A PROCEDURE ON STUDYING A CRITICAL FORCE ON FEM AS PLASTIC DEFORMATIONS ARISE WITHIN STRAIGHT TRUSSES

Transcription

1 11 th Nationa Congress on Theoretica and Appied Mechanics, -5 Sept. 009, Borovets, Bugaria A OCDU ON STUDYING A CITICAL FOC ON FM AS LASTIC DFOMATIONS AIS WITHIN STAIGHT TUSSS IVAN VAISILOV VSU L. Karaveov, 17 Sofia, 175 Suhodoska st., Bugaria LUBN LUBNOV VSU L. Karaveov, 17 Sofia, 175 Suhodoska st., Bugaria yubenov@vsu.bg ABSTACT. The paper refers to a step by step FM procedure. The aim is to cacuate the critica force for straight bars when pastic deformations arise. An intersectiona modue of easticity is used. The constitutive aw of the materia in the zone of pastic deformation is presented as a two-degree poynom depending on the zone s ength. KY WODS: Finite eement method, critica force, pastic deformation, step by step procedure. 1. Introduction. In [1], an anaytica procedure is shown on how to cacuate a critica force of a two-body-cut trusses fig. 1a, having different ratios between the engths of the strengthened and the weakened zone of the truss, since the type and the proportions of the cross sections varies, whie out of the pertaining physica features are being set in the diagram σ = σ(ε) of the materia. This particuar diagram represents the growth of pastic deformations which are shown in fig.1b where: σ - stress in imit of proportiona σ - faiure stress ε - strain in imit of proportiona ε - utimate strain of faiure

2 I.Vaisiov, L. Lyubenov The anaytica equation on is: σ = σ(ε ) Fig. 1. (1.1) σ = σ + ( ε ε ) ( ) ε ε δ where ε ε ε, and: (1.) δ = ε ε The change of the tangent modue within the zone of pastic deformation is: σ (1.) ( ε ) = = ( ε ε ) ε δ It can be seen that in case of ε = ε, (ε ) =, which makes for a smooth transition from the eastic zone towards the zone of pastic-deformations. If ε = ε and δ = ε ε from (1.1) we obtain:

3 A procedure on studying a critica force on FM (1.) δ σ = σ + δ = σ + δ That vaue does not overap with σ max of the curve on fig 1b. The contradiction is due to the fact that the unsetted pastic behaviour of the materia is determined by independent parameters - σ, σ,, δ = ε ε, however the square poynomia (1.1) is being defined ony of independent constants. In [1], with certain approaching, the reationship (1.1) is being formuated as we use: (1.5) ( σ σ δ = ) This case has been expored within the imit of inear-eastic behaviour of materia and it has aso been expored by FM [1], []. In a subsequent condensation of a net of discrete nods upon the truss of fig. 1a, the equation of bucking, has got to be used: G (1.) det [ K ] F[ K ] = 0 where: e [ ] = [ k ] e [ ] [ ] K is goba fexibe stiffness matrix, K G = k G is goba geometric stiffness matrix. [ k ] е 1 = I [ ] k e G F = 0

4 I.Vaisiov, L. Lyubenov As we enter into the zone of pastic deformations, the cacuation of the critica parameter F of (1.), must be organized as a step-by-step procedure. The changing tangent modue (ε ) is to be embodied within the stiffness e matrix eements [ k ], thus in [ K ], whereas [ K G ] remains invariabe. Another option for critica parameter study within an iteration procedure as the intersectiona modue of easticity has been introduced by B. Bankov in [1], yet with no gain of a resut expressed numerica. This is what wi be done in this paper.. Agorithm for cacuating the critica parameter of a truss oaded by its axis on FM whie pastic deformation arises and as the intersectiona modue of easticity is used. We sha imit the range of this case, as we sha organize a procedure for cacuating of F on FM of a truss whit a constant square cross-section, i.e. at cr 1 = и = 0 (fig.) aong with discretization of 0 truss finite eements Let s assume as a computation diagram of materia (1) to be the one shown as a graph on fig.. The goba matrix of cure is: [ K ( 1) ] = [ K1], 1 = tgα1. Fig.. 1 G = From the bucking equation det [ K ] F[ K ] 0 1, we gain initia parameter approaching, F cr = F1, σ F 1 1 = A, σ 1 ε 1 =. Having the cacuated deformation and 1 the ogged formua of anaytic ratio we do the foowing cacuation σ 1 = σ ( ε 1 ), σ 1 =. ε The goba stiffness matrix is [ ] [ K ] G = bucking equation: det [ ] F[ K ] 0 K. K = ( ) then again we appy the

5 A procedure on studying a critica force on FM Fig.. We cacuate F cr = F, σ F = A, σ ε =, σ = σ ( ε ), i1 i K = K ( ) and i.e. whenever eventuay obtein: F F. [ ] [ ] cr cr σ =, ε. Numeric exampe. Let for truss of fig. we have I = 1,.10 m, m σ =.10 kn / m, =,7.10 ( ) σ = σ δ =, Thus of (1.1) for σ we obtain: 7 =, a = 0, 0m, A = 0,0m, = 7,5.10 kn / m, σ 0.10 kn / m, = ε, ε =.10, σ = ( ε.7.10 ε.7.10 ) 1.,7.10 The vaue of bucking force by uer s formua, in the case of eastic behaviour of materias is: π I,1.1, F cr = = = 9 kn. 7

6 I.Vaisiov, L. Lyubenov The resuts of soutions obtain by step by step procedure described in this paper are shown in tabe1. Iteration [ kn] F i σ [ kn / m ] i Tabe 1. ε i [ kn / m ] i σ [ kn / m ] 1 7,97 1,8.10 8,5.10-0,.10 7, ,11 0,.10 8,5.10-5,9.10, ,11 5,9.10 8,5.10 -,.10, ,,.10 8,5.10 -,0.10, ,88,0.10 8,5.10 -,17.10, ,89, ,5.10 -,17.10, ,00, ,5.10 -,17.10, ,8, ,5.10 -,17.10, ,1, , , Concusion. Tabe 1 shows ceary a monotonous approximation to the critica vaue of the oad parameter F cr. We can assume that in or during the th iteration, the process has practicay ended. Thus, the iteration process, based on the use of the intersectiona modue i, might be appied successfuy in the event of zone of pastic deformations with evident inequaities σ < σ (in this case σ = 1, 7σ ). Further cacuations have to find out if σ = kσ, at what vaue of k the iteration process wi either sow down or discontinue, meaning there wi be no endcomputation. i F N C S [1] ВАЙСИЛОВ ИВ., Статическа устойчивост на осово натоварено едностъпални пръти и къси цилиндрични черупки при преход към състояние на физическа нелинейност, Докторска дисертация, ВСУ Л.Каравелов,София, 008 г. [] БАНКОВ БАНКО, Ю. ПАВЛОВА, Метод на крайните елементи в строителната механика, УАСГ, София, 199 г..