Restraining bars buckling by means of FRP wrapping: an analytical approach

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1 Restaining bas buckling by means o FRP wapping: an analytical appoach Giamundo V., Lignola G.P., Pota A. & Manedi G. Depatment o Stuctual Engineeing, Univesity o Naples Fedeico II, Italy SUMMARY: Intenal tansvese steel einocements (e.g. stiups) ae the main intenal devices that avoid the longitudinal steel bas buckling, but in most o all existing RC stuctues the quantity and the spacing between steel stiups ae inadequate. In these stuctues longitudinal bas buckling can be avoided by applying extenal einocement, in paticula, by means o Fibe Reinoced Polyme (FRP) wapping. A novel analytical appoach o the study o longitudinal bas buckling in column wapped with FRP is poposed. longitudinal bas has been consideed as axially loaded beam, while the mechanical eect o FRP wapping on the bas has been modelled by means o spings. The eect o elastic and inelastic behaviou has been taken into account by means o the educed modulus theoy. The well-known elations o steel stiups has been extended to the case o FRP wapping to popose an analytical omulation, valid both o cicula and noncicula column coss sections, o the evaluation o the FRP thickness needed to avoid the longitudinal bas buckling. Keywods: Bas buckling, Column, FRP wapping, Inelastic buckling, Reduced modulus 1. INTRODUCTION Pematue ailue modes due to buckling o compessed bas close to beam-column joints o in othe locations whee steel bas ae highly stessed (see Fig. 1.1) could educe the seismic capacity o existing einoced concete stuctues, and especially typical old-type einoced columns. Bas buckling may take place at the plastic hinge locations leading to even moe citical consequences, thus jeopadizing any capacity design pocedue. Buckling ailue mode is a pematue ailue mode elated to the instability o compessed slende einocing steel bas. Pooly detailed einocement pesents usually high spacing between stiups so that the estaint povided by the tansvese einocement is not suicient to avoid the instability o slende bas. The main paamete govening this phenomenon has been ecognized as the length/diamete atio; hence ecent building codes pescibe stict limitations to the atio between the stiups spacing and the diamete o einocing steel bas. Hence existing stuctues may need etoitting o deicient membes by means o extenal stengthening (e.g. FRP wappings). Figue 1.1. Typical ba buckling due to insuicient tansvese einocement

2 . ANTIBUCKLING Restaining ba buckling by means o intenal coninement was oiginally studied consideing steel stiups since second hal o last centuy. Howeve, the necessity to stengthen existing einoced concete (RC) stuctues lacking o einocement detailing aose nowadays. Today deep knowledge on extenal stengthening o such RC membes by means o extenal FRP wapping is equied. Pevious knowledge on steel stiups as anti-buckling device is epoted as a base to impove knowledge on FRP systems. Eiciency o estaining systems depends mainly on the shape o the coss section and, compaed to steel, FRP exhibits lowe lexual stiness due to educed thicknesses. In ectangula coss sections, slende FRP stips esult inadequate to estaint buckling phenomena..1. Anti-buckling equation o steel stiups Tansvese einocements (e.g. steel stiups) in a einoced concete column cay out the unction o conining device and then they avoid the longitudinal steel bas buckling. A pevious expeimental pogam by Pota and Cosenza (5) investigated the longitudinal steel bas behaviou unde axial load. The esults o these tests can be used to study the buckling mode o longitudinal bas between two consecutive layes o tansvese einocement. These tests highlighted that the compessive behaviou o the bas (and then the buckling mode) depends on the l/φ atio. Whee l is the spacing between two consecutive stiups and φ is the longitudinal bas diamete. In paticula vaying this atio the buckling can be elastic o inelastic. Consideing the estaint needed to avoid buckling ove a citical length involving seveal stiups o a longitudinal ba in the stain-hadening ange o axial compession, it is possible to evaluate the volumetic atio, ρ s, o the tansvese conining steel, accoding to Piestley et al. (1996), as:.45n s s (.1) EE h whee s is the expected steel stess in longitudinal bas, E = 4 E s E h /( E s + E) is the educed modulus o the longitudinal einocement at s, E h is the tansvesal einocement elastic modulus, E is the longitudinal einocement elastic modulus, n is the numbe o longitudinal bas and E s is the secant modulus om s to u, the ultimate stess. Consideing typical value o the paametes and lacking o deinitive analyses descibing the inteaction between coninement and buckling estaint the equation (.1) can be eplaced with:.13n (.) s whee the only signiicant paamete is the numbe o longitudinal bas, n... Theoy o educed modulus The linea elastic analysis is valid o slende bas and the well-known Eule s omula, that descibes the citical load o elastic buckling, is only valid o long ideal bas, The ultimate compession stength o the bas mateial is not geomety-elated and it is valid only o shot bas. Fo a ba with intemediate length, the axial buckling occus ate the ovecoming o the yield stess but beoe eaching o the ultimate stess o the bas mateial. This kind o situation is called inelastic buckling. Fo accuate analyses dieent inelastic buckling theoies ae commonly used (e.g. tangent modulus theoy, educed modulus theoy and Shanley (1947) theoy. Replacing the elastic modulus, E, in the Eule s omula with a modulus achieved by means o one o those theoies the inelastic citical load can be omulated. Using the educed modulus theoy the citical load is obtained by means o the concept o neutal equilibium. Theeoe, the citical load is deined as the axial load at which equilibium is possible both in the oiginal undeomed position and in an adjacent slightly bent coniguation. Stating om this assumption, accoding to Chajes (1974), the expession o the educed modulus, egading a ectangula section ba, is:

3 E 4EEt E E t (.3) whee E t is the tangent modulus at the buckling stess value. The expession o the educed modulus o a geneic section ba can be achieved by means o a suitable coeicient, c =E (cicula) /E (ectangula) that accounts o the dieent stess-stain esponse due to the dieent ba section, Papia et al. (1988). Fo cicula sections c is elated to E t /E atio in Fig..1 and it anges between 1.17 and 1. As discussed late, on sae side, lowest possible values o E should be used, i.e. E (cicula) /E (ectangula) = 1 1, E (cicula) /E (ectangula) 1,1 1,9,8,5,5,75 1 E t /E Figue.1. Cicula ba section coective coeicient tend 3. ANTIBUCKLING WITH FRP Longitudinal bas buckling can be avoided by applying uthe estaint to slende bas in between the estaints povided by the stiups widely spaced, in paticula by means o FRP wapping. To design a etoit intevention the main paamete needed is the FRP thickness, t, necessay to avoid the buckling. The poposed omulation has been evaluated by means o analytical models based on solid mechanic equations. The analytical model to achieve the citical load is based on a schematization o the longitudinal bas as an axially loaded beam with constant section (see Fig. 3.1). The mechanical eect (lateal pessue) o FRP wapping on the longitudinal bas has been modelled as spings, thus inceasing the citical load o these elements. P v l The equilibium dieential equation is: Figue 3.1. Analytical model schematization IV II EIv Fv kv (3.1) Whee E is elastic modulus, I is the moment o inetia o the ba section, k is the additional sping elastic constant and v is the displacement unction. The Eqn. 3.1 is a outh ode linea dieential equation with constant coeicients. As such, its geneal solution is: ' z ' z ' z ' z v Ae cos '' z Be sen'' z Ce cos '' z De sen'' z (3.)

4 The Eqn. 3. admit only one solution in which A=B=C=D=. Accoding with that esult, the citical load, F c, o n bas has been evaluated by means o: F c EIn kl 4 nw 4 l EIn nw (3.3) Whee n w = 1,,3, is the numbe o inlection waves. The elationship between F c and l is epesented by a cuve whose minimum is obtained o each n w : Fc kein (3.4) The citical load value achieved by the Eqn. 3.4 is the minimum, and it is independent on beam length. This value is the saest value to be used in design phase. The citical load tend vaying the length (o n w = 1,, 3) togethe with the Eule s cuve and the minimum citical load line ae shown in Fig. 3.. Figue 3.. Citical load tends vaying the length Stating om the knowledge o the minimum citical load omulation, a companion o Eqn..1 (speciic o the stiups) has been deived o the FRP case. Accoding to the consideation that the FRP wapping can be consideed as smeaed stiups, the FRP thickness to avoid the bas buckling has been deived. Fo columns with dieent coss section shapes, the FRP wapping assumes dieent stiness (i.e. dieent values o sping stiness). Theeoe it is needed to calibate the equivalent sping stiness o the analytical model depending on the coss section shape. In the ollowing, speciic cases o dieent column coss sections ae shown Cicula sections Fo cicula sections the lateal conining pessue due to FRP wapping is: E t k (3.5) c l c c Whee Δ c = ε c, E is the FRP elastic modulus, c is the adius o the column section, and ε is the adial deomation. In the Eqn. 3.5 the atio E t / c = k epesent the FRP stiness. The equivalent sping stiness has been obtained multiplying k o the length ove which the FRP is acting, i.e. the column cicumeential peimete: k = k π c. Replacing k in the Eqn. 3.6a with k π c, the minimum

5 citical stess, σ c, is simply evaluated as showed in Eqn. 3.6b, taking into account all the n bas distibuted inside the coss section: F ke ( 4 / ) In Et dein c a) c ; b) c (3.6) na na na b b b Whee d is the column diamete, A b is the bas aea and n is the numbe o bas. Equating σ cit to the bas yield stess, y, the FRP thickness to avoid the buckling is: t d y n (3.7) 4EE It is noted that Eqn. 3.6a is geneal and not only eeed to cicula coss sections. In Eqn. 3.7, omally compaable to Eqn..1, the steel yielding stess is equated to σ cit, howeve, the same appoach discussed in section.1 on the selection o steel stess can be epeated. Eqn. 3.7 clealy highlights that the lowe is E, the highe is t, so that, on sae side, lowest possible values o E should be used. 3.. Noncicula sections To achieve the equivalent FRP stiness o noncicula sections it is undamental to evaluate the oce exeted by the longitudinal ba on account o buckling in the diection coesponding to the plane in which this can occu. This oce must be evaluated in dieent modes accoding to the ba position inside the coss section. In the ollowing, the most two epesentative cases ae shown Cone bas Consideing the cone ba (see Fig. 3.3) the oce F exeted by the longitudinal ba can assume all the possible diections accoding to the inclination angle α. A paametic study has been peomed to undestand the inluence o the angle α on the equivalent sping stiness. Y X b/ h/ F Figue 3.3. Cone bas scheme Accoding to eeence system in Fig. 3.3 (whee, due to symmety, FRP wap is simply suppoted in the middle o coss section sides) the total displacement o FRP wap in the cone due to bas buckling is: sen cos Fb h Fb Et b Et (3.8) whee b and h ae, espectively, the width and height o the section (b h).

6 Consequently the equivalent sping stiness is: Et k (3.9) b Accoding to the unction ξ, the tend, vaying the oce inclination angle α, (see Fig. 3.4) pesents a maximum value o α =. This means that the lowest sping stiness is associated to α =, i.e. when the oce F is oiented in the diection othogonal to the minimum side o the section (i.e. b). ξ 1 h/b 45 9 Figue 3.4. ξ/α unction Theeoe the minimum value o the equivalent sping stiness, ecalling that b h, is: Et Et k k max bh ; b (3.1) Equating σ cit to the bas yield stess, y, and eplacing the equivalent sping stiness, k, achieved o the cone ba o the noncicula section in the Eqn. 3.6a, the FRP thickness to avoid the buckling is: α t b y n (3.11) EE This equation is still compaable to pevious Eqns. 3.7 and.1. Same comments can be epeated on the expected steel stess in longitudinal bas Cental bas Consideing the cental ba (see Fig. 3.5) the oce F exeted by the longitudinal ba can only assume the diection nomal to the consideed side (because it cannot ente concete, i.e. ente into the coe o the section). b/ F b/ Y h/ X Figue 3.5. Cental bas scheme

7 Accoding to eeence system in Fig. 3.5 the FRP the total displacement due to bas buckling is: 3 Fb Fb Et 16Et (3.1) This value is much highe than the pevious povided by Eqn. 3.8 mainly because it is well-known that FRP thickness is vey low and lexual stiness o FRP waps is commonly negligible. Consequently the equivalent sping stiness is: 16Et 16Et k max 8 ht b ;8bt h 3 3 k ht b (3.13) It is noted that even i b h, it is not possible to select a pioi the maximum value in the denominato o Eqn. 3.13, om an analytical point o view; howeve, analysing the two tems, it can be deived that i t >1.63h, obvious in pactical applications, the ome tem is always lage than the latte. In this case a closed om solution in tems o t is not staightowad; howeve the poblem can be solved numeically, as discussed in section APPLICATIONS OF PROPOSED MODEL In technical pactice it is useul to have a closed om solution. Intoducing the numeical constant μ the poposed model is heein modiied in the omat: t y nd ' (4.1) EE O, o moe simplicity, intoducing the constant μ * and assuming a value o y and E, in the omat: t * nd ' (4.) E Whee d epesents a chaacteistic dimension o the coss section (the diamete o cicula coss sections, o the maximum dimension in noncicula coss sections). The new paamete, μ *, intoduced in Eqn. 4., can be easily elated to the ome μ, μ * = μ y /E. Consideing, o the steel constitutive model (see Fig. 4.1a), the well-known Rambeg-Osgood (1943) elationship (Eqn. 4.3) the tends shown in igs. 4.1b, c, d espectively ee to peviously discussed secant, tangent and educed modulus. E K 1 n (4.3) Whee ε is the stain, σ is the stess and K and n ae constants that depend on the mateial being consideed. Assuming y = 45MPa, accoding to Rambeg-Osgood elationship, E = GPa hence μ * = 6.5 μ.

8 a) b) 6 4 σ [MPa] ε [mm/mm] E [GPa] Secant modulus ε [mm/mm] c) d) E [GPa] 1 6 Tangent modulus ε [mm/mm] E [GPa] 1 6 Reduced modulus ε [mm/mm] Figue 4.1. a) Rambeg-Osgood model. b, c, d) Secant, tangent and educed modulus tends 4.1. Cicula sections Fo cicula coss sections, accoding to Eqn. 3.7 (and given y and E accoding to section 4), maximum values ae μ =.5 and μ * = Howeve in eality (see Fig. 4.) the paametes depend on l/φ atios, and clealy o values o l/φ lowe than 6.5 thee is no need o FRP wapping because Eule citical stess, σ cit, is aleady highe than y (see Fig. 3.). μ* [MPa] n w =1 n w = n w = l/φ [-] Figue 4.. μ * /l/φ tend o cicula coss sections 4.. Rectangula sections Cone bas Fo cone bas in ectangula section, accoding to Eqn. 3.11, assuming d = min {b;h} (whee b and h ae, espectively, the width and height o the section), and given y and E as witten in section 4, maximum value ae μ = π/ and μ * = 6.5 π/ but as the cicula case it depends on l/φ atio, and clealy o values o l/φ lowe than 6.5 thee is no need o FRP wapping because Eule citical stess, σ cit, is aleady highe than y as shown in Fig. 4.3

9 1 n w =1 n w = n w =3 μ* [MPa] l/φ [-] Figue 4.3. μ*/l/φ tend o ectangula coss sections (cone bas) 4... Cental bas Fo cental bas in ectangula section a closed om, even o maximum values o μ and μ* is not simple, but the poblem can be solved numeically. Equating σ cit to the bas yield stess, y, and eplacing the equivalent sping stiness, k, achieved o the cental ba o the noncicula section in the Eqn. 3.6a, the FRP thickness to avoid the buckling is not povided in closed om (since Eqn is not linea in t ). Howeve inveting Eqn. 4. the tend o μ* is ound (see Fig. 4.4). It is highlighted that peliminay esults show that dependency on b, h/b and n is negligible and μ* n w =1 n w = n w =3 μ* [MPa] l/φ [-] Figue 4.4. μ * /l/φ tend o ectangula coss sections (cental bas) The incease o FRP thickness equied to avoid buckling o a cental ba is much highe than the thickness equied to avoid buckling o a cone ba. 5. CONCLUSIONS An analytical appoach to study the buckling o longitudinal bas was poposed, when an elastic device is used to conine and estain this phenomenon. Solid mechanics equations wee the basis o the poposed model, accounting o longitudinal bas as Eule beams estained by elastic spings along thei length. Pevious omulations to assess minimum atio o steel stiups to avoid longitudinal bas buckling wee discussed and compaed to pesent case o FRP wapping. These omulations account also o inelastic buckling, by means o educed modulus theoy. Beside analytical eined omulations, simpliied equations, pactitiones oiented, wee poposed based on ew paametes. They wee also analytically deived with the only exception o cental bas in noncicula coss sections. These omulations has shown that the eect o FRP wapping is negligible o l/φ atios less than about 6.5 o both cicula and ectangula coss sections. Fo ectangula coss sections the FRP is not able to avoid the cental bas buckling, mainly because o educed thickness o FRP waps (hence because o educed lexual stiness). Convesely it is eective o cone bas in noncicula sections and always in cicula sections.

10 ACKNOWLEDGEMENTS The authos acknowledge the inancial suppot povided by ReLUIS o the eseach pogam unded by the Depatment o Civil Potection Executive Poject 1-13 REFERENCES Chajes, A. (1974), Pinciples o Stuctual Stability Theoy, Pentice Hall, Uppe Saddle Rive, NJ. Cosenza E., Pota A. (6), Expeimental behaviou and numeical modelling o smooth steel bas unde compession. Jounal o Eathquake Engineeing. 1:3, Papia M., Russo G., Zingone G. (1988) Instability o longitudinalbas in R.C. columns. Jounal o Stuctual Engineeing ASCE. 114:, Piestley, M. J. N., Seible, F. and Calvi, G. M. (1996) Retoit Design, in Seismic Design and Retoit o Bidges, John Wiley & Sons, Inc., Hoboken, NJ, USA. Rambeg, W., Osgood W. R., (1943). Desciption o stess-stain cuves by thee paametes. Technical Note No. 9, National Advisoy Committee Fo Aeonautics, Washington DC. Shanley F.R. (1947). Inelastic column theoy. Jounal o Aeonautic Science.14:5,61-68.