Available online at www.sciencedirect.com ScienceDirect Transportation Research Procedia 14 (016 ) 411 40 6th Transport Research Arena April 18-1, 016 Resistance o reinorced concrete columns subjected to axial orce and bending Marek Lechman a, * a Building Research Institute, Filtrowa 1, 00-611 Warsaw, Poland Abstract The paper presents a method or determining the resistance o cro-sections o reinorced concrete (RC) columns subjected to the axial orce and bending. It takes account o the eect o concrete sotening in plastic range and the mean compreive strength o concrete cm. Such members are requently encountered in engineering practice (pillars, bridges, viaducts). The strestrain relationship or concrete in compreion or short term uniaxial loading is aumed according to Eurocode or nonlinear analysis. This stre-strain relation adequately represents the behaviour o the concrete by introducing our parameters. For reinorcing steel characterized by yield stre, linear-elastic model with hardening in plastic range is applied. In the derivation o the resistance o the cro-sections o columns under consideration the ollowing aumptions are introduced: plane cro-sections remain plane elasto-plastic stre/strain relationships or concrete and reinorcing steel are used the tensile strength o concrete is ignored the ultimate strains or concrete and reinorcing steel are determined a priori. The resistance o the RC cro-section is reached when either ultimate compreive strain in concrete or ultimate tensile strain in steel is reached anywhere in that section. The analytical ormulae or the resistance N Rm relating to the axial orce and M Rm relating to the bending moment are derived by integrating the equilibrium equations o the cro-section, taking account o physical and geometrical relationships as well as the condition o the ultimate limit state. On the basis o a combinatorial approach, twelve poible orms o the stre distribution in the section are considered. Using the derived ormulae the interaction curves with the values o the normalized, cro-sectional orces n Rm = N Rm /(b t cm ) and m Rm = M Rm /(b t cm ) or the rectangular cro-section have been obtained (b, t dimensions o the rectangle). The obtained ormulae describe the crosection under consideration in the phase o ailure. Replacing the mean values cm and by the corresponding design values cd * Marek Lechman. Tel.: +48--5796-117; ax: +48--5796-189. E-mail addre: m.lechman@itb.pl 35-1465 016 The Authors. Published by Elsevier B.V. This is an open acce article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility o Road and Bridge Research Institute (IBDiM) doi:10.1016/j.trpro.016.05.83
41 Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 and yd one obtains ormulae determining the design values o the normalized cro-sectional orces n R = N R /(b t cd ) and m R = N R /(b t cd ). For presentation o the proposed deormation model numerical calculations have been perormed. They are presented in the orm o interaction diagrams or rectangular cro-sections. Each curve reers to the corresponding value o the reinorcement ratio. The maximum compreive strain in concrete is calculated at the extreme ibre in the compreion zone o the section. The points located on the n Rm axis are related to pure compreion, while on the m Rm axis to pure bending. The occurrence o the tensile strains in the cro-section leads to the crack ormation in the concrete. Moreover, these solutions have been compared with those based on the parabolic-rectangular diagram or concrete under compreion and with those obtained experimentally by other authors. In a similar way one may obtain interaction diagrams or ring cro-sections. Based on this analysis conclusions are drawn concerning application poibilities o the proposed approach. 016 The Authors. Published by Elsevier B.V. This is an open acce article under the CC BY-NC-ND license 016The Authors. Published by Elsevier B.V.. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility o Road and Bridge Research Institute (IBDiM). Peer-review under responsibility o Road and Bridge Research Institute (IBDiM) Keywords: Reinorced; concrete; column; resistance; section 1. Introduction The resistance o cro-sections o reinorced concrete columns subjected to the axial orce and bending is considered as a theoretical problem as well as an experimental one. For determining the resistance o the crosections under consideration a variety o physical models o materials and methods are applied. Most oten a simpliied approach is used based on the rectangular stre distribution or concrete and represented by such design o cro-sections the bi-linear stre-strain relation or parabola-rectangle diagram or concrete may be also aumed. This diagram was used by Nieser and Engel in Commentary to German code DIN 1056 (1986) as well as in CICIND Model Code or Concrete Chimneys (001). For reinorcing steel in turn such models are used as linear elastic, linear elastic-ideal plastic and linear elastic-plastic with hardening. For analysis o the resistance o ring cro-sections deormation models combined with physical nonlinearity o concrete and reinorcing steel were proposed by Lechman & Stachurski (005) and Lechman (006, 011). The investigations o load-bearing capacity o RC columns under eccentric compreion have been conducted by many researchers, among others by Lloyd et al. (1996) and Trapko and Musia (011). Nomenclature cm the mean compreive strength o concrete cd the design strength o concrete in compreion yield stre o reinorcing steel yd the design yield stre o reinorcing steel E cm secant modulus o elasticity o concrete E s modulus o elasticity o steel E h coeicient o steel hardening c1 the strain at peak stre on the c - c diagram cu1, cu the ultimate strain or concrete su the ultimate strain or reinorcing steel c, c stre, strain in concrete s, s stre, strain in reinorcing steel N the axial orce M the bending moment n Rm the normalized ultimate axial orce m Rm the normalized ultimate bending moment m Rm t the thickne o cro-section b the width o cro-section
Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 413 t 1, t coordinates describing the locations o rebars x, coordinate describing the location o the neutral axis da c element o the concrete area A c F a1, F a areas o the steels in compreion and in tension, respectively s1 the compreive stre o the steel s the tensile stre o the steel 1, reinorcement ratios o steels in compreion and in tension, respectively. Derivation o ormulae or rectangular cro-sections.1. General aumptions The rectangular cro-section o a RC column is subjected to the axial orce N and the bending moment M. Fig. 1. The rectangular cro-section. Distribution o strain, strees in concrete c and strees in steel s acro the section. In the presented derivation the ollowing aumptions are introduced: plane cro-sections remain plane elasto-plastic stre/strain relationships or concrete and reinorcing steel are used the tensile strength o concrete is ignored the ultimate strains or concrete and reinorcing steel are determined a priori.
414 Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 According to Eurocode the stre-strain relation or concrete c - c in compreion or short term uniaxial loading is aumed as (Fig. ): k 1 ( k ) c cm (1) where: = c / c1, c1 the strain at peak stre on the c - c diagram, k = 1,05 E cm c1 / cm. Fig.. Representation o the stre-strain relation or non-linear structural analysis. For reinorcing steel characterized by yield stre, linear elastic model with hardening in plastic range is applied: s or - () E ) or (3) s h( E ) or - (4) s h(, Es (5) In urther considerations the corresponding dimensionle coordinates are used: = x / t,, = x / t, 1 = t 1 / t, = t / t. Due to the Bernoulli aumption one obtains: ' ' ( 1 ) (6)
Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 415 where: - the maximum compreive strain in concrete... Equilibrium equations The equilibrium equation o the axial orces in the cro-section takes the ollowing orm: x 0 da F c c s1 a1 F s a N 0 (7) The sectional equilibrium o the bending moments about the symmetry axis o the rectangle can be expreed in the orm: x 0 c ( 0.5t x' ) dac s1 Fa 1 (0.5t t1) s Fa (0.5t t) M 0 (8) The equations (7) (8) are strongly nonlinear and diicult to be integrated. Thereore, an analytical approach was chosen and applied to obtain a value-added solution. The resistance o the RC cro-section is reached when either ultimate compreive strain in concrete cu or ultimate tensile strain in steel su is reached anywhere in that section. On the basis o a combinatorial approach, all poible cases o stre distribution in concrete and reinorcing steel have been considered. Taking account o the physical and geometrical relationships (1) (6) in the equilibrium equations (7) and (8), ater integrating and some rearrangements one obtains the ormulae or determining the normalized ultimate bending moment m Rm and the normalized ultimate axial orce n Rm or the rectangular section: n 1 Rm cm (1/( k )) W 0.5k (1/( k )) ( W cm E k 1 h 1 E k11 h 1 ' ((1 ) ) k 1 ' ((1 ) ) k 11 / W )lnw ' 1 (1 ) 1 3 ' 1 (1 ) (9) m 1 Rm 0.5( W1 (1/( k ))) 0.5 W1 0.5k ((1/( k )) (1/( k )) 3 (1/ 3) k ( W /(( k ) W3 )) 0.5lnW ( W / W3 ) lnw cm cm E (0.5 1) k11 E (0.5 ) k 1 h h 1 ' ((1 ) ) k 1 ' ((1 ) ) k 11 ' 1 (1 ) 1 ' 1 (1 ) (10) N Rm nrm, bt cm M, N Rm, M Rm - the ultimate axial orce and bending moment, respectively, (11) Rm mrm bt cm
416 Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 (1 1) cu1,, = cu1, 1) su su cu1 1 (, (1) where: k = /( c1 ), W 1 =k-k, W =k(k-)+1, W 3 =(k-)k, W=1+(k-)k ; k = ((-1) k +1). The obtained ormulae (9) (1) describe the cro-section under consideration in the phase o ailure. Replacing in expreions (9) (11) cm and by design strength o concrete in compreion cd and design yield stre o steel yd, one obtains ormulae determining the design values o the normalized cro-sectional orces n R =N R /(bt cd )and m R =N R /(bt cd ). 3. Analysis o numerical calculations Using the derived ormulae the interaction diagrams with the normalized resistances n Rm - m Rm have been obtained or the ollowing data (Fig. 3): concrete grade C0/5, yield stre o steel =500 MPa, reinorcement ratios o steel in compreion and in tension 1 = =, t 1 /t = 0.1, E h = 0, the limiting value cu1 = -3.5 or -3.5 su 10. Each curve reers to the corresponding value o the substitute reinorcement ratio / cm. The two numbers c / s at each indication point are compreive strain in concrete and tensile strain in steel. The points located on the n Rm axis are related to pure compreion and on the m Rm axis to pure bending. The points denoted by c /0 can be interpreted as a transition rom the state c /( s <0) described as wholly in compreion (uncracked) to that c /( s >0) characterized by the occurrence o the tensile strains which cause the crack ormation in concrete (cracked). Fig. 3. Interaction diagrams with the normalized resistances n Rm - m Rm taking account o the eect o concrete sotening in the plastic range. Characteristic eature o the obtained interaction curves is the sign changing o the normalized ultimate bending moment m Rm within the state described as wholly in compreion, or the couples o strains -3.5 / c, -3.5 < c <-1 (Fig. 3). This act may be explained as a result o the eect o concrete sotening that makes the
Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 417 le compreive part o the cro-section become the more compreive and backwards. For the case when cu1 = c1 (elastic range) the values o the obtained interaction curves are very close to those based on the parabola-rectangle diagram. Fig. 4 presents the interaction curves resulting rom the relation (1) versus those obtained on the basis o the parabola-rectangle diagram or concrete in compreion ( c1 = -.0, cu1 = -3.5 ). The comparison presented in Fig. 4 indicates that the solutions based on the parabola-rectangle stre distribution in concrete may lead to overestimation o the section resistance. It is also apparent that the resistance o cro-sections is strongly inluenced by the concrete sotening interpreted as the descending part o the curve c - c (1). The increasing values o t 1 / t or t / t result in lower section resistance. It was veriied by calculations that the adequacy o the relation (1) describing the behavior o the concrete in compreion is limited to the concrete claes not higher than C35/45. In a similar way one may obtain ormulae determining the resistance o RC ring cro-sections o thin or thick thicknees in respect to the external radius o the ring. Fig. 4. Comparison o the solution based on the nonlinear relation c c (1) given in EC with that based on the parabola-rectangle diagramor concrete in compreion 4. Comparison o theoretical and experimental results The calculated results have been compared with those obtained by testing conducted by Lloyd at al. (1996) on RC columns under eccentric compreion, with rectangular cro-section o 175 mm x 175 mm and the height o 1680 mm. The longitudinal reinorcement o the column was symmetric: three rebars 1 mm, = 430 MPa, E s = 00 GPa; cm = 44,78 MPa, E cm = 3 GPa. The experimental setup is exhibited in Fig. 5. The ailure loads determined experimentally were as ollows: P 1 = 1476 kn, e 1 = 15 mm; P = 830 kn, e = 50 mm; P 3 = 660 kn, e 3 = 65 mm. The ultimate strain in concrete at ailure was aumed in calculations as -.4 which corresponds to the peak stre on the c c diagram (Fig. ). The comparison presented in Fig. 6 shows generally a good conormity between the analytical solution and the values o ailure loads. As it is seen, the theoretical values are lower than those obtained rom the experiment due to neglecting the eect o coninement in the section model o the column (stirrups).
418 Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 As the next example the results o testing conducted by Trapko at al. (011) on unstrenghtened RC columns under eccentric compreion, with rectangular cro-section o 00 mm x 00 mm and the height o 1500 mm are analyzed. The longitudinal reinorcement o the column was symmetric: two rebars 1 mm, steel grade A-IIIN = 608 MPa, E s = 4 GPa, the transverse reinorcement consisted o stirrups 6 mm made o steel grade A-I; cm = 31,9 MPa, E cm = 31 GPa. The ailure loads determined experimentally were as ollows: P 1 = 1548 kn, e 1 = 0 mm; P = 1386 kn, e = 16 mm; P 3 = 1098 kn, e 3 = 3 mm. The ailure mechanisms o the columns under consideration occurred in the orm o crushing o concrete in the upper part o the members and yielding o longitudinal reinorcing steel. The values collected in Table 1 conirm generally a good conormity between the calculated and experimental results. The dierences between the corresponding values o the section resistance are equaled 0, %, 4,9 % and 13,5 %, respectively. In author s opinion urther experimental work is needed concerning the post-critical behaviour o RC members subjected to the axial orce combined with bending or veriying the proposed section model. Fig. 5. Experimental setup o the RC column.
Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 419 n Rm - calculated; experimental m Rm Fig. 6. Comparison o the calculated and experimental results. Table 1. Comparison o the values o the cro-section resistance o columns. Failure load P i [kn]; eccentricity e i [mm] Normalized ailure loads n Normalized resistance n Rm P 1 = 1548; e 1 = 0 1,154 1,13 P = 1386; e = 16 1,086 1,1397 P 3 = 1098; e 3 = 3 0,8605 0,7441 5. Conclusions Using deormation model analytical ormulae have been derived or determining the resistance o RC rectangular cro-section subjected to bending and axial orce according to the stre-strain relationship given in Eurocode or non-linear analysis. The ormulation like this enables to analyze the behaviour o the cro-section o RC columns in the phase o ailure. The resistance o cro-sections is strongly inluenced by the concrete sotening represented as the descending part o the curve c - c or concrete in compreion. It has been proven that calculated results conorm to those obtained by testing on RC columns under eccentric compreion. The proposed approach results in more realistic evaluation o the resistance o the cro-section compared to that based on the parabolic-rectangular diagram or concrete in compreion. The obtained interaction diagrams can serve or determining the required reinorcement ratio and the thickne o rectangular cro-sections. The presented algorithm can be easily implemented and eectively used in structural design. Further experimental work is needed concerning the postcritical behavior o RC members subjected to the axial orce and bending or veriying the proposed section model. Reerences Kami -B-0364: 00 Knau M., Obliczanie konstrukc
40 Marek Lechman / Transportation Research Procedia 14 ( 016 ) 411 40 Knau M., Golubiska A., Knyziak P., Wydawnictwo Naukowe PWN, 013. Lechman M.,, Prace Naukowe Instytutu Techniki Budowlanej, Rozprawy, Wydawnictwa Instytutu Techniki Budowlanej, Warszawa 006. Lechman M., Stachurski A., Nonlinear section model or analysis o RC circular tower structures weakened by openings, Structural Engineering and Mechanics. An International Journal, Vol. 0, No, 005. Lechman, Instytut Techniki Budowlanej, Warszawa 011. Lloyd N. A., Rangan B. V., Studies on High-Strength Concrete Columns under Eccentric Compreion. ACI Structural Journal, Technical Paper, 93-S59, November-December 1996, 631-638. Nieser H., Engel V., Industrieschornsteine in Maivbauart (Structure o industrial chimneys), Kommentar zu DIN 1056, 1986 (in German). Trapko M., Musia M., The eectivene o CFRP materials strengthening o eccentrically compreed reinorced concrete columns, Archives o Civil and Mechanical Engineering, Vol. XI. No. 1, 011.