Ductility of slender reinforced concrete columns under monotonic flexure and constant axial load

Engineering Structures 40 (2012) 398–412

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Ductility of slender reinforced concrete columns under monotonic flexure and constant axial load A.C. Barrera, J.L. Bonet, M.L. Romero ⇑, M.A. Fernández Instituto de Ciencia y Tecnología del Hormigón (ICITECH), Universitat Politècnica de València, Spain

a r t i c l e

i n f o

Article history: Received 4 November 2011 Revised 17 February 2012 Accepted 1 March 2012 Available online 5 April 2012 Keywords: Slender column Curvature ductility Displacement ductility Plastic length Effective flexural rigidity

a b s t r a c t In this paper an investigation is presented based on tests results of reinforced concrete columns subjected to monotonic flexure and constant axial load from a previous paper [1]. The variables of the tests were strength of concrete (30, 60 and 90 MPa), shear span ratio (M/(V  h) = 7.5, 10.5 and 15), axial load level (0–45%), and transversal and longitudinal reinforcement ratio (1.4–3.2%). This paper is a continuation of the aforementioned but focuses on the deformation capacity, which depends on the method selected to idealize the response diagram comparing four of these methods. The results show that the parameters affecting ductility are greatly influenced by the different methods. It was observed that the ductility in displacements does not always decrease with axial load and column slenderness. In addition, a new equation is proposed to obtain the ultimate displacement for slender columns. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Ductility is an essential property of structures because it guarantees safety when these are subjected to accidental, blast, or seismic loads. In these cases, ductile behavior allows the appearance of plastic hinges in the critical parts of the structure without a serious loss of resistance capacity, and results in an internal force redistribution which avoids the overall failure of the structure. In order to guarantee ductile behavior, the design codes ACI-318 [2], EC-2 [3], and EC-8 [4] highlight that an overestimation of the resistance must be supplied to the brittle elements and regions of the structure for the ductile elements, thus enabling ductile failure mechanisms. The design of the structure must aim to make yielding appear first in the beams and later in the columns (weak beamstrong column). However, the ductility of the columns is also important since plastic hinges appear after an earthquake, and the connections – primarily column-foundation connections, bridge piers, buildings, and industrial facilities – require inelastic behavior [5,6]. The ductility l of a structural member is obtained from the idealization of the experimental or theoretical diagram response. The ductility factor l is obtained as the ratio between the ultimate value and the yielding value. The determination of such values in the response diagram depend on the method selected EC-8 [4], Lam et al. [7], Paultre et al. [8]. However, no bibliographical works were found which studied the differences in the results comparing the different methods. ⇑ Corresponding author. Tel.: +34 963877007x76742; fax: +34 963879679. E-mail address: [email protected] (M.L. Romero). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.03.012

Moreover, the use of high-strength concrete (HSC) in columns is becoming increasingly frequent and that a substantial reduction of the cross-section is obtained, producing an increase in slenderness for the same axial load and length and resulting in higher second order effects (P–D effect). Thus, when the ductile behavior of a structure with slender elements is studied, the reduction of the stiffness due to the second order effects must be taken into account. Bae and Bayrak [9] affirm that the second order effects have an effect on the deformation capacity of the columns. There are several tests in the literature (including [10–13]) that demonstrate that HSC columns are more brittle than NSC columns. These authors suggest further experimental research on the resistance capacity and deformation of HSC columns in order to analyze the reliability of the numerical models and simplified design methods. The ductility required in columns is guaranteed by specific design criteria and reinforcement arrangements defined in the different design codes: ACI-318 [2], EC-2 [3], and EC-8 [4], developed thanks to the many tests within the bibliography quantifying the deformation capacity of elements under flexure, for both monotonic and cyclic loads, PEER (Pacific Earthquake Engineering Research Center) [14] and F.I.B. (Fédération Internationale du Béton) [15]. Most of the experiments focus on a shear span ratio kV lower than 6.5 (kV = Ls/h = M/(V  h)), Panagiotakos and Fardis [16], where Ls is the distance between zero point and maximum moment M, V is the shear force, and h is the overall depth of the cross-section. Therefore, as the use of concrete with a higher capacity implies the construction of slender columns, the deformation capacity of the columns depends on the axial load applied and the slenderness. Since there are few experimental tests with a shear span ratio

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Nomenclature

lD lu ql qs DyI

uyI Du

uu kV h b

x Ac As fc fy M N Nuc V Vmax Mmax

ductility ratio in displacements ductility ratio in curvatures longitudinal reinforcement ratio = As/Ac confinement volumetric geometric reinforcement ratio = Wsc/Wc effective yielding displacement effective yielding curvature ultimate displacement of the column corresponding to 0.85 of the maximum load in the descending branch ultimate curvature of moment – curvature diagram corresponding to 0.85 of the maximum bending moment in the descending branch shear span ratio (kV = Ls/h = M/(V  h)) height of the cross-section width of the cross-section mechanical reinforcement ratio = (As  fy/Ac  fc) area of concrete cross section area of the longitudinal reinforcement cylinder compressive strength of concrete tension yield stress of longitudinal reinforcement bending moment axial load axial load for pure compression lateral force maximum vertical load maximum bending moment

higher than 6.5 it is necessary to study the ductility of slender reinforced concrete columns subjected to constant axial load and monotonic (or cyclic) lateral force. The aim of this paper is to study the deformation capacity of this type of column. The authors have used the results from their previous research [1], where the test parameters were strength of concrete (normal- and high-strength concrete), shear span ratio, axial load level, and longitudinal and transversal reinforcement ratio.

2. Experimental program Although this paper does not aim to describe the series of experiments carried out previously by the authors [1], it is necessary to present this experimental program so that the reader can understand the results. They were selected forty rectangular columns from this campaign. The lengths of the columns (L) were 3.3 m for all the specimens, which were subjected first to a constant axial load and later to a monotonic lateral force up to failure (Figs. 1 and 2). These specimens reproduce two semi-columns of two stories connected by a central element which represents the stiffener effect of an intermediate floor or the connection between a column and the foundation, represented by the stub element. This type of specimen has also been used by other authors, Yamashiro and Sies [17], and Priestley and Park [18] among others. The test parameters and the range of variables studied were the following:  Strength of concrete (fc): 30, 60 and 90 MPa (nominal strength).  Shear span ratio (kV = Ls/h = M/(V  h): 7.5, 10.5 and 15.  Axial load level (N/Nuc, where N is the axial load applied and Nuc is the axial load for pure compression). A variation between the null axial load and 45% of the ultimate axial load (Nuc) is studied, where:

xx Wsc Asi li Wc Acc st

a axs axe sli EIs

as EIe

ae lp

volumetric mechanical amount of confinement = (Wsc  fy)/(Wc  fc) volume of confining transverse reinforcement = RAsi  li area of each of the transverse confining reinforcements length of each of the transverse confining reinforcements volume of confined concrete = Acc  st area of concrete enclosed by the confining steel longitudinal spacing between the transverse confining reinforcements factor the takes account of the spacing between hoops and the configuration of the confining reinforcement = axs  axe factor that takes account of the effect of the longitudinal spacing between hoops = [1  st/(2  bc)]  [1  st/(2  hc)] if the core is rectangular, with dimensions bc, hc factor that takes account of effectiveness of the transverse reinforcement installed inside the confined area of the section = 1  Rs2li =6  Acc the spacing between the longitudinal reinforcements effective flexural stiffness of a cracked section effective flexural stiffness factor of the cross-section effective rigidity of the cracked RC member effective flexural stiffness factor of the cracked RC member plastic hinge length

Nuc ¼ b  h  fc þ rs  As

rs ¼ ecl  Es  fy

ð1Þ

where ec1 is the strain at peak stress of concrete following clause 3.1.5 of EC-2 [3].

ec1 ð‰Þ ¼ 0:7  fc0:31 < 2:8‰

ð2Þ

Es is the Young modulus of the longitudinal reinforcement (200 GPa) fy is the yielding stress of the longitudinal reinforcement  Confinement volumetric geometric reinforcement ratio (qs = Wsc/Wc, where Wsc is the volume of the confinement stirrups and Wc is the volume of the confined concrete): between 0.8% and 3.1%. In all cases the anchorage of the stirrups was arranged during compression. This anchorage was defined with a 90° angle and a length of 10  /t > 70 mm, Ut being the diameter of the transversal reinforcement (Clause 8.5 from EC-2 [2]).  Longitudinal reinforcement ratio (ql = As/(b  h), where As is the area of the longitudinal reinforcement and ‘h’ is the dimension of the section perpendicular to the bending axis): between 1.4% and 3.2% Table 1 lists the details of the 40 tests in the experimental program. The nominal cover of the longitudinal reinforcement (c) is 0.02 m in tests 1–31 and 0.023 m in tests 32–40 (see Fig. 2 and Table 1). 2.1. Material properties All columns were cast using concrete batched in the laboratory using Portland cement type CEM I 52,5R, following code UNE-EN 197-1:2000 [19]. Different additives were used depending on the strength of the concrete: BASF Glenium 300C with silica fume for 90 MPa and super-plasticizer Sika Cem for 60 MPa. The gravel used was calcareous, ranging between 4 and 7 mm in size, and the water-cement ratio was 0.63 for normal-strength concrete

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3.30 m 1.32 m

0.18

0.30

1.32 m

0.18

0.30

Fig. 1. Geometry of the specimens.

As

Ast

(30 MPa), 0.42 for 60 MPa, and 0.32 for 90 MPa respectively. All the columns were tested at 28 days, and the concrete compressive strength ‘fc’ was determined from 150  300 mm cylinders using standard tests. Table 2 presents the exact mix proportions used to manufacture each type of concrete in the experimental program. Four pieces of B 500 SD reinforcing steel were also tested for each diameter, where B stands for ‘steel bars’ and SD is a reference used to indicate special ductility characteristics. Fig. 3 shows the results obtained with quality control following European Standard UNE EN-10002-1 [20] with an elastic modulus of 200 GPa.

200 As

140

Ast

c c 150

150

(a)

(b) Fig. 2. Cross-sections (unit: mm).

Table 1 Experimental tests. Id

Specimen

h (m)

b (m)

fc (MPa)

kV

N (kN)

N/Nuc

Reinforcement Longitudinal As

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

N30-10.5-C0-2-00 N30-10.5-C0-2-15 N30-10.5-C0-2-30 N30-10.5-C0-2-45 N30-7.5-C0-2-30 N30-7.5-C0-2-45 N30-10.5-C0-1-30 N30-10.5-C0-1-45 N30-10.5-C0-3-15 N30-10.5-C0-3-30 H60-10.5-C0-2-00 H60-10.5-C0-2-15 H60-10.5-C0-2-30 H60-10.5-C0-2-45 H60-7.5-C0-2-30 H60-7.5-C0-2-45 H60-10.5-C0-1-15 H60-10.5-C0-1-30 H60-10.5-C0-3-15 H60-10.5-C0-3-30 H90-10.5-C0-2-00 H90-10.5-C0-2-15 H90-10.5-C0-2-30 H90-10.5-C0-2-45 H90-7.5-C0-2-30 H90-7.5-C0-2-45 H90-10.5-C0-1-15 H90-10.5-C0-1-30 H90-10.5-C0-3-15 H90-10.5-C0-3-30 N30-10.5-C3-2-30 N30-10.5-C3-2-45 N30-7.5-C3-2-30 N30-7.5-C3-2-45 H90-10.5-C3-2-30 H90-10.5-C3-2-45 H90-7.5-C3-2-30 H90-7.5-C3-2-45 H90-7.5-C2-2-30 H90-7.5-C2-2-45

0.14 0.14 0.14 0.14 0.20 0.20 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.20 0.20 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.20 0.20 0.14 0.14 0.14 0.14 0.14 0.14 0.20 0.20 0.14 0.14 0.20 0.20 0.20 0.20

0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15

32.2 31.8 31.6 34.5 30.1 33.0 42.2 35.2 33.5 29.5 55.8 54.1 60.5 63.9 63.0 67.7 57.8 58.5 58.3 61.6 85.7 90.5 90.1 93.2 100.4 94.0 90.3 96.2 89.6 94.4 41.0 34.2 35.8 35.0 93.5 92.0 86.4 78.2 95.7 89.2

10.5 10.5 10.5 10.5 7.5 7.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 7.5 7.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 7.5 7.5 10.5 10.5 10.5 10.5 10.5 10.5 7.5 7.5 10.5 10.5 7.5 7.5 7.5 7.5

0 123 255 381 350 533 228 440 142 280 0 208 432 676 637 947 220 412 238 470 0 329 636 972 914 1316 314 624 340 680 258 387 364 546 636 961 910 1360 910 1354

0.00 0.14 0.30 0.41 0.30 0.42 0.22 0.51 0.14 0.31 0.00 0.15 0.29 0.43 0.29 0.40 0.16 0.30 0.15 0.29 0.00 0.15 0.30 0.44 0.27 0.41 0.15 0.29 0.15 0.29 0.24 0.42 0.27 0.41 0.29 0.44 0.31 0.50 0.28 0.45

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

10 10 10 10 12 12 8 8 12 12 10 10 10 10 12 12 8 8 12 12 10 10 10 10 12 12 8 8 12 12 10 10 12 12 10 10 12 12 12 12

Transversal

ql (%)

Ast

2.2 2.2 2.2 2.2 2.3 2.3 1.4 1.4 3.2 3.2 2.2 2.2 2.2 2.2 2.3 2.3 1.4 1.4 3.2 3.2 2.2 2.2 2.2 2.2 2.3 2.3 1.4 1.4 3.2 3.2 2.2 2.2 2.3 2.3 2.2 2.2 2.3 2.3 2.3 2.3

/6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /6 /8 /8 /8 /8 /8 /8 /8 /8 /8 /8

@100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @100 mm @60 mm @60 mm @50 mm @50 mm @60 mm @60 mm @50 mm @50 mm @60 mm @60 mm

qS (%)

a  xx

1.0 1.0 1.0 1.0 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.8 0.8 1.0 1.0 1.0 1.0 3.1 3.1 3.0 3.0 3.1 3.1 3.0 3.0 1.9 1.9

0.03 0.03 0.03 0.02 0.03 0.02 0.02 0.02 0.03 0.03 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.04 0.04 0.04 0.04 0.02 0.02 0.02 0.02 0.02 0.02

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412 Table 2 Mix proportions (kg/m3). Concrete (MPa)

Cement

Sand

Gravel

Water

Super-plasticizer

Silica fume

90 60 30

570 425 348

705 918 1065

890 918 666

180 180 220

12 5 –

50 – –

700

bending plane in order to quantify geometric imperfections or the lateral instability of the column. During the tests, this effect was observed as negligible. The strains were measured in the reinforcing bars in 9 sections of one of the semi-columns (Fig. 5b). In order to ensure that the instrumented half of the column was the first to reach failure, the other semi-column was strengthened with an additional rebar both in the compressed part of the section (top) and the lower part of the cross-section (bottom) with a length higher than the potential plastic hinge length.

φ 12 mm

600 φ 8 mm

φ 10 mm

Stress (MPa)

500 400

φ 8 mm

φ 10 mm

531

537

538

ε y (mm/mm)

0.0026

0.0026

0.0026

ε sh (mm/mm)

0.0406

0.0503

0.0384

668

631

645

0.164

0.192

0.198

fy (MPa)

300 200

ft (MPa) ε u (mm/mm)

100

φ 12 mm

2.4. Test procedure

0 0

0.05

0.1

0.15

0.2

Strain (mm/mm) Fig. 3. Stress–strain behavior of reinforcing steel.

2.2. Test setup A special test frame was designed in order to perform all the tests (Fig. 4). The horizontal system for applying the axial load was made up of two external plates and four GEWI steel bars with a diameter of 36 mm. One of the plates was attached to a 2000 kN load cell and the second was fixed with the horizontal 2500 kN hydraulic jack which leant on a sliding support to enable horizontal displacement. The set-up for applying the vertical load was attached to an auxiliary framework which transmitted the vertical loads to a strong floor. This load was applied to the specimen with a different 500 kN hydraulic jack, and controlled by a 200 kN load cell and transmitted via a special assembly. A more in-depth description of the experiments can be obtained from Barrera et al. [1]. 2.3. Instrumentation Twelve LVDTs were used to measure the lateral displacement (Fig. 5a). LVDT 5 measures the displacement perpendicular to the

Initially, the axial load was applied and maintained constant during the tests. Later, a vertical load was applied up to failure of the column. The column was tested in a displacement control in order to measure post-peak behavior, with a view to measuring the displacement corresponding to 0.85 of the maximum vertical load (Vmax) in the descending branch. 3. Test results: deformation capacity 3.1. Idealization of the experimental diagram In order to perform a study of the experimental response, both at cross-section level and element level, it is necessary to assume an elastic–plastic idealization of the moment–curvature (M–u) or load–displacement (V–D) behavior respectively. This bi-linear diagram is defined in terms of two characteristic points: the effective yielding curvature or the effective yielding displacement (yI or yI), which denotes the change between elastic and plastic behavior; and the ultimate curvature or ultimate displacement (u or u). The experimental bending moment–curvature diagram (M–u) was obtained in the critical section of the column, which is located 5 cm from the ‘stub’ [1]. The curvature of the critical section was obtained from the top and bottom reinforcement strains of the cross-section divided by the distance between them, Fig. 5.

Fig. 4. Test framework.

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412

91.5

LVDT 4

LVDT 3

5 10 10 10

35 LVDT 1

LVDT 2

6

7

8

33 cm

24.5

LVDT 10

9

26 LVDT 11 LVDT 12

LVDT 5

Strain gauges

(a) PLAN VIEW 2 cm B

A

C1C2 C3 C4 C5 C6 C7

A’ Section A-A’

Section B-B’

5 5 5

Upper and lower placement of an additional rebar

5 5 10

C9

C8

25

B’

Strain gauges

(b) Fig. 5. Instrumentation: (a) LVDTs. (b) Arrangement of strain gauges.

3.2. Effective yield curvature or displacement The idealized diagrams were obtained from the experimental diagrams of each test applying four methods used habitually in the literature for comparison of results. Methods 1 and 2. The idealization of the real diagram followed the guidelines of Eurocode 8 (Annex E.3.2) [4], where the energy balance is obtained from the yielding point, Fig. 6a. The yielding state can be due to two possible factors: the yielding of the steel due to tension or the beginning of non-linear behavior of the concrete. Tension reinforcement yields (es 6 ey) if the strain es is lower than 2.65‰ for the all cases analyzed. However, there are many criteria that affect the definition of strain arising from the non-linear behavior of concrete. Paulay and Priestley [21] accept a strain of 1.5‰ as reference, Panagiatakos and Fardis [16] and Fardis and Biskinis [22] admit the strain 0.90fc/Ec, as reference where Ec is the elastic modulus of concrete. The fib Bulletin 24 [23] points out that non-linearity starts when strain is 0.75ec1, where ec1 is the strain at peak stress of the concrete following EC-2 [3] (see Eq. (2)). In this paper the two methods proposed in Eurocode 8 [4] are studied: (a) EC-8a: ey or 0.90ec1 (b) EC-8b: ey or 0.75ec1. When the experimental load–displacement (V–D) or moment– curvature (M–/) diagrams are converted to an ideal elastic–plastic diagram following Eurocode 8 [4] two values are obtained, the effective yielding displacement (DyI) and the effective yielding curvature (uyI) respectively. Method 3. In this method the idealization of the response diagrams (V–D or M–u) is performed through an energy balance, Fig. 6b, between the experimental curve and the ideal diagram up to ultimate load or ultimate bending moment, Lam et al. [7]. That is, the area below the experimental curve (OCD in Fig. 6b) is equal to the area below the ideal elastic–plastic curve (OBCD). The effective yielding deformation is obtained (uyI or DyI) by matching area A1 to area A2. This method is termed ‘method based on balance of energy (MBBE)’. Method 4. The yield point is approximated as the displacement or curvature corresponding to 75% of the maximum load Vmax or the maximum bending moment Mmax in the ascending branch of the real diagram, Paultre et al. [8], and Ho and Pam [12]. Thus, the elastic branch starts at origin, crosses the real diagram at yielding point and finishes at Vmax or Mmax, determining the coordinates

of the corresponding effective yielding deformations (uyI or DyI). The inelastic branch starts at point (DyI, Vmax) or (uyI, Mmax) and finishes at failure point (Du, Vu) or (uu, Mu). The vertical load Vu and the bending moment Mu are obtained when the energy of the ideal diagram is matched to the experimental diagram. This paper terms this procedure ‘approximated method’. 3.3. Ultimate curvature or displacement This factor is defined when a reduction of 15% of the maximum load or bending moment is reached in the descending branch [24,16,15,6,25,26]. Generally, a higher reduction of the load (i.e. 20%) was not achieved in the tests [4,22]. In the tests in which a long descending branch corresponding to a 15% reduction was not obtained, the last measurement was selected as the ultimate curvature or displacement (uu or Du). 3.3.1. Ductility factor The ductility factor in curvatures (lu = uu/uyI) and the ductility factor in displacements (lD = Du/DyI) are obtained as the ratio between the ultimate curvature or displacement (uu or Du) and the effective yielding curvature or displacement (uyI or DyI) respectively. 3.3.2. Yielding forces and deformations The yielding curvature and bending moment (uy, My), and the yielding displacement and load (Dy, Vy) corresponding to a real yielding point, Fig. 6, are obtained through the intersection between the elastic branch and the experimental response diagram (V–D or M–u). 3.3.3. Effective flexural stiffness Usually, the effective flexural stiffness or rigidity (EIs) of a cracked section is obtained using the ratio My/uy. However, this ratio does not take into account major effects such as tension stiffening, inclined cracking, or shear deformations. In order to introduce these effects, the effective rigidity of the cracked RC member to yielding (EIe) is obtained as (V y  L3s Þ=ð6  Dy Þ, where Ls is the distance between the support and the stub, and equals 1.5 m. Both stiffnesses are obtained from the experimental response diagrams. A stiffness factor for the cross-section (as = EIs/EcIc) and for the element (ae = EIe/EcIc) can also be defined as the ratio between

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412

18

(b) 18

Vmax

Lateral Load, V (kN)

16

B Vu = Vmax

16

Vu

14

< 0.15·Vmax

12 Vy

Yielding point (ε y or 0.75·ε c1) - EC-8a or Yielding point (ε y or 0.90·ε c1) - EC-8b

10 8 6 4 2

Δy

0 0

Lateral Load, V (kN)

(a)

C

12 Yielding point

10 8 6

A1

4 2

ΔyI

Δu

10

20

Δy

O

0 30

ΔyI

0

10

18

Lateral Load, V (kN)

16

Δu

D 20

30

Displacement, Δ (mm)

Displacement, Δ (mm)

(c)

< 0.15·Vmax

A2

14 Vy

Vmax Vu

< 0.15·Vmax

14 12

Vy=0.75·Vmax

10

Yielding point

8 6 4 2

Δy

0 0

ΔyI 10

Δu 20

30

Displacement, Δ (mm) Fig. 6. Definition of ultimate and ideal yield displacement (Test H90–10.5-C3–2-45): (a) method proposed by EC-8 (2008). (b) Method based on the balance of energy. (c) Approximated method.

the effective flexural stiffness and the uncracked gross section rigidity (Ec  Ic), where Ic is the moment of inertia of the gross concrete section on the centroidal axis, ignoring any reinforcement, and Ec is the secant Young modulus equal to 22000  (fcm/10)0.3 in MPa, following EC-2 [3]. In this study the value of the medium strength of concrete fcm is equal to the measured strength of concrete (fc) for each test, Table 1. Tables 3 and 4 list the results obtained with the idealization of the response diagrams of moment–curvature (M–u) and load–displacement (V–D). 3.4. Real yielding point In order to obtain the secant stiffness of the cross-section (EIs), the secant stiffness of the element (EIe) or the effective yielding curvature or displacement (uyI, DyI) it is necessary to find out the real yielding point, Fig. 6. If yielding is defined using a material deformation criterion, the yielding load is concurrent with the yielding bending moment My. This happens in the method proposed in Eurocode 8 – Annex E.3.2 [4] (termed as EC8a or EC8b), while the yielding load Vy is not concurrent with the yielding bending moment My in the methods based on the balance of energy (MBBE) – Method 3 [7] and the approximated method (AM) – Method 4 [8,12]. The methods proposed in Eurocode 8 – Annex E.3.2 [4] (termed as EC8a or EC8b) obtain the yielding point assuming a deformation criterion (ey or g  ec1 where g 6 1), while the approximated method (AM) obtains the yielding point assuming a force criterion (Vy = w1  Vmax or My = w2  Mmax where wi 6 1). Consequently, in order to compare the four methods (EC8a, EC8b, MBBE, and AM), deformations es and ec of the materials were studied in the critical section of the column, where es and ec represented the strain in the

most tensioned reinforcement and the strain of the most compressed fiber of concrete respectively. Ratios My/Mmax and Vy/Vmax were also studied in relation to the yielding point. In all tests, apart from the cases with null axial load, and for the four methods analyzed, it was observed that the most tensile reinforcement did not yield (es P 2.65‰). Logically, and as with cases with an axial load other than zero, the yielding point is defined by the beginning of the non-linear behavior of the concrete, and only the ratio ec/ec1 will be studied, where ec1 is the strain at peak stress of the concrete, Eurocode 2 [3], Eq. (2). For the idealization of the bending moment–curvature diagram (M–u) in terms of the different variables of the analysis, Table 5 shows the mean value, the variation coefficient (V.C.) for the ratio (ec/ec1) using the MBBE method and the approximated method (AM), and for the ratio (My/Mmax) using the EC8a, EC8b, and MBBE methods. Furthermore, Table 5 shows the same analysis with regard to the idealization of the load–displacement diagram (V–D) corresponding to ratio (ec/ec1) and ratio (Vy/Vmax). It can generally be observed that the MBBE and AM methods provide similar results in the determination of the yielding point for both experimental response diagrams (M–u and V–D). The mean value of the ratio ec/ec1 obtained with the idealization of the load–displacement diagram V–D is lower than that obtained with the idealization of the bending moment curvature diagram M–u, due to the consideration of the second order effects. Moreover, it was observed that when the approximated method or the MBBE method are applied in a test with high axial load, the ratio (ec/ec1) reaches values greater than unity at yielding point. The determination of the yielding point is very important both for the evaluation of effective flexural rigidity, and for the ductility ratios. The utilization of the method proposed by Eurocode 8 [4],

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412

Table 3 Idealization of the moment–curvature diagram (M–u). Id

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Reference

N30-10.5-C0-2-00 N30-10.5-C0-2-15 N30-10.5-C0-2-30 N30-10.5-C0-2-45 N30-7.5-C0-2-30 N30-7.5-C0-2-45 N30-10.5-C0-1-23 N30-10.5-C0-1-51 N30-10.5-C0-3-15 N30-10.5-C0-3-30 H60-10.5-C0-2-00 H60-10.5-C0-2-15 H60-10.5-C0-2-30 H60-10.5-C0-2-45 H60-7.5-C0-2-30 H60-7.5-C0-2-45 H60-10.5-C0-1-15 H60-10.5-C0-1-30 H60-10.5-C0-3-15 H60-10.5-C0-3-30 H90-10.5-C0-2-00 H90-10.5-C0-2-15 H90-10.5-C0-2-30 H90-10.5-C0-2-45 H90-7.5-C0-2-30 H90-7.5-C0-2-45 H90-10.5-C0-1-15 H90-10.5-C0-1-30 H90-10.5-C0-3-15 H90-10.5-C0-3-30 N30-10.5-C3-2-30 N30-10.5-C3-2-45 N30-7.5-C3-2-30 N30-7.5-C3-2-45 H90-10.5-C3-2-30 H90-10.5-C3-2-45 H90-7.5-C3-2-30 H90-7.5-C3-2-45 H90-7.5-C2-2-30 H90-7.5-C2-2-45

N N uc

0 0.14 0.3 0.41 0.3 0.42 0.22 0.51 0.14 0.31 0 0.15 0.29 0.43 0.29 0.4 0.16 0.3 0.15 0.29 0 0.15 0.3 0.44 0.27 0.41 0.15 0.29 0.15 0.29 0.24 0.42 0.27 0.41 0.29 0.44 0.31 0.5 0.28 0.45

M max 2 bh fc

0.13 0.19 0.23 0.21 0.27 0.23 0.15 0.16 0.23 0.26 0.08 0.15 0.15 0.19 0.17 0.18 0.12 0.16 0.17 0.19 0.06 0.11 0.14 0.14 0.13 0.14 0.09 0.13 0.14 0.15 0.18 0.22 0.22 0.21 0.14 0.13 0.15 0.14 0.13 0.13

uuh (‰)

20.01 19.03 14.91 6.85 16.38 15.17 21.21 3.31 18.71 11.61 18.07 15.52 7.22 6.66 12.96 11.59 15.99 12.15 9.11 13.19 17.47 16.36 7.70 5.50 10.81 11.11 16.44 10.37 11.34 7.52 16.83 11.09 21.93 15.65 6.89 6.31 8.28 9.96 10.76 9.64

lu

My/Mmax Method

as = EIs/EcIc

Method

Method

EC8a

EC8b

MBBE

EC8a

EC8b

MBBE

AM

EC8a

EC8b

MBBE

AM

0.92 0.83 0.61 0.74 0.58 0.64 0.67 0.68 0.68 0.55 0.94 0.71 0.69 0.64 0.69 0.66 0.82 0.71 0.69 0.62 0.71 0.77 0.82 0.74 0.71 0.70 0.78 0.78 0.90 0.76 0.65 0.62 0.54 0.58 0.80 0.62 0.72 0.51 0.69 0.61

0.92 0.71 0.54 0.62 0.51 0.53 0.60 0.52 0.60 0.50 0.94 0.61 0.60 0.46 0.61 0.52 0.74 0.62 0.59 0.52 0.71 0.66 0.70 0.57 0.57 0.53 0.69 0.70 0.74 0.64 0.59 0.53 0.47 0.48 0.67 0.47 0.60 0.37 0.58 0.44

0.84 0.79 0.69 0.75 0.71 0.76 0.70 0.73 0.67 0.70 0.94 0.69 0.70 0.74 0.74 0.76 0.73 0.75 0.63 0.76 0.81 0.70 0.70 0.79 0.82 0.83 0.72 0.76 0.63 0.70 0.71 0.74 0.77 0.76 0.70 0.72 0.72 0.76 0.72 0.80

3.73 3.26 3.31 2.37 3.60 4.70 4.57 1.70 3.04 2.79 3.51 2.27 1.81 2.16 2.75 3.01 2.49 3.14 1.50 3.06 2.78 2.73 1.66 1.97 2.54 3.26 3.08 2.57 1.67 1.67 3.38 3.39 4.75 4.61 1.55 2.31 1.92 3.30 2.36 3.02

3.73 4.10 3.52 2.92 4.17 5.36 5.47 2.08 3.27 3.25 3.51 2.61 2.21 2.61 3.21 3.49 2.96 3.86 1.67 3.40 2.78 3.15 2.01 2.36 2.82 3.62 3.65 3.32 1.91 2.05 3.99 3.99 5.48 5.34 1.95 2.80 2.39 3.93 2.79 3.46

3.84 3.59 2.02 2.04 2.89 3.73 4.07 1.47 2.98 2.10 3.56 2.31 1.53 1.72 2.33 2.57 2.82 2.74 1.46 2.21 2.69 2.90 1.83 1.57 2.13 2.70 3.23 2.56 2.06 1.70 2.93 2.64 3.00 4.46 1.65 1.78 1.75 2.25 2.14 2.30

4.05 3.78 2.31 2.00 2.69 4.03 3.73 1.43 2.72 1.97 3.70 2.12 1.39 1.74 2.27 2.64 2.65 2.71 1.34 2.31 2.73 2.70 1.72 1.69 2.30 2.99 3.01 2.63 1.83 1.57 2.80 2.57 3.14 3.45 1.54 1.75 1.70 2.29 2.02 2.56

0.30 0.38 0.59 0.85 0.67 0.86 0.46 1.07 0.46 0.72 0.27 0.39 0.65 1.11 0.67 0.97 0.33 0.73 0.48 0.80 0.22 0.45 0.72 1.14 0.81 1.03 0.41 0.81 0.51 0.80 0.50 0.81 0.57 0.80 0.80 1.13 0.80 1.06 0.74 0.96

0.30 0.48 0.67 1.04 0.76 0.98 0.54 1.24 0.49 0.82 0.27 0.43 0.77 1.30 0.77 1.07 0.39 0.88 0.52 0.87 0.22 0.51 0.85 1.33 0.88 1.13 0.48 1.02 0.58 0.95 0.58 0.93 0.66 0.91 0.96 1.32 0.96 1.24 0.85 1.09

0.31 0.43 0.50 0.82 0.57 0.72 0.42 1.02 0.47 0.60 0.27 0.41 0.64 0.98 0.61 0.85 0.39 0.67 0.51 0.63 0.22 0.49 0.84 1.05 0.72 0.90 0.47 0.85 0.64 0.88 0.45 0.67 0.39 0.73 0.91 0.97 0.79 0.80 0.71 0.79

0.33 0.46 0.45 0.81 0.53 0.77 0.39 0.99 0.43 0.56 0.28 0.37 0.59 0.99 0.60 0.87 0.37 0.67 0.47 0.66 0.22 0.45 0.79 1.13 0.78 1.00 0.44 0.88 0.57 0.81 0.43 0.65 0.41 0.63 0.85 0.96 0.77 0.81 0.67 0.88

where deformation criteria are applied, demonstrates two advantages: the determination of the initial yielding point from the materials in the experimental response diagrams, and the concurrence between yielding bending moment My and yielding load Vy. However, the selection of a force criterion, as used in the approximated method, provides reasonable results for cases in which the material deformations are unknown. If the beginning of the non-linear behavior of the concrete is selected for 0.75  ec1 as proposed by fib Bulletin 24 [23], in view of the results observed in Table 5 a rational value for the estimation of the yielding bending moment is 0.6  Mmax and 0.75  Vmax for yielding load for cases where the axial load is other than zero. For cases with null axial load these values will be My = 0.85  Mmax and Vy = 0.85  Vmax respectively. 3.5. Ductility factor in curvatures Table 3 shows the ductility factor in curvatures (lu) for each test for the four methods presented in this paper. It can be observed that the MBBE method and AM method show similar results and are slightly lower than the EC8a method. However, if the EC8b method is applied, higher values of lu are obtained because the effective yielding curvature obtained with this method (uyI) is lower than the other three. In order to simplify the study of factor lu, Fig. 7 presents it in terms of the parameters studied for the EC8b method. In addition,

Fig. 8 shows the idealization of the experimental relative moment– curvature diagrams for the tests with a span shear-ratio of 10.5, longitudinal reinforcement ratio 2.3%, and non-confined (qs 6 1%). The ductility in curvatures does not always decrease, Fig. 7a, and generally, higher ductility is observed for the axial load level of N/Nuc = 0.15. If the relative moment–curvature diagram (M– u  h) for different levels of axial load, Fig. 8a, is taken into account it can be deduced that the effective yielding curvature (uyI) and the ultimate curvature (uu) decrease with the level of applied axial load. However, as the effective yielding ductility decreases in a different way to that of the ultimate curvature, ductility in curvatures does not have to decrease with axial load. In general, ductility in curvatures decreases with the strength of concrete, Fig. 7. Accordingly, in Fig. 8b it can be observed that the factor lu decreases with the strength of concrete since the effective yielding curvature (uyI) has similar values for the three types of concrete, and the ultimate curvature (uu) decreases with the strength of concrete. This last statement is due to the ultimate deformation of the concrete decreasing with the strength of concrete (ecu1, Table 3.1. from EC-2 [3]). Ductility in curvatures is a cross-section parameter and independent of the slenderness of the column. However, large differences in the ductility were observed if the size of the cross-section was modified. This is because the relative cover rmec/h is not the same in all columns, where rmec is the distance between the center of gravity of the tensile reinforcements at the side of the

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412 Table 4 Idealization of the load–displacement diagram (V–D). Id

Reference

N N uc

M max 2 bh fc

Du/Ls(%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

N30-10.5-C0-2-00 N30-10.5-C0-2-15 N30-10.5-C0-2-30 N30-10.5-C0-2-45 N30-7.5-C0-2-30 N30-7.5-C0-2-45 N30-10.5-C0-1-23 N30-10.5-C0-1-51 N30-10.5-C0-3-15 N30-10.5-C0-3-30 H60-10.5-C0-2-00 H60-10.5-C0-2-15 H60-10.5-C0-2-30 H60-10.5-C0-2-45 H60-7.5-C0-2-30 H60-7.5-C0-2-45 H60-10.5-C0-1-15 H60-10.5-C0-1-30 H60-10.5-C0-3-15 H60-10.5-C0-3-30 H90-10.5-C0-2-00 H90-10.5-C0-2-15 H90-10.5-C0-2-30 H90-10.5-C0-2-45 H90-7.5-C0-2-30 H90-7.5-C0-2-45 H90-10.5-C0-1-15 H90-10.5-C0-1-30 H90-10.5-C0-3-15 H90-10.5-C0-3-30 N30-10.5-C3-2-30 N30-10.5-C3-2-45 N30-7.5-C3-2-30 N30-7.5-C3-2-45 H90-10.5-C3-2-30 H90-10.5-C3-2-45 H90-7.5-C3-2-30 H90-7.5-C3-2-45 H90-7.5-C2-2-30 H90-7.5-C2-2-45

0 0.14 0.3 0.41 0.3 0.42 0.22 0.51 0.14 0.31 0 0.15 0.29 0.43 0.29 0.4 0.16 0.3 0.15 0.29 0 0.15 0.3 0.44 0.27 0.41 0.15 0.29 0.15 0.29 0.24 0.42 0.27 0.41 0.29 0.44 0.31 0.5 0.28 0.45

0.025 0.028 0.024 0.022 0.061 0.048 0.018 0.020 0.031 0.027 0.015 0.019 0.015 0.015 0.035 0.036 0.014 0.014 0.020 0.014 0.011 0.012 0.012 0.010 0.026 0.026 0.009 0.012 0.015 0.013 0.016 0.019 0.045 0.042 0.012 0.009 0.029 0.026 0.025 0.023

4.97 3.04 2.94 1.94 2.22 1.52 2.27 1.13 4.50 3.01 4.85 3.11 2.33 2.24 1.86 1.51 2.70 2.77 3.69 3.33 4.47 2.89 2.38 1.83 1.72 1.32 3.21 2.29 3.30 2.38 3.12 2.32 2.63 1.68 2.55 1.86 1.76 1.43 1.92 1.30

lD Method

Vy/Vmax Method EC8a

EC8b

MBBE

EC8a

EC8b

MBBE

AM

EC8a

EC8b

MBBE

AM

0.92 0.87 0.85 0.95 0.66 0.69 0.87 0.82 0.76 0.78 0.87 0.84 0.92 0.89 0.77 0.73 0.97 0.95 0.80 0.94 0.71 0.93 1.00 0.98 0.78 0.77 0.99 0.99 0.97 0.97 0.93 0.91 0.62 0.67 0.99 0.93 0.80 0.61 0.79 0.71

0.92 0.76 0.77 0.85 0.58 0.57 0.81 0.63 0.67 0.70 0.87 0.76 0.82 0.71 0.68 0.58 0.92 0.90 0.72 0.86 0.71 0.87 0.95 0.85 0.64 0.60 0.96 0.96 0.89 0.90 0.89 0.83 0.54 0.56 0.93 0.81 0.67 0.43 0.66 0.53

0.94 0.69 0.77 0.79 0.73 0.73 0.79 0.75 0.68 0.77 0.87 0.74 0.78 0.74 0.77 0.73 0.78 0.84 0.72 0.76 0.79 0.72 0.73 0.76 0.74 0.73 0.77 0.73 0.76 0.72 0.78 0.76 0.76 0.73 0.76 0.77 0.75 0.72 0.74 0.72

2.36 1.81 3.06 2.28 2.36 2.00 2.83 1.88 2.11 2.64 2.18 1.83 2.04 2.21 1.84 1.64 1.89 2.28 1.88 2.54 1.80 1.77 1.62 1.86 1.62 1.47 2.11 1.96 1.32 1.78 2.53 2.61 2.87 2.17 1.68 2.36 1.63 1.82 1.74 1.71

2.36 2.02 3.21 2.65 2.75 2.36 3.19 2.75 2.56 3.07 2.18 2.30 2.49 2.48 2.09 1.91 2.34 2.81 2.28 3.13 1.80 2.27 2.26 2.12 1.88 1.72 2.64 2.46 1.92 2.25 3.02 3.11 3.30 2.58 2.21 3.19 1.90 2.38 2.03 2.14

2.19 2.15 3.04 2.93 1.91 1.74 3.48 2.01 2.28 2.52 2.52 2.33 2.71 2.67 1.69 1.54 3.51 3.21 2.09 3.53 3.53 2.99 3.14 2.72 1.57 1.41 4.18 3.34 2.31 2.97 3.70 3.37 2.15 1.75 3.18 3.33 1.60 1.61 1.69 1.52

2.39 1.92 3.32 3.29 1.86 1.69 3.87 2.02 2.02 2.66 2.13 2.30 2.78 2.68 1.74 1.48 3.92 3.56 1.95 3.55 1.71 3.03 3.02 2.88 1.56 1.39 4.30 3.21 2.37 3.00 4.04 3.40 2.09 1.80 3.18 3.45 1.60 1.57 1.72 1.50

0.28 0.36 0.56 0.62 0.65 0.67 0.59 0.78 0.32 0.49 0.23 0.37 0.45 0.55 0.58 0.72 0.33 0.39 0.34 0.38 0.20 0.34 0.37 0.50 0.59 0.67 0.26 0.51 0.28 0.47 0.36 0.50 0.54 0.62 0.40 0.51 0.58 0.71 0.53 0.65

0.28 0.40 0.59 0.75 0.74 0.77 0.67 1.10 0.38 0.56 0.23 0.46 0.54 0.67 0.65 0.80 0.41 0.48 0.40 0.47 0.20 0.43 0.52 0.57 0.67 0.76 0.33 0.63 0.39 0.59 0.43 0.59 0.61 0.72 0.51 0.68 0.66 0.87 0.61 0.77

0.27 0.44 0.59 0.84 0.58 0.64 0.74 0.89 0.37 0.50 0.23 0.47 0.62 0.71 0.58 0.71 0.66 0.59 0.39 0.56 0.20 0.60 0.73 0.77 0.62 0.70 0.55 0.88 0.49 0.81 0.54 0.66 0.44 0.55 0.74 0.77 0.61 0.64 0.57 0.64

0.29 0.41 0.64 0.95 0.57 0.63 0.85 0.90 0.33 0.53 0.24 0.47 0.64 0.70 0.60 0.70 0.72 0.65 0.37 0.56 0.20 0.60 0.71 0.81 0.61 0.68 0.57 0.86 0.51 0.81 0.59 0.67 0.44 0.56 0.76 0.79 0.61 0.62 0.56 0.62

section and h is the height of the section, perpendicular to the bending axis. The relative cover rmec/h in the non-confined columns (qs 6 1%) is: (a) rmec/h = 0.16 for the tests with a slenderness of kV = 7.5, where h = 0.20 m, (b) rmec/h = 0.22 for the tests with a slenderness of kV = 10.5, where h = 0.14 m. Columns with lower relative cover are more ductile because the mechanical arm is higher, Fig. 7b. Finally, the ductility in curvatures decreases with the longitudinal reinforcement ratio, Fig. 7c, because the failure of the cross-section is usually due to the concrete. Fig. 7d shows how generally this parameter also increases with the confinement ratio. As an example Fig. 9 shows the ductility ratio in curvatures for the tests with an effective confinement ratio of the transversal reinforcement (axx) of 0.01 (Table 1); the experimental ductility ratio in curvatures is obtained using the EC8b method as opposed to that proposed in Eurocode 8 – part 1 [27] in terms of the relative axial load md:

lu;EC8 ¼

a  xx þ 0:035 bo  30  v d  esy;d bc

ae = EIe/EcIc Method

ð3Þ

where a is the effective confinement ratio; xx is the confinement volumetric mechanical reinforcement ratio (Eurocode 8 – part 1 [27], clause 5.4.3.2.2. (8)); md is the relative axial load (md = N/ (b  h  fc)); esy,d is the design value of the yielding strain of the

longitudinal reinforcement; bc is the width of the cross-section; and bo is the width of the confined part of concrete (measured between the axis of the central confining stirrups). The safety factors adopted are equal to one. Fig. 9b displays the relationship between the experimental ductility in curvatures from EC8b (lu,test,EC8b) and those obtained from Eurocode 8 – part 1 [27], Eq. (3), in terms of the axial load level (N/ Nuc). In both graphs, Fig. 9a and b, the ductility for the cases with null axial load is not included, because it was not possible to measure curvatures until the end of the test, underestimating the ultimate curvature of the cross-section uu, and thus, the ductility in curvatures (lu). It can be observed in Fig. 9 that the proposal from Eurocode 8 – part 1 [27] is conservative, except for the cases where the axial load level is lower than 0.2. Table 6 lists the average errors and the variation coefficient for the different methods. It can be observed that if the MBBE or AM methods are used to idealize the moment–curvature diagram M– u, for low axial load levels, slender columns and high-strength concrete, the method from Eurocode 8 – part 1 [27] is unsafe. However, if it is idealized according to EC8a or EC8b, the proposal is on the unsafe side for low axial load levels. It is possible to observe that for the longitudinal reinforcement ratios equal to 1.4% and 3.2% the average value is below unity. It is worth noting that these columns were tested with low axial load levels, 0.15 and 0.30. In all cases the variation coefficient is very high.

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412

5.99 0.76 18.7 0.76 13.5 0.85 27.8 0.71 25.3 0.73 7.8 0.74 19.3 0.60 0.70 0.93

23.3

0.96

24.0

14.2

6.36 6.61 6.43 0.76 0.73 0.75 18.3 24.2 20.1 0.78 0.59 0.71 13.6 15.7 14.5 0.86 0.75 0.81 31.3 23.7 27.1 0.68 0.90 0.79 28.1 22.4 24.6 0.70 0.88 0.80 7.7 7.5 7.8 0.74 0.76 0.74 20.4 25.1 24.4 0.63 0.51 0.52 0.73 0.65 0.63

All

1 2 3

qs (%)

30 2 8

0.90 1.02 1.03

26.3 23.3 22.9

0.93 1.02 1.05

27.3 25.0 24.2

14.2 15.7 16.3

4.68 6.39 6.57 0.74 0.76 0.74 13.1 19.6 18.1 0.79 0.73 0.79 11.4 14.2 13.5 0.87 0.82 0.87 17.2 28.6 30.4 0.70 0.75 0.70 16.1 25.8 28.6 0.69 0.76 0.69 7.5 7.6 8.4 0.68 0.76 0.68 14.1 21.7 21.3 0.60 0.59 0.60 0.70 0.69 0.70 1.4 2.2 3.2

ql (%)

6 28 6

0.92 0.95 0.92

25.3 25.0 25.8

1.01 0.95 1.01

21.5 26.6 25.0

17.3 14.8 14.6

1.96 6.69 0.74 0.77 12.2 11.2 0.59 0.83 9.4 8.6 0.72 0.90 16.3 19.5 0.94 0.62 16.6 17.8 0.93 0.64 5.2 8.6 0.76 0.73 13.5 18.0 0.52 0.63 0.64 0.73 12 28 7.5 10.5 kv

1.07 0.88

19.9 22.3

1.05 0.92

19.5 25.2

11.2 13.3

8.77 5.11 3.75 2.84 0.86 0.73 0.76 0.74 13.0 12.6 17.3 21.7 0.83 0.82 0.78 0.66 13.0 9.8 13.5 15.7 0.83 0.89 0.86 0.80 12.3 22.9 28.3 18.2 0.44 0.63 0.72 0.84 28.2 16.9 27.0 16.5 0.55 0.61 0.74 0.84 7.6 7.6 5.0 4.1 0.87 0.70 0.73 0.76 14.8 8.8 11.3 12.8 0.86 0.67 0.59 0.50 0.86 0.77 0.68 0.65 3 9 16 12 0 0.15 0.3 0.45 N/Nuc

0.51 0.78 1.00 1.05

18.8 8.2 21.6 7.6

0.42 0.85 1.04 1.04

16.6 8.9 21.3 7.9

14.8 9.8 12.1 10.4

V.C.(%) Mean

0.76 0.77 0.74 18.0 14.0 21.8 0.72 0.78 0.77 13.8 9.5 14.9 0.81 0.87 0.87 27.5 24.0 28.1 0.80 0.67 0.67 22.8 19.8 26.5 0.83 0.69 0.67 6.3 10.6 7.6 0.74 0.75 0.74 20.3 21.6 17.6 0.58 0.62 0.60 16.0 13.2 12.7 0.66 0.72 0.73 22.5 22.6 20.1 1.09 0.93 0.86 21.6 20.7 22.2 1.05 0.90 0.85 14 10 16 30 60 90 fc (MPa)

MBBE

V.C. (%) Mean

EC8b

V.C. (%) EC8a

Mean Mean Mean Mean V.C. (%) Mean Mean

MBBE

V.C.(%)

AM

V.C. (%)

EC8b

V.C.(%)

MBBE

V.C. (%)

Mean

V.C. (%)

Mean

V.C. (%)

Vy/Vmax

AM MBBE EC8a

Load – displacement idealization

ec/ec1 My/Mmax

ec/ec1

Bending moment – curvature idealization No. of data Table 5 Yielding point.

7.97 6.05 2.86

3.6. Ductility factor in displacements Table 4 shows the ductility factor in displacements (lD) for each test for the four methods presented in this paper. Of all the methods studied, it can be seen how the MBBE method and AM approximated method show similar results, obtaining the maximum values of lD. Fig. 10 presents the ductility factor lD in terms of the parameters for the idealization of the diagram in accordance with the EC8b method. As an example, Fig. 11 shows the idealization of the loaddrift diagrams (V–D/Ls) for the tests with a slenderness equal to 10.5, longitudinal reinforcement ratio of 2.3%, and non-confined (qs 6 1%), where D is the displacement of the section close to the stub, and L is the distance between the support and the stub equal to 1.5 m. The ductility ratio in displacements does not always decrease with the axial load, Fig. 10a, as can also be inferred from Fig. 11a where the effective yielding drift (DyI/Ls) and the ultimate drift (Du/Ls) decrease with the axial load level applied but not in the same proportion. For this reason, in columns with a slenderness (kV) of 10.5 the maximum ductility is observed for an axial load level equal to 30%. However, previous research [28,24,29,30] demonstrated that ductility decreases with the axial load level. Although the second order effects cause a significant reduction of the resistance capacity of the column, it has been observed that the ductility ratio in displacements lD does not always decrease with slenderness, Fig. 10b. In this way for instance, for the nonconfined column with 30 MPa with longitudinal reinforcement 2.3% and for the axial load levels N/Nuc of 30% and 45% (Fig. 11b), the ductility also increases with slenderness. This occurs because the ultimate drift (Du/Ls) increases more than the effective yielding drift (DyI/Ls). Evidently, as was stated by Menegotto [31], the decrease of second order effects reduces the strength capacity of R the column and the dissipated energy (W = VdD, where V is the lateral load and D the displacement). It is worth noting that the experimental tests of this study are cases with medium slenderness (kV of 7.5 and 10.5), the behavior of which is influenced by second order effects (P–D effect). Previous research had focused analysis on columns with a shear span ratio lower than 6.5. The analysis of the influence of the slenderness and the axial load level in ductility requires a more exhaustive study, given the limited experimentation available in tests with slenderness higher than 6. As was expected, and with the conclusions obtained for the ductility factor in curvatures (lu), the ductility ratio in displacements (lD) decreases with the strength of the concrete, Fig. 10a, and with the longitudinal reinforcement ratio, Fig. 10c, and it increases with the level of confinement, Fig. 10d. 3.7. Effective flexural stiffness factor For each test Tables 3 and 4 list the effective flexural stiffness factor obtained for the four methods applied both for the moment–curvature (M–u) diagram, and for the load–displacement (V–D) diagram. As was expected the effective flexural stiffness factor of the cross-section as is higher than the element ae reaching differences of up to 50%. In view of the results from My/Mmax or Vy/Vmax, Table 5, for the four methods, the maximum values of as are obtained with the EC8b method and the maximum values of ae are obtained with the approximated method (AM) or the MBBE method. The same tendency was obtained for both effective flexural stiffness factors (as and ae) in terms of the different variables analyzed. Fig. 12 presents factor ae in terms of the different parameters for the EC8b method. The parameter with the most influence on factor ae is the axial load level, and when the axial load level in-

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412

(a)

(b) 6.00

6.00

fc 30 MPa fc 60 MPa fc 90 MPa

5.00

4.00

μϕ

4.00

μϕ

ρl =2.3%; ρs ≤1%

5.00

3.00

3.00

2.00

2.00

1.00

1.00

N/Nuc=0.30; λv=7.5 N/Nuc=0.45; λv=7.5 N/Nuc=0.30; λv=10.5 N/Nuc=0.45; λv=10.5

λ V=10.5; ρ l =2.3%; ρ s ≤1% 0.00 0.00

0.00 0.10

0.20

0.30

0.40

0

0.50

25

50

N/Nuc 6.00

(d) 6.00

5.00

5.00

4.00

4.00

μϕ

μϕ

(c)

3.00

fc 30 MPa fc 60 MPa fc 90 MPa

2.00 1.00

1.00

100

3.00 2.00

λV=7.5; ρ l =2.3% fc 30 MPa; N/Nuc=0.30 fc 90 MPa; N/Nuc=0.30 fc 30 MPa; N/Nuc=0.45 fc 90 MPa; N/Nuc=0.45

1.00

λ V=10.5; N/Nuc=0.15; ρ s ≤1% 0.00 0.00

75

fc (MPa)

2.00

3.00

0.00 0.00

4.00

1.00

2.00

3.00

4.00

ρ s (%)

ρ l (%) Fig. 7. Ductility in curvatures (EC8b).

(a) 0.15

(b) 0.25

Experimental

N/N uc =0.30

0.2

0.1

N/Nuc =0.15

M/(b·h 2·f c)

M/(b·h 2·f c)

N/Nuc =0.44

Experimental Idealized (EC8b)

0.05

N/N uc=0

0 0.00

5.00

N/N uc

ϕ yI·h

ϕ u·h

μϕ

0.00 0.15 0.30 0.44

6.28 5.19 3.82 2.33

17.47 16.36 7.70 5.50

2.78 3.15 2.01 2.36

10.00

15.00

fc=63.90

0.15

fc=93.20

0.1

fc (MPa) 34.50 63.90 93.20

0.05

0 0.00

20.00

fc=34.50 MPa

Idealized (EC8b)

2.00

ϕ yI·h

ϕ u·h

μϕ

2.34 2.55 2.33

6.85 6.66 5.50

2.92 2.61 2.36

4.00

6.00

8.00

ϕ·h (‰)

ϕ·h (‰)

f c ≈90 MPa; λ V =10.5; ρ l =2.3%; ρ s ≤1%

N/Nuc ≈0.45; λ V=10.5; ρ l =2.3%; ρ s ≤1%

Fig. 8. Experimental bending moment curvature diagram. Idealization with EC8b.

creases ae also increases, Fig. 12a. Furthermore, it is possible to observe that for low axial load levels (N/Nuc = 0 or 0.15) factor ae is independent of the strength of concrete, while for medium or higher axial load levels (N/Nuc = 0.30 or 0.45) factor ae decreases when the strength of concrete increases. This behavior occurs because for low axial load levels the yielding point is basically defined by the yielding of the steel, while for higher axial load levels it is due to the beginning of the non-linear behavior of concrete. It can also be observed that the stiffness factor ae decreases with slenderness as a result of second order effects, Fig. 12b. Factor ae does not depend substantially on the longitudinal and transversal reinforcement ratios, Fig. 12b and Fig. 12c.

3.8. Plastic hinge length Usually, the calculation of the ultimate displacement (Du) in a column with height Ls can be expressed assuming that all the behavior develops as a result of flexure using the concept of plastic hinge and the plastic hinge length lp, where inelastic deformations of the entire column are concentrated and distributed uniformly:

Du ¼

uyI  L2s 3

þ ðuu  uyI Þ  lp  ðLs  0:5  lp Þ ¼ Dy þ Dp

ð4Þ

The first term of Eq. (4) is the elastic displacement (Dy) and the second term is the plastic displacement (Dp). As stated by Panagiatakos

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(a)

(b) μ ϕ ,test, EC8b / μ ϕ ,EC8

14.00 EC8

12.00

Test

μϕ

10.00

α · ω ω ≈ 0.01

8.00 6.00 4.00 2.00 0.00 0.00

0.20

0.40

0.60

0.80

3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0.00

0.20

0.40

0.60

N/N uc

ν = N/(b·h·f c )

Fig. 9. Verification of the ductility ratio in curvatures proposed by Eurocode 8.

Table 6 Verification of the ductility in curvatures between EC-8 and the experimental tests. Variables

No.

lu,test,EC8a/luEC8

lu,test,EC8b/lu,EC8

lu,test,MBBE/lu,EC8

Mean

VC (%)

Mean

VC (%)

Mean

VC (%)

lu,test,AM/lu,EC8 Mean

VC (%)

fc (MPa)

30 60 90

14 10 16

1.30 1.05 1.12

42.79 48.72 49.28

1.51 1.24 1.32

42.25 48.56 47.33

1.07 0.90 0.98

41.17 42.95 35.70

1.03 0.89 0.89

43.89 46.73 52.19

N/Nuc

0 0.15 0.3 0.45

3 9 16 12

– 0.56 1.06 1.72

– 21.56 24.73 27.46

– 0.65 1.25 2.01

– 23.30 23.62 25.15

– 0.60 0.90 1.38

– 21.46 20.11 27.48

– 0.56 0.87 1.28

– 23.81 21.95 45.36

kv

7.5 10.5

12 28

1.61 0.95

36.21 37.63

1.86 1.13

35.38 38.35

1.30 0.84

35.73 28.20

1.19 0.82

52.35 31.26

ql (%)

1.4 2.2 3.2

6 28 6

0.81 1.28 0.81

52.82 44.41 52.82

0.92 1.50 0.92

52.85 43.08 52.85

0.71 1.06 0.71

35.95 39.22 35.95

0.67 1.00 0.67

40.72 49.16 40.72

qs (%)

1 2 3

30 2 8

1.11 1.34 1.30

48.24 47.06 43.77

1.30 1.55 1.54

47.11 45.17 42.44

0.97 1.09 1.03

40.34 35.95 40.10

0.96 0.39 0.99

46.20 139.13 36.65

1.16

46.20

1.37

45.12

0.99

39.03

0.94

47.05

and Fardis [16] the advantages of this formulation are: (a) it represents a mechanical and physical model (that of lumped inelasticity) and (b) uu and uyI can be determined from a sectional analysis. Other effects like shear deformation, bond slip, or tension stiffening will be taken into account in the calculation of the length lp, using the following expressions as can be observed here:

lp ¼ 0:10  Ls þ 0:0015  fy  db

proposed by Eurocode 8-part 2; ½4

ð5Þ

Dy ¼ DyI  d ¼

lp ¼ 0:18  Ls þ 0:025  asl fy  db

proposed by the CEB-FIP; Bulletin 25 ½15

the columns used in this paper, with a slenderness kV of 7.5 and 10.5, subjected to medium and high axial load levels (N/Nuc) of 0.30 and 0.45. In this case, Eq. (4) must include the increment of the displacements resulting from the performance of equilibrium in the deformed shape. It is necessary to reformulate the proposed equations of the plastic hinge length and to calculate the elastic displacement taking into account the second order effects from the following expression, Chen and Lui [32]:

ð6Þ

where: fy is the yielding stress of the steel in MPa; db is the diameter of the reinforcement in mm; and the coefficient asl equals 1 if slippage of longitudinal steel from its anchorage zone beyond the end section is possible, or 0 if it is not. The equation proposed from Eurocode 8-part 2, [4] is for cyclic loading while the equation proposed by the CEB-FIP, Bulletin 25 [15] is for monotonic loading. It has been verified experimentally that the plastic hinge length is higher for monotonic loading than for cyclic loading [16]. The first term of Eqs. (5) and (6) takes into account the flexural behavior of the column while the second term represents the increment of the plastic length due to the slip of the reinforcement in the area of the fixed support. Eqs. (4)–(6) are valid for columns in which the shear-span ratio (kV) is lower than 6.5 [16,15] and where second order effects are negligible. However second order effects cannot be neglected for

uyI  L2s 3

 

1 1  N=Ncr

 ð7Þ

where N is the axial load applied in the column; Ncr is the Euler critical axial load and is equal to p2EIe =L2s ; and EIe is the effective flexural stiffness of the column. The amplification factor d increases first order elastic displacement DyI due to the P–D effect. Therefore, the following equation is proposed to calculate the ultimate displacement:

Du ¼

uyl  L2s 3  ð1  N=Ncr Þ

þ ðuu  uyl Þ  lp  ðLs  0:5  lp Þ ¼ Dy þ Dp

ð8Þ

From Eq. (8) it is possible to obtain the ratio (lp/h) between the plastic hinge length lp and the height of the section h. Table 7 shows the ratio lp/h (termed as relative plastic hinge length) for each test obtained using the four idealization methods of the response diagram. The value of the results obtained depends notably on the method selected to idealize the response diagram.

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(a)

(b)

4.00

4.00

ρ l =2.3%; ρ s ≤1% 3.00

μΔ

μΔ

3.00

2.00

fc 30 MPa fc 60 MPa fc 90 MPa

1.00

2.00

N/Nuc=0.30; λv=7.5 N/Nuc=0.45; λv=7.5 N/Nuc=0.30; λv=10.5 N/Nuc=0.45; λv=10.5

1.00

λ V =10.5; ρ l =2.3%; ρ s ≤1% 0.00 0.00

0.10

0.20

0.30

0.40

0.00 0

0.50

25

50

(c)

100

(d)

4.00

4.00

3.00

3.00

μΔ

μΔ

75

fc (MPa)

N/Nuc

2.00

fc 30 MPa fc 60 MPa fc 90 MPa

1.00

2.00

λ V =7.5; ρ l =2.3% fc 30 MPa; N/Nuc=0.30 fc 90 MPa; N/Nuc=0.30 fc 30 MPa; N/Nuc=0.45 fc 90 MPa; N/Nuc=0.45

1.00

λ V=10.5; N/Nuc =0.15;ρ s ≤1% 0.00 0.00

1.00

2.00

3.00

4.00

0.00 0.00

1.00

2.00

3.00

4.00

ρ s (%)

ρ l (%) Fig. 10. Ductility ratio in displacements (EC8b).

0.014 N/Nuc =0.30

0.1

N/Nuc =0.15 Experimental

0.09

0.012

N/Nuc =0.30; λV=7.5

0.08

0.01 N/Nuc =0.44 Idealized (EC8b) N/N =0

0.008

uc

0.006

0.07

V/(b·h·f c)

V/(b·h·f c)

Idealized (EC8b)

0.06 0.05 0.04

N/Nuc =0.40; λV =7.5

0.03

0.004

N/Nuc =0.30; λ V =10.5

0.02

0.002

N/N uc =0.41; λ V =10.5

0.01

0 0.00

1.00

2.00

3.00

4.00

5.00

0 0.00

1.00

2.00

Δ /L s (%) f c ≈90 MPa ; λ V=10.5; ρ l =2.3%; ρ s ≤1% N/Nuc 0.00 0.15 0.30 0.44

ΔyI/Ls (%) Δu/Ls (%) 2.48 1.27 1.05 0.86

4.47 2.89 2.38 1.83

3.00

4.00

Δ /L s (%) c

30 MPa; ρ l=2.3%; ρ s ≤ 1%

μΔ

λV

1.80 2.27 2.26 2.12

7.50 10.50

N/Nuc

ΔyI/Ls (%)

Δu/Ls (%)

μΔ

0.30 0.4 0.30 0.41

0.81 0.65 0.92 0.73

2.22 1.52 2.94 1.94

2.75 2.36 3.21 2.65

(b)

(a) Fig. 11. Experimental load-drift diagram. Idealization EC8b.

It can be inferred that the maximum values of lp/h are obtained using the EC8b method. A relative plastic hinge length lp/h was obtained from the experimental measured values for the cases of null axial load and the cases with low axial load ratio (N/Nuc = 0.15), and this was higher than expected (lp/h  0.5). This occurred since in these

experimental tests it was not possible to measure the moment– curvature diagram completely, underestimating the ultimate curvature of the cross-section uu, and consequently, the ductility ratio in curvatures (lu). Fig. 13 displays the ratio lp/h in terms of the different parameters for the EC8b method of idealization. Furthermore, the proposals

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412

(a)

(b)

1.00

1.00

fc 30 MPa fc 60 MPa fc 90 MPa

0.80

α e =EIe /EcIc

α e =EIe /EcIc

0.80 0.60 0.40

0.60

ρ l =2.3%; ρ s ≤1%

0.40

0.20

N/Nuc=0.30; λv=7.5 N/Nuc=0.45; λv=7.5 N/Nuc=0.30; λv=10.5 N/Nuc=0.45; λv=10.5

0.20

λ V=10.5; ρ l =2.3%; ρ s ≤1% 0.00 0.00

0.00 0.10

0.20

0.30

0.40

0

0.50

25

N/Nuc

(c)

75

100

(d)

1.00

1.00

fc 30 MPa fc 60 MPa fc 90 MPa

0.80

α e =EIe /EcIc

0.80

α e =EIe /EcIc

50

fc (MPa)

0.60 0.40 0.20

0.60 0.40

λ V=7.5; ρ l =2.3%

0.20

fc 30 MPa; N/Nuc=0.30 fc 90 MPa; N/Nuc=0.30 fc 30 MPa; N/Nuc=0.45 fc 90 MPa; N/Nuc=0.45

λ V=10.5; N/Nuc=0.15; ρ s ≤1% 0.00 0.00

1.00

2.00

3.00

0.00 0.00

4.00

1.00

2.00

3.00

4.00

ρ s (%)

ρ l (%)

Fig. 12. Behavior at element level: effective flexural stiffness factor ae (EC8b).

Table 7 Relative plastic hinge length (lp/h). Id

Reference

EC8a

EC8b

MBBE

AM

Id

Reference

EC8a

EC8b

MBBE

AM

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

N30-10.5-C0-2-00 N30-10.5-C0-2-15 N30-10.5-C0-2-30 N30-10.5-C0-2-45 N30-7.5-C0-2-30 N30-7.5-C0-2-45 N30-10.5-C0-1-23 N30-10.5-C0-1-51 N30-10.5-C0-3-15 N30-10.5-C0-3-30 H60-10.5-C0-2-00 H60-10.5-C0-2-15 H60-10.5-C0-2-30 H60-10.5-C0-2-45 H60-7.5-C0-2-30 H60-7.5-C0-2-45 H60-10.5-C0-1-15 H60-10.5-C0-1-30 H60-10.5-C0-3-15 H60-10.5-C0-3-30

2.34 0.62 1.23 2.12 0.93 0.58 0.30 2.92 1.80 1.94 2.67 0.46 2.10 2.46 0.77 0.62 0.10 1.38 5.31 1.67

2.34 0.92 1.27 2.52 1.04 0.66 0.47 3.56 1.93 2.19 2.67 0.86 2.84 3.03 0.94 0.78 0.55 1.75 5.61 1.95

2.36 0.78 0.76 1.93 0.72 0.41 0.19 2.51 1.81 1.35 2.68 0.59 1.78 2.05 0.52 0.41 0.54 1.34 5.67 1.21

2.39 0.84 0.51 1.93 0.63 0.47 0.08 2.39 1.65 1.16 2.70 0.33 1.19 2.09 0.48 0.45 0.43 1.35 5.60 1.33

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

H90-10.5-C0-2-00 H90-10.5-C0-2-15 H90-10.5-C0-2-30 H90-10.5-C0-2-45 H90-7.5-C0-2-30 H90-7.5-C0-2-45 H90-10.5-C0-1-15 H90-10.5-C0-1-30 H90-10.5-C0-3-15 H90-10.5-C0-3-30 N30-10.5-C3-2-30 N30-10.5-C3-2-45 N30-7.5-C3-2-30 N30-7.5-C3-2-45 H90-10.5-C3-2-30 H90-10.5-C3-2-45 H90-7.5-C3-2-30 H90-7.5-C3-2-45 H90-7.5-C2-2-30 H90-7.5-C2-2-45

2.22 0.36 0.40 1.61 0.92 0.50 0.81 0.84 0.70 1.06 0.93 1.25 0.87 0.65 1.98 1.42 1.64 0.89 1.16 0.66

2.22 0.72 1.97 2.50 1.07 0.62 1.17 1.44 1.71 2.27 1.19 1.51 0.95 0.75 3.51 2.27 1.96 1.06 1.39 0.83

2.18 0.65 2.05 1.44 0.63 0.26 1.13 1.11 2.09 2.00 0.81 0.92 0.49 0.56 3.70 1.07 1.48 0.31 1.02 0.23

2.20 0.51 1.77 1.96 0.78 0.40 1.02 1.15 1.77 1.64 0.74 0.87 0.54 0.42 3.53 1.05 1.41 0.34 0.91 0.41

from Eurocode 8, part 2 [4] (Eq. (5)) and from the CEB-FIP, Bulletin 25 [15] (Eq. (6)) are presented. The second term of Eqs. (5) and (6) is deleted because in these cases the longitudinal reinforcement is continuous (Fig. 5) and there is no sliding in the fixed support. The factor lp/h increases when the axial load level, the longitudinal reinforcement ratio, the confinement ratio, or the slenderness increase, Fig. 13. There is not a clear trend depending on the strength of concrete. Other authors, such as Pam and Ho [33], Paul-

tre et al. [8] and Bae and Bayrak [9] have verified that this length depends on the axial load level applied. However, authors such as Mendis [34] or Priestley and Park [18] point out that this length is not dependent on the axial load level. The plastic hinge length is a fictitious distance in which the inelastic deformations are developed in a uniform but concentrated way thanks to a specific curvature profile. Thus, the plastic hinge length increases with the axial load because the yielding

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A.C. Barrera et al. / Engineering Structures 40 (2012) 398–412

(a)

(b)

(c)

3.50

3.50

3.50

λ V =10.5; ρ l =2.3%; ρs ≤1% fc 30 MPa fc 60 MPa fc 90 MPa

2.00

fib 25

1.50

fib 25

1.50

EC8 0.50

0.50

0.00 0.00

0.00

0.10

0.20

0.30

0.40

25

(d)

(e)

3.50

3.50

2.50

2.00

fib 25

1.50 1.00

50

75

100

0

2.00

25

50

75

100

fc (MPa)

λ V=7.5; ρ l =2.3%

fib 25

1.50 1.00

EC8

0.50 0.00 0.00

λV=10.5; ρ l =2.3%; ρs ≤1%

fc 30 MPa; N/Nuc=0.30 fc 90 MPa; N/Nuc=0.30 fc 30 MPa; N/Nuc=0.45 fc 90 MPa; N/Nuc=0.45

3.00

lp /h

lp /h

2.50

EC8

0.00

f c (MPa)

fc 30 MPa fc 60 MPa fc 90 MPa

fib 25

1.50

0.50

N/N uc

λV =10.5; N/Nuc=0.15; ρ s ≤1%

2.00

1.00

EC8 0

0.50

N/Nuc=0.45

2.50

λ V=7.5; ρ l =2.3%; ρ s ≤1%

2.00

1.00

1.00

3.00

3.00

N/Nuc=0.45

2.50

lp /h

lp /h

2.50

N/Nuc=0.30

N/Nuc=0.30

3.00

lp /h

3.00

EC8

0.50 0.00 1.00

2.00

3.00

4.00

0.00

1.00

2.00

3.00

4.00

ρ s (%)

ρ l (%)

Fig. 13. Plastic hinge length. Idealization EC8b.

deformation (uu–uyI) and the effective yielding displacement (DyI) decrease (Figs. 8a and 11a) more than the ultimate displacement (Du). The same behavior appears with the increment of the longitudinal reinforcement ratio. The increase in the ultimate deformation of the concrete with the confinement results in an increase in the damaged zone of the columns, Barrera et al. [1], with a greater plastic hinge length required to produce an increment in the ultimate displacement maintaining the rest of parameters constant. Finally, second order effects amplify both the first order elastic displacement and the plastic displacement. For this reason, the increment of the plastic hinge length with the slenderness highlights the amplification in the plastic displacement as a result of second order effects. As was expected, the equation proposed by Eurocode 8, Part 2 [4] to obtain the plastic hinge length underestimates the factor lp/h since it is valid for cyclic loading, while the equation proposed by the CEB-FIP, Bulletin 25 [15] underestimates the estimation of lp/h for the columns with a slenderness of 10.5 and high axial load levels (more than 0.15). 4. Conclusions The deformation capacity of 40 reinforced concrete slender columns subjected to monotonic flexure and constant axial load has been studied. The test parameters were strength of concrete (normal- and high-strength concrete), shear span ratio, axial load level and the longitudinal reinforcement ratio. The following conclusions can be summarized: 1. It has been demonstrated that the ductility factor in curvatures or in displacements, the effective stiffness factor, and the plastic hinge length depend notably on the criteria adopted to idealize the response diagram.

2. The determination of the yielding point is very important for the determination of the deformation capacity. The implementation of the method proposed by Eurocode 8 part 2 Annex E.3.2. [4] shows the following advantages: (a) the determination is based on deformation criteria, (b) the yielding bending moment and the yielding load are concurrent. However, if the material deformations are unknown, it is reasonable to apply a method based on the forces to obtain an approximated yielding point. If the beginning of the non-linear behavior of the concrete is adopted from that proposed by the fib Bulletin 24, [23], 0.75  ec1; it is proposed that for rectangular columns with symmetrical reinforcement and with axial load other than zero, the approximated yielding bending moment can be 0.6Mmax and the yielding load 0.75  Vmax. 3. The ductility factor in curvatures does not always decrease with the axial load. It decreases with the strength of concrete, the reinforcement ratio and the relative cover of the longitudinal reinforcement and it increases with confinement level. 4. If the bending moment–curvature diagram is idealized using the approximated method (AM) or the method based on energy balance (MBEE), the proposal from Eurocode 8 part 1 [27], Eq. (3) is unsafe for low axial load levels, slender columns, and high-strength concrete. However, if the method proposed by Eurocode 8 part 2 (Annex E.3.2.) [4] is used (EC8a or EC8b), Eq. (3) is not conservative for low axial load levels. 5. The ductility ratio in displacements does not always decrease with the axial load and the slenderness of the column due to second order effects. Due to the scarcity of tests in the bibliography for columns with slenderness higher that 6, a more indepth study is required. As was expected, the ductility ratio in displacements decreases with the strength of concrete and with the longitudinal reinforcement ratio and increases with the confinement level.

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6. The parameter with the most influence on the effective flexural stiffness factor is the axial load level, meaning that when the axial load level increases the rigidity factor decreases. For low axial load levels, the factor does not depend on the strength of concrete, while for higher axial load levels, the factor decreases when the strength is increased. Moreover, the stiffness factor decreases with the slenderness and does not depend notably on the transversal and longitudinal reinforcement ratios. 7. A new equation has been proposed to obtain the ultimate displacement (Eq. (8)) taking into account the computation of the total elastic displacement including second order effects. The increment of the plastic displacement due to second order effects is taken into account in the estimation of the plastic hinge length. 8. The ratio between the plastic hinge length and the height of the cross-section increases with axial load level, longitudinal reinforcement ratio, confinement level, and column slenderness. In view of the results there is not a clear trend with regard to the strength of the concrete. 9. The equation for the monotonic loading proposed by the CEBFIP, Bulletin 25 [15] (Eq. (5)) underestimates the calculation of factor lp/h for columns with a slenderness of 10.5 and high axial load levels (above 0.15).

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