High strain rate response of S355 at high temperatures

Materials and Design 94 (2016) 467–478

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High strain rate response of S355 at high temperatures Daniele Forni a,b,⁎, Bernardino Chiaia b, Ezio Cadoni a a b

DynaMat Lab, University of Applied Sciences of Southern Switzerland, 6952 Canobbio, Switzerland Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Torino, Italy

a r t i c l e

i n f o

Article history: Received 5 November 2015 Received in revised form 26 December 2015 Accepted 29 December 2015 Available online 7 January 2016 Keywords: Fire induced progressive collapse Robustness High strain rates at high temperatures Extreme combined effect SHTB S355 structural steel Material constitutive law Dynamic strain ageing

a b s t r a c t In this paper the high strain rate behaviour in tension and in a wide range of elevated temperatures of the S355 structural steel is presented. A Split Hopkinson Tensile Bar for the mechanical characterisation at high strain rates, equipped with a water-cooled induction heating system is used. These data are collected with the purpose of evaluating the extreme combined effect of dynamic loadings and elevated temperatures (200 °C, 400 °C, 550 °C, 700 °C and 900 °C), e.g., a fire load followed by an explosion. The reduction factors for the main mechanical properties are reported. The novelty of our data is the addition of the strain rate dependency to the temperature. High strain rate tests at 550 °C highlighted the phenomenon known as blue brittleness where an increase of strength and a decrease of ductility were ascribed to the dynamic strain ageing. Focusing the attention on the thermal softening parameter, m, the widely used constitutive law proposed by Johnson and Cook during the eighties is critically reviewed highlighting some weaknesses. The results can be of great interest for the assessment of robustness in structures where a fire induced progressive collapse should be evaluated focusing the attention to the extreme combined effects. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The usual definition of robustness [1], as the ability of a structure to withstand events like fire, explosions, impact or the consequences of human error, without being damaged to an extent disproportionate to the original cause, takes undoubtedly into account the fire load. This is because an extended exposure to elevated temperatures may seriously influence the performance of the steel framed structures, leading to possible fire induced progressive collapses. Furthermore, a progressive collapse could also be triggered by other different extreme accidental actions, such as for example impact loadings, earthquake forces and blast loads. During the last decades, a significant amount of research has been carried out to assess the fire resistance and to predict the blast response of steel structures. In spite of that, in many of these studies only uncorrelated effects of high temperatures and dynamic loadings have been considered [2]. Only after the 9/11 World Trade Center tragedy [3,4] the international community raised significant questions on fire safety and on disproportionate collapse as a result of local failures due to impacts or blasts. As a consequence the mechanical response of steel structures subjected to both high temperatures and impact loadings cannot be ignored. Nevertheless, many aspects regarding the

⁎ Corresponding author at: DynaMat Lab, University of Applied Sciences of Southern Switzerland, 6952 Canobbio, Switzerland. E-mail address: [email protected] (D. Forni).

http://dx.doi.org/10.1016/j.matdes.2015.12.160 0264-1275/© 2016 Elsevier Ltd. All rights reserved.

mechanical behaviour of steel structures subjected to extreme combined effects has still criticisms open to investigation. In order to assess the robustness in case of fire loading, Liew et al. [5] proposed different methodologies of advanced analysis techniques for studying the large displacement inelastic behaviour of building frames subjected to localised fire. Fang et al. [6] presented the key issues that should be addressed in the robustness assessment of steel-composite structures subjected to localised fire and proposed robustness assessment approaches that offer a practical framework for the consideration of such issues. The same authors [7] presented also a simplified energybased robustness assessment approach, based on a Temperature Independent Approach (TIA) where the core idea was a sudden column loss (typically employed for blast loading) and in which the maximum temperature is assumed to be unknown. Sun et al. [8] developed a robust static-dynamic procedure to model the dynamic and static behaviour of steel buildings during both local and global progressive collapse under fire conditions. Kucz et al. [9] demonstrated that the boundary conditions play an important role in describing the real behaviour of elements under fire conditions. Porcari et al. [10] in their extensive literature review of the mechanisms involved in fire induced progressive collapse of steel building structures, concluded that the robustness is influenced also by the durability of fireproofing when subjected to blast or other sources of stress. Other authors have tried to study the combined effect of fire and blast. Izzuddin et al. [11,12] made one of the first attempt to perform an integrated fire and blast analysis. Liew [2] proposed a numerical model for analysing steel frame structures subjected to localised

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damage caused by blast load and subsequently investigating their survivability under fire attack. They considered the strain rate effect due to blast load by multiplying the static strength by dynamic increase factors [13], while for the temperature effect they considered the strength and stiffness reduction curves of steel at elevated temperatures [14]. Song et al. [15] and Izzuddin et al. [12] considering both the strain rate and the temperature effects proposed and verified a method for integrated analysis of steel frames subjected to explosion followed by a fire loading. Even if [2,15,12] studied the influences of blast loading on the fire resistance of columns and steel frames, some aspects are still open to criticism. For example Song et al. [15] considered the effect of dynamic loads followed by fire loading and not acting at the same time. But, in case of a fire induced progressive collapse or where a dynamic event is triggered by high temperatures, the coupled effect of fire and dynamic loadings should be considered. In addition, the detailed knowledge of the high strain rate and high temperature properties of structural steel can be of great interest, e.g., in forensic investigations where the appearance of damage can witness the dynamics of the explosions and fire. As an example, on December 28, 2014, a strong fire occurred in two garage decks of the Italian motorship Norman Atlantic during navigation in the Adriatic sea. Relatively high temperatures were attained especially in the open deck, and local explosions of inflammable truck content caused high strain rates upon the structural steel grids. The structure was severely damaged, however, an overall robust behaviour was observed. Only the material properties at high strain rates and in a wide range of temperatures can explain the resilient response under the coupled effect of local dynamic events triggered by high temperatures. The so-called “blue brittleness” effect can be clearly evidenced, as it will be explained in the following sections of this paper. In this paper a typical structural steel, namely S355, has been deeply investigated in a wide range of strain rates and temperatures up to 900 °C by means of a Split Hopkinson Tensile Bar (SHTB) installed at the DynaMat Laboratory of the University of Applied Sciences of Southern Switzerland. The main mechanical properties as well as different strain energy densities have been evaluated. A critic review of the Johnson-Cook constitutive law [16] has been reported focusing on the thermal softening parameter, m. The novelty is the evaluation of this parameter at high strain rates and at different temperatures. In future developments, these data could contribute to the improvement of numerical simulations [17], leading to a more robust analysis of progressive collapse of steel structures subjected to extreme combined effects. The paper is organised as follow: (i) state of the art of structural steel properties at high strain rates and at elevated temperatures, (ii) description of the material characteristics and sample preparation, (iii) description of the experimental techniques at high strain rates and elevated temperatures, (iv) results and discussion, (v) review of the Johnson–Cook constitutive law and (vi) conclusions. 2. Outlook of the structural steel properties at high strain rates and at elevated temperatures For the evaluation of the material properties at elevated temperatures two methods are available [18], namely steady-state and transient-state. In the steady-state test, that is an isothermal process, the specimens is brought into thermal equilibrium at a predefined temperature and then loaded until it fails keeping the same temperature. On the other hand, in the transient-state test, that is an anisothermal process, the specimens are under constant load while the temperature is increased at a given heating rate. Outinen et al. [19] tested in quasi-static conditions the structural steel S355 at elevated temperatures by using the transient-tensile test method and reported three equations in function of the temperature for the elastic modulus, the proportional limit and the yield stress. The same authors [20] tested in the same conditions three additional

structural steels by using different heating rates. They found that the highest heating rate gives the higher strength properties. In this technical report [18] is given guidance on available, but uncorrelated, elevated temperature and high strain rate material property data for high strength steels used specifically for offshore structures. Chen et al. [21] tested at high temperatures and in quasi-static steady and transient state modes two different materials, the BISPLATE80, similar to the S690 structural steel and the XERPLATE Grade 350. They reported the thermal reduction factors for the main mechanical properties and compared them with some current standards [22,14]. Heidarpour et al. [23] studied in steady-state conditions the quasi-static tensile behaviour of a very high strength (VHS) steel at elevated temperatures. Qiang et al. [24, 25] performed quasi static tests on specimens from three structural steels (S960, S690 and S460) after cooling down from temperatures up to 1000 °C, finding that the post-fire performances were correlated with the steel grade. The same authors [26], using both steady and transient state methods at temperatures ranging from 20 °C to 700 °C, carried out experimental quasi-static characterisation of the S690 structural steel. They obtained the reduction factors at different temperatures for the elastic modulus as well as for the yield strength at various strain levels and the ultimate tensile strength, finding a good agreement with the data proposed by AISC [27] and Eurocode 3 [14]. Only in few studies the influence of strain rate and temperature were considered as a combined effect. Krabiell and Dahl [28] set the temperatures between 77 K and 295 K while the strain rates between 10−4 and 100 s−1. Knobloch et al. [29] tested in steady state conditions the influence of the strain rate (1.67 · 10−5–1 s−1) on the material properties of the S355 structural steel. This is another confirmation that there is a lack of data on the mechanical properties of structural steels with the coupled effect of elevated temperatures and high strain rates. 3. Material: S355 The material used in this research is the S355 [30] low-alloy structural carbon steel, widely used in the construction field due to its good strength characteristics as well as good welding properties. At room temperature and in quasi-static conditions, the nominal tensile properties of this steel are: elastic modulus of 205 GPa, yield strength of 355 MPa and ultimate strength of 510 MPa [31]. In quasi-static conditions and at temperatures up to 1200 °C, the reduction factors for the stress–strain relationship of carbon steels are provided in [14]. On the other hand, also other authors [18,26,21] have evaluated these reduction factors in their researches. In particular, Knobloch et al. [29] evaluated the behaviour of the same structural steel used in this research by studying the influence of temperature and strain rate in quasi-static tensile conditions and at medium strain rates in compression. Finally, in a previous study by the current authors [32], where the strain rate behaviour in tension has been studied, the strain rate sensitivity and the strain hardening capacity were demonstrated. The specimens for the mechanical characterisation were obtained in the longitudinal direction from an hot-rolled wide-flange section HE A [29,32]. Starting from the initial section geometry and with the purpose of avoiding any influence of the machining operations, wire electrical discharge machining (WEDM) was used in order to obtain small prismatic samples with 7 × 7 mm cross section. These small samples were then turned in order to obtain the usually adopted geometry for dynamic testing with the Split Hopkinson Tensile Bar, consisting of round samples with 3 mm in diameter and 5 mm of gauge length. A full description of the geometry is reported in [33], while the uniformity of the material properties within the section has been demonstrated in a previous research [32]. 4. Mechanical testing at high strain-rates and elevated temperatures It is broadly known that the most accurate testing method for the measurements and for assuring results of high precision of the dynamic

D. Forni et al. / Materials and Design 94 (2016) 467–478

mechanical properties of materials, is the Split Hopkinson Bar technique. By means of this technique it is possible to generate a loading pulse well controlled in rise time, amplitude and duration, giving rise to the propagation of an uniaxial elastic plane stress wave [34]. This method is based on the elastic stress wave propagation theory developed by [35,36,37]. The Split Hopkinson Tensile Bar (SHTB) used in this research is a tensile version of the modified Hopkinson Bar developed during the seventies by Albertini et al. [38,39]. The SHTB consists of two bars made of high strength steel called the input and output bars, 9 m and 6 m long and 10 mm in diameter. The specimen is mounted with threads between the bars. The first 6 m of the input bar are used as a pretensioned bar. The length of the pretensioned bar has been chosen in order to generate a tensile pulse of 2.4 ms allowing the specimen deformation until fracture at a constant strain rate, while the output bar length has been chosen in order to allow the specimen deformation under the clean and controlled loading resulting only from the incident tension pulse and avoiding the superposition of the wave reflections. A detailed description of the functioning of the SHTB has been given in [34], while the scheme has been depicted in Fig. 1. Two basic assumptions should be fulfilled in order to obtain an accurate measurement of the mechanical properties of a material subjected to a dynamic loading. Firstly, the specimen should be short enough so that the time taken by the wave to propagate through the specimen is short compared to the total time of the test, allowing many reflections inside the specimen necessary for reaching an homogeneous stress and strain distribution along the specimen gauge length and leading to an equilibrium of the forces acting on both ends of the specimen. Secondly, the bar diameter should be small in comparison with the pulse length, leading to the consequence that the stresses and the velocities at the interfaces are propagated down through the bars with negligible dispersions. The superposition of the signals acquired (Fig. 2) from the input and output strain gauges, namely incident plus reflected (ϵI + ϵR) and transmitted (ϵT) pulses, respectively, is the confirmation of the achievement of force equilibrium within the sample. As a result, the reflected pulse is obtained by subtracting the incident pulse (rectangular wave) from the input signal. The pulses used in the analysis have been reported in [32]. Having achieved the force equilibrium within the sample and being the two bars elastically loaded, the one-dimensional elastic plane stress wave propagation theory can be applied [34,40], and the stress (1), the strain (2) and the strain rate (3) versus time within the specimen can be evaluated: σ ðt Þ ¼ E0 

A0  ϵT ðt Þ A Z

t

∫ ðt Þ ¼ −

2C 0 L

ϵ_ ðt Þ ¼ −

2C 0  ϵR ðt Þ L

0

ϵR ðt Þ

ð1Þ

ð2Þ

ð3Þ

469

Fig. 2. Incident, reflected and transmitted pulses.

where E0 is the elastic modulus of the bars, A0 is the cross section of the input and output bars, A is the cross section of the specimen within the gauge length L, C0 is the bar elastic wave speed, and ϵT and ϵR are the transmitted and reflected pulses, respectively. In order to perform the tests at elevated temperatures, an Ambrell compact EASYHEAT induction water-cooled heating system with maximum power of 2.4 kW was used. This non-contact induction heating is ideal for heating parts of many geometries and compositions with precise power control within 25 W resolution and is able to supply energy only to the part and the zone to be heated. In Fig. 3, the setup for the high strain rate tests at elevated temperature is shown. It is possible to observe, the input ① and the output bars ② of the SHTB, the heating system ③, the water-cooled induction coil ④, the sample to be tested ⑥ connected by means of a thermocouple to a thermal controller ⑤ and the cooling system ⑦ for the input and output bars, respectively. In Fig. 1, the positioning of the thermal system in the SHTB is depicted. Using the previous Eqs. (1) and (3), both stress and strain rate as a function of time have been evaluated and depicted in Fig. 4. The averaged strain rate has been evaluated after the yield and up to the ultimate tensile strength. It can be noted that the strain rate remains nearly constant in this zone. The tests were performed as follow: firstly (i) the sample was mounted between the input and output bars, (ii) and then the thermocouple was welded in the middle part of the gauge length of the sample. Secondly, using the heating systems, (iii) the sample was heated at a constant heating rate (2.78 °C/s) to the desired temperature that (iv) was kept constant for 10 min in order to reach an homogeneous distribution along the whole specimen and in particular in the gauge length

Fig. 1. Schematic of the SHTB used for the combined high strain rate and temperatures testing.

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Fig. 3. Setup for the high strain rate tests at elevated temperatures.

part (Fig. 5). Lastly, (v) the test was performed in steady-state conditions with the well-known procedure [34]. In order to understand the effect of the temperature on the dynamic mechanical properties of the S355 structural steel, the same testing conditions adopted at room temperature were used [32]. With these testing conditions and at 20 °C, the obtained averaged strain rates (Fig. 4) were approximately 300 s−1, 500 s−1 and 850 s−1. The corresponding input bar velocities are reported in the following Table 1. Five temperature were considered: 200 °C, 400 °C, 550 °C, 700 °C and 900 °C. Three tests for each testing condition and temperature pair were performed. 5. Results and discussion In order to make a full comparison between the elevated temperature data and data obtained at room temperature, the tensile properties, such as the ultimate tensile strength and uniform strain, the fracture strength and strain, the proof strength, the effective yield strength as well as the reduction of area at fracture were evaluated. The proof strength (fp,0.2%) was determined as the stress corresponding to the intersection point of the stress–strain curve and the initial slope line offset by 0.2%. On the other hand, the effective yield strength (fy , x%) was

Fig. 5. Distribution of the temperature along the whole specimen at 550 °C.

obtained as the stress at level of total strain of 0.5%, 1.0%, 2.0%, 5.0%, 10% and 15% corresponding to the intersection point of stress–strain curve and a vertical line at these strain levels. For the sake of completeness the mechanical tensile property data are summarised as averaged and standard deviations in Tables 2-4. The best and easy way of comparing the tensile properties at different temperatures is by means of a reduction factor determined as the ratio of the value at elevated temperatures to the corresponding value at room temperature: f ðT Þ ¼

f x;θ f x;20

ð4Þ

where, fx,20 and fx,θ are the mechanical properties under consideration at room and high temperature, respectively. This ratio is widely used in literature [19,20,26,29] in order to give a fast way to evaluate the reduction of a pre-established quasi-static property at increasing temperatures. The first crucial point is the effective strain rate behaviour evaluated for increasing temperatures. In Fig. 6 a noticeable increase in the averaged strain rate is observed at different temperatures. This is undoubtedly due to the different mechanical properties of steel at elevated temperatures, which lead to an increase of the reflected pulses (cfr. Eq. (3)). This is demonstrated in Fig. 7, where different incident and reflected pulses are depicted at different temperatures. As a consequence, the reduction factors for some of the above mentioned mechanical properties are reported comparing data at the same testing condition (Fig. 8). It is possible to observe that the reduction factors seem not to be strongly different for the three testing conditions, although the strain rates are noticeably dissimilar (Fig. 6). On the other hand, the reduction factor proposed by the Eurocode 3 [14] for the proportional limit in quasi-static conditions is not applicable for the high strain-rate tests. In particular, the proposed reduction factor clearly overestimates those obtained at high strain rates.

Table 1 Testing conditions.

Fig. 4. Stress and strain rate versus time at 200 °C.

Testing condition (−)

Input bar. velocity (m/s)

Averaged ϵ_ at 20 °C (s−1)

1 2 3

v1 = 2.30 v2 = 2.90 v3 = 4.00

300 500 850

D. Forni et al. / Materials and Design 94 (2016) 467–478

471

Table 2 Averaged experimental results (v1 = 2.30 m/s, 300 s−1 at 20 °C). Temperature, T (°C)

20 [32]

200

400

550

700

900

Reduction of area, Z (%) Upper yield strength, fy,up (MPa) Lower yield strength, fy, low (MPa) Proof strength, fp,0.2% (MPa) Eff. yield strength, fy,0.5% (MPa) Eff. yield strength, fy,1.0% (MPa) Eff. yield strength, fy,2.0% (MPa) Eff. yield strength, fy,5.0% (MPa) Eff. yield strength, fy,10.0% (MPa) Eff. yield strength, fy,15.0% (MPa) Ultimate tensile strength, fu (MPa) Uniform strain, ϵu (%) Eng. fracture strength, ff (MPa) Eng. fracture strain, ϵf (%) Strain energy, Uu (MJ/m3) Strain energy, Uf (MJ/m3) True fracture strain, εf,true (-) True fracture strength, ff,true (MPa)

76 ± 1 741 ± 175 584 ± 17 614 ± 41 617 ± 40 598 ± 18 595 ± 15 631 ± 14 665 ± 11 677 ± 9 679 ± 10 17 ± 1 351 ± 13 48 ± 2 107 ± 8 1410 ± 73 1.43 ± 0.01 1217 ± 42

78 ± 2 493 ± 19 400 ± 32 473 ± 34 466 ± 37 465 ± 40 465 ± 24 498 ± 19 528 ± 17 533 ± 14 536 ± 16 14 ± 2 272 ± 15 46 ± 4 68 ± 12 1206 ± 124 1.53 ± 0.09 1017 ± 72

73 ± 2 471 ± 45 309 ± 26 476 ± 52 476 ± 47 329 ± 48 392 ± 18 477 ± 13 477 ± 6 476 ± 6 481 ± 6 13 ± 2 297 ± 5 34 ± 2 57 ± 6 961 ± 37 1.31 ± 0.07 926 ± 29

76 ± 4 387 ± 13 350 ± 25 392 ± 5 391 ± 12 341 ± 25 405 ± 8 476 ± 4 526 ± 4 531 ± 5 534 ± 5 13 ± 1 264 ± 11 39 ± 1 63 ± 4 929 ± 70 1.44 ± 0.18 931 ± 191

88 ± 1 339 ± 34 285 ± 29 338 ± 32 145 ± 6 286 ± 4 297 ± 23 362 ± 10 392 ± 51 393 ± 53 396 ± 52 12 ± 1 103 ± 10 57 ± 8 41 ± 2 1205 ± 75 2.14 ± 0.11 714 ± 18

95 ± 1 110 ± 7 108 ± 6 110 ± 8 71 ± 1 111 ± 7 120 ± 7 150 ± 7 179 ± 5 195 ± 5 211 ± 2 32 ± 3 59 ± 2 78 ± 5 59 ± 5 1665 ± 195 2.96 ± 0.09 914 ± 111

In addition, from Fig. 8 it is possible observe a significant decrease of strengths up to 400 °C, while in the range of 400–550 °C a slight increase is noted before a new marked decrease noticed up to 900 °C. This atypical behaviour, highlighted in particular for the ultimate tensile strength fu ratio (Fig. 8b), is a common occurrence for carbon steels, known as blue brittleness, where in specific ranges of strain rates and temperatures an increase of strength and a reduction of ductility are ascribed to the dynamic strain ageing (DSA). This phenomenon is caused mainly by the interaction of nitrogen atoms with dislocations or, in other words based on the dislocation motion and its dependence on the interstitial atoms [41,42]. Also [43,44,45] have highlighted this phenomenon, pointing out that the DSA shifts to higher temperatures with increasing strain rate. In this research we found a strength increase as well as a reduction of total elongation at 550 °C. In addition, this embrittlement seems to be smaller for increasing strain rates (Tables 2-4). This is an important consideration for the S355 mechanical properties at high strain rates and high temperatures. The temperature effect on the mechanical properties of the S355 structural steel can also be represented by depicting the engineering stress versus engineering strain plots. In Fig. 9, where a representative test for each temperature is depicted, it is possible to highlight the effect

of the dynamic strain ageing at 550 °C, where an increase of tensile strength as well as a decrease of uniform and fracture strains are obtained in comparison to the tests performed at a lower temperature (400 °C). Another important aspect that should be analysed in depth at high temperatures is the ductility. Different ductility indices could be considered. For example the ratio between the ultimate tensile strength (fu) and the proof strength (fp,0.2%), as well as the reduction of area at fracture (Z). The reduction of the area at fracture was measured by means of a post-mortem examination of the specimens by measuring the diameter and the meridional radius of curvature at the reduced section (Fig. 10). Furthermore, these data were used for the evaluation of the true stress–strain diagram (Fig. 11) by using the Bridgman formulae [46]. The ratio between the ultimate tensile strength (fu) and the proof strength (fp,0.2%) is shown in Fig. 12, where the marked increase of the ratio at a very high temperature (900 °C) is visible. In addition, at 550 °C the effect of dynamic strain ageing is noticeable. With the intention of fulfilling one of the key aspects in the progressive collapse analysis, considering also the temperature effect, the strain energy has been evaluated as the area under the true stress versus strain

Table 3 Averaged experimental results (v2 = 2.90 m/s, 500 s−1 at 20 °C). Temperature, T (°C)

20 [32]

200

400

550

700

900

Reduction of area, Z (%) Upper yield strength, fy,up (MPa) Lower yield strength, fy, low (MPa) Proof strength, fp,0.2% (MPa) Eff. yield strength, fy,0.5% (MPa) Eff. yield strength, fy,1.0% (MPa) Eff. yield strength, fy,2.0% (MPa) Eff. yield strength, fy,5.0% (MPa) Eff. yield strength, fy,10.0% (MPa) Eff. yield strength, fy,15.0% (MPa) Ultimate tensile strength, fu (MPa) Uniform strain, ϵu (%) Eng. fracture strength, ff (MPa) Eng. fracture strain, ϵf (%) Strain energy, Uu (MJ/m3) Strain energy, Uf (MJ/m3) True fracture strain, εf,true (-) True fracture strength, ff,true (MPa)

76 ± 1 628 ± 27 600 ± 25 626 ± 28 626 ± 26 609 ± 32 613 ± 3 650 ± 14 680 ± 13 686 ± 11 688 ± 11 15 ± 2 347 ± 4 47 ± 4 101 ± 9 1393 ± 41 1.45 ± 0.04 1218 ± 55

79 ± 2 556 ± 33 394 ± 21 546 ± 22 557 ± 28 471 ± 38 432 ± 7 494 ± 10 526 ± 10 532 ± 12 535 ± 12 14 ± 2 276 ± 11 50 ± 1 71 ± 11 1263 ± 87 1.55 ± 0.07 1072 ± 42

74 ± 1 511 ± 17 325 ± 20 518 ± 20 515 ± 23 319 ± 24 399 ± 15 453 ± 12 481 ± 8 481 ± 8 487 ± 6 13 ± 3 294 ± 15 35 ± 4 58 ± 12 960 ± 22 1.43 ± 0.03 927 ± 30

76 ± 5 394 ± 39 360 ± 19 401 ± 41 390 ± 35 355 ± 24 390 ± 8 461 ± 4 512 ± 6 627 ± 11 529 ± 11 15 ± 1 250 ± 48 44 ± 3 71 ± 7 924 ± 54 1.44 ± 0.23 860 ± 38

87 ± 2 347 ± 29 300 ± 40 343 ± 22 148 ± 3 291 ± 9 277 ± 45 348 ± 42 373 ± 43 371 ± 41 375 ± 42 12 ± 2 110 ± 23 46 ± 5 39 ± 2 1116 ± 55 2.05 ± 0.16 692 ± 38

94 ± 1 125 ± 16 121 ± 11 124 ± 15 74 ± 2 123 ± 14 119 ± 7 147 ± 6 176 ± 8 193 ± 12 211 ± 13 33 ± 1 56 ± 8 78 ± 6 60 ± 5 1295 ± 16 2.83 ± 0.17 758 ± 81

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Table 4 Averaged experimental results (v3 = 4.00 m/s, 850 s−1 at 20 °C). Temperature, T (°C)

20 [32]

200

400

550

700

900

Reduction of area, Z (%) Upper yield strength, fy,up (MPa) Lower yield strength, fy, low (MPa) Proof strength, fp,0.2% (MPa) Eff. yield strength, fy,0.5% (MPa) Eff. yield strength, fy,1.0% (MPa) Eff. yield strength, fy,2.0% (MPa) Eff. yield strength, fy,5.0% (MPa) Eff. yield strength, fy,10.0% (MPa) Eff. yield strength, fy,15.0% (MPa) Ultimate tensile strength, fu (MPa) Uniform strain, ϵu (%) Eng. fracture strength, ff (MPa) Eng. fracture strain, ϵf (%) Strain energy, Uu (MJ/m 3) Strain energy, Uf (MJ/m 3) True fracture strain, εf,true (-) True fracture strength, ff,true (MPa)

75 ± 1 668 ± 18 591 ± 43 653 ± 44 655 ± 36 633 ± 3 624 ± 25 656 ± 27 687 ± 21 693 ± 12 695 ± 15 14 ± 2 352 ± 11 44 ± 6 95 ± 10 1301 ± 43 1.38 ± 0.01 1147 ± 37

77 ± 1 583 ± 17 408 ± 54 566 ± 29 578 ± 28 494 ± 59 466 ± 18 514 ± 20 543 ± 25 548 ± 30 550 ± 28 14 ± 2 282 ± 15 45 ± 4 72 ± 14 1187 ± 89 1.49 ± 0.05 1019 ± 37

73 ± 4 501 ± 65 316 ± 24 508 ± 39 494 ± 14 383 ± 132 415 ± 15 464 ± 6 497 ± 17 500 ± 14 506 ± 21 13 ± 1 305 ± 33 36 ± 5 58 ± 8 884 ± 72 1.31 ± 0.16 930 ± 43

74 ± 2 422 ± 45 341 ± 14 423 ± 44 413 ± 39 377 ± 32 389 ± 19 460 ± 19 514 ± 21 533 ± 25 534 ± 25 16 ± 1 258 ± 36 43 ± 4 76 ± 7 957 ± 103 1.36 ± 0.08 827 ± 94

86 ± 1 374 ± 43 331 ± 28 381 ± 46 146 ± 13 285 ± 8 324 ± 32 356 ± 27 385 ± 26 385 ± 20 389 ± 23 13 ± 1 132 ± 24 45 ± 6 44 ± 1 1142 ± 64 1.96 ± 0.11 759 ± 68

94 ± 1 132 ± 9 130 ± 9 131 ± 10 74 ± 2 130 ± 9 128 ± 10 156 ± 4 189 ± 5 208 ± 5 230 ± 4 34 ± 2 67 ± 6 82 ± 3 67 ± 3 1520 ± 58 2.76 ± 0.04 866 ± 51

curve, up to the determined value of strains. The total mechanical energy consumed by the material during straining to a particular value is evaluated as follows: Z U ðT Þ ¼

ϵ 0

σ ðϵ; T Þdϵ

ð5Þ

where the value of ϵ⁎ has been chosen in order to evaluate the modulus of resilience (ϵ⁎ = ϵy), and the strain energy in correspondence of the true uniform strain (ϵ⁎ = ϵu,true). The modulus of toughness has been evaluated up to the true fracture strain (ϵ⁎ = ϵf,true), evaluated as follow: ϵ f ;true ¼ 2  ln

D0 Dfract

ð6Þ

where D0 is the initial diameter, while Dfract is the diameter of the reduced section. Furthermore, beyond the point of ultimate tensile strength in the engineering stress–strain curve, the one-dimensional true stress–strain curve has been reconstructed, by calculating the true stress and the true strain using the Bridgman formulae [46], which introduces the correction for the triaxial stress state.

Fig. 6. Effective strain rate at increasing temperatures.

In Fig. 13 the strain energy up to the ultimate tensile strength has been reported in function of the testing conditions. It is possible to observe a marked decrease in the strain energy up to 400 °C, while an almost constant behaviour with approximately 50% of the strain energy capacity evaluated at room temperature is noted at the higher temperatures. The modulus of toughness has been reported in Fig. 14. It is possible to observe a significant decrease up to 550 °C, while a marked increase is noted up to 900 °C.

6. Johnson–Cook constitutive law The structural performance evaluation in response of a coupled effect of fire and blast loading may be performed by a variety of approaches ranging from elastic static to inelastic dynamic. The choice of the approach to perform a numerical simulation strongly influences the analysis, and as a consequence a critical evaluation of the results is therefore fundamental. This is because the greater the simplification of the analysis in representing the response of the structure, the more difficult it will be to evaluate the real performance of the structure from the calculated results. The real mechanical properties of the materials subjected to a coupled effect of dynamic and fire loadings are then required. Thanks

Fig. 7. Comparison of reflected pulses at increasing temperatures.

D. Forni et al. / Materials and Design 94 (2016) 467–478

Fig. 8. Reduction factors for: proof strength (a), ultimate tensile strength (b), effective yield strength at 0.5% (c), 1.0% (d), 2.0% (e) and 5.0% (f).

473

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Fig. 9. Comparison of engineering stress–strain curves at different temperatures.

Fig. 11. Comparison of true stress–strain curves at different temperatures.

to its simplicity, one of the most commonly implemented and easy-touse constitutive relationships in the finite element programs is the constitutive model proposed by Johnson and Cook [16]. This model, which has been widely used by numerous authors in order to modelling different materials [47,48,49,50], is based on three phenomena, which are isotropic hardening, strain-rate hardening as well as thermal softening. The flow stress can be expressed as:

thermal softening sensitivity, respectively. In a previous research [32] the parameters representing the strain hardening as well as the strain rate sensitivity were evaluated. In this research the attention was focused on the thermal softening parameter, m, which was evaluated for two predetermined strain rates. To do this, tests at different temperatures and with comparable effective strain rate of 450 s−1 and 550 s−1 were used. Even if this parameter is strain rate independent in the original Johnson-Cook constitutive law, the m dependency on both strain rate and temperature is evaluated. This parameter, that is the only one governing the thermal effect, could be evaluated for a fixed strain rate by solving the following equation:

      ϵ_  1−T m σ ¼ A þ B  ϵnp  1 þ c  ln ϵ_ 0

ð7Þ

where, ϵp is the true plastic strain, ϵ_ is the considered strain rate, ϵ_ 0 is the reference strain rate (taken as 1 s−1) and T⁎ is a dimensionless temperature, valid for Tr ≤ T ≤ Tm and defined as: T−T r T ¼ T m −T r 

ð8Þ

   ϵ_ −1  ¼ 1−T m R ¼ R  1 þ c  ln ϵ_ 0

ð9Þ

where, T is the current temperature, Tm is the melting temperature and Tr is the reference temperature. T⁎ is equal to zero for T b Tr, while T⁎ is equal to 1 for T N Tm. The parameters to find by means of the experimental data are A, B and n, representing the strain hardening effects of the material in quasi-static conditions, c and m representing the strain rate and the

Fig. 10. Reduction of area of samples tested at increasing temperatures.

Fig. 12. Trend of the ratio between the ultimate tensile strength and the proof stress for increasing temperatures.

D. Forni et al. / Materials and Design 94 (2016) 467–478

475

where the upper integration limit, ϵp⁎, was set in order to have the same integration interval in Eqs. (11) and (12). The values of ΦR ð_ϵ; T R Þ were evaluated by means of the results at different strain rates obtained in a previous research [32], while the values of ΦH; j ð_ϵ; T H; j Þ were evaluated with the current data. In order to have an adequate amount of data at the same strain rate, additional tests were performed compensating the preload. Lastly, applying the logarithm, the thermal softening parameter can be evaluated from Eq. (9) as:

mðϵ_ ; T Þ ¼

Fig. 13. Strain energy (UTS) for different temperatures.

where c is the strain rate softening parameter equal to 0.02476 [32], while R could be defined as the thermal softening reduction factor, evaluable as the ratio: Rðϵ_ ; T Þ ¼

ΦR ðϵ_ ; T R Þ   ΦH; j ϵ_ ; T H; j

T H; j NT R ¼ T room :

ð10Þ

For each strain rate (450 and 550 s−1), five pairs of ΦR(_ϵ,TR) and ΦH,j(ϵ_ ,TH,j) could be evaluated for fixed room temperature (TR) and five elevated temperatures (TH,j, j = 1,…,5), respectively. These terms are defined as the area under the true stress versus true plastic strain curve and are evaluated as follow: Z ΦR ðϵ_ ; T R Þ ¼

ϵp 0

  ΦH; j ϵ_ ; T H; j ¼

σ p ðϵ_ ; T R Þdϵp

Z

ϵp 0

  σ p ϵ_ ; T H; j dϵp

Fig. 14. Modulus of toughness for different temperatures.

ð11Þ

ð12Þ

logð1−R Þ : logðT  Þ

ð13Þ

In Fig. 15, the variation of the thermal softening parameter is shown at different strain rates as a function of temperature. It is possible to observe a noticeable variation at different temperatures, ranging from m ≈ 0.45 to 1.00. The thermal softening reduction factor is shown as a function of temperature. Fig. 16 shows a comparison of the experimental data from Eq. (9) in terms of R*. A single averaged value of m obtained by comparing only two sets of data from room temperature and a single high temperature, could lead to considerable errors. This is confirmed by plotting (1− T⁎m) evaluated at a prescribed temperatures of 900 °C (m≈ 0.45) at 450 s−1. These results confirm that if the coupled effect of high temperature and dynamic loading would be evaluated, the effect of the extreme variation of the thermal softening would not be neglected. In addition, the use of a single averaged value of m in the Johnson–Cook relationship is a limit and could lead to significant errors. 6.1. Modification of the Johnson–Cook constitutive law Even with these limitations, the proposed variation of the thermal softening parameter (Fig. 15) should be interpreted as an aid to obtain a more robust analysis of coupled effect of dynamic and fire loadings, for example by considering a step-by-step numerical simulation in which the thermal softening sensitivity term is modified in function of the temperature. For that reason, with the data at disposal, it is possible to propose a simple modification to the thermal softening sensitivity term (1 − T⁎m). In the classic formulation this term is simply a decreasing

Fig. 15. The thermal softening parameter (m) of the Johnson–Cook model at different temperatures and strain rates.

476

D. Forni et al. / Materials and Design 94 (2016) 467–478 Table 5 Parameters for the definition of the dimensionless temperature TD. i

TD,i

αi

Ti

T range

0 1 2 3 4

0 0.1216 0.2095 0.1892 0.2804

1 0.65 −0.20 0.90 2.80

20 °C 200 °C 400 °C 550 °C 700 °C

20–199 °C 200–399 °C 400–549 °C 550–699 °C 700–900 °C

A comparison between the traditional dimensionless temperature and the modified version is shown in Fig. 17. It is possible to observe (Fig. 16) how by applying Eq. (14) the experimental data at different temperatures are well fitted.

7. Conclusions

Fig. 16. Comparison of the thermal softening factors obtained experimentally, with fixed single values of m and by applying the proposed modification of the Johnson–Cook constitutive law.

exponential function for increasing temperatures (Fig. 17), mainly influenced by the dimensionless temperature (T⁎). With the experimental data we can propose a more suitable expression for the thermal softening sensitivity term:   1−T D f

ð14Þ

where f is a coefficient to be determined and TD is a dimensionless temperature step-by-step defined for different temperature ranges: T D ¼ T D;i þ α i 

T−T i T m −T R

ð15Þ

where Tm is the melting temperature, TR is the room temperature and the other parameters are reported in Table 5. The value of f could be adopted equal to 0.58, corresponding to the averaged value of m in correspondence of T1.

Fig. 17. Comparison between the traditional and the modified homologous temperature.

With the intention of setting down a basis for an enhancement of fire induced progressive collapse analysis, a widely used structural steel, namely S355, has been investigated in a wide range of strain rates and temperatures, up to 900 °C. A Split Hopkinson Tensile Bar equipped with a water-cooled induction heating system has been used for the mechanical characterisation. These results can be of great interest for the assessment of robustness in structures where a fire induced progressive collapse should be evaluated focusing the attention to the extreme combined effects of fire and dynamic loads. A widely used way of comparing the mechanical properties at different temperatures is by means of reduction factors determined as the ratio between the values at elevated temperatures and the corresponding values at room temperature. In this research the reduction factors are reported comparing data at the same testing conditions. A noticeable increase in the effective strain rate was observed at elevated temperatures (Fig. 6). On the other hand the tensile properties do not seem to be significantly influenced by the different testing conditions (Fig. 8b–f), while a marked difference between quasi-static and high strain rate reduction factors (Fig. 8a) has been demonstrated. An important consideration of the mechanical behaviour of the structural steel S355 was highlighted at 550 °C, where a strength increase as well as a reduction of total elongation with respect to tests performed at 400 °C has been observed (Fig. 9). This is a common occurrence for carbon steels, known as blue brittleness, ascribed to the dynamic strain ageing and caused mainly by the interaction of nitrogen atoms with dislocations. As a future development, and with the intention of better understanding the effect of nitrogen atoms with dislocations, a TEM analysis should be performed. Furthermore, additional tests in the neighbourhood of 550 °C could lead to a better identification of the dynamic strain ageing at high strain rates. This could be helpful also for the study of a physical-based constitutive relationship. The effect of blue brittleness seems to influence also the strain energy evaluated up to the uniform strain. A marked decrease in the strain energy (Uu) up to 400 °C is observed, whereas an almost constant behaviour is highlighted at the higher temperatures (Fig. 13). On the other hand, a significant decrease in the modulus of toughness (Uf) up to 550 °C is noted, while a marked increase is noted up to 900 °C (Fig.14). A critical review of the Johnson–Cook constitutive law highlighted a perceptible variation of the thermal softening parameter, ranging from m ≈ 0.45 to 1.00. This parameter is strongly influenced by the temperature and the strain rate. This lead to the conclusion that if a coupled effect of temperature and dynamic loading would be evaluated, the effect of the extreme variation of the thermal softening could not be neglected. The use of a single averaged value of m is a limit and could lead to considerable errors. For these reasons and following a fitting approach, a modification of the dimensionless temperature (T⁎) was proposed. This modification, due to its step-by-step definition for

D. Forni et al. / Materials and Design 94 (2016) 467–478

increasing temperatures (Fig. 17), is also able to take into consideration the blue brittleness effect at 550 °C. In a next planned step, a numerical simulation will be implemented in Ls-Dyna considering the material properties of the structural steel S355 at high temperatures and high loading rates. The modified Johnson–Cook thermal softening will be adopted. Acknowledgements This work is part of research project Behaviour of structural steels under fire in a wide range of strain rate founded by the State Secretariat for Education, Research and Innovation of the Swiss Confederation (project C12.0051), in the frame of COST ACTION TU0904 — Integrated Fire Engineering and Response (IFER). A special acknowledgement goes also to Matteo Dotta for his precious collaboration in performing the tests and Gianmario Riganti for his help in interpreting the constitutive model results. References [1] Eurocode 1, Actions on structures, Part 1–7: General Actions — Accidental Actions, 2006. [2] J.R. 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