EXPERIMENTAL BEHAVIOUR AND NUMERICAL MODELLING OF SMOOTH STEEL BARS UNDER COMPRESSION

Transcription

1 Journal of Earthquake Engineering, Vol. 10, No. 3 (2006) c Imperial College Press EXPERIMENTAL BEHAVIOUR AND NUMERICAL MODELLING OF SMOOTH STEEL BARS UNDER COMPRESSION EDOARDO COSENZA and ANDREA PROTA Department of Structural Analysis and Design University of Naples Federico II Via Claudio 21, 80125, Naples, Italy Received 25 January 2005 Reviewed 25 July 2005 Accepted 1 September 2005 Many existing reinforced concrete buildings located in seismic regions are characterised by internal steel reinforcement made of smooth bars and stirrups with inadequate spacing. These bars could be subjected to significant compression and eventually buckle. This paper deals with a comprehensive experimental campaign investigating the compressive behaviour of smooth bars for different values of the ratio L/D, L being the restraints distance and D the bar diameter. The stress-strain relationship is then modelled ranging from an elastic-plastic behaviour identical to that in tension (L/D = 5)to the elastic buckling behaviour (L/D > 20). The comparison between the experimental results and the outcomes of the model confirms the accuracy of the proposed stress-strain relationship. Keywords: Buckling; compressive behaviour; smooth bars; stress-strain relationship. 1. Introduction The analysis of the behaviour of compressive bars is of particular relevance in order to achieve a reliable assessment of existing Reinforced Concrete (RC) buildings. This issue is important even for RC buildings subjected only to gravity loads because there are many under-designed structures where the high compression in the columns and insufficient or absent stirrups increase the likelihood that buckling of compressive steel bars could occur. The situation becomes worse for old buildings where the corrosion of steel bars causes the spalling of the concrete and then reduces both the gross cross section of the column and the restraint provided to the longitudinal bars by the concrete cover. This problem is certainly more important for buildings subjected to horizontal actions due to seismic events that determine a cyclic amplification of compressive stresses in the columns and could cause a sudden loss of the cover. Such condition is typical of unconfined beam-column joints (i.e. exterior or corner joints) where the construction practice does not place stirrups resulting in spacing values in the order of the beam height. Figure 1 depicts two examples of buckling of steel bars in unconfined joints after a severe earthquake. 313

2 314 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 when performing 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 existing RC 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 on smooth steel bars tested monotonically at the Department of Structural Analysis and Design of the University of Naples Federico II. The experimental results are discussed and the proposed analytical constitutive law is presented depending on the 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, σ y. It is pointed out that the experimental activity involves cyclic tests which are very important because, when buckling occurs, the cyclic response is dramatically altered with severe consequences on member performance. Due to space constraints the discussion of cyclic test results will be presented in a separate manuscript. Studies have been presented in literature about the cyclic behaviour of ribbed bars [Monti and Nuti, 1992; Dodd and Restrepo-Posada, 1995; Gomes and Appleton, 1997; Albanesi et al., 2001]. The definition of an accurate theoretical constitutive law that can be used within a wide range of L/D values for smooth bars is important in order to be able to model all different cases that could be found in real buildings, where the lack of seismic provisions at the time of their construction or the low quality control of the execution could have determined that the spacing of the stirrups is largely inadequate. In particular, the obtained results could be useful for many applications such as, for instance, the study of the interaction of the longitudinal bars with concrete and stirrups and the design of the strengthening of columns in order to increase their confinement. Therefore, the research presented was planned to cover a much broader range of L/D values than that dealt with by Monti and Nuti [1992] with respect to ribbed bars. That study aimed at assessing the L/D threshold that, if overcome,

3 Numerical Modelling of Smooth Steel Bars under Compression 315 could cause a significant decrease of load carrying capacity of the member due to compressive bars. Its outcomes have allowed defining practical design criteria regarding stirrup spacing, which nowadays has been included in all modern codes. 2. Experimental Program The experimental work has concerned monotonic tensile and compressive tests on smooth steel bars with diameters of 8 mm, 12 mm, 14 mm and 16 mm (i.e. denoted in 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 performed for each diameter with every fixed ratio L/D. The former has been used to determine the mechanical tensile properties that have been found to vary slightly depending on the diameter even though the samples had the same commercial nominal properties. The compressive tests have allowed assessing the repeatability of the results and have then provided a reliable experimental database to be used for the calibration of the analytical constitutive laws. The tests have been carried out in a displacement control mode with a head speed of the used MTS 810 machine equal to 5 mm/sec; the pressure of the hydraulic controlled grips was kept constant at 20 MPa. Before proceeding with the planned test matrix, preliminary tests have been conducted in order to optimise the set up for the compressive tests. In fact, the instrumentation generally used for tensile tests based on an extensometer could not be adopted for the compressive tests due to the following reasons: The values recorded by that device have no meaning as soon as the bar starts buckling (Fig. 2) and the values meaningful for compressive tests concern the average deformations over the entire length, L, of Fig. 2. Deformometer after bar buckling (wrong setup).

4 316 E. Cosenza & A. Prota the bar rather than local deformations measured over the undisturbed length (i.e. typically 40 or 50 mm) 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 range of L values (i.e. between 65 mm and 600 mm) did not allow installing the LVDTs directly on the grips and required the adoption of steel plates that were bonded on the machine at one end and supported each LVDT at the other. In order to optimise the set up, some calibration tests have been carried out to assess whether any slip occurred 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 the machine 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 become basically equal as soon as the bar overcomes the elastic range and that the deformations provided by the LVDTs placed on the grips are in general more reliable than those obtained from the LVDTs located on the heads. In addition, it is appropriate to take the readings of the deformometer within the elastic range in order not to account for the slips between the bar and the grips that, even small, could have a percent influence in that range (i.e. small deformations). Finally, other preliminary tests have been conducted in order to assess the influence 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 on the grips have been selected to do this (Fig. 4). These tests have confirmed that the flexural deformation of the steel plates is basically negligible and that the readings provided by LVDTs placed at their ends are reliable. Fig. 3. Instrumentation used during the calibration tests.

5 Numerical Modelling of Smooth Steel Bars under Compression 317 Fig. 4. Transducers placed directly on the grips and on the heads. In conclusion, based on the described calibration the data analysis for all the tests discussed beneath was performed as follows: For the tensile tests only the deformometer readings were used; the stress-strain curves of compressive bars were obtained based on the deformometer readings up to the yielding and based on the average deformation given by the two LVDTs connected to the grips through the steel plates from the yielding to the rupture. 3. Experimental Results The tests have confirmed that the tensile behaviour of the bars slightly changes as L/D varies (Figs. 5 8); it is underlined that all constitutive relationships represented in this paper refer to engineering stresses [Dodd and Retrepo-Posada, 1995]. The mean mechanical properties obtained by testing bars with L/D ranging between 5 and 15 are summarised for each diameter in Table 1 where the following notation is used: ε y, mean strain at yielding and f y corresponding mean stress; ε sh, strain at hardening and f sh corresponding mean stress; ε t,meanstrainatmaximum load and f t corresponding mean stress. The values reported in the table point out that D8, D12 and D16 bars have similar properties, whereas D14 bars showed higher strength values; however, both classes are representative of typical reinforcing steel bars used during the 1960s to build RC structures. For all the bars, the

6 318 E. Cosenza & A. Prota L/D = 5 15 σ (MPa) ε (mm/mm) Fig. 5. Experimental tensile stress-strain curves for D8 bars L/D = 5 15 σ (MPa) ε (mm/mm) Fig. 6. Experimental tensile stress-strain curves for D12 bars.

7 Numerical Modelling of Smooth Steel Bars under Compression L/D = σ (MPa) ε (mm/mm) Fig. 7. Experimental tensile stress-strain curves for D14 bars L/D = 5 15 σ (MPa) ε (mm/mm) Fig. 8. Experimental tensile stress-strain curves for D16 bars.

8 320 E. Cosenza & A. Prota Table 1. Experimental mean tensile properties of tested bars. ε y f y ε sh f sh ε t f t (mm/mm) (N/mm 2 ) (mm/mm) (N/mm 2 ) (mm/mm) (N/mm 2 ) D D D D mean Young modulus was about N/mm 2 ; it was also noticed that, after yielding, the surface of the bars exhibited a surface flaking that increased up to rupture; an example is depicted in Fig. 4. The experiments have highlighted that the stress-strain relationship of compressive 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 curves of bars with different diameters characterised by L/D = 10. It is underlined that values on the x-axis on this and following figures are represented up to 100 in order to allow observing the asymptotic trend of the curves; the portion of the x-axis useful for assessment purposes is certainly limited to lower values. Figure 10 allows for the observation of how the effects of the buckling in the plastic range becomes more significant as L/D increases from 5 to 20; the instance given in the figure refers to D12 bars and does not include L/D ratios beyond 20 since that was found to be the threshold after which elastic buckling occurred regardless of the bar diameter D16 D8 σ/σy D14 D ε/εy Fig. 9. Normalised compressive stress-strain curves for different diameters with L/D = 10.

9 Numerical Modelling of Smooth Steel Bars under Compression 321 σ/σy 1.5 tension ε/εy 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 and 13 for D8, D14 and D16 bars, respectively. It is recalled that Figs show the mean stress-strain curves obtained on three tests for each L/D. Figure 14 shows the D16 samples after the compressive tests. 4. Analysis of Test Results The analysis of the experimental outcomes allows the identification of four threshold values 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 the bar in compression is so large that its compressive behaviour is very similar to that 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 still exhibits a certain level of hardening before buckling; based on the experimental results, 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 starts buckling close to the yielding and does not exhibit any hardening; the obtained results suggest to assume (L/D) y = 20;

10 322 E. Cosenza & A. Prota σ/σy tension ε/εy Fig. 11. Normalised compressive stress-strain curves for D8 bars. σ/σy 1.5 tension ε/εy Fig. 12. Normalised compressive stress-strain curves for D14 bars.

11 Numerical Modelling of Smooth Steel Bars under Compression 323 σ/σy 1.5 tension ε/εy Fig. 13. Normalised compressive stress-strain curves for D16 bars. (iv) (L/D) e (i.e. e stands for elastic) as the value beyond which elastic buckling occurs. Such threshold could be theoretically found recalling that: Therefore, if the following notation is used: N cr = π 2 EI L 2. (1) o N cr = σ cr πd 2 /4; I = πd 4 /64; L o = β L, (2) and assuming that E = N/mm 2 and taking the critical stress, σ cr,equal to the yielding stress, σ y, the following expression for (L/D) e can be derived: ( ) L = =40 (3) D e β σ y σ y 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 longitudinal geometrical imperfections, and variability of the mechanical properties of the material, as it will be analysed in the following sections.

12 324 E. Cosenza & A. Prota 5. Modelling of the Constitutive Relationship of Compressive Bars Considering the experimental results and the above defined thresholds of L/D, the following cases can be defined towards the modelling of the stress-strain relationship of compressive bars: First case: (L/D) 5: Since the ductility of the compressive bar is very large, its behaviour in compression can be assumed equal to that in tension. Second case: (L/D) =6 7: The compressive behaviour can be safely assumed to be elasticplastic ending at a conventional strain value ε u to be experimentally determined depending on L/D. In particular, such value could be obtained from Figs by intersecting the relevant experimental curve (on the descending branch) with a horizontal line overlapped to its plateau. If the experimental curves are not available, the experimental results suggest that it can be safely recommended to assume ε u /ε y equal to at least 65 and 30 (i.e. ε u 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 of the plastic branch since the ductility provided by such simplified method is already very large and the remaining portion of the stress-strain relationship could be not too 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 nonlinear softening. The transition from the plateau to the beginning of the softening occurs at a strain value herein defined as ε s (i.e. s stands for softening) and included between ε y and ε h ; such value can be derived from the experimental results that suggest obtaining ε s from the following expression: ε s =1+c 1 ε h ε y e ( c2 L D ), (4) ε y ε y where the statistical analysis of test data indicates that c 1 =43.3 andc 2 =0.47. The expression (4) provides ε s ε y for L D =20andε s ε h for L D =8.The softening branch can be described by the following expression: σ = σ + [(σ y σ ) e c3 ( ε 1)] εs, (5) where the asymptotic value of all curves, σ, and the parameter c 3 should be derived from the experimental results. The statistical analysis of test data suggest assuming c 3 =0.2 and adopting the following expression for σ : σ = σ y c 4 L/D, (6)

13 Numerical Modelling of Smooth Steel Bars under Compression 325 Fig. 14. D16 bars after the compression tests. where c 4 =2.8 is obtained from laboratory outcomes. This is strongly different from the values of 5 [Albanesi et al., 2001] or 6 [Monti and Nuti, 1992] defined with reference to ribbed bars. The outcomes of the proposed model are depicted in Fig. 15 that shows the theoretical curves for L/D ranging between 8 and 20; its effectiveness can be appreciated in Figs. 16 and 17 where two comparisons with 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 a ratio of ε h /ε y that is equal to 16. It is pointed out that, for values of L/D ranging between 15 and 20, neglecting the horizontal plateau would not imply a significant error from a technical stand point; within this range of L/D values, the maximum error corresponds to L/D = 15 for which the plateau ends at about 1.5 ε y. Fourth case: (L/D) > 20: The compressive bar exhibits an elastic buckling and therefore σ cr < σ y. In particular, the experimental results underlined that, for L/D ranging between 20 and 30, some bars were able to attain the yielding and then started buckling before a plateau could be developed; for L/D larger than 30, all tested bars buckled before yielding. The value of σ cr should be determined based on the experimental results.

14 326 E. Cosenza & A. Prota 1,1 1,0 0,9 0,8 σ/σy 0,7 0,6 0,5 0,4 L/D=8 L/D=9 L/D=10 0,3 L/D=13 L/D=12 L/D=11 0,2 0,1 L/D=20 0, ε/εy Fig. 15. Theoretical normalised compressive stress-strain curves in the range of L/D = 8 20 (third case) D16 D12 L D model = 10 σ/σy D8 D ε/εy Fig. 16. Experimental-theoretical comparison for L/D = 10.

15 Numerical Modelling of Smooth Steel Bars under Compression 327 σ L D = D8 σ/σy D model D12 D ε/εy Fig. 17. Experimental-theoretical comparison for L/D = 15. However, this case can be effectively studied by analyzing how the ratio σ cr /σ y varies when the non-dimensional slenderness λ changes; such slenderness is the parameter that all modern codes like Eurocode 3 [1992] and Eurocode 4 [1992] use to define the buckling of compressive members. In particular λ can be expressed as: λ = λ Npl σy = = = 4β σy L λ c N cr σ cr π E D =25 σy L 320 D, (7) where σ y has dimensions of N/mm 2. According to Eurocode 3, the buckling of all steel cross-sections is described by the five curves (i.e. imperfection factors a 0, a, b, c, d) reported in Fig. 18 along with that related to elastic buckling. The expression of this curve can be obtained according to the linear elastic theory (i.e. ideal member) which, considering (7) gives: σ cr = 1 σ y ( λ). (8) 2 The comparison shown in Fig. 18 between the theoretical buckling curves and the experimental results highlights that the buckling behaviour of all tested bars with L/D ranging between 22 and 40 could be safely predicted by the curve a 0 that Eurocode 3 suggests for rolled sections with thickness less than 40 mm, and for hot finished hollow sections. It is then suggested that for the assessment of RC members with 20 < L D 40, the value of σ cr is determined using the equation of curve a 0 of Eurocode 3.

16 328 E. Cosenza & A. Prota a0 a experimental D8 experimental D12 experimental D14 experimental D16 σcr/σy b elastic buckling c d σ y L 320 D λ/λc Fig. 18. Buckling curves: Experimental-theoretical comparison. Figure 18 shows that curve a 0 could not be safe for the ratios 50, 60 and 70; the available data suggests that, for the assessment of RC members with L D > 40, the value of σ cr is determined using the equation of curve c recommended by Eurocode 3 for cold formed hollow sections, U- and T-sections, and solid sections (including circular). The comparison of Fig. 18 confirms also that the elastic buckling curve, based on the assumption of ideal member, would overestimate the experimental σ cr in all cases. 6. Conclusions The analysis of the compressive behaviour of smooth bars is basic for the assessment of existing RC structures. In fact, it is very likely that columns and/or external and corner beam-column joints of these structures have stirrups with inadequate spacing. In many cases, such components are also subjected to high levels of compression because the existing frame is either deteriorated due to durability problems or underdesigned. As far as the authors know, the study presented in the paper is the first where this problem is tackled in a comprehensive way from both an experimental and a theoretical stand point. In particular, after the presentation of the experimental campaign, a numerical model for the compressive stress-strain relationship of smooth bars, covering a wide range of L/D ratios, has been developed; such model could be very useful in order to perform the assessment of existing underdesigned RC structures by means of a push-over analysis, to study the interaction of the longitudinal bars with concrete and stirrups and eventually design the strengthening of columns in order to provide them with the needed confinement level.

17 Numerical Modelling of Smooth Steel Bars under Compression 329 The study highlights that depending on the actual structural situation the compressive 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 ductile. As the spacing increases, the ductility is reduced; for L/D overcoming about 20, the bar could buckle before yielding. These outcomes are of particular relevance since L/D ratios in the order of 30 or 40 can be found in many columns of existing buildings and the buckling of the longitudinal bars could control their seismic behaviour and strongly contribute to the failure of either columns or joints. Acknowledgements The authors would like to thank Mr. Pietro Paolo Coppola for his undergraduate thesis on the research topic presented in the paper and for his collaboration to the development of the related experimental activities. References Albanesi, T., Biondi, S. and Nuti, C. [2001] Influenza dell instabilità delle armature longitudinali sulla risposta d elementi in c.a., CD-ROM Proc. of the 10th Italian Conference on Earthquake Engineering, Potenza,Italy. ATC40 [1996] Seismic evaluation and retrofit of concrete buildings, California Seismic Safety Commission, Report SSC Eurocode 3 [1992] Common unified rules for steel structures, European Committee for Standardization (CEN), ENV Eurocode 4 [1992] Common unified rules for composite steel and concrete structures, European Committee for Standardization (CEN), ENV Eurocode 8-Part 3 [2004] Assessment and retrofitting of buildings, European Committee for Standardization (CEN), ENV , 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 Management Agency. Dodd, L. L. and Restrepo-Posada, J. I. [1995] Model for predicting cyclic behavior of reinforcing steel, Journal of Structural Engineering 121(3), Gomes, A. and Appleton, J. [1997] Nonlinear cyclic stress-strain relationship of reinforcing bars including buckling, Engineering Structures 19(10), Monti, G. and Nuti, C. [1992] Nonlinear cyclic behavior of reinforcing bars including buckling, ASCE Journal of Structural Engineering 118(12), Ordinanza n [2003] Primi elementi in materia di criteri generali per la classificazione sismica del territorio nazionale e normative tecniche per le costruzioni in zona sismica, Supplemento Ordinario 72 alla Gazzetta Ufficiale no. 105.