI EI (5) At the frst stage a statcally determnate cantlever beam wth removed unlateral constrants s consdered. By solvng the system of equatons (5) we can fnd the lnear dsplacements of nodes 2,,4 (table 1). Accordng to the frst teraton, node 2 alone s dsplaced toward the unlateral constrant. Thus, the next step wll ntroduce the unlateral constrant n node 2. Accordng to the second teraton, nodes and 4 are dsplaced toward the unlateral constrants. The dsplacement of node 4 exceeds that of node, whch means the unlateral constrant n node 4 wll be nvolved nto the work earler than the constrant n node. The analytcal model for the thrd teraton ncludes the record of operaton for the two unlateral constrants n nodes 2 and 4. Accordng to the thrd teraton, unlateral constrant n node s not nvolved nto the work as the node s dsplaced n the drecton opposte to the unlateral constrant. Table 1. The results of the calculaton Dsplacement of nodes Node 2 Node Node 4 Iteraton m m m Iteraton m m Iteraton m 0 The responses n the unlateral constrants n nodes 2 and 4 are determned basng on the equlbrum condton: R P q q = t, R P q t (6) The effcency of the suggested analyss method s assessed through the comparson of the obtaned solutons wth the analytcal ones gven n [19, 20]. Table. 2. The comparson of the calculaton results Classcal mxed FEM analyss Analytcal soluton Accuracy, % Constrant response n node t.7872 t 0.52 Dsplacement of node m m 1.69 Constrant response n t t 1.18 node 4

6 A.V. Ignatyev et al. / Proceda Engneerng 150 ( 2016 ) Conclusons Thus, the results of the problem soluton through the classcal mxed FEM are accurate and requre fewer fnte elements to be consdered as compared to the dsplacement-based FEM. Accordng to [20], the soluton of ths problem wth the use of the dsplacement-based FEM reled on 1 nodes and 15 FEs, whle the soluton of the same problem through the classcal mxed FEM requred the consderaton of 4 nodes and FEs. References [1] A.V. Perelmuter, O.V. Kabantsev, Analyss of structure wth varable analytcal model, SCAD SOFT, ASV Publshng House, Moscow, [2] A.V. Perelmuter, V.I. Slvker, Desgn models of structures and a possblty of ther analyss, Publsher SKAD SOFT, Moscow, [] I.M. Rabnovch, Fundamentals of the analyss of cable and bar systems, Strozdat, Moscow, [4] I.M. Rabnovch, Certan ssues of the theory of structures wth unlateral constrants, Engneerng works collecton, Vol. 6, Moscow, [5] I.M. Rabnovch, To the problem of statstcally ndetermnate systems wth unlateral constrants (proof of unqueness of soluton), Research on the theory of structures. 10 (1961). [6] A.V. Perelmuter, Fundamentals of the analyss of cable and bar systems, Stroyzdat, Moscow, [7] V.P. Alenn, Iteratve methods for the analyss of systems wth external and nternal unlateral constrants, Dr. dss., VolgGASA Publ., Volgograd, [8] V.P. Alenn, P.V. Alenn, Drect methods for the analyss of structurally nonlnear systems, Sfera, Omsk, [9] V.P. Alenn, V.A. Ignatev, Practcal methods for the analyss of slabs on Wnkler foundaton bed, Investgatons on the theory of structural analyss and desgn, SPI Publ., Saratov, [10] V.A. Ignatyev, A.V. Ignatyev, A.V. Zhdelev, Mxed Form of Fnte Element Method n Problems of Structural Mechancs, VolgGASU Publ., Volgograd, [11] A.D. Lovtsov, Lnear problem of compatblty n structural mechancs of systems wth unlateral constrants, Izdatelstvo Tkhookeanskogo unversteta, Khabarovsk, 201. [12] A.A. Lukashevch, Buldng and mplementng fnte element method models for contact problems soluton, Izvestya vuzov: Constructon. 12 (2007) [1] A.A. Lukashevch, L.A. Rozn, On the decson of contact problems of structural mechancs wth unlateral constrants and frcton by stepby-step analyss, Magazne of Cvl Engneerng. 6 (201) [14] L.A. Rozn, M.S. Smrnov, Solutons of contact problems of Theory of Elastcty wth flexblty of unlateral constrants, Izvestya vuzov: Constructon. 5 (2000) [15] V.N. Bukhartsev, A.A. Lukashevch, Analyss of structures wth respect to successve erecton and exstence of a unlateral constrants on the contacts, Scentfc and techncal Gazette of SPbGPU. 89 (2009) [16] V.A. Ignatev, Analyss of regular and quas-regular bar systems wth blateral and unlateral constrants though combned varables method, Research n structural mechancs of bar systems, SPU Publ., Saratov, [17] E.M. Tkhonov, Analyss of beams wth unlateral constrants through combned varables method, SPI Publ., Saratov, [18] V.A. Ignatyev, A.V. Ignatyev, Mxed fnte element method n structural mechancs problems, VolgGASU Publ., Volgograd, [19] V.A. Ignatyev, A.V. Ignatyev, V.V. Galshnkova, E.V. Onshchenko, Nonlnear structural mechancs of bar systems, Theoretcal fundamentals, Analyss cases, VolgGASU Publ., Volgograd, [20] Verfcaton report on the LIRA-SAPR software package, Volume II, Moscow, 201.