Application of the theory of compound cores for the assessment of stress pattern in the cross section of a strengthened beam column

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1 IOP Conferene Series: Mterils Siene nd Engineering PAPER OPEN ACCESS Applition of the theory of ompound ores for the ssessment of stress pttern in the ross setion of strengthened bem olumn To ite this rtile: R F Frdiev et l 8 IOP Conf Ser: Mter Si Eng 4 View the rtile online for updtes nd enhnements This ontent ws downloded from IP ddress 4858 on //8 t 7:44

2 IOP Conf Series: Mterils Siene nd Engineering (8) doi:88/ x/4// Applition of the theory of ompound ores for the ssessment of stress pttern in the ross setion of strengthened bem olumn R F Frdiev, L S Sbitov,, N F Kshpov nd I R Gilmnshin,4 OOO PKF KARKAS, Kzn, 44, Russin Federtion Kzn Federl University, 8 Kremlyovsky street, Kzn, 48, Russin Federtion Kzn Stte Power Engineering University, 5 Krsnoselsky street, Kzn, 466, Russin Federtion 4 Kzn ntionl reserh tehnil university nmed fter A N Tupolev, K Mrx St, Kzn, 4 Ttrstn, Russi lsbitov@bkru Abstrt The rtile onsiders the results of theoretil studies of the stress-strin stte of n eentrilly ompressed element reinfored with reinfored onrete ge The theory of omposite rods is used s theoretil bsis As result of the solution of the theoretil problem, nlytil expressions re obtined for determining the norml stresses in the rosssetion of the reinfored element nd tngentil stresses long the ontt sem The results of the lultions for the proposed pproh re ompred with the previously performed reserh results To ssess the strength of the reinfored element, the importnt point is to determine the mgnitude nd nture of the distribution of fores pplied to the ge nd to the min element To dte, there is deformtion method for determining stresses nd fores [, ], but it is bsed on the hypothesis of flt setions, whih n be broken for the setion reinfored with reinfored onrete rosssetion To solve this problem, in the se of eentri ompression for long elements, it is proposed to use the theory of ompound rods [], introduing some hnges relted to the omplex stress-strin stte of the reinfored element Content from this work my be used under the terms of the Cretive Commons Attribution liene Any further distribution of this work must mintin ttribution to the uthor(s) nd the title of the work, journl ittion nd DOI Published under liene by Ltd

3 IOP Conf Series: Mterils Siene nd Engineering (8) doi:88/ x/4// Fig The lultion sheme An eentrilly ompressed element in the ross setion is represented in the form of pket of three ores (Figure ): rod from the min reinfored element, ore from the prt of the ge loted to the left of the zero setion line, rod from the prt of the ge loted to the right of the zero setion line The oordinte of the zero line is preliminrily determined from the known eqution of the ourse of the resistne of mterils: x нл h t, () e where е o-ordinte of fore pplition N, h setion height The onsidered system is lulted by the method of fores As the min system, ore devoid of sher bonds whose tion is repled by unknowns T nd T The totl rigidity of rod devoid of sher bonds will be equl to: EI E I E I E I () y y y he tngentil fores use bending moments in the onstituent ores: T M T T T, T T M T b T T T T T T T, T M T T T ()

4 IOP Conf Series: Mterils Siene nd Engineering (8) doi:88/ x/4// The totl bending moment ording to the method of fores will be equl to the sum of the bending moment from the externl lod nd the moments in eh of the ores: M M N e M M M Ne Ts Ts, (4) where s nd s the distne between the enters of grvity of the ores Longitudinl fores in the onstituent ores will be equl: N T, N N T T, N T (5) The differentil eqution of the urved xis of the pk of rods will be written in the sme wy s for the bent element: M x'' (6) EI The totl shift of the outer fibers of the two ores will be equl to the differene in sher of the two djent outermost fibers loted on either side of the ontt sem The first derivtive of the bsolute elongtion long the length of the sem is equl to the reltive elongtion Tking this expression into ount (8) tkes the form: ' ' ' (7) i i i i The reltive elongtion of the outermost fibers of the ores is determined from the expressions: раст t Mt / x '' - растянутых волокон -го стержня; EI сжат N M x '' - сжатых волокон -го стержня; E A EI N M раст x '' - растянутых волокон -го ст EA EI сжат t Mt / '' - сжатых волокон -го стержня x EI ержня; (8) The totl shift of the outermost fibers of the first nd seond ore ording to expression (7) tking into ount (8) will be equl to: N M раст сжат ' - EA EI (9) раст сжат N M ' - второго и третьего стержня E A In the expressions (9) the following nottion is introdued: EI первого и второго стержня;

5 IOP Conf Series: Mterils Siene nd Engineering (8) doi:88/ x/4// t /, t / () In order for the bsi system to beome equivlent to the speified one, it is neessry to equte the sher in the sems with the sher sher fores divided by the stiffness oeffiient of the sem: i Ti' i, Ti '' i ' () The oeffiient of stiffness of the sem ξ depends on the ompline of the joint of the prefbrited nd monolithi onrete, the presene of keyed joint, the trnsverse reinforement tht rosses the ontt sem, the presene of tretment of the ontt surfe To obtin equtions with the known solution for three ores, we find the sum nd differene of the shifts Δ nd Δ : T'' T'' M EI T '' T'' N M EA EI () From equtions () it follows tht: h t, () We introdue the nottion: T is the ntisymmetri prt of the set of fores, T is the ntisymmetri prt of the set of efforts: T TT, (4) T T T Tking into ount (4), the system of equtions () tkes the form: h tt s T s T '' EI T'' 4T N, E A E A Ne, EI (5) where the expression in prentheses of the i th eqution n be represented in the form: T s s T s s T s s T s T s (6) Tking into ount (6), equtions (5) tke the form: 4

6 IOP Conf Series: Mterils Siene nd Engineering (8) doi:88/ x/4// h ts s T '' Ne EI EI T '' 4 N T T E A E A T T (7) where γ, δ, γ, δ, oeffiients defined from expressions: h t s s Ne, EI EI 4 N, E A E A (8) The system of equtions (7) hs known solution: h x T, h l h x T hl (9) The tngentil stresses re determined from expressions: sh x T', h l sh x T', hl () where shλx hyperboli sine, hλx hyperboli osine, defined from expressions: h sh x x e e x x e e x x, ; () λ, λ oeffiients defined by expressions: () With suffiiently rigid ontt sem t ξ>, expressions () tke the form: 5

7 IOP Conf Series: Mterils Siene nd Engineering (8) doi:88/ x/4// x e hx shx () In this se, the system of equtions () tkes the form: e xl xl e (4) For known τ nd τ, the required stresses τ nd τ n be expressed from expressions (4):, (5) Unknown stresses in eh of the ores n be determined from expression: Ni M ixi i, A I (6) i where N i the fores determined in ordne with expression (5), M i -bending moments in the ores, determined from expression: i M Mi x'' Ei Ii Ei Ii (7) EI Tking equtions (5, 6) into ount, the expressions for determining the unknown stresses for the omponent ores will tke the form: T A A T A N T T T T Ex, EI T T Ex EI T T T T T T Ex, EI (8) With known tngentil fores, the refined position of the zero line is determined from expression (5): x нл N T T EI A E T T T T T T (9) The lultion is mde by suessive pproximtions with the refinement of the position of the zero line until the required ury of the determintion of effort is hieved 6

8 IOP Conf Series: Mterils Siene nd Engineering (8) doi:88/ x/4// To verify the nlytil expressions obtined, the norml stresses in the ross setion of the reinfored element were ompred (see Figure ) from the nlytil expressions obtined nd ording to the pproh bsed on the pieewise stress distribution lw [4, 5] For the lultion, n element m long with ross-setion of 5 m, reinfored with reinfored onrete ge of m thikness Fig Grph of the dependene of norml stresses in the ross setion of the element: t z =, ording to the proposed method (liner dependene) nd ording to the previously proposed pproh (nonliner dependene) The nlysis of the obtined results showed tht the proposed lultion pproh is rther well orrelted with the results of previous studies Aording to the epted lultion sheme, the tngentil stresses between the ores must be mximl t z =, nd miniml t z =, whih is onfirmed by the obtined grphs (Figure ) 7

9 IOP Conf Series: Mterils Siene nd Engineering (8) doi:88/ x/4// Fig Grph of the dependene of tngentil stress long the ontt sem: - between the st nd nd elements, b - between the nd nd rd elements Dt on the nture of the distribution of stresses on the ontt sem llow one to evlute its strength by known methods [6, 7, 8] to perform test of the strength of the ross setion Thus, on the bsis of the theory of ompound ores, solution ws obtined to determine the nture of the distribution of fores between the ge nd the olumn in the se of reinforement of n eentrilly loded element with reinfored onrete ge Referenes [] Bbih V I, Kohkrev D V 4 Clultion of reinfored onrete strutures by deformtion Conrete nd reinfored onrete No pp -6 [] Zlesov A S, Chistykov E A, Lrihev I Y 997 New methods of norml setion lultion on the bsis of deformtion design model Conrete nd reinfored onrete No 5 pp -4 [] Rzhnitsyn A R 986 Compound brs nd pltes (M: Stroiizdt) p 6 [4] Frdiev R F, Kyumov R A, Mustfin I I Clultion of bem olumn strengthened by reinfored onrete ollr with llowne for previous lod history nd nonliner properties of onrete Izvesiy KGASU pp 9-4 [5] Frdiev R F, Mustfin A I The study on the reinfored onrete bem olumns strengthened by ollr with llowne for stress stte before strenthening Integrtion, prtnership nd innovtion in onstrution siene nd edution pp 5-57 [6] Vleev G S, Ftkhullin V S 994 Applition of prefbrited reinfored onrete in reonstrution of buildings nd strutures News of Higher Edutionl Institutions Constrution pp 4-7 [7] Grozdov V T, Sergeev S L 996 The question of llowne for strength of the ontt zone under the lultion of reinfored onrete bending strutures strengthened by the extension of ross setions News of Higher Edutionl Institutions Constrution No 4 pp 4-8 [8] Sungtullin Y G 975 Experimentl nd theoretil bsis of the lultion of sher resistne of reinfored nd non-reinfored ontts of st-in-ple nd prest strutures Cst-in-ple nd prest strutures: olletion of reserh ppers pp