Effects of the compressive reinforcement buckling on the ductility of RC beams in bending

Engineering Structures 37 (2012) 14–23

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Effects of the compressive reinforcement buckling on the ductility of RC beams in bending Adelino V. Lopes a,1, Sérgio M.R. Lopes a,2, Ricardo N.F. do Carmo b,⇑ a b

Department of Civil Engineering, F.C.T.U.C. – Polo II – University of Coimbra, 3030-290 Coimbra, Portugal Department of Civil Engineering, Instituto Superior de Engenharia de Coimbra, Rua Pedro Nunes – Quinta da Nora, 3030-199 Coimbra, Portugal

a r t i c l e

i n f o

Article history: Received 29 September 2011 Revised 21 December 2011 Accepted 23 December 2011 Available online 2 February 2012 Keywords: Structural concrete Beams Types of failure Bar buckling

a b s t r a c t This paper presents a study on the structural behavior of 15 reinforced concrete beams. The beams were 3 m long and the rectangular cross section measured 0.20  0.30 m2. They were subjected to two concentrated forces, applied at a third of the span until the failure occurred. The beams had different amounts of longitudinal and transverse reinforcement and were divided into five series. Their strength, deformation and type of failure were analyzed. The main results are presented and discussed in terms of load-displacement, moment–curvature, ductility factor and plastic rotation capacity. In the light of the results it was concluded that beams with high tensile reinforcement ratios suffer premature failure due to the buckling of the compressive bars. These results indicate the importance of the stirrups spacing in RC beams. The influence of the tensile reinforcement ratio and the transverse reinforcement on the plastic rotation capacity and on the general beam behavior was analyzed. This experimental study made it possible to define maximum values for q and minimum values for Asw/s to ensure a certain structural energy dissipation capacity. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The behavior of simple RC structures is now well known [1]. It is nonetheless useful to see how certain parameters influence both the members’ failure and their behavior during several phases. The tensile reinforcement ratio, q, and the transverse reinforcement, Asw/s, have a significant influence on a beam’s bending behavior, particularly on its ductility [2–4]. The ductility of RC structures is certainly desirable even if it is not indispensable for structures safety [5,6]. So, the way elements are designed should consider not only strength but also the ability to support plastic deformations without significant loss of strength. Section failure is defined by the deformation limits assumed for the materials, i.e. the ultimate compressive strain in the concrete [7–9]. Depending on the materials’ properties and quantities used, specifically, the tensile reinforcement ratio, the section can have several types of failure. When a low longitudinal tensile reinforcement ratio is used a brittle failure may occur after the first crack appears because in this case the steel has less strength than the force supported by the tensile concrete. But when a high ⇑ Corresponding author. Tel.: +351 239 790200; fax: +351 239 790201. E-mail addresses: [email protected] (A.V. Lopes), [email protected] (S.M.R. Lopes), [email protected], [email protected] (R.N.F. do Carmo). 1 Tel.: +351 239 797100. 2 Tel.: +351 239 797253/100; fax: +351 239 797123/242. 0141-0296/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.12.038

longitudinal tensile reinforcement ratio is used, the ultimate strain in the concrete is reached before the steel reaches yielding, and this may originate a brittle failure by concrete crushing. Therefore, it is important to choose the appropriate q since it is a relevant parameter to consider in the beam’s design, both to avoid an undesirable brittle failure and to use the material’s strength and deformation capacity, mainly that of the steel, effectively. This parameter also gives relevant information on production cost. Consideration of the tensile reinforcement ratio makes it possible to understand if a more reasonable design can be accomplished [10]. The compressive reinforcement ratio, q0 , also affects the beam’s bending behavior. The longitudinal compressive reinforcement is used to share the compression stress with the concrete. These bars increase the strength capacity in the compression area. However, the hypothesis must be considered that these bars may suffer lateral instability for a load lower than the maximum load sustained by concrete. In these cases the buckling of the compressive reinforcement causes the concrete failure, usually by spalling of the concrete cover. The buckling of the compressive reinforcement is to some extent influenced by the spacing of the transverse reinforcement (stirrups) and by the diameter of compressive bars. The closer the stirrup spacing the higher the impediment to the compressive bars’ suffering lateral instability. The confining effect on concrete provided by transverse and compressive reinforcement, and the stirrups’ role as factor for lateral stability in compressive reinforcement, has been particularly well studied in columns

A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

15

Nomenclature d fc fym ftm q qo s As A0s Asw L M

displacement at mid-span section concrete compressive strength mean value of yield strength of reinforcement mean value of tensile strength of reinforcement behavior factor basic behavior factor stirrup spacing cross-sectional area of tensile reinforcement cross-sectional area of compressive reinforcement cross-sectional area of transverse reinforcement beam length bending moment

[11–13]. In relation to beams the subject is still not thoroughly understood. This study will try to show whether some additional recommendations related to the transverse reinforcement detailing should apply. Particularly when beams have certain tensile and compressive ratios the engineer should take some care with stirrup spacing. The reserve of strength in sections could be very important in extreme situations and this is also addressed in this work. Reducing the spacing between stirrups adds to the confining effect on the concrete core which increases not only the concrete’s strength but also its deformation capacity, i.e., the ultimate compressive strain in the concrete is higher [14]. So, in some design situations, although a certain value is assumed for concrete strength it is actually higher. This could cause a significant difference between the maximum moment resistance and moment at the yielding steel. Another consequence that should be highlighted is that the member’s maximum deformations could also be improved. The moment redistribution capacity, and consequently the safety of the structure, is related to the plastic rotation capacity in critical regions. Plastic deformation without significant loss of strength affects the ability to exploit the additional resistance of hyperstatic structures. Designs based on linear analysis with moment redistribution and plastic analysis, which allows a design optimization, depend on the member’s ductility [15]. The design of concrete buildings for seismic action should provide a structure with an adequate ductility without substantial reduction of its overall resistance to horizontal and vertical loading. In this context, the importance of dissipating energy is crucial for structure stability, to avoid premature brittle failures. When an elastic structural analysis is performed, the seismic effects are reduced by introducing the behavior factor q. As mentioned on Eurocode 8 – 1, Section 3.2.2.5, this reduction occurs because ‘‘the capacity of structural systems to resist seismic actions in the non-linear range generally permits their design for resistance to seismic forces smaller than those corresponding to a linear elastic response’’ [16]. The overall ductility assumed when calculating the seismic actions should be commensurate with the deformation demands in critical regions. These considerations make clear that is useful to establish a relationship between the q value and the critical region’s ductility. It would be positive for engineering practice, at the design and member detailing stages, to relate the q value to q and to Asw/s. Finally, in recent years steel has seen some technological developments and steels with high ductility have appeared on the market. The use of these steels does not necessarily imply that the reinforced concrete member has higher ductility. Depending on the geometrical and mechanical proprieties of the member, the increase in steel ductility may be reflected in the member’s plastic

My Mmax Pmax V

q q0 £ hpl

l/ ld

bending moment corresponding to the yielding steel maximum bending moment maximum load shear force tensile reinforcement ratio compressive reinforcement ratio diameter of a reinforcing bar plastic rotation capacity curvature ductility factor displacement ductility factor

behavior, but in some situations there is no additional benefit. This study, based on experimental results, sets out to shed some light on this situation. 2. Experimental program 2.1. Overview This study was carried out on 15 beams until failure was reached. All beams were 3 m long, had a span of 2.75 m and were subjected to a symmetrical load system consisting of two concentrated forces applied 0.90 m from the supports (Fig. 1). The central part of the span with 0.95 m long was theoretically submitted to pure flexure (region without shear force), so the main tensile and compressive stress trajectories are parallel (on state I) to the longitudinal beam axis. The main stress trajectories have a direct influence on cracking, whose pattern is roughly at right angles to the direction of the principal tensile stresses. The action was applied by an electromechanical actuator with a maximum capacity of 1000 kN. The force from an electromechanical actuator was applied to the metal beam and it was then transmitted to the beam tested through the supports. The action was gradually increased by controlling the deformation at a rate of 0.02 mm/s (piston’s displacement). Around 10 pauses at specific loads were predicted for each test. After each load increment, the readings were recorded and the crack pattern was highlighted. Although the main aim of the tests was to study the ultimate behavior of the beams, other relevant aspects of their behavior at different load levels were recorded. The supports’ reactions were measured by load cells placed under the beam supports and the vertical displacements were measured through LVDTs. The experimental program was divided into five series, each of which studied the influence of one parameter on the beam’s structural behavior, that is, on its strength, maximum deformations and types of failure. 2.2. Test specimens In all five series, the beams had a rectangular cross-section of 0.20  0.30 m2. The beams were labeled for identification purposes; for example S1B1, the first letter means series, first number identifies the series, second letter means beam and the second number identifies the beam. The geometrical and mechanical properties of all the beams tested are given in Table 1 and Fig. 2. 1st series – role of q: This series contains 4 beams and q ranges from 0.05% to 2.38%. The transverse reinforcement was arranged to ensure that failure

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A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

Fig. 1. Test setup.

Table 1 Mechanical properties of tested beams. As (cm2)

A0s ðcm2 Þ

Asw/s (cm2/ m)

q

q0

(%)

(%)

S1B1 S1B2 S1B3 S1B0 S2B1 S2B2 S2B3 S3B1 S3B2 S3B3 S4B1 S4B2 S4B3 S5B1

0.25 12.57 12.06 6.03 6.03 6.03 6.03 12.57 12.57 12.57 1.01 1.57 6.79 6.79

0.25 0.57 4.02 1.01 0 1.01 1.01 0 0.57 0.57 1.57 1.57 1.57 1.57

2.10 11.40 12.67 3.00 0 6.33 12.66 0 6.33 12.66 3.17 3.17 3.17 0

0.05 2.38 2.34 1.13 1.13 1.13 1.13 2.38 2.38 2.38 0.17 0.26 1.13 1.13

0.05 0.11 0.78 0.19 – 0.19 0.19 – 0.11 0.11 0.26 0.26 0.26 0.26

30.6 29.9 38.3 38.6 38.9 38.9 40.0 39.8 24.4 24.3 23.8 24.3

S5B2 S5B3

6.79 6.79

1.57 1.57

3.17 9.5

1.13 1.13

0.26 0.26

23.8 23.9

Beam

fcm, cube (MPa)

Detailing description and failure expected

29.9

Fragile failure (very low tensile reinforcement) High tensile reinforcement ratio with low compressive reinforcement ratio High tensile reinforcement ratio with high compressive reinforcement ratio Standard beam Average tensile reinforcement ratio without transverse reinforcement Average tensile reinforcement ratio with intermediate transverse reinforcement Average tensile reinforcement ratio with high transverse reinforcement High tensile reinforcement ratio without transverse reinforcement High tensile reinforcement ratio with intermediate transverse reinforcement High tensile reinforcement ratio with high transverse reinforcement Very low tensile reinforcement ratio with high ductility steel Low tensile reinforcement ratio with high ductility steel Average tensile reinforcement ratio with high ductility steel Average tensile reinforcement ratio with high ductility steel and without transverse reinforcement Average tensile reinforcement ratio with high ductility steel Average tensile reinforcement ratio with high ductility steel and high transverse reinforcement

occurred by flexure and not prematurely by shear force. This series analyzed the effect of the amount of longitudinal reinforcement on the ultimate behavior, specifically, how different ratios of the tensile reinforcement may lead to different types of failure. 2nd and 3rd series – role of Asw/s: The 2nd series contains three beams where q is 1.13% and Asw/s ranges from 0 to 12.66 cm2/m. The 3rd series, similar to the 2nd, also has three beams with similar Asw/s but now with a different q, which is 2.38%. The goal of these tests was to study the influence of compressive reinforcement on the flexural beam behavior. It was found in some previous experimental tests that the compressive reinforcement suffered lateral instability, buckling, when the critical sections had high curvature values. These results raised the possibility that concrete crushing might have been premature because of the lateral forces resulting from the reinforcement buckling. Several tests were defined, varying Asw/s, to study this phenomenon. The purpose was to examine how the reinforcement used to resist the shear force could be used to confine the compressive reinforcement and thus prevent its lateral instability. 4th series – role of q using high ductility steel: This series has three beams with different q: 0.17%, 0.26% and 1.13%. The reinforcement of these beams had higher ductility. The aim was to evaluate the effectiveness of the increased steel

ductility on the ductility of RC beams and to see if there is any relationship with their resistance. The beams with low q values were designed to have a failure on the tensile side and the beam with higher q value, 1.13%, was supposed to have failure in the compressed concrete. 5th series – role of Asw/s using high ductility steel: This series contains three beams but one of them, S5B2, is the same as the S4B3 of the 4th series. These beams have a normal q value of 1.13% and low q0 value of 0.26%. As the steel used has a great deformation capacity, beam rupture was expected to occur on the compressed zone. The objective of these tests was to study the influence of the higher steel ductility on confining the concrete and the compressive reinforcement. The transverse reinforcements’ Asw/s values were: 0, 3.17 and 9.50 cm2/m.

2.3. Materials’ properties Three concrete types were used, two with a medium compressive strength, about 24 and 30 MPa, and other with strength around 39 MPa (see Table 1). The compressive strength was determined using cubic specimens with 150 mm. In all, 30 cubes were tested during the course of the study.

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A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

3Ø16

Ø6//0,05

2Ø16

0.30 m

Ø4//0,12

2Ø6

0.30 m

Ø6//0,19

2Ø4

0.30 m

0.30 m

2Ø8

Ø6//0,045

4Ø20

2Ø4

6Ø16

0.20 m

variable

2Ø6 variable

3Ø16

Beam S1B2 0.30 m

0.30 m

2Ø8

Beam S1B1

0.30 m

Beam S1B0

4Ø20

2Ø10

2Ø10

Ø6//0,18

Ø6//0,18

0.20 m

Beams S3B1-2-3

2Ø10

2Ø10

2Ø10

Ø6//0,18

without stirrups

Ø6//0,06

6Ø12

6Ø12

6Ø12

0.30 m

Beams S2B1-2-3

0.20 m

Beam S4B3 =

Beam S4B1

Beam S4B2

0.20 m

Beam S5B1

S5B3

2Ø10

2Ø8 0.20 m

0.20 m

Beam S1B3

Beam S5B3

Fig. 2. Details of the reinforcement.

Table 2 Average values of fym and ftm. Beams

Steel

Diameter £ (mm)

fym (MPa)

ftm (MPa)

1st series

S500NR

4 6 8 16

642 587 526 531

815 665 632 645

2nd and 3rd series

S500NR

6 8 16 20

571 599 596 619

612 706 694 723

4th and 5th series

S500NRSD

8 10 12

532 551 541

635 648 632

The longitudinal and transverse reinforcement used was hot rolled steel, class S500NR (ribbed 500 MPa yield strength with normal ductility – class B according Eurocode 2 annex C) and class S500NRSD (ribbed 500 MPa yield strength with high ductility – class C according Eurocode 2 annex C). The steel’s proprieties, i.e. fy and ft, were checked according to European Standard NP EN 10002-1 [17]. At least three specimens were tested for each bar diameter used in the beams. The average values for yielding and maximum strength are presented in Table 2. 3. Experimental results and analysis 3.1. Maximum load and type of failure Table 3 shows the maximum load and the load for steel yielding, recorded experimentally. The steel yielding did not occur

simultaneously in all bars. When the yielding strain is recorded at the steel’s centroid it means that only part of the steel reached the yield stress. The yielding of all bars occurs when the steel fiber closest to the neutral axis reaches yield strain. Considering this situation the yielding point was defined by the intersection of the lines corresponding to the horizontal line at the maximum load and the line resulting from the beam’s stiffness at state II (see Fig. 3). This procedure was used for all beams and based on Eurocode 8, Part 1, annex B. Table 3 also presents the maximum load values predicted theoretically. These values were obtained considering average strength of the materials and the small deviations in the cross-section dimensions. The bending moment corresponding to the maximum load was computed according to Eurocode 2 rules and recommendations. The unreinforced beam S1B1 (q = 0.05%) had an abrupt failure, without warning, immediately after the appearance the first crack, as predicted. Analyzing the values presented for beams S1B2 and S1B3, it is found that the experimental values are almost 10% lower than those expected theoretically. These results can be explained by the buckling of the compressive reinforcement (Fig. 4). The lateral instability of the compressed bars causes the concrete to lose resistant capacity because it is subject to a tension stress in the perpendicular direction (particularly relevant in the sections between the transverse reinforcements). On the collapse of beam S1B0, even though the maximum experimental load was higher than foreseen, buckling of the compressive bars was seen. This might support the idea that the maximum load could be higher if the phenomenon of bars buckling had not occurred. All beams in the 2nd and 3rd series were designed to fail by flexure. However, beams S3B1 and S3B2 had a shear failure because of a wrong estimate of the maximum shear strength. Beam

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A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

Table 3 Comparison of the theoretical and the experimental values. Beam

Pteo (kN)

Pmax,

S1B1 S1B2 S1B3 S1B0 S2B1 S2B2 S2B3 S3B1 S3B2 S3B3 S4B1 S4B2 S4B3 S5B1 S5B2 S5B3

20 319 326 177 194 196 196 359 365 365 26 49 191 185 191 191

17 275 303 185 194 202 198 274 329 345 36 54 192 183 192 186

exp

(kN)

Pyp,

exp

Pmax;exp Pteo

(kN)

– 209 259 160 171 178 178 226 262 272 32 48 170 162 170 157

0.85 0.86 0.93 1.05 1.00 1.03 1.01 0.76 0.90 0.94 1.38 1.10 1.00 0.99 1.00 0.97





Pmax Pyp exp

– 1.32 1.17 1.16 1.13 1.13 1.11 1.21 1.26 1.27 1.13 1.13 1.13 1.13 1.13 1.18

200

P (kN)

150

YP

0.85Pmax

100

50

0 0

20

40 d (mm)

60

80

Fig. 3. Determination of the yielding point (Beam S1B0).

S2B1 collapsed due to concrete crushing. It should be noted that this beam had neither compressive bars nor transverse reinforcement in the pure bending zone. Failure of beam S2B2 occurred in the compression area and lateral instability of the compressive bars was also observed (Fig. 5). Beam S2B3 had a failure similar to beam S2B1, concrete crushing, but after the beam suffered considerable deformation. No buckling of the compressive reinforcement was observed in this beam. Such behavior is due, without any doubt, to the narrow stirrup spacing. Despite this beam having a maximum load similar to beam S2B2, it was found that it had a much greater capacity to suffer plastic deformations. Spalling of the compressive concrete that covered whole length of the region with pure bending occurred in beam S3B3. Lateral instability of the compressive reinforcement was also observed, although this was very incipient. However, this might be the most plausible reason why P max;exp =P teo has a value of 0.94, i.e. less than 1.

Type of failure Fragile failure Concrete crushing and spalling (with bars buckling)

Concrete crushing and spalling Concrete crushing and bars buckling Large deformation with consequent concrete crushing and without bars buckling Shear failure Concrete crushing and spalling Large deformation with consequent concrete crushing Concrete crushing and spalling Concrete crushing and spalling Large deformation with consequent concrete crushing

The beams of the 4th series had ductile failure, especially beams S4B1 and S4B2, and after suffering large deformations there was concrete crushing and spalling. For beam S4B1 the difference between the maximum load recorded experimentally and that predicted theoretically is very large. The opposite is true of beam S4B3 where the maximum load recorded experimentally is practically equal to the one predicted. In the 5th series the maximum load recorded was very similar for all beams, and not very different from the value predicted theoretically. Fig. 6 shows the relationship between Mmax/My, which quantifies the extra capacity in bending with respect to the design moment. It can be seen that the increase in the tensile reinforcement ratio causes a slight tendency to increase the section’s reserve of resistance, the relation between the moment at the steel yielding and the maximum moment supported. The relationship between Mmax/My and Asw/s can also be seen in Fig. 6.b. Greater transverse reinforcement does not imply a clear increase in the ratio between Mmax and My. Considering the results obtained for beams S1B2 and S1B3, it can be concluded that, beams with high areas of tensile steel, specifically when q is about 2.4%, there is a higher probability of lateral instability occurring in the compressive bars before the maximum moment predicted theoretically is reached. This phenomenon was not observed for sections with lower values of q, of about 1.1%. Due the balance of the internal forces, a larger area of tensile steel corresponds to a higher area of compressive concrete and so the strain at the level of the compressive steel will be higher. Therefore, in these beams the compressive bars will have higher trend to suffer buckling problems if they are not suitably braced, using a small gap between stirrups. Considering the results obtained, it can also be concluded that high ductility steel does not change the type of failure. Comparing

Fig. 4. Buckling of the compressive bars (1st series).

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A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

Fig. 5. Second and 3rd series beams’ failures.

1.500

1.500 S1B0

S1B2

S1B3

S2B1

S2B2

S2B3

S4B1

S4B2

S4B3

S3B1

S3B2

S3B3

S5B1

S5B2

S5B3

1.375

M max/M y

M max/M y

1.375

1.250

1.125

1.125

1.000 0.00

1.250

1.000 0.75

1.50

2.25

3.00

0

ρ (%)

3.5

7

10.5

14

Asw/s (cm2/m)

(a)

(b) Fig. 6. Mmax/My  q and Mmax/My  Asw/s relationship.

the 1st and 4th series, and the 2nd and 5rd series, it can be seen that differences are not significant. 3.2. Load–displacement relationship Based on the results obtained during the tests, the load applied (P) is related to the displacement at the mid-span section (d). It was found for the 1st series’ beams, excepted beam S1B1, that after the cracking phase and before the steel yield, the flexure stiffness is higher the greater the area of the tensile steel (Fig. 7a). After the steel yields, the flexure stiffness falls again and the beam ductility is higher the larger the curve post-peak. Beams S1B2 and S1B3 have a similar q, and it is about the double that of beam S1B0. From Fig. 7a it can be seen that S1B0 have a little more capacity to support plastic deformations. Beams S1B2 and S1B3 have a different behavior post-peak and the reason is probably the diameter and area of the compressive bars. In the 2nd and 5th series of tests, beams with a q of 1.13%, it can be clearly observed that the higher the transverse reinforcement

ratio the greater the deformation capacity (Figs. 7b and e). In the test of beam S2B3 it was not possible to achieve 85% of the maximum supported load in the post-peak phase, although the deformations on the last recording are very high. The transverse reinforcement enables the compressive bars to suffer larger strains without lateral instability occurring. In the 3rd series, beams with a q of 2.38%, the effect of the transverse reinforcement variation is quite different. The increase of deformation with Asw/s was practically nil for beams S3B1 and S3B2 and was not particularly high for beam S3B3 (Fig. 7c). This different behavior occurs because these beams had a shear failure and concrete crushing. The failure was not influenced by the buckling bars. As with the 1st series, the 4th series’ beams have higher flexure stiffness the greater the area of the tensile steel. Beams S4B1 and S4B2 had a very ductile failure and the loss of strength was practically nil after steel yielding. Beam S4B3 had a quite different curve post-peak, with a significant loss of strength although there is some plastic deformation (Fig. 7d). The high ductility of the steel could be the reason for these significant plastic rotations, but it

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A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

220

S1B0

320

0.90Pmax

S1B2 S1B3

165

YP 0.85Pmax

S2B1

P (kN)

P (kN)

240

110

160

S2B2 S2B3

80

55

0

0

YP 0.85Pmax

0

20

40 d (mm)

(a)

60

0

80

30

60

90

120

d (mm)

varying ρ

(b)

360

varyingA sw/s (ρ = 1.13%) S4B1

200

S4B2 S4B3

150

YP

P (kN)

P (kN)

270 S3B1 S3B2 S3B3 YP 0.85Pmax

180 90

50

20

40

60

0.9Pmax 0.95Pmax

0 0

0.85Pmax

100

80

0 0

30

60 d (mm)

d (mm)

(c)

90

120

(d) varying ρ using high ductility steel

varyingA sw/s (ρ = 2.38%) 200

P (kN)

150 100

S5B1 S5B2 S5B3

50

YP 0.85Pmax

0 0

(e)

20

40 d (mm)

60

80

varyingA sw/s using high ductility steel Fig. 7. P–d curves.

might also be that the tensile reinforcement ratio has a major influence on the beam behavior. 3.3. Moment–curvature relationship The moment–curvature, M  1/r, curves were obtained based on the experimental P–d curves. Knowing the vertical displacement at mid-span and integrating the bending moment diagram of a unit load with the bending moment diagram resulting from the load applied, it is possible to determine the actual value of flexure stiffness, EI, in the pure bending zone and throughout the test. After calculating the EI value, the average curvature is easily determined at every load stage. It should be stressed that at a new load step, some sections (those in the central zone of the beam) had bending moments higher that any other moments in previous load steps. However, other sections (nearer to supports) had lower moments and such moments could be compared with moments of other sections in previous load steps. Therefore, at a new load step, moment curvature relationships were already known for some sections. Part of the deformation of the beam

was then fixed and the deflection was only used for evaluating the curvature of sections in the central zone, which would be used in the next load steps. Ten elements were used in lateral zone and five elements were used in the central zone of the beam. The curve M  1/r relationship for all series was very similar to respective P–d curves. The curve M  1/r curves of the 1st and 4th series confirm experimentally that the amount of tensile steel has a significant influence on flexure stiffness, and that it is higher the greater the area of the tensile steel. After steel yielding the curve M  1/r relationship had different trends. In some beams the curve post-peak, corresponding to the plastic phase, is descending and in others it is almost horizontal. Once again, it is possible to conclude that q has a significant influence on the section behavior, in both phase II and in the plastic stage. This is observed for the 1st and 4th series, and comparing the 2nd and 5th series with 3th series (Fig. 8). The variation of the transverse reinforcement effect on section behavior is quite important, as the higher Asw/s is the greater the plastic deformations are. This is clear in the 2nd and 5th series, with sections where q is 1.13%. But in the 3rd series, with sections

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A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

where q is 2.38%, the effect of the transverse reinforcement variation was not so extensive, although there was an increase in plastic deformations with higher Asw/s. These results should be regarded with caution because some beams had a shear failure.

3.4. Ductility Ductility is defined as the capacity of a material, section or structure to undergo considerable plastic deformation without significant loss of strength capacity [18]. The quantification of significant loss of strength capacity may vary from researcher to researcher. For this study a 15% loss was adopted, which complies with Eurocode 8, Part1, Section 5.2.3.4 (3) – Local ductility condition [16]. The displacement (or curvature) ductility factor is defined as the ratio of the post-ultimate strength displacement 140

(or curvature) at 85% of the moment of resistance, to the displacement (or curvature) at yield. Fig. 9a shows the relationship between the displacement ductility factor ld and q. Beams S1B2 and S1B3 have almost the same ductility factor, ld ffi 2.1, which means that the variation of the compressive reinforcement ratio has a small effect on the section ductility capacity. The displacement ductility factor is almost the same for beams S1B0 and S4B3, around 3, so it might be concluded, based on this result, that high ductility steel is not relevant to the section ductility for this level of tensile reinforcement ratio, with a q of 1.13%. Of course this conclusion should be backed up with more experimental tests. Beams S4B1 and S4B2 have a much higher ld than the other beams, above 10. This could be because the beams’ characteristics allow all the steel’s deformation capacity to be used. In these cases the high ductility steel could now explain this high ductility. Analyzing Fig. 9a, the decline of the ductility 100

S1B0 S1B2

75

S1B3

M (kN.m)

M (kN.m)

105

YP 0.85Mmax

70

0.90Mmax

50

S2B1 S2B2 S2B3

25

35

YP 0.85Mmax

0

0 0

50

100

0

150

70

1/r (x10-3 m -1)

(a)

140

210

1/r (x10-3 m -1)

varying ρ

(b)

160

varyingA sw/s (ρ = 1.13%)

100 S4B1 S4B2

75

S4B3

M (kN.m)

YP S3B1

80

S3B2 S3B3

40

0.85Mmax

50

25

YP

0.90Mmax

0.85Mmax

0.95Mmax

0

0 0

25

50

75

0

75

1/r (x10-3 m -1)

(c)

150 1/r (x10-3 m -1)

varyingA sw/s (ρ = 2.38%)

(d)

varying ρ using high ductility steel

100

75 M (kN.m)

M (kN.m)

120

50

S5B1 S5B2 S5B3

25

YP 0.85Mmax

0 0

55

110 1/r (x10-3 m -1)

(e)

varyingA sw/s using high ductility steel Fig. 8. The curve M  1/r relationship.

165

225

22

A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

12 S1B2

S1B3

S4B1

S4B2

S4B3

9

8

6

μδ

S2B1 S3B1 S5B1 2nd series

10 S1B0

μδ 3

S2B2 S3B2 S5B2 5th series

S2B3 S3B3 S5B3 3rd series

5

3

0

0

0

0.75

1.5

2.25

3

0

3.5

ρ (%)

7

10.5

14

Asw/s (cm2/m)

(a)

(b) Fig. 9. ld  q and ld  Asw/s relationship.

12 Experimental values EC8 relationship

9

μδ 6 3

0 0

4

8

12

μφ

16

Fig. 10. Displacement and curvature ductility factor relationship.

factor with the increase of q is noted, and this reduction is very significant when moving from values of around 0.3–1.1%. The ductility reduction is much less important when moving from q of 1.1% to a value of 2.3%. Fig. 9b shows that as the area of transverse reinforcement increases, the higher the beam’s ductility. Comparing the 2nd and 5th series it appears that the high ductility steel used in 5th series is not reflected in an increase in beam ductility, meaning that, in these cases, the ductility of the S500NR steel used to make the stirrups is enough. Eurocode 8, Part 1, Section 5.2.3.4 (3) presents a relationship between the curvature ductility factor l/ and the displacement ductility factor ld, l/ = 2ld  1, which is normally a conservative

approximation for concrete members. Fig. 10 presents the ld  l/ relationship obtained from experimental results and the expression recommended by Eurocode 8. As expected, the Eurocode 8 expression gives conservative results but when the ductility factors are less than 4 the results are very close. Based on the results obtained, it is also possible to verify that the Eurocode 8 expression is more conservative for higher ductility factors. Based on a linear analysis, the behavior factor q could be used to quantify the seismic effects. To have a specific value of q, enough structural ductility must be ensured because it contributes directly to the structural energy dissipation capacity. The q factor is related to the basic value of the behavior factor qo, which depends on the type of the structural system and on its regularity in elevation. To achieve the required overall ductility of the structure the potential regions for plastic hinge formation should have high plastic rotational capacity [16]. Which means that for a given value of qo a minimum value of the ductility factor is required. According to Eurocode 8 the curvature ductility factor l/ of these regions should be at least equal to the following values:

l/ ¼ 2qo  1 if T 1 P T C and l/ ¼ 1 þ 2ðqo  1ÞT C =T 1 if T 1 < TC

where T1 is the fundamental period of the building and TC is the period at the upper limit of the constant acceleration region of the spectrum, according to 3.2.2.2(2)P of Eurocode 8. Fig. 11a shows the relationship between l/, q and the value qo for 2 situations: where qo is 1.5 and where qo is 3.0. Considering the S2B1 S3B1 S5B1 2nd series

14

16 S1B0

S1B2

S1B3

S4B1

S4B2

S4B3

ð1Þ

12

11

S2B2 S3B2 S5B2 5th series

S2B3 S3B3 S5B3 3rd series

q0 = 3 and Tc/T1 = 2

q0 = 3 e Tc/T1 = 2

μφ

μφ

8

7 q0 = 3 and T1 ≥

q0 = 3 e T1 ≥

4

q0 = 1.5 e Tc/T1 = 2

q0 = 1.5 and Tc/T1 = 2

4

q0 = 1.5 e T1 ≥

q0 = 1.5 and T1 ≥

0 0

0.75

1.5

ρ (%)

2.25

3

0 0

3.5

7

Asw/s (cm 2/m)

(a)

(b) Fig. 11. l/  q and l/  Asw/s relationship.

10.5

14

A.V. Lopes et al. / Engineering Structures 37 (2012) 14–23

60

ρ 45

ρ



θ 30



15



ρ 0 0

3.5

7

10.5

14



Asw/s (cm2 /m) Fig. 12. Plastic rotation capacity and Asw/s relationship.



results of this experimental study, it is possible to define maximum values of tensile reinforcement ratios for a certain qo value. For example, where qo is 3.0 q should be less than 0.70% (considering Tc/T1 = 2) or less than 1.10% (considering T1 > Tc). A similar relationship was established for l/, Asw/s and the qo value (Fig. 11b). In this case a minimum value of Asw/s is determined for a certain qo value. For instance when qo is 3, in the 2nd and 5th series the minimum value of Asw/s should be at least 2.2 and 6.3, respectively (considering T1 > Tc). For the same situation, this requirement cannot be met for the 3rd series beams. As mentioned above, when structures are subject to seismic actions it is important that critical sections have high plastic rotational capacity. The plastic rotation capacity is important not only because it provides a higher structural energy dissipation capacity but also because it is essential for performing linear analysis with moment redistribution and plastic analysis on beams and slabs. Fig. 12 presents the relationship between plastic rotation capacity, hpl, and Asw/s in terms of q. The plastic rotation was calculated for a region 300 mm long (beams’ height), the length considered for the plastic hinge by several researchers [15,18]. The graph clearly shows that the higher q is, the lower hpl is, but when Asw/s increases, then the hpl also increases. This effect is greater when q takes small values. 4. Conclusion The main conclusions are:  Some beams did not have the resistance expected and the most plausible reason for this was the buckling of the compressive reinforcement. These experimental results enhance the importance of the stirrup spacing in beams in order to avoid that reduction.  Comparing the beam’s behavior when there is a large area of tensile steel (q = 2.4%) and when the area of tensile steel is smaller (q = 1.1%), it is possible to conclude that in the first case there is more likelihood of the compressive bars will suffer buckling.  These experimental results demonstrate that increasing the tensile reinforcement ratio increases the beam stiffness and also generates a small increase in the section reserve of resistance, i.e. the ratio Mmax/My.  It was confirmed experimentally that using high ductility steel does not change the type of failure. It was also demonstrated that using high ductility steel to make the stirrups is not relevant to section ductility for beams with these characteristics. Only for a very low level of tensile reinforcement ratios

23

(q < 0.3%) it was verified an additional benefit in section’s ductility when is used high ductility steel for longitudinal reinforcement. This conclusion should be borne out with more experimental tests. In the curve M  1/r relationship, the curve post-peak could be very different depending on q and Asw/s values. The section and beam ductility is higher the higher Asw/s is, and the lower q is, mainly when q is less than 1.5%. The relationship between the displacement and curvature ductility factors obtained experimentally confirms that the relationship recommend by Eurocode 8 is conservative, especially when the ductility factors are high. Based on the results of this experimental study, it was possible to define maximum values for q and minimum values for Asw/s to ensure that a certain value of the basic behavior factor qo is achieved. Based on the experimental results a relation was established between hpl and Asw/s in terms of q.

Acknowledgements The authors would like to express their gratitude to the Department of Civil Engineering of the University of Coimbra (FCTUC) for providing the conditions to carry out this study, especially Sofia Pires, Filipa Fernandes, Patrı´cia Ferreira, Pedro Amaral, Vitor Carvalho, Salomé da Costa, João Fernandes and Rodrigo da Costa. References [1] Park R, Paulay T. Reinforced concrete structures. A Wiley-Interscience publication, John Wiley & Sons; 1975. [2] Carpinteri A, Corrado M, Mancini G, Paggi M. Size-scale effects on plastic rotational capacity of reinforced concrete beams. ACI Struct J 2009;106:887–96. [3] Carpinteri A, Corrado M. Dimensional analysis approach to the plastic rotation capacity of over-reinforced concrete beams. Eng Fract Mech 2010;77:1091–100. [4] Carpinteri A, Corrado M. Lower and upper bounds for structural design of RC members with ductile response. Eng Struct 2011. [5] Carmo RN, Lopes SMR. Ductility and linear analysis with moment redistribution in reinforced high strength concrete beams. Can J Civ Eng 2005;32(1):194–203. [6] Ko M-Y, Kim S-W, Kim J-K. Experimental study on the plastic rotation capacity of reinforced high strength concrete beams. RILEM, Mater Struct 2001;34:302–11. [7] ACI Committee 318. Building Code Requirements for Reinforced Concrete (ACI 318-05) and Commentary (ACI 318R-05). Reported by ACI Committee 318, American Concrete Institute, Farmington Hills, Mich; 2005. [8] Eurocode 2. EN 1992-1-1, Design of Concrete Structures, Part 1-1, General rules and rules for buildings. European Committee for Standardisation, Brussels, Belgium; 2004. [9] CEB-FIP. Model code for concrete structures. Comité Euro-Internacional du Béton – Fédération Internationale de la Précontrainte, Thomas Telford Services Ltd., Lausanne, Switzerland; 1990. [10] Rashid MA, Mansur MA. Reinforced high-strength concrete beams in flexure. ACI Struct J 2005;102(3):462–71. [11] Tanaka H. 1990. Effect of lateral confining reinforcement on the ductile behaviour of reinforced concrete columns. PhD thesis, Civil Engineering, University of Canterbury. [12] Barrera AC, Bonet JL, Romero ML, Miguel PF. Experimental tests of slender reinforced concrete columns under combined axial load and lateral force. Eng Struct 2011. [13] Pallarés L, Bonet JL, Miguel PF, Fernández Prada MA. Experimental research on high strength concrete slender columns subjected to compression and biaxial bending forces. Eng Struct 2008;30(7):1879–94. [14] Hadi MNS, Jeffry R. Effect of different confinement shapes on the behaviour of reinforced HSC beams. Asian J Civ Eng 2010;11(4). [15] CEB. Comité Euro-Internacional du Béton. Bulletin d’ information no 242, Ductility of Reinforced Concrete Structures. Lausanne, Switzerland; 1998. [16] Eurocode 8. EN 1998-1, Design of structures for earthquake resistance, Part 1: General rules, seismic actions and rules for buildings. European Committee for Standardisation, Brussels, Belgium; 2004. [17] CEN. EN 10002-1, Tensile testing of metallic materials – Part 1: method of test at ambient temperature, European Committee for Standardisation; 2001. [18] Lopes S, Carmo R. Deformable Strut and Tie Model for the calculation of the plastic rotation capacity. Comput Struct 2006;84(31–32):2174–83.