Vidmantas Jokūbaitis a, Linas Juknevičius b, *, Remigijus Šalna c

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1 Availale online at Procedia Engineering 57 ( 203 ) t International Conference on Modern Building Materials, Structures and Tecniques, MBMST 203 Conditions for Failure of Normal Section in Flexural Reinforced Concrete Beams of Rectangular Cross-Section Vidmantas Jokūaitis a, Linas Juknevičius, *, Remigijus Šalna c a,,c Department of Reinforced Concrete and Masonry Structures, Faculty of Civil Engineering, Vilnius Gediminas Tecnical University, Saulėtekio ave., Vilnius, LT-0223, Lituania Astract Te stress in longitudinal tensile reinforcement is one of te main important parameters wile examining te tecnical state of underreinforced concrete structures. Te most important issue is to determine weter te external loads cause te close to yield stress in te main reinforcement. Yield stress in tensile reinforcement could e treated as te start of incipient failure of te flexural structure. Te state of tensile reinforcement of flexural reinforced concrete structures could e examined y oserving te properties of te normal cracks. Te application of fracture mecanics of solids could e used for determining te actual damage to te structure y knowing only te measured eigt of te normal crack. 203 Te Autors. Pulised y Elsevier y Elsevier Ltd. Open Ltd. access under CC BY-NC-ND license. Selection and and peer-review under under responsiility of te of Vilnius te Vilnius Gediminas Gediminas Tecnical Tecnical UniversityUniversity. Keywords: concrete; crack, reinforcement; stress; defomation; deflection.. Introduction Te assessment of stress state in longitudinal tensile reinforcement is igly important wile examining te tecnical state of under-reinforced concrete structures. Te most important issue is to determine weter te external loads cause te close to yield stress in te main reinforcement. Te appearance of yield stress in tensile reinforcement could e treated as te start of incipient failure of te flexural structure [-4]. Te propagation of cracks in flexural reinforced concrete eams is investigated extensively ut suc researc usually is limited to te serviceaility stage, i. e. efore te failure starts [5-7]. Altoug it is also important to know te caracteristics of te critical macro-crack wic cause te actual failure of te memer, e.g. critical dept of normal crack, wic cause te yield stress in tensile reinforcement. Te availaility of suc researc data in scientific literature is limited [8]. Te relationsip etween te dept of normal crack and stress state witin te cross-section of te eam is proven y teoretical and experimental researc many years ago [5], [9-2]. Te stress in main reinforcement could e determined y using te data from experimental researc, namely te dept of normal crack wic could e measured relatively easy in most cases. Tus te stress in tensile reinforcement could e determined for te examined eams witout unloading. 2. Influence of te critical dept of te crack on te stress in te tensile reinforcement Calculation model for te crack development (Fig. ) is ased on te rules provided y te fracture mecanics of solids [3], [3-6]. According to tis teory, te two tips of eac crack could e determined. One of tem causes te propagation * Corresponding autor. address: lj@vgtu.lt Te Autors. Pulised y Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsiility of te Vilnius Gediminas Tecnical University doi: 0.06/j.proeng

2 Vidmantas Jokūaitis et al. / Procedia Engineering 57 ( 203 ) of te crack towards te neutral axis of flexural memer. Te position of te oter tip coincide wit te level of tensile reinforcement. Te widt of te crack tip wic is close to te neutral axis is critical and generally govern te furter crack development. Te ond forces etween concrete and reinforcing steel resist to te crack development. Fig.. Te model for calculation of normal crack development Te parts of te memer separated y te crack rotate around te point wic is an intersection etween crack plane and neutral axis. Distance etween te crack surfaces witin cr is proportional to te distances to neutral axis, see Fig.. Te following formula could e written for calculation of stress in tensile reinforcement ased on model sown in Fig. : ( 0.75 ) 2 Mtot cr + ct P cr σ s = y I A α σ t, () were σ s actual stress in tensile reinforcement; M tot = M Pe 0p ΔM; M ending moment; P prestress force; e 0p eccentricity of prestress force; ΔM = P(y cr ct ) te increase of ending moment caused y matcing te geometrical center and neutral axis in te design cross-section; y distance from most tensile fire of cross-section to geometrical axis (center of gravity); dept of te crack; ct eigt of te tensile zone aove te crack; A and I area of equivalent cross-section and second moment of area of equivalent cross-section respectively (wen matcing te geometrical center and neutral axis); widt of cross-section; A S and A S2 areas of tensile and compressive reinforcement respectively; a dimensionless adjustment function wic depends on te dept of te crack and geometry of te cross-section (usually te cr A s ratio ); t = ; σ y yield stress in tensile reinforcement. Wen yield stress σ y is reaced in tensile reinforcement, te tensile zone of concrete aove te crack is insignificant and may e neglected, i.e. ct = 0. Also ecause of significant plastic deformations in te tensile reinforcement te prestress force could e neglected too. Terefore Eq. () could e written in te following form: 0.59M u cr, lim cr, lim I* t 2 σy α, (2) ere M u and cr,lim ending moment and critical dept of te crack respectively, wen stress in tensile reinforcement reac its yield limit σ y (σ 0.2 or σ 0. ). Te second moment of area of equivalent design cross-section could e calculated according to te following formula: 3 * ( 0.5 cr ) 0.5 e s cr e s2( cr 2) I = + + α A +α A d, (3) 2

3 468 Vidmantas Jokūaitis et al. / Procedia Engineering 57 ( 203 ) were te eigt of equivalent design cross-section ( ) 2 2 = k + k + k ; k 2 ( αeas2 cr) = (wen A S2 = 0, k = cr ); 2α e( As2d2 + As2cr 0.5Ascr) αeascr k2 = (wen A S2 = 0, k2 = ); dept of te crack cr = cr,lim ; α e ratio etween te modulus of elasticity of reinforcement and concrete; d 2 distance from most compressive fire of cross-section to te center of gravity of compressive reinforcement. It is ovious tat te precision of Eq. (2) directly depends on te adjustment function α. Tis function depend on many parameters of normal crack ut te greatest influence is related to ratio cr = and reinforcement ratio ρ. 3. Influence of normal crack parameters on te adjustment function Te teory of relationsip etween te dept of normal cracks and stress state in cross-section is ased on numerous experimental researc data [], [5]. Tis teory allows considering te case wen te slip etween tensile reinforcement and concrete occurs after te crack appearance, i.e. ypotesis of plane sections is not valid. In suc a case te deformations in concrete and reinforcement are not equal anymore ecause of damaged ond etween tem. Tus te stress in tensile reinforcement could e calculated y using te following equation system: 3 ct ( I Scr ) ( S Acr ) d 2 ct = M P + S + A d 2fct 2f ct, (4) ( S Acr ) P 2 fct A ct 2 f ct σ s = As Were A, S, I and M respectively area of cross-section, first moment of area, second moment of area and ending moment in respect to te edge of cross-section sujected to te greatest tension. Te area of cross-section witin te dept of te crack (including area of tensile reinforcement A S ) is neglected. Te values of te stress in tensile reinforcement calculated according to Eq. (4) were similar to te ones otained from te experimental researc on flexural eams of rectangular and tee cross-sections, wen eams were loaded y 40 to 80% of te ultimate load [5], [2], [7-8].. Wen calculating te stress in reinforcement according to Eqs. (4) te use of expression is avoided and tus tese equations are suitale for determination of adjustment factor α itself. Te stress in tensile reinforcement σ S could e calculated y using Eqs. (4) and relationsip etween cr and M otained from experimental researc. Ten adjustment function could e calculated y putting M, cr and σ S values (determined according to Eqs (4)) to Eq. (2) and assuming tat ct = 0. On te next stage te influence of ratio cr / and oter parameters on adjustment function α could e determined. Wen stress in tensile reinforcement is close to yield state te strengt of compressive concrete remains partially unused. Tus in suc stress state te triangular design diagram of stress distriution witin te compressive zone is most relevant. Fig. 2. Design state of stress witin te cross-section wen tensile reinforcement yields

4 Vidmantas Jokūaitis et al. / Procedia Engineering 57 ( 203 ) Wen performance of tensile concrete aove te crack is neglected carrying capacity of te eam increases. On te oter and, suc increase of carrying capacity sould e reduced ecause of ignorance of te compressive reinforcement and plastic deformations in compressive concrete. Wen taking into account tese assumptions (Fig. 2) and te condition of static equilirium etween te moments of internal and external forces, we can write te following expression for calculation of carrying capacity of te eam: M u σ yas ( cr, lim + 2 3d). (5) 3 Wen calculating te stress σ y, Equations (2), (4) and (5) results te same values ecause of insignificant influence of te tensile concrete aove te crack on te stress state. Te data of experimental researc on 28 eams of rectangular cross-section was used to analyze te adjustment function α. In 26 of tese experimental eams te various pre-stress degree and low reinforcing ratio was present. Reinforcing ratio in te remaining 2 eams was significantly iger [7-8]. Te main parameters of experimental eams are given in Tale. All tested eams failed in pure ending zone wic was middle one tird of te eam span. Te span for all eams was.80 m wit exception of two S group eams wic span was.20 m. One eam in eac series (including eams S and S2) was loaded in steps 0. M u2 all te way to te incipient failure. Te remaining test eams were loaded in steps 0. M u2 until te (.3.5)M cr (ere M u2 and M cr ultimate and cracking moments of te eams respectively) and ten unloaded. Later te eams were loaded until te (.75 2.)M cr and unloaded again. Finally te eams were loaded in steps 0.2 M u2 until te incipient failure. Te depts of te cracks witin te pure ending zone were measured y 24 times magnifying microscope. Dept of one normal crack at te concrete failure point in compressive zone was additionally monitored y measuring longitudinal deformations. Te duration of eac loading step was 20 to 30 minutes. Tale. Main parameters of experimental eams Test Dimensions of Caracteristics of tensile reinforcement Concrete strengt eam Quantity cross-section f group, mm c,cue, MPa Quantity of rears and ρ, % P, kn σ teir diameter, mm y, MPa A B (ard wire) C D AI BI (deformed rears) 587 CI S (deformed rears) S (even rears) Te values of adjustment function α were determined y using te tecnique descried aove and te relationsips etween te depts of normal cracks and ending moments. Also te clear influence of te ratio cr,lim / and reinforcing ratio on te adjustment function α (wen σ s = σ y ) was determined. Te relationsip etween te measured dept of te crack cr,lim and dept ct otained from Eq. (4), wen σ s = σ y, was determined y analyzing experimental researc data, see Tale 2. According to te analysis te adjustment function α could e calculated y te following formula (coefficient of correlation ): 5.53 cr.4 α=, (6) ρψ

5 470 Vidmantas Jokūaitis et al. / Procedia Engineering 57 ( 203 ) were ρ = A s / (d)00 reinforcing ratio; d design dept of cross-section; factor ψ =.6 wen ρ = 0.37%, ψ = wen ρ = (0.79.0)% and ψ = 0.65 wen ρ =.82%. Te intermediate values of product ρψ could e otained y interpolating. Tale 2. Relation etween adjustment function α and geometrical caracteristics of te eams Test eam Test eam group code cr ct α cr ct A A A A A B B B B B C C C D D D D D AI AI AI AI AI BI BI BI BI BI CI CI CI CI CI S S S2 S Te eigt of tensile concrete zone aove te crack could e calculated y te following formula (coefficient of correlation 0.966): ct cr =, (7) 0ηω were η =, wen deformed ars are used for reinforcing and η =.2, wen deformed wires are used; ω = 25ρ, wen ρ < 0.8% and ω =, wen ρ 0.8%. According to te criteria of crack propagation known in fracture mecanics te following empirical relationsip etween parameters of normal crack could e written [4]:

6 Vidmantas Jokūaitis et al. / Procedia Engineering 57 ( 203 ) δiccr w =. (8) ct 3 Here critical widt of crack tip δ = d ( diameter of tensile reinforcement). Eqs. (7) and (8) could e used to calculate te dept of te crack: Ic cr 0.8w = 0.2w +δ ηω. (9) Suc teoretical-empirical expression assist te more reliale control of measurements of normal crack parameters wen investigating te structures. 4. Conclusions. Te possiilities offered in fracture mecanics Eqs. () and (2) could e used for analysis of stress state in flexural reinforced concrete memers togeter wit known section metod wen writing te equations of static equilirium etween internal and external forces Eqs. (4) and (5). 2. Adjustment function α allows te evaluation of geometrical caracteristics of reinforced concrete memer. Eq. (6) is valid only for te tested eams descried in tis paper. It sould e refined for te eams of different cross-section sape and wit different (especially iger) reinforcing ratio. Adjustment function could e refined y eiter using te metod presented in tis paper or directly y experimental researc. It is not enoug to know te ratio cr / te reinforcing ratio sould e estimated also. 3. Te formulas presented in tis paper (e.g. Eq. (9)) allow te accurate enoug representation of relationsip etween te various parameters of normal crack of flexural eam. It also allow te more reliale estimation of actual state of flexural eam during te on-site investigation. c References [] Alam, S. Y., Lenormand, T., Loukili, A., Regoin, J. P., 200. Measuring crack widt and spacing in reinforced concrete memers, Proceedings of te 7t International conference on Fracture Mecanics of Concrete and Concrete Structures, Korea, Seoul, pp [2] Gilert, R. I., Control of Flexural Cracking in Reinforced Concrete, Structural Journal 05(3), pp [3] Jokūaitis, V., Pukelis, P., Kaminskas, K. A., 993. Stress Assessment of Reinforced Concrete Structures wit Cracks, Proceedings of IABSE Colloquium Copenagen. Remaining Structural Capacity Report, pp [4] Kovacs, T., 200. Crack-related damage assessment of concrete eams using frequency measurements. PD tesis. Budapest, p. 70. [5] Niemen, V. N., 967. Experimental researc on deformations of flexural reinforced concrete memers sujected to static loading. Summary of doctoral tesis. Kaunas, p. 25 (in Russian). [6] Sagar, R. V., 20. Damage assessment reinforced concrete eams using acoustic emission tecnique, Proceedings of te National Seminar & Exiition on Non-Destructive Evaluation, NDE 20, Decemer 8-0, 20, pp [7] Saraf, H., Soudki, K., Strengt Assessment of Reinforced Concrete Beams wit Deonded Reinforcement and Confinement wit CFRP Wraps, Proceedings of 4t Structural Speciality Conference of te Canadian Society for Civil Engineering, Montreal, Queec, Canada, June 5-8, 2002, p. 0. [8] Murty, A. R. C., Palani, G. S., Iyer, N. R., State-of-te-art review on fracture analysis of concrete structural components. Sadana 34(2), pp [9] Gerdžiūnas, P., Rozenliumas, A., 973. Deformations in compressive zone of flexural reinforced concrete memers wit flanges, Reinforced Concrete Structures 5, pp (in Russian). [0] Jokūaitis, V., 967. Influence of consistent and accidental cracks on reinforced concrete eams sujected to sort-term loading. Doctoral tesis. Kaunas, p. 235 (in Lituanian). [] Rozenliumas, A., 966. Calculation of reinforced concrete structures y considering te tensile stress in concrete, Researc on Reinforced Concrete, pp (in Russian). [2] Židonis, I., 973. Researc on stress and strain in reinforced concrete wit various tensile zone and sujected to static sort-term loading, Reinforced Concrete Structures 5, pp (in Russian). [3] Baluc, M. H., Azad, A. K., Aswavi, W., 992. Fracture mecanics application to reinforced concrete memers in flexure in Application of Fracture Mecanics to Reinforced Concrete, Carpinteri, A. (Ed.), London, pp [4] Jokūaitis, V., Kamaitis, Z., Cracking and repair of reinforced concrete structures. Monograp. Tecnika, Vilnius, p. 55 (in Lituanian). [5] Jokūaitis, V., Pukelis, P., Influence of longitudinal reinforcement on development of normal cracks, Journal of Civil Engineering and Management (), pp [6] Raczuk, T., Belytscko, T., Application of particle metods to static fracture of reinforced concrete structures, International Journal of Fracture 37, pp

7 472 Vidmantas Jokūaitis et al. / Procedia Engineering 57 ( 203 ) [7] Girnys, M., Te analysis of longitudinal reinforcement stresses calculation metods in cracked reinforced concrete eams. MSc tesis. Vilnius, p. 42 (in Lituanian). [8] Kupetauskas, A., Connection etween parameters of cracks and position of tensile reinforcement in cross-section. MSc tesis. Vilnius, p. 65 (in Lituanian).