The elastoplastic behaviour of steel used for structural member fabrication has received attention to facilitate a mechanical-resistant design. New Zealand and South African standards have adopted various theoretical approaches to describe such behaviour in stainless steels. With respect to the building industry, describing the tensile behaviour of steel rebar used to produce reinforced concrete structures is of interest. Differences compared with the homogenous material described in the above mentioned standards and related literatures are discussed in this paper. Specifically, the presence of ribs and the TEMPCORE^{®} technology used to produce carbon steel rebar may alter the elastoplastic model. Carbon steel rebar is shown to fit a Hollomon model giving hardening exponent values on the order of 0.17. Austenitic stainless steel rebar behaviour is better described using a modified Rasmussen model with a free fitted exponent of 6. Duplex stainless steel shows a poor fit to any previous model.
When designing a building or a structure, it is increasingly important to have analytical descriptions of the elastoplastic material behaviour. Exact knowledge is needed to supply numerical methods to produce precise results or to find analytical solutions to simple load cases. Hollomon (
The durability and ductility of structures are also of paramount importance in modern design and can be described using the approaches mentioned above. An interesting case that has received little attention is rebar steels. Beginning in the nineties, international standards introduced weldable steels with improved ductility. EN 1992-1-1 (
Currently, high ductility can also be obtained using stainless steel rebar that is hot or cold rolled. Due to its patterned shape and specific production routes, stainless steel rebar structures are far from the homogeneous materials that are used to validate the previously mentioned formulations. Nevertheless, it is well known that the ductility of reinforced concrete structures is of paramount importance when designing in areas of high seismicity. The description of plastic behaviour has been accomplished using experimental formulations (Doñate Megías
Five different grades of steel were tested: three carbon grades and two stainless grades, of which one was austenitic and the other was duplex. The carbon grades were produced using the TEMPCORE^{®} process, but one of the grades was rolled to a plain circular section, whereas the other two grades exhibited the conventional ribs of reinforcing rebar. The stainless steel ribs were produced using cold rolling as the final shaping process. All of the ribs were selected to have a 14 mm nominal diameter (
Mechanical properties measured (mean values)
Material | ε_{max} | ε_{u,5} | ||||
---|---|---|---|---|---|---|
TEMPCORE^{®} 1 (round) | 521.22±8.23 | 627.45±2,57 | 1.21±0.02 | 0.106±0.003 | 0.261±0.008 | 195 |
TEMPCORE^{®} 2 (rebar) | 522.32±9.99 | 647.37±1,27 | 1.24±0.02 | 0.154±0.009 | 0.259±0.017 | 200 |
TEMPCORE^{®} 3 (rebar) | 545.55±4.05 | 678.22±2.68 | 1.24±0.01 | 0.123±0,004 | 0.174±0.011 | 187 |
AISI 304 | 752.35±11.82 | 878.52±6.00 | 1.17±0.01 | 0.175±0.008 | 0.314±0.010 | 197 |
Duplex 2205 | 983.54±19.35 | 1103.45±6.88 | 1.12±0.02 | 0.032±0.021 | 0.171±0.027 | 195 |
Reinforcing bars used in this study. (Left) Carbon round, carbon rebar and stainless steel rebar; (Right) Tempcore^{®} structure of the carbon steel.
Ten samples prepared from each of the grades were tested. The free length between the clamping heads was fixed at approximately 110 mm. The tensile tests were conducted according to European Standards UNE-EN-ISO 6892-1 (
Stress-strain data obtained by testing the five materials are represented in
Experimental results plotted as σ vs. ε. Both the engineering and the real results are shown. From top to bottom: (a) TEMPCORE^{®} 1 (round), (b) TEMPCORE^{®} 2 (rebar), (c) TEMPCORE^{®} 3 (rebar), (d) AISI 304 and (e) Duplex 2205.
Empirical models of elastoplastic behaviour of metals are typically defined by Ramberg and Osgood (
Hollomon (
The elastoplastic behaviour model proposed by Ramberg and Osgood (
Mirambell and Real (
Additionally, the plastic strain is referred to as the stress in excess of the yield strength (ε^{p} = K” (σ-f_{y})^{m}). The value of m could be computed from a simple equation: m = 1 + 3.5 (f_{y}/f_{s}). To apply the Rasmussen model to carbon steel rebar produced by hot rolling and the TEMPCORE^{®} process, some adjustments must be made. Again, the initial strain hardening point (ε^{p}_{0}, σ^{p}_{0}) was considered to correspond to the experimental point where the deformation was 0.4 mm in excess of the last minimum of the yield stress. Rewriting the equation results in the following (
which is valid for the interval (ε^{p}_{0}, σ^{p}_{0}) → (σ_{max}, f_{s}), where ε_{up} = ε_{max} - ε_{0} - f_{y}/E_{0}, and m could be computed from m = 1 + 3.5 σ_{0}/f_{s}.
Log-log graphs and the best fit to the Hollomon equation. From the top to the bottom: (a) TEMPCORE^{®} 1 (round), (b) TEMPCORE^{®} 2 (rebar), (c) TEMPCORE^{®} 3 (rebar), (d) AISI 304 and (e) Duplex 2205.
Hollomon strain hardening exponent values
Material | n | R^{2} |
ε_{max,t} |
---|---|---|---|
TEMPCORE^{®} 1 (round) | 0.176±0.003 | 0.988 | 0.101±0.026 |
TEMPCORE^{®} 2 (rebar) | 0.179±0.003 | 0.995 | 0.143±0.074 |
TEMPCORE^{®} 3 (rebar) | 0.164±0.002 | 0.995 | 0.118±0.039 |
AISI 304 | 0.087±0.003 | 0.945 | 0.161±0.007 |
Duplex 2205 | 0.109±0.008 | 0.885 | 0.032±0.002 |
When fitting all 10 test data samples to a single Hollomon equation. Corresponding n values shown in
Comparing the results of n to ε_{max,t}, it is evident that there is not a clear correlation even though, in all cases, ε_{max,t}<n except for AISI 304. The criteria given by Considerè (1885), n=ε_{max,t}, may be considered a condition derived from the fact that the Hollomon curve should include the coordinate (ε_{max,t}, f_{s,t}) even though this results in a poorer fit. Coming back to the experimental data curves in
Comparing the n values to those found in the literature is not an easy task because, as shown by Bergström (
Hollomon strain hardening exponent from the literature (Dieter,
Very scarce data on the applicability of the Hollomon model to stainless steels are available at the literature. Castro
Experimental data have also been fitted to the modified Rasmussen model. In this case, statistical regression to fit a single Rasmussen curve to the data cloud collected from all of the tests for each specimen results in a very low statistical significance. This is due to differences in the values of (ε^{p}_{0}, σ^{p}_{0}) used for each sample. With respect to the Rasmussen model, such a data pair is not only the initial data point to be fitted, as in the Hollomon model, but also modifies the parameters in the
ε-σ graphs: Rasmussen model and best fit to the modified Rasmussen equation (free m value). From the top to the bottom: (a) TEMPCORE^{®} 1 (round), (b) TEMPCORE^{®} 2 (rebar), (c) TEMPCORE^{®} 3 (rebar), (d) AISI 304 and (e) Duplex 2205.
The Rasmussen model does not describe the experimental strain hardening as well as the Hollomon model does.
Rasmussen m values (according to the model and freely computed from the best fits)
Material | m_{free} | R^{2} | m_{Rasm} | R^{2} |
---|---|---|---|---|
TEMPCORE^{®} 1 (round) | 2.86±0.18 | 0.90±0.02 | 4.06±0.01 | 0.85±0.01 |
TEMPCORE^{®} 2 (rebar) | 3.14±0.47 | 0.91±0.04 | 4.04±0.02 | 0.86±0.06 |
TEMPCORE^{®} 3 (rebar) | 2.84±0.19 | 0.92±0.03 | 4.10±0.02 | 0.86±0.02 |
AISI 304 | 6.22±0.92 | 0.99±0.04 | 4.00±0.04 | 0.81±0.09 |
Duplex 2205 | 1.65±0.14 | 0.91±0.02 | 4.11±0.05 | 0.71±0.06 |
In every case, the m value computed as proposed by the Rasmussen model results in a less accurate description of the elastoplastic behaviour. Fitting is improved when the value of m is computed from the statistical methods. Generally, the values computed from the statistical methods are less than the values obtained from the Rasmussen equations. However, in the case of AISI 304 steel, where the statistically computed value was greater, a very good fit with the experimental data is obtained. This result is in agreement with the original claim from Rasmussen (
Comparing the values obtained in the work to the data reported in the literature is not straightforward. Mirambell and Real (
In this paper, the effect of the anisotropic structures of rebar steels along with the patterned shape on their elastoplastic hardening behaviour was investigated. Both carbon and stainless steels were tested. Experimental elastoplastic behaviour has been traditionally fitted to several well-known formulations that are typically potential relations of stress and strain. Some of these relationships have been considered by various standards to aid in structural design.
The Hollomon model was found to precisely describe the carbon steel rebar strain hardening behaviour once extensive yielding was complete. The values found (n = 0.16-0.18) compare well to, although slightly less than, the values reported in the literature for hot rolled steels with comparable carbon content. No clear effect can be drawn resulting from the presence of ribs.
The Rasmussen model does not describe the experimental strain hardening as well as the Hollomon model. Fitting is improved when the value of m is computed using statistical methods.
Generally, the values computed using statistical methods are less than the values obtained using the Rasmussen equations. However, in the case of AISI 304 steel, where the statistically computed value was greater (m=6), a very good fit to the experimental data was obtained. A satisfactory fit for the behaviour of duplex rebar steel was not obtained using any of the tested models.
The authors would like to acknowledge the contribution to the experimental work carried out by J. Pinto from Superior Technical School of Engineering and R. Sánchez-Matos from the Superior Polytechnic School both at the University of Seville.
E: Elastic modulus
σ: Engineering normal stress
ε: Engineering strain
L_{0}: Gage length, which is 5 times greater than the nominal diameter
f_{y}: Engineering yield strength computed to a 0.002 permanent elongation
ε_{y}: Engineering strain at f_{y}
f_{y,t:} True yield strength
f_{s}: Rupture strength computed as the maximum engineering stress
ε_{max}: Engineering strain at f_{s}
ε_{u,5}: Engineering strain after break measured for the gage length
σ_{t}: True normal stress
ε_{t}: True strain
f_{s,t}: Rupture strength computed as the maximum true stress
ε_{max,t}: True strain at f_{s,t} measured for the gage length
σ^{p}: Engineering plastic normal stress
ε^{p}: Engineering plastic strain
ε^{e}: Engineering elastic strain
σ^{p} _{t}: True plastic normal stress
ε^{p} _{t}: True plastic strain