experimental behaviour and numerical modelling …experimental behaviour and numerical modelling of...

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Journal of Earthquake Engineering, Vol. 10, No. 3 (2006) 313–329c© Imperial College Press

EXPERIMENTAL BEHAVIOUR AND NUMERICAL MODELLINGOF SMOOTH STEEL BARS UNDER COMPRESSION

EDOARDO COSENZA and ANDREA PROTA

Department of Structural Analysis and DesignUniversity of Naples Federico II

Via Claudio 21, 80125, Naples, Italy

Received 25 January 2005Reviewed 25 July 2005

Accepted 1 September 2005

Many existing reinforced concrete buildings located in seismic regions are characterisedby internal steel reinforcement made of smooth bars and stirrups with inadequate spac-ing. These bars could be subjected to significant compression and eventually buckle.This paper deals with a comprehensive experimental campaign investigating the com-pressive behaviour of smooth bars for different values of the ratio L/D, L being therestraints distance and D the bar diameter. The stress-strain relationship is then mod-elled ranging from an elastic-plastic behaviour identical to that in tension (L/D = 5) tothe elastic buckling behaviour (L/D > 20). The comparison between the experimentalresults and the outcomes of the model confirms the accuracy of the proposed stress-strainrelationship.

Keywords: Buckling; compressive behaviour; smooth bars; stress-strain relationship.

1. Introduction

The analysis of the behaviour of compressive bars is of particular relevance inorder to achieve a reliable assessment of existing Reinforced Concrete (RC) build-ings. This issue is important even for RC buildings subjected only to gravity loadsbecause there are many under-designed structures where the high compression inthe columns and insufficient or absent stirrups increase the likelihood that bucklingof compressive steel bars could occur. The situation becomes worse for old build-ings where the corrosion of steel bars causes the spalling of the concrete and thenreduces both the gross cross section of the column and the restraint provided tothe longitudinal bars by the concrete cover.

This problem is certainly more important for buildings subjected to horizontalactions due to seismic events that determine a cyclic amplification of compressivestresses in the columns and could cause a sudden loss of the cover. Such conditionis typical of unconfined beam-column joints (i.e. exterior or corner joints) wherethe construction practice does not place stirrups resulting in spacing values in theorder of the beam height. Figure 1 depicts two examples of buckling of steel barsin unconfined joints after a severe earthquake.

313

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314 E. Cosenza & A. Prota

Fig. 1. Examples of buckling of steel bars in joints without stirrups.

The modelling of the monotonic behaviour of longitudinal bars is essential whenperforming a nonlinear static seismic analysis (i.e. push-over analysis) of buildings,that is in general necessary if one attempts to perform the assessment of existingRC buildings according to modern codes such as ATC40 [1996], Eurocode 8-Part 3[2004], FEMA-310 [1998] and the lately issued Italian seismic code [Ordinanza 3274,2003].

The paper presents the outcomes of an extensive experimental campaign onsmooth steel bars tested monotonically at the Department of Structural Analysisand Design of the University of Naples Federico II. The experimental results arediscussed and the proposed analytical constitutive law is presented depending onthe ratio L/D (i.e. where L is the spacing between two consecutive stirrups and D

is the bar diameter) and the yield strength of the bars, σy . It is pointed out that theexperimental activity involves cyclic tests which are very important because, whenbuckling occurs, the cyclic response is dramatically altered with severe consequenceson member performance. Due to space constraints the discussion of cyclic testresults will be presented in a separate manuscript. Studies have been presented inliterature about the cyclic behaviour of ribbed bars [Monti and Nuti, 1992; Doddand Restrepo-Posada, 1995; Gomes and Appleton, 1997; Albanesi et al., 2001].

The definition of an accurate theoretical constitutive law that can be used withina wide range of L/D values for smooth bars is important in order to be able tomodel all different cases that could be found in real buildings, where the lack ofseismic provisions at the time of their construction or the low quality control ofthe execution could have determined that the spacing of the stirrups is largelyinadequate. In particular, the obtained results could be useful for many applicationssuch as, for instance, the study of the interaction of the longitudinal bars withconcrete and stirrups and the design of the strengthening of columns in order toincrease their confinement.

Therefore, the research presented was planned to cover a much broader rangeof L/D values than that dealt with by Monti and Nuti [1992] with respect toribbed bars. That study aimed at assessing the L/D threshold that, if overcome,

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Numerical Modelling of Smooth Steel Bars under Compression 315

could cause a significant decrease of load carrying capacity of the member dueto compressive bars. Its outcomes have allowed defining practical design criteriaregarding stirrup spacing, which nowadays has been included in all modern codes.

2. Experimental Program

The experimental work has concerned monotonic tensile and compressive tests onsmooth steel bars with diameters of 8mm, 12mm, 14mm and 16mm (i.e. denotedin the following as D8, D12, D14 and D16 bars, respectively) for ratios, L/D,ranging between 5 and 70. One tensile and three compressive tests have been per-formed for each diameter with every fixed ratio L/D. The former has been used todetermine the mechanical tensile properties that have been found to vary slightlydepending on the diameter even though the samples had the same commercial nom-inal properties. The compressive tests have allowed assessing the repeatability ofthe results and have then provided a reliable experimental database to be used forthe calibration of the analytical constitutive laws.

The tests have been carried out in a displacement control mode with a headspeed of the used MTS 810 machine equal to 0.05mm/sec; the pressure of thehydraulic controlled grips was kept constant at 20MPa. Before proceeding with theplanned test matrix, preliminary tests have been conducted in order to optimisethe set up for the compressive tests. In fact, the instrumentation generally used fortensile tests based on an extensometer could not be adopted for the compressivetests due to the following reasons: The values recorded by that device have nomeaning as soon as the bar starts buckling (Fig. 2) and the values meaningful forcompressive tests concern the average deformations over the entire length, L, of

Fig. 2. Deformometer after bar buckling (wrong setup).

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the bar rather than local deformations measured over the undisturbed length (i.e.typically 40 or 50mm) where the deformometer is mounted.

Therefore, it has been necessary to use linear variable differential transformer(LVDT) transducers placed along the entire length of the bar; however, the rangeof L values (i.e. between 65mm and 600mm) did not allow installing the LVDTsdirectly on the grips and required the adoption of steel plates that were bonded onthe machine at one end and supported each LVDT at the other. In order to optimisethe set up, some calibration tests have been carried out to assess whether any slipoccurred between the heads and the grips, and between the bar and the grips;for this reason the deformometer, two LVDTs directly placed on the heads of themachine and two LVDTs placed on the grips (Fig. 3) were used at the same time.

The analysis of these calibration tests highlighted that the three readings becomebasically equal as soon as the bar overcomes the elastic range and that the deforma-tions provided by the LVDTs placed on the grips are in general more reliable thanthose obtained from the LVDTs located on the heads. In addition, it is appropriateto take the readings of the deformometer within the elastic range in order not toaccount for the slips between the bar and the grips that, even small, could have apercent influence in that range (i.e. small deformations).

Finally, other preliminary tests have been conducted in order to assess theinfluence of the deflection of the steel plates (i.e. working as cantilever beams)on the LVDTs readings; ratios L/D that allowed mounting the LVDTs directly onthe grips have been selected to do this (Fig. 4). These tests have confirmed that theflexural deformation of the steel plates is basically negligible and that the readingsprovided by LVDTs placed at their ends are reliable.

Fig. 3. Instrumentation used during the calibration tests.

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Fig. 4. Transducers placed directly on the grips and on the heads.

In conclusion, based on the described calibration the data analysis for all thetests discussed beneath was performed as follows: For the tensile tests only thedeformometer readings were used; the stress-strain curves of compressive bars wereobtained based on the deformometer readings up to the yielding and based on theaverage deformation given by the two LVDTs connected to the grips through thesteel plates from the yielding to the rupture.

3. Experimental Results

The tests have confirmed that the tensile behaviour of the bars slightly changesas L/D varies (Figs. 5–8); it is underlined that all constitutive relationships rep-resented in this paper refer to engineering stresses [Dodd and Retrepo-Posada,1995]. The mean mechanical properties obtained by testing bars with L/D rangingbetween 5 and 15 are summarised for each diameter in Table 1 where the followingnotation is used: εy, mean strain at yielding and fy corresponding mean stress; εsh,strain at hardening and fsh corresponding mean stress; εt, mean strain at maxi-mum load and ft corresponding mean stress. The values reported in the table pointout that D8, D12 and D16 bars have similar properties, whereas D14 bars showedhigher strength values; however, both classes are representative of typical reinforc-ing steel bars used during the 1960s to build RC structures. For all the bars, the

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0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

450.0

500.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

ε (mm/mm)

σ(M

Pa)

L/D = 5 ÷ 15

Fig. 5. Experimental tensile stress-strain curves for D8 bars.

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

450.0

500.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

ε (mm/mm)

σ(M

Pa)

L/D = 5 ÷ 15


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0.0

50.0

100.0

150.0

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250.0

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500.0

550.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

ε (mm/mm)

σ(M

Pa)

L/D = 5 ÷ 15


0.0

50.0

100.0

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450.0

500.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

ε (mm/mm)

σ(M

Pa)

L/D = 5 ÷ 15


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Table 1. Experimental mean tensile properties of tested bars.

εy fy εsh fsh εt ft

(mm/mm) (N/mm2) (mm/mm) (N/mm2) (mm/mm) (N/mm2)

D8 0.0018 358.0 0.037 336.2 0.215 448.5D12 0.0016 327.4 0.032 315.1 0.231 439.2D14 0.0021 410.8 0.024 409.0 0.166 540.9D16 0.0016 321.1 0.030 305.5 0.222 423.8

mean Young modulus was about 2.05× 105 N/mm2; it was also noticed that, afteryielding, the surface of the bars exhibited a surface flaking that increased up torupture; an example is depicted in Fig. 4.

The experiments have highlighted that the stress-strain relationship of compres-sive bars depends on the ratio L/D, whereas it is not influenced by the bar diameter.Figure 9 shows a comparison between experimental normalised stress-strain curvesof bars with different diameters characterised by L/D = 10. It is underlined thatvalues on the x-axis on this and following figures are represented up to 100 in orderto allow observing the asymptotic trend of the curves; the portion of the x-axis use-ful for assessment purposes is certainly limited to lower values. Figure 10 allows forthe observation of how the effects of the buckling in the plastic range becomes moresignificant as L/D increases from 5 to 20; the instance given in the figure refers toD12 bars and does not include L/D ratios beyond 20 since that was found to bethe threshold after which elastic buckling occurred regardless of the bar diameter.

0.0

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0.7

0.8

0.9

1.0

1.1

0 10 20 30 40 50 60 70 80 90 100ε/εy

σ/σy

D14

D12

D8D16

Fig. 9. Normalised compressive stress-strain curves for different diameters with L/D = 10.

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0 10 20 30 40 50 60 70 80 90 100

ε/εy

σ/σy

6

5

7

8

910

14

15 1213

11

tension

20

18

Fig. 10. Normalised compressive stress-strain curves for D12 bars.

Figure 10 also allows the comparison of the compressive behaviour for different L/D

ratios to that average in tension. The same curves are depicted in Figs. 11, 12 and13 for D8, D14 and D16 bars, respectively. It is recalled that Figs. 9–13 show themean stress-strain curves obtained on three tests for each L/D. Figure 14 showsthe D16 samples after the compressive tests.

4. Analysis of Test Results

The analysis of the experimental outcomes allows the identification of four thresholdvalues of the L/D ratio as follows:

(i) (L/D)p (i.e. p stands for plastic) as the value below which the ductility of thebar in compression is so large that its compressive behaviour is very similar tothat in tension; based on the performed tests, it can be stated that (L/D)p = 5;

(ii) (L/D)h (i.e. h stands for hardening) as the value below which the bar stillexhibits a certain level of hardening before buckling; based on the experimentalresults, it can be deduced that (L/D)h = 8;

(iii) (L/D)y (i.e. y stands for yielding) as the threshold beyond which the bar startsbuckling close to the yielding and does not exhibit any hardening; the obtainedresults suggest to assume (L/D)y = 20;

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0.0

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1.1

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0 10 20 30 40 50 60 70 80 90 100

ε/εy

σ/σy

tension

8

9

10

14

15

1213

11

20

18


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0 10 20 30 40 50 60 70 80 90 100

ε/εy

σ/σy

6

5

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8

9

10

14

15

12

13

11

tension

20

18


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0.0

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0 10 20 30 40 50 60 70 80 90 100

ε/εy

σ/σy

6

5

7

8

9

10

1415

12

1311

tension

20

18


(iv) (L/D)e (i.e. e stands for elastic) as the value beyond which elastic bucklingoccurs. Such threshold could be theoretically found recalling that:

Ncr = π2 EI

L2o

. (1)

Therefore, if the following notation is used:

Ncr = σcr · πD2/4; I = πD4/64; Lo = β · L, (2)

and assuming that E = 2.05× 105 N/mm2 and taking the critical stress, σcr, equalto the yielding stress, σy, the following expression for (L/D)e can be derived:

(L

D

)e

=20β

√320σy

= 40

√320σy

(3)

if the coefficient β is equal to 0.5 as for a beam fixed at both ends.However, the actual critical stress of steel elements is lower due to residual

stresses generated during the manufacturing process, transverse and longitudinalgeometrical imperfections, and variability of the mechanical properties of the mate-rial, as it will be analysed in the following sections.

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5. Modelling of the Constitutive Relationship of Compressive Bars

Considering the experimental results and the above defined thresholds of L/D, thefollowing cases can be defined towards the modelling of the stress-strain relationshipof compressive bars:

First case:

(L/D) ≤ 5: Since the ductility of the compressive bar is very large, its behaviourin compression can be assumed equal to that in tension.

Second case:

(L/D) = 6 ÷ 7: The compressive behaviour can be safely assumed to be elastic-plastic ending at a conventional strain value εu to be experimentally determineddepending on L/D. In particular, such value could be obtained from Figs. 10–13by intersecting the relevant experimental curve (on the descending branch) with ahorizontal line overlapped to its plateau. If the experimental curves are not avail-able, the experimental results suggest that it can be safely recommended to assumeεu/εy equal to at least 65 and 30 (i.e. εu equals to at least 13% and 6%) for L/D

equal to 6 and 7, respectively. No effort is devoted to define a more accurate law ofthe plastic branch since the ductility provided by such simplified method is alreadyvery large and the remaining portion of the stress-strain relationship could be nottoo useful since the likelihood that it is attained in real columns is very low.

Third case:

8 ≤ (L/D) ≤ 20: Three ranges of the stress-strain relationship can be identified,namely: An elastic behaviour up to the yielding, a plateau, and then a nonlinearsoftening. The transition from the plateau to the beginning of the softening occursat a strain value herein defined as εs (i.e. s stands for softening) and includedbetween εy and εh; such value can be derived from the experimental results thatsuggest obtaining εs from the following expression:

εs

εy= 1 + c1 ∗ εh − εy

εye(−c2∗ L

D ), (4)

where the statistical analysis of test data indicates that c1 = 43.3 and c2 = 0.47.

The expression (4) provides εs � εy for LD = 20 and εs � εh for L

D = 8. Thesoftening branch can be described by the following expression:

σ = σ∞ +[(σy − σ∞) ∗ e−c3∗( ε

εs−1)

], (5)

where the asymptotic value of all curves, σ∞, and the parameter c3 should bederived from the experimental results. The statistical analysis of test data suggestassuming c3 = 0.2 and adopting the following expression for σ∞:

σ∞ = σyc4

L/D, (6)

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Fig. 14. D16 bars after the compression tests.

where c4 = 2.8 is obtained from laboratory outcomes. This is strongly differentfrom the values of 5 [Albanesi et al., 2001] or 6 [Monti and Nuti, 1992] definedwith reference to ribbed bars. The outcomes of the proposed model are depictedin Fig. 15 that shows the “theoretical” curves for L/D ranging between 8 and20; its effectiveness can be appreciated in Figs. 16 and 17 where two comparisonswith the experimental curves are shown for L/D equal to 10 and 15, respectively.Theoretical curves shown in these three figures have been calculated considering aratio of εh/εy that is equal to 16. It is pointed out that, for values of L/D rangingbetween 15 and 20, neglecting the horizontal plateau would not imply a significanterror from a technical stand point; within this range of L/D values, the maximumerror corresponds to L/D = 15 for which the plateau ends at about 1.5 εy.

Fourth case:

(L/D) > 20: The compressive bar exhibits an elastic buckling and therefore σcr <

σy. In particular, the experimental results underlined that, for L/D ranging between20 and 30, some bars were able to attain the yielding and then started bucklingbefore a plateau could be developed; for L/D larger than 30, all tested bars buckledbefore yielding. The value of σcr should be determined based on the experimentalresults.

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0 10 20 30 40 50 60 70 80 90 100

ε/εy

σ/σy

L/D=8

L/D=9

L/D=10

L/D=11L/D=12L/D=13

L/D=20

Fig. 15. Theoretical normalised compressive stress-strain curves in the range of L/D = 8 ÷ 20(third case).

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0 10 20 30 40 50 60 70 80 90 100

ε/εy

σ/σy

D14

D16 D12

D8

model 10=D

L

Fig. 16. Experimental-theoretical comparison for L/D = 10.

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σ

0.0

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0 10 20 30 40 50 60 70 80 90 100

ε/εy

σ/σy

model

D8

D16

D14D12

15=D

L

Fig. 17. Experimental-theoretical comparison for L/D = 15.

However, this case can be effectively studied by analyzing how the ratio σcr/σy

varies when the non-dimensional slenderness λ changes; such slenderness is theparameter that all modern codes like Eurocode 3 [1992] and Eurocode 4 [1992] useto define the buckling of compressive members. In particular λ can be expressed as:

λ =λ

λc=

√Npl

Ncr=

√σy

σcr=

4β

π

√σy

E

L

D= 0.025

√σy

320L

D, (7)

where σy has dimensions of N/mm2. According to Eurocode 3, the buckling of allsteel cross-sections is described by the five curves (i.e. imperfection factors a0, a, b,c, d) reported in Fig. 18 along with that related to elastic buckling. The expression ofthis curve can be obtained according to the linear elastic theory (i.e. ideal member)which, considering (7) gives:

σcr

σy=

1(λ)2

. (8)

The comparison shown in Fig. 18 between the theoretical buckling curves andthe experimental results highlights that the buckling behaviour of all tested barswith L/D ranging between 22 and 40 could be safely predicted by the curve a0

that Eurocode 3 suggests for rolled sections with thickness less than 40mm, andfor hot finished hollow sections. It is then suggested that for the assessment of RCmembers with 20 < L

D ≤ 40, the value of σcr is determined using the equation ofcurve a0 of Eurocode 3.

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0

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1

1.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 λ/λc

σ cr /σ

y

8 16 24 32 40 48 56 64 72 80 88 96 104 112

experimental D12

experimental D14

experimental D16

elastic buckling

a0

a

b

c

d

0

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1

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 λ/λc

σ cr /σ

y

8 16 24 32 40 48 56 64 72 80 88 96 104 112

elastic buckling

a0

a

b

c

d

experimental D8

experimental D12

experimental D8

experimental D14

experimental D16

σ y

320LD

Fig. 18. Buckling curves: Experimental-theoretical comparison.

Figure 18 shows that curve a0 could not be safe for the ratios 50, 60 and 70; theavailable data suggests that, for the assessment of RC members with L

D > 40, thevalue of σcr is determined using the equation of curve c recommended by Eurocode3 for cold formed hollow sections, U- and T-sections, and solid sections (includingcircular). The comparison of Fig. 18 confirms also that the elastic buckling curve,based on the assumption of ideal member, would overestimate the experimental σcr

in all cases.

6. Conclusions

The analysis of the compressive behaviour of smooth bars is basic for the assess-ment of existing RC structures. In fact, it is very likely that columns and/or externaland corner beam-column joints of these structures have stirrups with inadequatespacing. In many cases, such components are also subjected to high levels of com-pression because the existing frame is either deteriorated due to durability problemsor underdesigned.

As far as the authors know, the study presented in the paper is the first wherethis problem is tackled in a comprehensive way from both an experimental anda theoretical stand point. In particular, after the presentation of the experimen-tal campaign, a numerical model for the compressive stress-strain relationship ofsmooth bars, covering a wide range of L/D ratios, has been developed; such modelcould be very useful in order to perform the assessment of existing underdesignedRC structures by means of a push-over analysis, to study the interaction of the lon-gitudinal bars with concrete and stirrups and eventually design the strengtheningof columns in order to provide them with the needed confinement level.

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The study highlights that depending on the actual structural situation the com-pressive behaviour of the reinforcing smooth bars can strongly change. In particular,for small stirrups spacing (up to L/D = 7) the compressive behaviour is very duc-tile. As the spacing increases, the ductility is reduced; for L/D overcoming about20, the bar could buckle before yielding.

These outcomes are of particular relevance since L/D ratios in the order of 30or 40 can be found in many columns of existing buildings and the buckling of thelongitudinal bars could control their seismic behaviour and strongly contribute tothe failure of either columns or joints.

Acknowledgements

The authors would like to thank Mr. Pietro Paolo Coppola for his undergraduatethesis on the research topic presented in the paper and for his collaboration to thedevelopment of the related experimental activities.

References

Albanesi, T., Biondi, S. and Nuti, C. [2001] “Influenza dell’instabilita delle armaturelongitudinali sulla risposta d’elementi in c.a.,” CD-ROM Proc. of the 10th ItalianConference on Earthquake Engineering, Potenza, Italy.

ATC40 [1996] “Seismic evaluation and retrofit of concrete buildings,” California SeismicSafety Commission, Report SSC 96-01.

Eurocode 3 [1992] “Common unified rules for steel structures,” European Committee forStandardization (CEN), ENV 1994-1-1.

Eurocode 4 [1992] “Common unified rules for composite steel and concrete structures,”European Committee for Standardization (CEN), ENV 1994-1-1.

Eurocode 8-Part 3 [2004] “Assessment and retrofitting of buildings,” European Committeefor Standardization (CEN), ENV 1998-3, Draft No 7.

FEMA-310 [1998] “Handbook for the seismic evaluation of buildings — A prestandard,”prepared by the American Society of Civil Engineers for Federal Emergency Manage-ment Agency.

Dodd, L. L. and Restrepo-Posada, J. I. [1995] “Model for predicting cyclic behavior ofreinforcing steel,” Journal of Structural Engineering 121(3), 433–445.

Gomes, A. and Appleton, J. [1997] “Nonlinear cyclic stress-strain relationship of reinforc-ing bars including buckling,” Engineering Structures 19(10), 822–826.

Monti, G. and Nuti, C. [1992] “Nonlinear cyclic behavior of reinforcing bars includingbuckling,” ASCE Journal of Structural Engineering 118(12), 3268–3285.

Ordinanza n. 3274 [2003] “Primi elementi in materia di criteri generali per la classifi-cazione sismica del territorio nazionale e normative tecniche per le costruzioni inzona sismica,” Supplemento Ordinario 72 alla Gazzetta Ufficiale no. 105.

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