ABOUT THE DESIGN OF PRESTRESSED FRAMES WITH THE INFINITELY RIGID BEAM, SUBJECTED TO HORIZONTAL FORCES

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NTERNTONL SCENTFC CONFERENCE CBv 00 November 00, Braşov BOUT THE DESGN OF PRESTRESSED FRES WTH THE NFNTELY RGD BE, SUBJECTED TO HORZONTL FORCES Dan PRECUPNU*, Codrin PRECUPNU** *Prof., Universiy G.

1 NTERNTONL SCENTFC CONFERENCE CBv 00 November 00, Braşov BOUT THE DESGN OF PRESTRESSED FRES WTH THE NFNTELY RGD BE, SUBJECTED TO HORZONTL FORCES Dan PRECUPNU*, Codrin PRECUPNU** *Prof., Universiy G. saci aşi **a., Coege C. Negruzzi aşi... Corresponding auor: Dan PRECUPNU, e-mai: bsrac: design meod of frames wi e infiniey rigid beam and presressed coumns by ig sreng ie bars, is presened. fer a brief reminder of e equivaen secion meod e cacuaion formuas of dispacemen siffness for e basic sysems are esabised. Te concusion is reaced a e dispacemen siffness of ese bars can be deermined by e formua k p k ρ were k is e siffness known for unpresressed bars (wiou ie bars) and ρ a correcion facor, wic depends on e presressing degree and ype of basic sysem. Te oer parameers wic appear in e cacuaion reaionsip of e facor ρ, ave a very sma imporance. Te reaionsips corresponding o ese correcion facors of e dispacemen siffness of e presressed coumns are given. Key words: buiding-in momen, equivaence coefficien, presressing, ie-bar, siffness, sress.,generltes. CLCULTON HYPOTHESES Te coumns of e frames wi an infinie bending siffness of beams, subjeced o imporan orizona forces, can be conceived as presressed eemens by e ig-sreng ie-bars (Fig. ).

2 0 Dan PRECUPNU, Codrin PRECUPNU Fig. Exampes of Presressed Frames wi e nfiniey Rigid Beam is convenien o make e design of ese srucures by aking advanage of e equivaen secion meod. s sown in, is design meod is based on e foowing principe: a presressed secion (Fig. a.) can be repaced in e cacuaion, from e poin of view of sreng condiion by unpresressed secion capabe of e same bending momen (Fig. b). Beween ese wo secions ere are e foowing dependence reaionsips Fig. a). Presressed Secion; b). Equivaen Secion e we w e e were (a) α e R α (b), (c) So, e unpresressed equivaen secion mus be dimensioned a e oa bending momen from e cacuaion secion (overaken by e bar ie bar assemby): b x (d) ccording o is meod, e design of e secion of e presressed srucures is made wi e foowing cacuaion sages:.te oa bending momen is deermined..te necessary equivaen secion is dimensioned using e cacuaion know o unpresressed srucures (using e reaionsips of opima dimensioning of unpresressed secion ).. ccording o e quaiy of e maeria in e bar and ie-bar, e equivaence coefficien is cacuaed (re. (c)). By coosing e raio e / e (usuay ) e equivaen area of e ie-bar, e, and en is area (re. (b)) are deermined.

3 bou e design of presressed frames wi e infiniey rigid beam, subjeced o orizona forces 4. Te ensioned area of e equivaen secion is reduced wi e and ere resus e presressed secion wic as:, e w we, e e (a ) Te secion and ie bar us obained exacy saisfy e sreng condiion under e service forces. s i foows, ony e cecking of e srucure a e presressing of ie-bar is necessary. 5.Te oa effor in e ie-bar,, resus from: - e given effor a is presressing, 0, and -e effor produced by e service forces, 0 (e) is e effor of e srucure subjeced o e service forces, aving e ie-bars assembed bu unpresressed..ou of e reaionsips (e) we ge 0 (f) wic mus ceck e srucure. For deermining e oa bending momen of e srucure, necessary in e firs sage of cacuaion: - e momen of perfec buiding-in, and - e dispacemen siffness of e basic sysems mus be known. Te firs probem is soved in, were i is sown a e momen of perfec buiding-in, in a suc sysem is given by e reaion: c, C ' ', were (g), (g ), ' are e momen of perfec buiding-in of e and, respecivey of e bar wiou e ie bar c, c correcion coefficiens wic can be abuaed according o e presressing degree of e secion and e ype of basic sysem-given by e reaionsips: c () ' (4 ) K c η c ( ) K (i) η for e oading case for basic sysem ' η for e oading case ' c ( ) for basic sysem K for bo previous oading cases ( 0.5)

4 Dan PRECUPNU, Codrin PRECUPNU s i foows, e second probem is anayzed for e basic sysem of ese srucures: Fig. Basic Sysem; a). Presressed Bar a One of Ends b). Presressed Bar a Bo Ends c). Presressed Bar in e Span Te foowing cacuaion ypoeses are admied: a). for e srucure - e beam of e frame as an infinie siffness; - e coumns are bui-in a is bo ends; - e service forces are orizona and may be disribued by any aw; b). for e basic sysem: - e bar is sraig and as a consan secion aong; - e presressing ie-bar is reciinear and parae o e ongiudina axis; - e maeria of e bar and ie bar obeys e ypoeses of inear easic cacuaion.. THE ESTBLSHNG OF CLCULTON RELTONSHPS Bar wi ie bar paced a one of is ends Considering e bar wi assembed, bu unpresressed ie bar, we appy a uniary dispacemen beween is bo ends (Fig. 4a).From e compaibiiy condiion of dispacemens, expressed on e sysem in fig. 4b.

5 bou e design of presressed frames wi e infiniey rigid beam, subjeced o orizona forces Fig. 4 Presressed Bar a One of s Ends a). Deformed xis b). Cacuaion Sysem [δ]{}{δ 0 } were () [ ] ) ( E δ, { } ', { } Δ Δ 0 () E E γ Δ e E E E E α γ we ge: () [ ] γ γ E 4 ) ( [ ] [ ] γ γ E ' 4 (4), (5) [ ] γ E 4 () Noing: k ρ - e dispacemen siffness of e exremiy (wi ie bar)

6 4 Dan PRECUPNU, Codrin PRECUPNU k ρ ' - e dispacemen siffness of e exremiy (wiou ie bar) and aving in view a E / presens e dispacemen siffness of e unpresressed bar (wiou ie bar), k, from (4) (5) reaionsips, i resus: k P k ρ k P' k ρ ' (7),(8) ρ and ' ρ being e correcion facors, wose expressions are: ρ ( ) γ [ ( )] γ (9),(0) [ ( 4 )] γ ρ ' [ ( 4 )] γ Bar wi e ie bars a bo ends (Fig. 5) n e same way o e preceden case, we ge: Fig. 5 Deformed xis of Presressed Bar a Bo Ends k k k ρ (7 ) P P' e correcion facor ρ aving e expressions: ρ γ [ ] γ (9 )

7 bou e design of presressed frames wi e infiniey rigid beam, subjeced o orizona forces 5. DSCUSSON OF THE CORRECTON FCTORS OF STFFNESS Ou of e definiion reaionsips (9), (0), (9 ), i foows a e correcion facors ρ ij, i,, j,, depend on e degree of presressing of e secion and on e quaiy of e maeria in e bar and ie bar (/, γ) on e eng of e bar () as we as e disance beween e ie bar and e axis of e bar ( ). 4. CONCLUSSONS Ou of e resus obained e foowing may be concuded:.te dispacemen siffness of e bui-in presressed bar is bigger en a of e unpresressed bar wi e same secion. Te grow of siffness increases wi e presressed degree, reacing up o 4% for e presressed bar a bo ends. For e bars presressed in e span, e siffness cange is smaer, maximum %..Te ower e resisance of e presressing ie-bar, e iger is e siffness of e presressed bar. f e resisance of e ie bar is beween Pa e modificaion of e correcion facors is under %..Te modificaion of facor (and ) produced by e cange of e disribuion aw of cross oads as aso a sma infuence on e dispacemen siffness ( max 4.5 %). s a concusion, if e presressing is acieved by e ig-sreng ie-bars, paced on e face of e bar, e correcion facors of siffness pracicay depend ony on e presressing degree and on e ype of e basic sysem. f e presressing degree is beow 0.5, e siffness of e presressed bar differs insignificany from a of e unpresressed bar. Noaions: area of e bar secion; area of e ie-bar secion; e equivaen area of e ie-bar secion; E bending siffness of e bar secion; E siffness of e ie bar; - ie-bar eccenriciy; k - dispacemen siffness; - oa bending momen produced by e service forces on e srucure; b - bending momen in e ie bar secion; - oa effor in e ie-bar; - buiding-in momen a e end ; - buiding-in momen a e end ; α - equivaence coefficien regarding e sresses; ρ - correcion facors of unpresressed bar siffness.

8 Dan PRECUPNU, Codrin PRECUPNU REFERENCES. PRECUPNU, D., Dimensionnemen des poures méaiques ypersaiques à une seue ravée préconraines par des irans à aue resisance. nnaes de nsiu Tecnique du Bâimen e des Travaux Pubics, Paris 49, pag. 0-,984.. PRECUPNU, D., Poures méaiques préendues dans e domaine éasique. Rev. Consrucii 9. Bucuresi,97.. PRECUPNU, D., Design of Secion of Doube Bui-in Beams, presressed a One End, by Hig- Sreng Tie Bars. Rev. Consrucii, Bucuresi, PRECUPNU, D., La forme opimae de a secion du profie en doube T, Buein P asi, ome, fasc. -4,97. Received Sepember 0, 00

CONSIDERATIONS REGARDING THE OPTIMUM DESIGN OF PRESTRESSED ELEMENTS

CONSIDERATIONS REGARDING THE OPTIMUM DESIGN OF PRESTRESSED ELEMENTS Bullein of e Transilvania Universiy of Braşov CIBv 5 Vol. 8 (57) Special Issue No. - 5 CONSIDERTIONS REGRDING THE OPTIU DESIGN OF PRESTRESSED ELEENTS D. PRECUPNU C. PRECUPNU bsrac: Engineering educaion