Curvature ductility of biaxially loaded reinforced concrete short columns

Engineering Structures 200 (2019) 109669

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Curvature ductility of biaxially loaded reinforced concrete short columns ⁎

T

Marco Breccolotti , Annibale Luigi Materazzi, Bruno Regnicoli Benitez Department of Civil and Environmental Engineering, Perugia, Italy

A R T I C LE I N FO

A B S T R A C T

Keywords: RC columns Seismic action Biaxial bending Curvature ductility

Modern structural design codes strongly rely on the dissipative capacity of structural elements to resist the relevant seismic actions at ultimate limit state. The full exploitation of dissipative capacities can, however, be achieved only in presence of adequate local and global ductility. Local ductility in cast-in situ and precast reinforced concrete buildings refers to curvature ductility of the critical part of the structural elements where plastic hinges are expected to occur. Bidirectionality of the seismic action is taken into account by the standards, in a more or less conventional manner, for the evaluation of stresses on the structure. Nevertheless, there are no specific indications of how to take biaxial bending into account in verifying the curvature ductility of reinforced concrete short columns. On the basis of this observation, a specific parametric investigation was planned and performed, with different values of angle between neutral axis and section principal axis, longitudinal reinforcement ratio, transverse reinforcement ratio, normalized axial force and section aspect ratio. Its results allowed for verification of the influence of bending moment directionality and of other parameters on the deformation capacities of reinforced concrete short columns beyond their own elastic limits.

1. Introduction Directionality of ground shaking in relation to the main directions of buildings and their structural elements and, therefore, biaxial loading, are of great importance in designing new buildings or in the assessment of existing ones towards seismic actions. Since 1975 Park and Paulay [1], commenting the special provision for the seismic design of ACI 318-71 and the capacity design method, observed that seismic loading not applied along a principal axis of a building will require column strengths considerably greater than the ACI requirement. This observation was also confirmed by other more recent investigations [2–4]. Current standards, such as the EN 1998 [5], provide different methods of investigation to take into account the possible combination of bending actions in the two main directions of the structural elements. For instance, in linear dynamic analysis with response spectrum, the seismic action along one of the principal direction is combined with a reduced seismic action in the orthogonal direction (30%). This takes into account the possibility of biaxial loading induced by earthquake. In this context, energy dissipation due to the ductile hysteretic behaviour of structural elements is accomplished by reducing the elastic response spectrum with the behaviour factor q evaluated only for the principal directions of the structural elements. A similar approach is adopted by ASCE 7-16 [6], and consequently by ACI 318 [7], for structures assigned to Seismic Design Category C.

⁎

The same standard requires special provisions for structures assigned to Seismic Design Category D, E or F. Attention is placed on columns or walls that forms part of two or more intersecting seismic force-resisting systems subjected to seismic axial load equaling or exceeding 20% of the axial design strength. These elements shall be designed for the most critical load effect due to application of seismic forces in any direction. Also for ASCE 7-16 energy dissipation for the excursion in the plastic range of materials can be taken into account by reducing the elastic seismic base shear with the response modification factor R. It is therefore, evident that the current standards provide, with different levels of complexity and severity, the free directionality of the seismic action for the evaluation of the stresses in the structural elements. Conversely, they do not require specific checks to assess the element ductility for a generic angle of attack of the seismic action and accepting ductility evaluations only in the main directions of the structural element. This can, probably be ascribed to the absence of general agreement on the effect of earthquake directionality on the ductility of structural elements even if most of the reference found in the literature report a negative influence. For instance, Tsuno and Park [8] studied the behaviour of reinforced concrete bridge piers subjected to bi-directional quasi-static loading. The results of their experimental investigations indicate that the maximum displacement of a pier when it reaches the ultimate state in a bi-directional cyclic loading, is smaller than that of

Corresponding author. E-mail addresses: [email protected] (M. Breccolotti), [email protected] (A.L. Materazzi).

https://doi.org/10.1016/j.engstruct.2019.109669 Received 23 May 2019; Received in revised form 9 September 2019; Accepted 9 September 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Structures 200 (2019) 109669

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Nomenclature

α cc

α β ∊tol γc μϕ νd ωwd yc, ij

ys, k ′ ,c ϕyd

′ ,s ϕyd ρl σ2 (=σ3)

σl σc, ij σs, k εc 2d εcud

design value of tension steel strain at yielding εsy, d As, k area of the k-th steel rebar Ast , x , Ast , y total cross-section of transversal reinforcements in the x and y direction stirrup cross-section Ast bc gross cross-sectional width depth of confined core (to the centreline of the hoops) ho distance between two consecutive constrained longbi itudinal rebars dimensions of the concrete confined core in the x and y bx , b y direction diameter of the concrete confined core D0 EEdx , EEdy action effects due to the application of the seismic action along the horizontal axes x and y of the structure fcd design value of concrete compressive strength f yd design value of steel yield strength f yk, st stirrups characteristic yielding stress hc gross cross-sectional depth parallel to the horizontal direction in which the value of μϕ is evaluated width of confined core (to the centreline of the hoops) bo ′ MRd section bending strength ′ ′ Myd bending moment corresponding to ϕyd n number of longitudinal rebars with lateral constraint by stirrups or ties q0 basic value of the behaviour factor from Table 5.1 of EN 1998 [5] s stirrups spacing T1 fundamental period of the building within the vertical plane in which bending takes place period at the upper limit of the constant acceleration reTC gion of the spectrum

coefficient taking account of long term effects on the compressive strength of concrete confinement effectiveness factor [5] coefficient equal to 2 for single circular stirrups and equal to 1 for spiral stirrups parameter for tolerance check on equilibrium conditions partial safety factor for concrete curvature ductility factor normalised design axial force (νd = NEd /(Ac ·fcd ) ) mechanical volumetric ratio of confining hoops within the critical regions distance between the centroid of the concrete element placed at the i-th row and j-th column and the centroid of the concrete gross section distance between the centroid of the k-th steel rebar and the centroid of the concrete gross section curvature at the attainment of εc 2d strain in the compressed concrete curvature at the yielding of the tensile reinforcement longitudinal reinforcement ratio effective lateral compressive stress at the ULS due to confinement confining pressure exerted by the transversal reinforcements stresses in the concrete element placed at the i-th row and j-th column stresses in the k-th steel rebar strain at reaching the maximum concrete compressive strength in the parabola-rectangle diagram ultimate compressive strain in the concrete

action on the curvature ductility of square and rectangular reinforced concrete short columns, the parametric analysis described in the following paragraphs was carried out, after having briefly recalled the indications provided by current standards.

the same column subjected to the standard uni-directional loading pattern, thus highlighting lower ductility. Similar results have also been obtained for reinforced concrete (RC) columns with smooth rebars [9,10]. Campione et al. [11] have well documented the reduction of curvature and ductility produced by the combinations of axial load and biaxial bending. Del Zoppo et al. [12] carried out an experimental campaign aimed at investigating the effects of biaxial bending on the seismic behavior of existing RC columns with design characteristics non-conforming to modern seismic codes. The results of the experimental investigations highlighted the influence of the load path direction on the lateral strength at peak load of the columns and their deformation capacity at conventional failure. Ho and Pam [13] investigated ultimate curvature for deformability evaluation of highstrength reinforced concrete columns proposing an equation for estimating the ultimate curvature of high-strength reinforced concrete columns. Ho [14] analyzed the flexural ductility performance of high strength RC columns with concrete cylinder strengths from 50 to 96 MPa. Curvature ductility factors close to 10 have been obtained for levels of compressive axial load from 0.1 to 0.65. On the other side, Giannopoulos and Vamvatsikos [15] affirm that in non-linear time histories analysis, the record-to-record variability clearly overrides the influence of the incident angle. Consequently, these researchers suggested, it is by far more important and produces more reliable results using as many different records as possible than reducing the number of records to employ different orientations of each. The issue of earthquake directionality is obviously of less importance for circular columns, for which relevant increases of curvature ductility can always be obtained by increasing confinement on the concrete core for every angle of attack of the seismic action [16]. To better understand the role of the directionality of the seismic

2. Directionality of ground shaking in current standards Directionality of the seismic action in the current standards is generally addressed in a conventional manner. According to EN 1998-1, plan irregular buildings shall be analysed using a spatial model but two independent analyses with lateral loads applied in one direction only may be performed. Subsequently, the combination of the two horizontal components of the seismic action taken as acting simultaneously may be accounted for in the following different ways:

• the structural response to each component shall be evaluated se•

parately. The maximum value of each action effect on the structure due to the two horizontal components of the seismic action may then be estimated by the square root of the sum of the squared values of the action effect due to each horizontal component; the action effects due to the combination of the horizontal components of the seismic action may be computed using the two load combinations EEdx “ + ” 0.30EEdy and 0.30EEdx “ + ” EEdy .

Such combinations must also be used when performing non-linear static (pushover) analysis considering the forces and deformations due to the application of the target displacement in the x and y directions as EEdx and EEdy , respectively. The internal forces resulting from these combinations should not exceed the corresponding capacities. ASCE 7-16 requires that the design ground motions shall be assumed to occur along any horizontal direction of a building structure. 2

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moment:

Simplified rules are allowed for buildings belonging to Seismic design category B and for regular buildings of Seismic design category C. Buildings of Seismic design category C with horizontal structural irregularity and buildings of seismic design categories D through F shall be designed for the most critical load effect due to application of seismic forces in any direction. This can be achieved by using one of the following procedures:

ϕu = min [ϕ (εcu );ϕ (M85%)]

The conventional curvature of first yielding ϕyd is expressed by the following relation:

bc − 0.035 bo

(6)

where

ωwd =

volume of confining hoops f yd · volume of concrete core fcd

(7)

3.2. Italian NTC 2018 The recently issued Italian standard NTC 2018 [21] foresees an approach very similar to that of EN 1998-1. In cast-in situ structure the explicit verification of the curvature ductility can be avoided provided that the condition of Eq. (6) is met. Nevertheless, whereas precast structures behave similarly to cast-in situ ones [22,23], for precast buildings with columns fixed to the base and horizontal elements connected to them by hinges, the standard requires the ductility verification, regardless of the adopted construction details. In fact, for this kind of structures, energy dissipation takes place only in the critical region at the base of the columns and a specific check, as that presented in par. 3.1 is, thus, necessary to ensure the effectiveness of energy dissipation. In this case the local ductility (curvature ductility μϕ ) must satisfy the following requirement:

The existence of a correlation between the columns curvature ductility capacity and the global displacement ductility of RC structures [19] is well known. Global ductility can, thus, be achieved only if local ductility is available. In the following paragraphs the requirement of several standards on local (curvature) ductility are briefly recalled. 3.1. European EN 1998-1 Eurocode 8 [5] defines the curvature ductility μϕ as the ratio of the post-ultimate strength curvature at 85% of the moment of resistance, to the curvature at yield, provided that the limiting strains of concrete, ∊cu , and steel, ∊su, k , are not exceeded. The standard requires the curvature ductility factor of critical regions for both ductility classes medium (DCM) and high (DCH), to be at least equal to the following values:

μϕ ⩾

⎧1.2(2q0 − 1) ⎨1.2 ⎡1 + 2(q0 − ⎩ ⎣

if T1 ⩾ TC T 1) TC ⎤ 1⎦

if T1 < TC

(8)

thus 20% higher than that required in Eq. (1) by the corresponding European standard. 3.3. Other national standards

(1)

This value, valid for critical regions of primary seismic elements with longitudinal reinforcement of steel class C, should be increased at least 1.5 times, when using longitudinal reinforcement of steel class B. The exploitation of the higher mechanical properties of confined concrete is allowed in the critical region at the base of primary seismic columns if concrete strains larger than εcu = 0.0035 are needed to reach the required value of the curvature ductility factor μϕ . By doing so, a compensation for the loss of resistance due to spalling of the concrete can be achieved by means of adequate confinement of the concrete core, on the basis of the properties of confined concrete in EN 1992-1-1 [20]. The capacity in terms of curvature ductility factor μϕ can be calculated as the ratio between the ultimate curvature ϕu and the conventional curvature of first yielding ϕyd :

ϕu ϕyd

(5)

αωwd ⩾ 30μϕ ·νd·εsy, d·

3. RC columns ductility requirements of International and National standards

μϕ =

′ = min (ϕyd ′ , s , ϕyd ′ ,c) ϕyd

According to EN 1998-1, there is no need to explicitly check Eq. (1) if the following relation is satisfied:

Also Australian standard AS 1170.4 [17] and New Zealand standard NZS 1170.5 [18] consider interactions from the horizontal axes by summing 100% of the design actions on one axis and 30% of the design actions on the axis at right angles. Nevertheless, the commentary of NZS 1170.5 requires that where the force resisting systems are not located on axes at right angles to each other the forces have to be applied on a range of axes to determine the critical directions.

2q0 − 1 if T1 ⩾ TC ⎧ TC ⎨1 + 2(q0 − 1) T if T1 < TC 1 ⎩

(4)

with

tions. The members and their foundations are designed for 100% of the forces for one direction plus 30% of the forces for the perpendicular direction. The combination requiring the maximum component strength shall be used; the structure is analyzed using the linear or the nonlinear response history procedure with orthogonal pairs of ground motion acceleration histories applied simultaneously.

μϕ ⩾

MRd ·ϕ′ ′ yd Myd

ϕyd =

• the loading is applied independently in any two orthogonal direc•

(3)

The current edition of the American ACI 318 [7] does not foresee any specific control of local or global ductility. The reduction of the seismic forces through the response modification coefficient R is justified by prescriptive requirements on quantity and spacing of transverse reinforcement. This allows for adequate attainment of concrete confinement and to avoid longitudinal reinforcement buckling after concrete cover spalling. Columns brittle failure is also addressed for elements made of high strength concrete and having high relative axial loads. A similar approach is adopted by the New Zealand standard NZS 3101-1 for RC structures [24]. According to this standard, columns containing plastic regions designed for ductility in earthquakes (great sectional ductility with maximum curvature greater than 19 times yielding curvature) must satisfy specific detailing prescription but no specific check is required for the value of curvature ductility.

(2)

4. Confined concrete

The ultimate curvature ϕu is equal to the minimum between the curvature corresponding to the attainment of the concrete ultimate strain εcu and that corresponding to a reduction of 15% of the bending

In order to carefully assess the actual curvature ductility of RC elements, the effect of confining pressure on the concrete mechanical 3

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• Rotation of the reference system by the angle θ (Fig. 2, right); • Computation of the χ − M curvature - bending moment relationship; • Determination of the curvature ductility μ .

properties cannot be ignored. Confinement of concrete results in a modification of its stress-strain relationship with higher strength and higher critical strains. An exhaustive collection of design equations for confining reinforcement used around the world can be found in the recent work by Eid et al. [25]. According to the standards used in this investigation [5,21], in the absence of more precise data, the stressstrain relationships with the following increased characteristic strength and strains may be used:

fck ·(1.0 + 5.0·σ2/ fck ) for σ2 ⩽ 0.05fck fck, c = ⎧ f ⎨ ⎩ ck ·(1.125 + 2.5·σ2/ fck ) for σ2 > 0.05fck fcd, c = α cc ·fck, c / γc

ϕ

Each point of the χ − M curve has been obtained by checking that the difference between the external axial load Next and the internal stress resultant Nint is less or equal to a suitable tolerance value:

|Nint (yn , θ) − Next | ⩽ ∊tol (9)

with: nx

(10)

εc 2d, c = εc 2d·(fck, c / fck )2

(11)

εcud, c = εcud + 0.2·σ2/ fck

(12)

Nint (yn , θ) =

(13)

σl, x ·σl, y

σl, x =

n⩾

(14)

Ast , x ·f yk, st by ·s

; σl, y =

Ast , y ·f yk, st bx · s

(15)

M (yn , θ) =

(16)

∑

bi2 /(6·bx ·by ) αs = [1 − s /(2·bx )]·[1 − s /(2·by )]

n

(17)

j=1

∑

σs, k (yn , θ) As, k ys, k

k=1

(22)

• curvature and moment values corresponding to the yielding of the tensile reinforcement (cyan point); • curvature and moment values corresponding to the attainment of the (confined) ε strain value (red point); • curvature and moment values corresponding to the attainment of the (confined) ε strain value (green point); • curvature and moment values corresponding to a 15% reduction of

(18)

for circular sections. The effect of confinement on the design stress-strain relationship of C45/55 concrete is shown in Fig. 1 (compressive strain shown positive) for several values of the effective lateral compressive stress σ2 .

c 2d, c

cud, c

the internal bending moment (red cross).

5. Numerical algorithm for biaxial bending in RC short columns

35

The calculation of the curvature - bending moment relationships has been carried out by means of a numerical algorithm developed in a Matlab environment. This is a different approach to what has been done by other researchers [26]. In this investigation there is no interest in checking the inclination of the neutral axis θ against the ratio of principal bending moments My / Mx . It is in fact, well known that for sections with the plane of loading not parallel to an axis of symmetry, the neutral axis will not be parallel to the resultant moment vector. Taking into account the free directionality of the seismic ground motion, it is assumed that a combination of bending moments Mx and My which corresponds to inclinations of the neutral axis θ always exists. The attention is, thus, placed on the value of θ rather than on the values of the moments Mx and My . The general scheme of the numerical algorithm, represented in the flowchart of Fig. 3, is based on the following key steps:

30

Stress (MPa)

25 20 15 10 5

• Discretization of the cross section in elements of sufficiently small •

ns

σc, ij (yn , θ) yc, ij dxdy +

Fig. 4 shows an example of the results of the numerical evaluation plotted as curvature - bending moment relationships for a square RC section obtained for several values of the inclination of the neutral axis θ . In each curve, the following particular points are also plotted, when available:

for rectangular sections and

αn = 1αs = [1 − s /(2·D0 )]β

(20)

(21)

ny

∑∑ i=1

for circular sections. The confinement effectiveness factor α = αn·αs can be evaluated as follows:

αn = 1 −

σs, k (yn , θ) As, k

k=1

log(b − a) − log(∊tol ) log(2)

nx

2Ast ·f yk, st D0 ·s

j=1

∑

where a and b are two values of internal axial force Nint with opposite signs obtained for two tentative values of the neutral axis depth yn, a , yn, b . Once Eq. (19) is satisfied the corresponding internal bending moment M has been calculated as:

for rectangular sections and

σl =

ns

σc, ij (yn , θ) dxdy +

A value ∊tol equal to 0.05 kN has been used for the investigated cross-sections. An iterative algorithm based on the bisection method has been used to adjust the neutral axis depth yn in order to satisfy Eq. (19) and obtain the equilibrium conditions. The number n of iteration performed for each value of the curvature χ is equal to:

with:

σl =

ny

∑∑ i=1

The confining lateral stress σ2 can be determined as follows:

σ2 = α·σl

(19)

0 0

0.002

0.004

0.006

0.008

2

=0

2

= 0.01 fck

2

= 0.02 fck

2

= 0.03 fck

2

= 0.04 fck

0.01

0.012

Strain

size such as to assume a constant stress distribution within each element without introducing relevant errors (Fig. 2, left); Definition of the inclination of the neutral axis θ ;

Fig. 1. Stress-strain laws for a C45/55 concrete with different levels of confining stresses. 4

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Fig. 2. Original (left) and rotated (right) reference systems.

In the same graph are also reported the values of the curvature ductilities μϕ obtained for the investigated values of the inclination of the neutral axis θ . 5.1. Verification of the numerical model The numerical model has been verified by comparing its provision with the results of several experimental tests found in the literature [27–29]. However, in order to reduce the forces to be applied to the specimens and simplify the experimental setup, the available experimental data related to biaxial bending tests on RC columns are generally referred to reduced scale samples with small cross-sections. For these tests it is thus reasonable that the tolerances on the manufacture of the samples can produce errors on overall dimensions of the cross section, on reinforcement position and quantity and on other parameters. These errors would cause some inaccuracies on the mechanical behaviour of the RC elements. The comparison between the theoretical bending moment Mth calculated with the proposed algorithm and the experimental bending moment at collapse Mexp calculated as vector sum of the principal moments Mx and My is shown in Table 1. The relative errors between the predicted and the experimental valuesare also shown in the same table. The accuracy of the proposed algorithm is considered satisfactory for the purposes of the present investigation. 6. Parametric investigation A parametric investigation has been planned and carried out to evaluate whether the orientation of horizontal seismic action influences significantly or not the response and ductility of cast-in situ and precast RC columns. Curvature ductility μϕ has been evaluated for different values of the following parameters:

• angle between neutral axis and section principal axis θ; reinforcement ratio ρ ; • longitudinal • transverse reinforcement ratio ω ; • normalized axial force ν ; • section aspect ratio λ .

Fig. 3. Flowchart for determination of the χ − M curvature - moment relationship in a biaxially loaded RC column.

l

wd

d

•

A symmetrical steel layout has been assumed even if in some cases (high axial load) has been proven that unsymmetrical design reinforcements can provide increased ductility [30]. Based on the provision of EN 1998 [5] and on the indication provided by Yuen et al. [31,32] the following ranges for the investigated parameters have been analyzed:

•

• Total longitudinal reinforcement ratios ρ : from 0.01 to 0.04 with l

increments of 0.005. These lower and upper values correspond to the minimum and maximum longitudinal reinforcement ratios that

5

can be used in RC column according to the investigated standards [5,21]; Normalized axial force νd : from 0.15 to 0.65 with increments of 0.10. The upper limit corresponds to the maximum normalized axial force allowed by the investigated standards for RC columns in DCM ductility class. The lower one is a typical value for precast industrial buildings with columns fixed to the base and horizontal elements connected to them by hinges; Transverse reinforcement mechanical ratio ω wd : from 0.08 to 0.32 with increments of 0.04. The lower limit corresponds to the minimum value of the transverse reinforcement mechanical ratio required by the investigated standards for the validity of Eq. (6) in RC columns designed in DCM ductility class. The upper value

Engineering Structures 200 (2019) 109669

M. Breccolotti, et al. 800

700

600

Moment [kNm]

curvature moment relationships have been determined. From these, the values of the ultimate curvatures ϕu and the curvature of first yielding ϕyd have been extracted to determine the curvature ductility μϕ .

Ductility values =12.42 ( 0°) = 8.85 (15°) = 7.2 (30°) = 6.71 (45°)

7. Comments on the results of the parametric investigation Due to the significant amount of data, it is not possible to report the full details of results of the parametric investigation. However, the most significant information will be illustrated in the following paragraphs. The comments on the results of the parametric investigations will be carried out for each parameter used in the analysis.

Curvature values ) syd

500

)

c2d

400

)

cud

x (85%M Rd )

300

7.1. Influence of normalized axial force The effect of the normalized axial force νd on the curvature ductility μϕ is well known and it is explicitly taken into account by simplified formulas provided by the standards such as Eq. (6). The results of the parametric investigation show that, generally, the higher the axial load, the steeper the descending part of the moment-curvature curve and the lower the flexural ductility. Similar results have been obtained by Bai and Au [33] and by Arslan [34]. Nevertheless, it should be noted that due to the high axial stress states allowed by the standards (up to νd equal to 0.65 and 0.55 for DCM and DCH, respectively) excessively

Neutral axis inclination

200

= 0° =15° =30°

100

=45° 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

[1/m] Fig. 4. Example of moment curvature relationships for a square section with different inclinations of the neutral axis θ .

Table 1 Validation of the proposed algorithm with experimental data found in literature. Reference

[27] [28] [29]

Sample

U-3 RC-1 HC0 HC1 HC2

Mexp

Mth

Error

(kNm)

(kNm)

(%)

4.59 55.44 11.39 10.61 11.36

4.68 51.30 10.12 9.78 10.69

+1.9 −7.5 −11.1 −7.8 −5.9

represents the maximum amount of transversal reinforcement that can be used in RC columns for practical reasons. Four different cross sections have been assumed (Fig. 5):

• square section with dimension b = h = 250 mm; • square section with dimension b = h = 300 mm; • square section with dimension b = h = 600 mm; • rectangular section with dimension b = 300 mm, h = 600 mm (aspect ratio h/ b = 2.0 ).

In each case, a concrete cover of 40 mm has been assumed. The square cross-section with dimension 250 mm has been chosen since it is the minimum dimension allowed by the standards [5,21] for RC columns in seismic zone. The dimension of 300 mm has been chosen since in low-rise residential buildings, it is the most used dimension for square RC columns as well as for the smallest dimension of rectangular RC columns. The square section with 600 mm side has been chosen since it is the most used dimension for RC columns in precast industrial buildings with columns fixed to the base and horizontal elements connected to them by hinges. The rectangular Section 300 × 600 mm is a combination of these two latter values. The inclination of the neutral axis θ for the square sections has been investigated in the range between 0° and 45° with increments of 15°. For the rectangular sections the investigations have been extended up to values of θ equal to 90° with an increment of 30°. A total of 5586 (7 × 6 × 7 × 4 × 3 + 7 × 6 × 7 × 7 × 1)

Fig. 5. Cross sections investigated in the parametric analysis. 6

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the post-peak branch decreases slower allowing greater curvature values to be reached. Reduction of the curvature ductility for increasing inclination of the neutral axis can also be observed in Figs. 8 c), 9b) and

reduced values of curvature ductility are achieved. This is evident by comparing, for instance, Figs. 6 and 7. 7.2. Influence of transverse reinforcement mechanical ratio

50

It is well known that the confinement of concrete produces benefits in terms of strength and ductility of its core, maintaining the resisting moment up to a much larger curvature at the post-peak stage. The results of the parametric investigations confirm this trend as obtained in similar investigations by Bai and Au [33] and Shin et al. [35]. This effect is clearly visible comparing Fig. 7 c) with Fig. 8 c). Nevertheless, it may be possible that the RC section is not able to dissipate energy in the plastic region of the two materials (concrete and steel) before a 15% reduction of the bending moment occurs for the concrete cover spalling. This phenomenon occurs for high normalized axial forces and in presence of strong confinement reinforcements, such in the cases depicted in Fig. 8a) and b) for an inclination of the neutral axis of 0°. In these cases, in fact, the concrete core is not able to reach the εc 2d, c strain. The steel in tension is also unable to reach the εsy, d strain before the section is considered collapsed due to the excessive reduction in flexural strength.

40

Moment [kNm]

35

)

c2d

25

)

cud

x (85%M Rd )

20

Neutral axis inclination = 0° =15° =30° =45°

10 5 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

90

The influence of the total longitudinal reinforcement ratio ρl can be observed by looking at Figs. 8 and 9, where the moment curvature relationships are plotted for the three square sections with the same normalized axial forces and transverse reinforcement mechanical ratio but with different longitudinal reinforcement ratios. Contrary to what was observed by Arslan [34] but similarly to the findings of Taheri et al. [36], it is evident that the total longitudinal reinforcement ratio ρl generally plays a positive role in increasing the curvature ductility μϕ . Nevertheless, this influence is not reflected by simplified formulas provided by the standards such as Eq. (6), especially for small dimension cross-sections. This behaviour is produced by the following reasons:

Ductility values =3.47 ( 0°) =4.88 (15°) =4.16 (30°) =5.77 (45°)

80

70

Moment [kNm]

60

Curvature values ) syd

50

)

c2d

)

cud

40

x (85%M Rd ) 30

Neutral axis inclination

• the

•

Curvature values ) syd

30

15

7.3. Influence of total longitudinal reinforcement ratio

•

Ductility values =4.47 ( 0°) =4.71 (15°) =3.32 (30°) =3.28 (45°)

45

increase in flexural strength due to the greater amount of longitudinal reinforcements causes the flexural strength loss resulting from the concrete cover spalling to be less than 15% of the maximum bending strength; a higher number of well-distributed longitudinal reinforcements produces a larger confined concrete area and more uniform distribution of the confining pressure; a greater amount of longitudinal reinforcements in the compression zone of the RC section delays the attainment of the (confined) ultimate strain in the concrete εcud, c .

20

= 0° =15° =30°

10

=45° 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

800

Ductility values =12.42 ( 0°) = 8.85 (15°) = 7.2 (30°) = 6.71 (45°)

700

600

Moment [kNm]

Similar results have been obtained for the rectangular cross section (Fig. 10b) and c)). 7.4. Influence of neutral axis inclination The influence of the neutral axis inclination θ is, generally, detrimental but its variability is affected by many factors. For low values of the normalized axial load and small amount of longitudinal reinforcement, the curvature ductility decreases significantly for increasing rotations of the neutral axis. This happens for instance, in the square section with 250 mm side (Fig. 6a)) where the curvature at collapse is always calculated at the attainment of 15% reduction of the internal bending moment. A similar trend can also be observed for the square section with 600 mm side (Fig. 6 c)). However, in this case the curvature at collapse is always calculated at the attainment of the (confined) εcud, c strain value. Between these two situations an opposite trend can be observed for the square section with 300 mm side. The curvature at collapse is still evaluated at 15% reduction of the internal moment but

Curvature values ) syd

500

)

c2d

400

)

cud

x (85%M Rd )

300

Neutral axis inclination

200

= 0° =15° =30° =45°

100

0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

[1/m]

Fig. 6. Moment - curvature relationships for square cross sections with νd = 0.15, ωwd = 0.08, ρl = 0.01, 40 mm cover and different neutral axis inclinations: a) dimension 250 mm, b) 300 mm and c) 600 mm. 7

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70

Ductility values =1.28 ( 0°) =1.44 (15°) =1.56 (30°) =1.62 (45°)

60

50

Curvature values ) syd

40

Moment [kNm]

Moment [kNm]

50

)

c2d

) cud

30

x

20

Ductility values = NaN ( 0°) =1.01 (15°) =1.13 (30°) =1.16 (45°)

60

(85%M Rd )

Curvature values ) syd

40

)

c2d

)

cud

30

x

20

Neutral axis inclination

Neutral axis inclination

= 0° =15° =30° =45°

10

0 0

0.02

0.04

0.06

0.08

0.1

0.12

0 0

0.02

0.04

0.06

0.08

0.1

Ductility values =1.18 ( 0°) = 1.4 (15°) =1.48 (30°) =1.48 (45°)

100

0.14

Ductility values = NaN ( 0°) =0.88 (15°) =1.13 (30°) = 1.2 (45°)

100

80

80

Moment [kNm]

Curvature values ) syd )

c2d

60

)

cud

x

(85%M Rd )

40

Curvature values ) syd )

c2d

60

)

cud

x

(85%M Rd )

40

Neutral axis inclination

Neutral axis inclination

= 0°

= 0°

=15° =30°

20

=15° =30° =45°

20

=45° 0 0

0.12

120

120

Moment [kNm]

= 0° =15° =30° =45°

10

0.14

(85%M Rd )

0.02

0.04

0.06

0.08

0.1

0.12

0 0

0.14

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1200

1000

Ductility values =3.87 ( 0°) =3.96 (15°) =4.05 (30°) = 4.3 (45°)

900 800

Ductility values =7.71 ( 0°) =7.55 (15°) =7.32 (30°) =7.23 (45°)

1000

700

Moment [kNm]

Moment [kNm]

800

Curvature values ) syd

600

)

c2d

500

)

cud

x

400

(85%M Rd )

Curvature values ) syd )

c2d

600

)

cud

x

(85%M Rd )

400

300

= 0°

= 0°

200

=15° =30°

200

=15° =30°

100 0 0

Neutral axis inclination

Neutral axis inclination

=45°

=45° 0.02

0.04

0.06

0.08

0.1

0.12

0 0

0.14

0.02

0.04

0.06

0.08

0.1

0.12

0.14

[1/m]

[1/m]

Fig. 7. Moment - curvature relationships for square cross sections with νd = 0.65, ωwd = 0.08, ρl = 0.01, 40 mm cover and different neutral axis inclinations: a) dimension 250 mm, b) 300 mm and c) 600 mm.

Fig. 8. Moment - curvature relationships for square cross sections with νd = 0.65, ωwd = 0.28, ρl = 0.01, 40 mm cover and different neutral axis inclinations: a) dimension 250 mm, b) 300 mm and c) 600 mm.

c). In very few cases an opposite trend can be noted. This happens, for instance, for small square sections having high values of the normalized axial load and great amount of longitudinal reinforcements (Fig. 9a)).

The gain in curvature values is due to increasing εcud, c strain values and decreasing εc 2d, c strain values thus producing with increasing μϕ . The influence of neutral axis inclination on curvature ductility of 8

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120

Ductility values =4.01 ( 0°) =4.49 (15°) =5.36 (30°) =5.91 (45°)

100

Ductility values =6.47 ( 0°) =5.28 (30°) =4.32 (60°) =7.14 (90°)

300

250

Curvature values ) syd

Moment [kNm]

Moment [kNm]

80

)

c2d

60

) cud x

(85%M Rd )

Curvature values ) syd

200

)

c2d

)

cud

150

x

(85%M Rd )

40

100

Neutral axis inclination = 0° =15° =30° =45°

20

0 0

0.05

0.1

0.15

0.2

0.25

=90° 0 0

0.3

160

0.04

0.06

0.08

0.12

0.14

0.16

Ductility values =1.04 ( 0°) = 4.2 (30°) = 4.7 (60°) = 0.8 (90°)

400 350

Moment [kNm]

Curvature values ) syd

120

)

c2d

100

)

cud

x

80 60

(85%M Rd )

= 0°

0.05

0.1

0.15

0.2

0.25

)

c2d

250

)

cud

x

200

(85%M Rd )

Neutral axis inclination = 0°

100

=15° =30° =45°

20

Curvature values ) syd

300

150

Neutral axis inclination

40

=30° =60°

50 0 0

0.3

2250

=90° 0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

900

Ductility values =8.18 ( 0°) =8.43 (15°) =7.81 (30°) = 7.8 (45°)

2000

1750

Ductility values =4.54 ( 0°) =4.71 (30°) =4.58 (60°) = 5 (90°)

800

700

1500

600

Moment [kNm]

Curvature values ) syd

1250

)

c2d

)

cud

1000

x

(85%M Rd )

750

Curvature values ) syd

500

)

c2d

)

cud

400

x

(85%M Rd )

300

Neutral axis inclination 500

Neutral axis inclination

= 0° =15° =30° =45°

250

0 0

0.1

450

140

Moment [kNm]

0.02

500

Ductility values =7.08 ( 0°) =6.31 (15°) = 6 (30°) =6.81 (45°)

180

Moment [kNm]

= 0° =30° =60°

50

200

0 0

Neutral axis inclination

0.05

0.1

0.15

0.2

0.25

200

= 0°

100

=30° =60° =90°

0 0

0.3

[1/m]

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Fig. 10. Moment - curvature relationships for the rectangular cross sections with: a) νd = 0.15, ωwd = 0.08, ρl = 0.01; b) νd = 0.65, ωwd = 0.28, ρl = 0.01; c) νd = 0.65, ωwd = 0.28, ρl = 0.04.

Fig. 9. Moment - curvature relationships for square cross sections with νd = 0.65, ωwd = 0.28, ρl = 0.04, 40 mm cover and different neutral axis inclinations: a) dimension 250 mm, b) 300 mm and c) 600 mm.

9

Engineering Structures 200 (2019) 109669

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30

Confining hoops mechanical ratio ωwd • 0.08 • 0.12 • 0.16 • 0.20 • 0.24 • 0.28 • 0.32

20

25

Curvature ductility μθ

Curvature ductility μθ

25

15

10

5

0

Neutral axis inclination θ • 0° • 15° • 30° • 45°

20

15

10

5

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.7

0

0.1

0.2

Normalized axial force ν

0.3

a) Normalized axial force ν • 0.65 • 0.55 • 0.45 • 0.35 • 0.25 • 0.15

20

• 0.040 • 0.035 • 0.030 • 0.025 • 0.020 • 0.015 • 0.010

25

Curvature ductility μθ

Curvature ductility μθ

0.7

Longitudinal reinforcement ratio ρl

15

10

5

20

15

10

5

0

5

10

15

20

25

0

30

0

Approximated curvature ductility μθ

5

10

15

20

25

30

Approximated curvature ductility μθ

c)

d)

30

30

Neutral axis inclination θ • 0° • 15° • 30° • 45°

Confining hoops mechanical ratio ωwd • 0.08 • 0.12 • 0.16 • 0.20 • 0.24 • 0.28 • 0.32

25

Curvature ductility μθ

25

Curvature ductility μθ

0.6

30

25

20

15

10

5

0

0.5

b)

30

0

0.4

Normalized axial force ν

20

15

10

5

0

5

10

15

20

25

0

30

Approximated curvature ductility μθ

0

5

10

15

20

25

30

Approximated curvature ductility μθ

e)

f)

Fig. 11. Comparison between the curvature ductilities obtained with the numerical investigation μϕ, n and those provided by the approximated Eq. (6) of EN 1998-1 μϕ, s for a confined DCM 250 × 250 mm section.

10

Engineering Structures 200 (2019) 109669

M. Breccolotti, et al. 30

30

Confining hoops mechanical ratio ωwd • 0.08 • 0.12 • 0.16 • 0.20 • 0.24 • 0.28 • 0.32

20

25

Curvature ductility μθ

Curvature ductility μθ

25

15

10

5

0

Neutral axis inclination θ • 0° • 15° • 30° • 45°

20

15

10

5

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.7

0

0.1

0.2

Normalized axial force ν

0.3

a)

20

• 0.040 • 0.035 • 0.030 • 0.025 • 0.020 • 0.015 • 0.010

25

Curvature ductility μθ

Curvature ductility μθ

0.7

Longitudinal reinforcement ratio ρl

15

10

20

15

10

5

5

0

5

10

15

20

25

0

30

0

5

10

15

20

25

30

Approximated curvature ductility μθ

Approximated curvature ductility μθ

c)

d)

30

30

Neutral axis inclination θ • 0° • 15° • 30° • 45°

Confining hoops mechanical ratio ωwd • 0.08 • 0.12 • 0.16 • 0.20 • 0.24 • 0.28 • 0.32

25

Curvature ductility μθ

25

Curvature ductility μθ

0.6

b)

Normalized axial force ν • 0.65 • 0.55 • 0.45 • 0.35 • 0.25 • 0.15

25

20

15

10

5

0

0.5

30

30

0

0.4

Normalized axial force ν

20

15

10

5

0

5

10

15

20

25

0

30

Approximated curvature ductility μθ

0

5

10

15

20

25

30

Approximated curvature ductility μθ

e)

f)

Fig. 12. Comparison between the curvature ductilities obtained with the numerical investigation μϕ, n and those provided by the approximated Eq. (6) of EN 1998-1 μϕ, s for a confined DCM 300 × 300 mm section.

11

Engineering Structures 200 (2019) 109669

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30

30

Confining hoops mechanical ratio ωwd • 0.08 • 0.12 • 0.16 • 0.20 • 0.24 • 0.28 • 0.32

20

25

Curvature ductility μθ

Curvature ductility μθ

25

15

10

5

0

Neutral axis inclination θ • 0° • 15° • 30° • 45°

20

15

10

5

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.7

0

0.1

Normalized axial force ν

0.2

0.3

a) Normalized axial force ν • 0.65 • 0.55 • 0.45 • 0.35 • 0.25 • 0.15

20

• 0.040 • 0.035 • 0.030 • 0.025 • 0.020 • 0.015 • 0.010

25

Curvature ductility μθ

Curvature ductility μθ

0.7

Longitudinal reinforcement ratio ρl

15

10

5

20

15

10

5

0

5

10

15

20

25

0

30

0

Approximated curvature ductility μθ

5

10

15

20

25

30

Approximated curvature ductility μθ

c)

d)

30

30

Neutral axis inclination θ • 0° • 15° • 30° • 45°

Confining hoops mechanical ratio ωwd • 0.08 • 0.12 • 0.16 • 0.20 • 0.24 • 0.28 • 0.32

25

Curvature ductility μθ

25

Curvature ductility μθ

0.6

30

25

20

15

10

5

0

0.5

b)

30

0

0.4

Normalized axial force ν

20

15

10

5

0

5

10

15

20

25

0

30

Approximated curvature ductility μθ

0

5

10

15

20

25

30

Approximated curvature ductility μθ

e)

f)

Fig. 13. Comparison between the curvature ductilities obtained with the numerical investigation μϕ, n and those provided by the approximated Eq. (6) of EN 1998-1 μϕ, s for a confined DCM 600 × 600 mm section. 12

Engineering Structures 200 (2019) 109669

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30

30

Confining hoops mechanical ratio ωwd • 0.08 • 0.12 • 0.16 • 0.20 • 0.24 • 0.28 • 0.32

20

25

Curvature ductility μθ

Curvature ductility μθ

25

15

10

5

0

Neutral axis inclination θ • 0° • 15° • 30° • 45° • 60° • 75° • 90°

20

15

10

5

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.7

0

0.1

0.2

Normalized axial force ν

0.3

a) Normalized axial force ν • 0.65 • 0.55 • 0.45 • 0.35 • 0.25 • 0.15

20

• 0.040 • 0.035 • 0.030 • 0.025 • 0.020 • 0.015 • 0.010

25

Curvature ductility μθ

Curvature ductility μθ

0.7

Longitudinal reinforcement ratio ρl

15

10

5

20

15

10

5

0

5

10

15

20

25

0

30

0

Approximated curvature ductility μθ

5

10

15

20

25

30

Approximated curvature ductility μθ

c)

d)

30

30

Neutral axis inclination θ • 0° • 15° • 30° • 45° • 60° • 75° • 90°

20

Confining hoops mechanical ratio ωwd • 0.08 • 0.12 • 0.16 • 0.20 • 0.24 • 0.28 • 0.32

25

Curvature ductility μθ

25

Curvature ductility μθ

0.6

30

25

15

10

5

0

0.5

b)

30

0

0.4

Normalized axial force ν

20

15

10

5

0

5

10

15

20

25

0

30

Approximated curvature ductility μθ

0

5

10

15

20

25

30

Approximated curvature ductility μθ

e)

f)

Fig. 14. Comparison between the curvature ductilities obtained with the numerical investigation μϕ, n and those provided by the approximated Eq. (6) of EN 1998-1 μϕ, s for a confined DCM 300 × 600 mm section. 13

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results obtained with the parametric investigation and those of the equation of EN 1998-1 is carried out. If the results match, the points should be aligned along the bisector of the first quadrant. However, it is possible to note a significant scattering of the results, with several values of the curvature ductility estimated with the simplified normative formula which are well above the corresponding one determined by the numerical analysis. Also in this case, the results are plotted with different colors to highlight the effect of the parameters νd, ρl , θ and ω wd . For example, looking at Fig. 12e) for the square section with dimensions 300x300 mm, the influence of the cover/height ratio is evident. In fact, very low curvature ductilities have been obtained for a neutral axis inclination θ equal to 0° whereas much higher values have been obtained with the equation of EN 1998-1. This behaviour is caused by the reduction of the bending moment, bigger than 15%, produced by the concrete cover spalling, that prevents the exploitation of the plastic resources of the section beyond this limit. By disregarding the corresponding very high normalized axial force values for the square section with dimension 600x600 mm, such effects are no longer observed, and results are plotted in Fig. 13. The influence of the neutral axis inclination θ is clearly ignored by the simplified equation. Nevertheless, its effect is clearly visible, for example, in Fig. 13e): the actual curvature ductility is significantly reduced as the angle of inclination increases while the curvature ductility foreseen by the simplified equation does not change. The same observation can also be drawn looking at Figs. 11e) and 12e), by disregarding the lower left part of the graphs, where the data is influenced by the combination of cover/height ratio and longitudinal reinforcement ratio values. Many of these effects are similarly present in a rectangular section. For instance, for the analyzed cross section, the effect of the spalling of the concrete cover that produces a relevant reduction of the bending strength is visible by looking at the yellow points in the lower part of Fig. 14e). These points refer to an angle of rotation of the neutral axis of 90° with the concrete cover, along the major dimension of the section that reaches its ultimate deformation and produces an abrupt reduction of the inner bending moment. For the rectangular section it is evident that the unconservative estimation of the curvature ductility is achieved by the simplified Eq. (6) of EN 1998-1. In this case, it must be pointed out that the higher ratio between bc / b0 and hc / h 0 has been used in Eq. (6) regardless of the inclination of the neutral axis, thus achieving a lower value of the curvature ductility μϕ . Curvature ductility is overestimated in rectangular cross sections, especially for low values of the normalized axial force ν (Fig. 14 c)) and for bending moment around the weak axis of the cross section (points with θ = 90° in Fig. 14e)). The overestimation of curvature ductility tends to be reduced for lower values of the aspect ratio h/ b . The influence of other parameters, such as longitudinal and transverse reinforcement ratios, is similar to that already observed for square sections. The differences between the results of the parametric investigations and that of the simplified equation of EN 1998-1 are due to the following reasons:

rectangular section is described at par. 7.6. 7.5. Influence of cover/height ratio The ratio between concrete cover and the overall height of the section plays a role in the evaluation of the curvature ductility of a RC section. Its effect can be observed, for instance, in Fig. 6 where the curvature ductility decreases gradually from the biggest to the smallest section with the same concrete cover (40 mm). In fact, in sections having a great value of cover/height ratio, the spalling of the concrete cover is generally responsible for the reduction of internal moment, greater than 15%, thus preventing in the calculation of the curvature ductility the attainment of the ultimate concrete strain εcud, c . This tendency, is somewhat evident in its presence in other analyzed cases (Figs. 7 and 8). When a high longitudinal reinforcement ratio is present in the section, the attainment of the ultimate concrete strain εcud, c is reached before a 15% internal moment reduction occurs, regardless of the value of cover/height ratio (Fig. 9). Nevertheless, in these cases higher values of the curvature ductility are obtained in sections having smaller values of cover/height ratio. 7.6. Influence of cross section shape The influence of cross section shape can be inferred comparing Figs. 6–9 with Fig. 10. For low values of the axial load and low values of the transversal and longitudinal reinforcement (Fig. 10a)) the rectangular cross section exhibit the highest value of the curvature ductility μϕ for an inclination of the neutral axis θ equal to 90°. In this case the rectangular cross section behaves similarly to the square cross section with 600 mm side. A smaller value of the curvature ductility is obtained for θ equal to 0° but this value is definitely higher than that obtained for the square cross sections with similar reinforcements. Smaller values of the curvature ductility μϕ have been obtained for intermediate inclinations of the neutral axis (θ equal to 30° and 60°). An opposite trend can be observed for high axial load, high amount of transversal reinforcement and low longitudinal reinforcement (Fig. 10b)). For these conditions very small curvature ductilities have been obtained for the θ values of 0° and 90° whilst higher values have been calculated for θ values of 30° and 60°. It can, therefore be observed that the rectangular cross sections:

• behaves similarly to the corresponding square section with side 300 mm (Fig. 8b)) for inclination of the neutral axis θ of 0° and 90°; • behaves better than the corresponding square section with side 300 mm (Fig. 8b)) for inclination of the neutral axis θ of 30° and 60°; • behaves always worse than the corresponding square section with side 600 mm (Fig. 8 c)) for any inclination of the neutral axis.

Finally, the curvature ductility of rectangular sections with high axial load and great amount of transversal and longitudinal reinforcement (Fig. 10 c)) is always smaller than that of the corresponding square sections with side 300 and 600 mm (Figs. 9b) and 9 c), respectively).

• the positive effect of longitudinal reinforcement in the curvature •

8. Comparison between EN 1998-1 provisions and investigations results The significant heterogeneity of the results obtained from the parametric investigation led to compare these results with the indications provided by the simplified Eq. (6) of EN 1998-1. The results of the comparison are shown in Figs. 11–14 for the investigated cross sections with different shapes and dimensions. In the top part of these figures the results of the parametric investigation are plotted in a νd − μϕ reference system with colors according to the values of ω wd and θ . In the central and lower parts of the figures the comparison between the

•

ductility is not taken into account, in contrast to the New Zealand standard NZS 3101-1; the influence of the cover/height ratio is disregarded. Other standards take this aspect into account by introducing the ratio between the gross and the core area of the column in the equation to calculate the volumetric ratio of transverse reinforcement given a required curvature ductility [24]; the effect of biaxial bending is neglected.

9. Summary and conclusions In this research, a parametric analysis to investigate the curvature ductility of biaxially loaded square and rectangular RC short columns have been carried out. The investigation was aimed at: 14

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M. Breccolotti, et al.

• verifying the existence of different behavior in terms of local duc•

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tility between RC short columns subjected to bending along their principal axes or along any other direction; verifying the applicability of Eq. (6) of EN 1998-1 for the case of biaxially loaded RC columns. The main conclusions of the parametric investigation are as follow:

• biaxial bending in axially loaded RC short columns produces not-

• •

•

negligible modifications in the curvature ductility of RC columns. In some cases, biaxial bending produces higher values of the curvature ductility but in many other cases it tends to reduce the ductility. Especially for low longitudinal reinforcement ratios ρl and low normalized axial force νd ; the loss of concrete cover in sections with high values of cover/ height ratio may be responsible of abrupt reductions in the resisting bending moment with the corresponding reduction in curvature ductility; the simplified Eq. (6) of EN 1998-1 sometimes provides unconservative results even for bending along the principal direction of the cross section. This mainly happen for sections with low normalized axial force νd , high values of cover/height ratio and for rectangular section. This latter behaviour can be partially ascribed to the non-uniform state of confinement in the two main directions of the investigated section; sections with high longitudinal reinforcement ratios ρl generally possess increased curvature ductility. This dependence could be inserted in the design equations to obtain more accurate estimates of the curvature ductility.

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