Prediction of the Tensile Strength of Normal and Steel Fiber Reinforced Concrete Exposed to High Temperatures

The tensile strength of concrete has a great impact on the performance of concrete structures, especially for members exposed to high temperatures. The inclusion of steel fibers in concrete is one of the measures to retrieve the loss of tensile strength. The previous equations for the prediction of the tensile strength, are valid for conventional concrete and can predict the tensile strength after high-temperature exposure. Therefore, they are unsatisfactory for forecasting the tensile strength of plain and steel fiber reinforced concrete under high-temperature exposure. To establish a model that can effectively simulate the tensile strength of plain concrete, specimens with compressive strengths of 20–80 MPa are tested. Then by performing tensile strength tests on the specimens containing various content of steel fiber, an equation for prediction of the tensile strength at the ambient temperature is proposed. Meanwhile, the tensile strength tests are conducted at temperatures of 100–800 °C to develop a model for high-temperature exposure. The results indicate that an increase of compressive strength from 20 to 84 improves the tensile strength by 169.4%, and the incorporation of 0.25 and 0.5% of steel fibers can improve the tensile strength of normal concrete by 58.48 and 80.29% on average at the tested temperatures, respectively. Moreover, the proposed model is able to predict the tensile strength of normal and steel fiber reinforced concrete exposed to high temperatures accurately. This equation would help a wider application of the steel fibers in the construction industry with the risk of a fire accident.


Introduction
One of the widely used construction materials is concrete which enjoys a higher compressive strength in comparison with tensile strength. The relatively low tensile strength of concrete breeds a weak resistance to crack propagation. Propagation of cracks damages the concrete members, which is intensified for members exposed to high temperatures. Concrete may be exposed to high temperatures such as the chimneys, the furnaces, the runway of airports, factories where melting of metals occur, and structures with the risk of fire. High temperatures can induce a severe physical and chemical transformation to their microstructure, affecting their mechanical and durability properties (Moghadam & Izadifard, 2020a). The behavior of concrete after exposure to high temperatures is critical to evaluate the residual strength and the reliability of structure for rehabilitation or destruction. However, investigating the specification of concrete exposing to high temperatures is crucial for the safety design of the members and providing a deep understanding of their behavior during the exposure to high temperatures. If the necessary measures in the design process are not taken, irreparable loss of life and property will be possible during exposure to high temperatures. Experimental studies are essential to monitoring the exact behavior of concrete elements in these situations, however, need considerable time, cost, and may cause injury during the test. The experimental results indicated that the effect of high temperature on the mechanical properties of concrete was marginal at temperatures below 200 °C. On the other hand, significant changes were observed at temperatures above 400 °C (Uysal et al., 2012). A comparison that was made between the tensile strength and compressive strength illustrated that the vulnerability of the tensile strength owing to high-temperature exposure is greater than the compressive strength (Khaliq & Waheed, 2017). Therefore, improvement and prediction of the tensile strength are crucial. The tensile strength of concrete after high-temperature exposure, which illustrates the residual tensile strength of concrete was investigated in previous studies (Khaliq & Waheed, 2017;Uysal et al., 2012). However, a few studies focused on the tensile strength of concrete during exposure to high temperatures (Bamonte & Gambarova, 2012;Izadifard et al., 2021a;Novak & Kohoutkova, 2018;Novák & Kohoutková, 2017). The results of the tensile strength test on cooled specimens showed that no changes were observed for the tensile strength up to 200 °C (Abdelrahim et al., 2021). In contrast, the experimental test on the tensile strength of the normal, high strength, and pozzolanic concrete during exposure to high temperatures showed severe damage at temperatures below 200 °C (Izadifard et al., 2021a, b;Mehrdad Abdi Moghadam, 2019). The vulnerability of the tensile strength of concrete to high temperatures has led researchers to make efforts to improve this characteristic of concrete exposed to high temperatures. One of the solutions to improve concrete behavior during exposure to high temperatures is the inclusion of fibers in concrete (Moghadam & Izadifard, 2020b). Steel fiber possesses a high melting point and supports the structure of concrete at high temperatures by bonding effect. Mehdipor et al. demonstrated that increasing the amount of steel fiber in the mix leads to an increase in tensile strength (Mehdipour et al., 2020). Rao and Narayana reported that steel fiber improved the compressive, tensile, and flexural strength in the range of 6-10, 0-12, and 0-20%, respectively, at temperatures ranging from 50 to 250 °C (Rao & Narayana, 2013). One of the factors affecting the productivity of the fiber on the specifications of the concrete after exposure to high temperatures is the dosage of fiber (Ahmad et al., 2019;Liu et al., 2020). Ali et al. incorporated a range of 0.05-2% of steel fibers to investigate the compressive and tensile strength of concrete at ambient temperature. They showed that the optimum content for compressive strength was 0.25% (Ali et al., 2020).
As mentioned earlier, the study of the tensile strength of concrete at the hot state has been less studied. Novak and Kohoutkova illustrated that the tensile strength at 400 and 600 °C is 0.6 and 0.3 of the tensile strength at ambient temperature (Novák & Kohoutková, 2017). The influence of steel and polypropylene fiber on the properties of normal concrete subjected to the ISO 834 fire was studied by Yermak et al. (2017). They revealed that specimens containing a higher dosage of steel fibers experienced spalling. Drzymała et al. studied the effect of high temperature on the mechanical and physical properties of concrete at temperatures of 300, 450, and 650 °C (Jackiewicz-Rek et al., 2016). The results showed that the compressive strength at a temperature of 300 °C was higher than that of at the ambient temperature. In this regard, Abdi Moghadam and Izadifard (Mehrdad Abdi Moghadam, 2019) reported an increase of the compressive strength at temperatures of 400 °C for normal and pozzolanic concrete. A deep investigation on the microstructure of mortar showed that the internal curing effects of the mortars at these temperatures caused this rise of strength (Moghadam & Izadifard, 2020a).
Lack of knowledge in the properties of concrete during high temperatures exposure could result in financial damages and life-threatening events. Therefore, there is a need for a comprehensive model, to predict the tensile strength of concrete. An efficient and reliable model for estimating the tensile strength of concrete with the risk of high-temperature exposure can help to a safe and reliable structural design. In this regard, the estimation of the mechanical properties after high-temperature exposure was introduced by Marques et al. (Marques et al., 2013). In this study, specimens were tested at temperatures of 400, 600, and 800 °C after being heated under the ISO 834 time-temperature curve. In another study that tested the concrete under high temperature, a mathematical equation for the prediction of the compressive and tensile strength of concrete was proposed. They studied the performance of concrete at elevated temperatures in the temperature range of 23-800 °C and showed that an increase in temperature causes a loss of strength. (Khaliq & Waheed, 2017). Furthermore, a model for the prediction of the splitting tensile strength of concrete after being exposed to elevated temperatures was proposed (Gao et al., 2012). In another study by Abdi , an equation for the estimation of the shear strength of normal and steel fiber reinforced concrete at temperatures range of 28 to 800 °C was proposed. This equation was able to predict the shear strength of concrete at the ambient temperature more accurately than the previous equation, and predict the shear strength of plain and steel fiber reinforced concrete at each temperature.
By reviewing the literature, it is concluded that the experimental data on the tensile strength of steel fiber reinforced concrete during exposure to various temperatures is rare. Although experimental efforts have been carried out in the literature to provide data on the post-heating behavior of plain concrete and there are some models, an efficient model to predict the tensile strength of plain and steel fiber reinforced concrete (SFRC) under high-temperature exposure was not proposed. Hence, the main purpose of this study is to add robust results to the literature and propose an efficient equation to predict the tensile strength of concrete in these situations. For this purpose, a comprehensive investigation on the effects of high temperatures on the tensile strength of SFRC through experiment is conducted at temperatures of 28-800 °C, and equations based on the experimental findings to predict the tensile strength of plain and SFRC under high temperatures are developed. Finally, the proposed models are validated using test data obtained from other literature.

Material Properties
The cementitious material in this study was ordinary type 2 Portland cement, and the characteristics of cement are shown in Table 1. Fine aggregates were river sand with a maximum size of 4 mm and a specific gravity of 2.81.
Natural coarse aggregates were calcareous and had a maximum particle diameter of 19 mm. The steel fibers were hooked end with a length of 30 mm, an aspect ratio of 37.5, and a tensile strength of 1100 MPa. The shape of fibers at temperatures of 28, 500, 650, and 700 °C, is illustrated in Fig. 1. It is seen that the appearance of steel fibers does not change at temperatures below 500 °C. An increase in temperature causes the oxidation and loss of the surface area of fibers, which results in loss of interfacial adhesion of fibers.

Experimental Procedure
In this study, the effect of steel fibers on the tensile properties of normal strength concrete (N) was investigated to develop an equation for the prediction of the tensile strength of steel fiber reinforced concrete exposing to high temperatures. To this purpose, the inclusion of steel fibers was studied and compared with plain concrete (N). The volume fraction of steel fiber in concrete's mix was 0.25% (St25) and 0.5% (St50) (equal to 19.62 and 39.24 kg/m3, respectively). The properties of the mixture with the corresponding labels are presented in Table 2, in which they were mixed and cured following ASTM C192 (2012). In the first stage of the mixing plan, the aggregates and cement were added and mixed with one-third of the mixing water for 2 min. Then, the fibers were added gradually to the running mixer and mixed for 3 min. Finally, the remained water was poured into the mixture and mixed for a further 2 min to achieve a homogeneous mix. For each mixture, three cubic specimens for the compressive strength and three cylindrical specimens for the tensile strength were cast, and the mean of those was presented. All specimens were cast, demolded after 24 h, and then submerged in the water bath at 24 ± 2 °C for 28 days. Before setting out the heating procedure, all the specimens were stored in a laboratory at a temperature of 25-30 °C and 50-60% relative humidity for 14 days to avoid explosive spalling as a result of excessive moisture.

Heating Regimes
Because higher content of free water in specimens causes spalling during the heating process, they were cured in Table 1 Characteristics of the cement.

Mineralogical
Composition of Clinker  1 Changes in the shape of steel fibers exposed to high temperature.
Page 4 of 16 Abdi Moghadam and Izadifard Int J Concr Struct Mater (2021) 15:47 the laboratory environment for 14 days after removing them from the water tank. This technique is confirmed by previous researchers (A. C192 & C192M, 2012;Farzadnia et al., 2013;Khaliq & Waheed, 2017;Liu et al., 2020;Marques et al., 2013). The heating of specimens was conducted after 42 days of casting, and tests were conducted at a hot state. The rate of heating was 1-3 °C/ min. To ensure a steady-state thermal condition throughout the specimens, when the temperature reached the target temperature, they were maintained in the furnace for 180 min. All of the tests were performed immediately after removing them from the furnace, and these tests were carried by a 3000 KN testing machine. Although a temperature drop occurs after leaving the furnace, this is negligible for the core of specimens (Moghadam & Izadifard, 2020b). In a laboratory study by Mydin and Wang, concrete temperatures throughout the testing period as reported almost constant. The results showed that the temperature drop of the concrete core after 1 min was 0.5 °C (Mydin & Wang, 2012). Novak and Kohoutkova suggested that to avoid excessive temperature loss, the time of the test after removing specimens from the furnace should be under 10 min (Novák & Kohoutková, 2017). References (Rao & Narayana, 2013) and (Faiyadh & Al-Ausi, 1989) also examined the characteristics of concrete in hot conditions by this procedure. In this study, this duration is in the range of 1-3 min.

Testing Method
The compressive and tensile strength tests were conducted according to British standard (2009) and ASTM C 496 (2004), respectively. To prevent temperature drop, immediately after finishing the heating procedure, the specimens were removed from the furnace by a steel paddle and placed on a moving table to transfer near the jack. Then, they were placed under the jack using hightemperature resistant gloves and a trowel. Afterward, compressive and tensile strength was performed. Fig. 2 shows the detail of the apparatus and test process.

The Effect of Steel Fiber on the Mechanical Properties
The experiments on the specimens at the ambient temperature are conducted after 28 and 42 days of curing. The experimental results which are the average of three replicate, the coefficient of variation (C.o.V.), and the standard deviation (St. Dev.) are listed in Table 3. It is seen that the incorporation of 0.25% and 0.5% of fibers can provide an increase of 24.53% and 22.37% for the tensile strength of 28 days. The corresponding improvement Page 5 of 16 Abdi Moghadam and Izadifard Int J Concr Struct Mater (2021) 15:47 for 42 day specimens is 28.15% and 26.32%, respectively. Steel fibers limit hair cracks expansions and improve tensile strength. During the splitting of specimens, the stress transfers from the cement matrix to fibers and improves the tensile strength of concrete. On the other hand, the incorporation of steel fibers has provided a loss of compressive strength. The results show that the compressive strength of St25 after 28 and 42 curing days is 5.75 and 1.53% lower than normal concrete. The incorporation of 0.5% of steel fibers decreases the 28 day and 42 day compressive strength of normal concrete by 6.79 and 1.96%. The addition of steel fibers causes a reduction in workability, which leads to inadequate compaction. In addition, a balling effect resulting from the non-uniform distribution of fibers leads to detachment of the concrete structure. These issues seem to be the reasons for the loss of compressive strength. Moreover, it is observed that an increase of the compressive strength from 40 to 80, has increased the tensile strength by 86.51%.

Tensile Strength of Plain Concrete
To propose an equation for prediction of the tensile strength, a specimen with a compressive strength of 20.1 and a tensile strength of 1.93 MPa was prepared. All of the cubic compressive strength were converted to the standard cylindrical compressive strength and are plotted in Fig. 3. In this figure, the correlation between the tensile strength and compressive strength is plotted. Using regression analysis, an equation is proposed with a high coefficient of determination ( R 2 = 0.88 ). This shows a strong relationship between the tensile strength ( f t ) and the compressive strength ( f c ).
To verify the accuracy of Eq. 1, 72 experimental data from 27 previous studies (Bani-Yasin, 2004;Craig et al., 1986;El-Niema, 1991;El-SAYAD & EFFECT, 2005;Kwak et al., 2002;Marar & Celik, 2002;Mitchell et al., 1996;Nadiya & Saffar, 2006;Narayanan & Kareem-Palanjian, 1983;Patel et al., 2017;Ramadoss, 2014;Rao, 2016;Rjoub & Muhammad, 2006;Sarbini et al., 2013;Sharma, 1986;Singh et al., 2016;Song & Hwang, 2004;Srikar & Kalyan, 2018;Sumathi & Saravana, 2014;Vairagade et al., 2012;Wadekar & Pandit, 2014;Yan et al., 2013) are collected ( f t exp . The lowest and highest compressive strengths of the selected data are 18.1 and 98.92 MPa, respectively. Using the experimental compressive strength ( f c ) and Eq. 1, f t calc is calculated. To evaluate the accuracy of this equation, the correlation of the predicted tensile strength with the experimental results of previous researchers is shown in Fig. 5. In this figure, the horizontal axis shows the experimental tensile strength and the vertical axis shows the calculated tensile strength. Using regression analysis, a strong equation with a high coefficient of determination ( R 2 = 0.93 ) is obtained. This graph illustrates a strong correlation between the evaluated and experimental results. Examination of these 72 cases show that the average difference between the evaluated and experimental tensile strength is 7.53%, which describes the accuracy of the equation.

Tensile Strength of Steel Fiber Reinforced Concrete
To determine the tensile strength of the steel fiber reinforced concrete, the contribution of fibers ( f t(f ) ) is summed to the tensile strength of plain concrete (f t(c) ) . Equation (2) illustrates the tensile strength of the steel fiber reinforced concrete: (2) f t(FRC) = f t(c) + f t(f )  Page 8 of 16 Abdi Moghadam and Izadifard Int J Concr Struct Mater (2021) 15:47 By conducting the regression analysis on the experimental data presented in Table 3, f t(f ) is obtained in the form of the following equation: where V f is the volume fraction of steel fiber. By substituting Eq. (3) in Eq. (2), the tensile strength of steel fiber reinforce concrete is calculated by the following equation: To evaluate the validity of this equation, the experimental results from previous studies are gathered (Abbass et al., 2018;Bošnjak et al., 2019;El-SAYAD & EFFECT, 2005;Gündüz et al., 2016; N. V, J.K. Dattatreya & S. Suresh, 2015;Ramadoss, 2014;Salwan & EFFECT, 2016;Sarbini et al., 2013;Singh et al., 2016;Song & Hwang, 2004;Srikar & Kalyan, 2018;Sukumar & John, 2014;Vairagade et al., 2012;Wadekar & Pandit, 2014) and compared with the calculated tensile strength using Eq. (4). In this aim, the selected data are for specimens containing a wide range of volume fractions (0.25 ≤ V f ≤ 4%). To evaluate the accuracy of this equation, Fig. 6 which illustrates the difference between the evaluated and experimental results is presented. The horizontal axis shows the volume fraction of fibers, and the vertical axis shows the difference between the calculated and experimental data (∆). The difference between the calculated tensile strength and experimental results of specimens containing various content of steel fibers is in the range of 0.95-28.76%, and the average deviation is around 11%. Based on these results it is observed that the deviation of the calculated and experimental tensile strength is in an acceptable range. The deviation of the proposed model from experimental results of Wadekar and Pandit, (2014) for volume fractions of 1, 2, 2.5, 3, 3.5, and 4 is 4. 83, 8.39, 8.11, 9.41, 15.77%, 25%, and 24%, respectively. This comparison with the experimental results of Ramadoss (2014) (1.72%), Vairagade (2012) (1.57%), and Bošnjak (2019) (3.06%) is marginal. Overall, the average deviation of the model from 62 test data is 11%, which indicates the high accuracy of Eq. 4. These differences are attributed to the specifications of fibers (type and aspect ratio), aggregate size, and the compressive strength of reference concrete. ( Ramadoss, 2014;Sarbini et al., 2013;Singh et al., 2016;Song & Hwang, 2004;Srikar & Kalyan, 2018;Sukumar & John, 2014;Wadekar & Pandit, 2014) are used to verify Eq. 5. The deviation between the evaluated and experimental results is plotted in Fig. 7. The average deviation of this model from experimental data is 11%, which indicates the high accuracy of Eq. (5).

The Tensile Strength of Normal Concrete Exposed to High Temperatures
To develop Eq. (5) for high temperatures, the tensile strength of normal concrete at high temperatures is linked to the tensile strength at ambient temperature using T function in the form of the following equation: where f t(T ) is the stensile strength of normal concrete at high temperature, f t(a) is the tensile strength of normal concrete at ambient temperature, and T is a function of temperature which considers the effect of temperature on the tensile strength of normal concrete. Regression analysis was carried out on the f t(T ) f t(a) .
(Listed in Table 4) and temperatures to obtain T which is expressed by Eq. (7). The experimental results with the corresponding (C.O.V.) and (St. Dev.) are shown in Table 5.
A comprehensive review of past studies has shown that limited equations have been proposed for the prediction of the tensile strength at high temperatures. These equations are presented in Table 6 and are plotted in Fig. 8. In this table, the temperature ranges in which these relationships are valid and the type of the test is provided (hot or cooled state). In Fig. 8, the experimental results of the normalized tensile strength which is as T are also presented. Blue and red values illustrate the cooled and hot state conditions, respectively.
It can be seen that at temperatures below 300 °C, the experimental tensile strength at the hot state is lower than the residual tensile strength. This can be attributed to the effect of internal pressure resulting from the evaporation of free water. Furthermore, it is seen that an increase of the tensile strength at temperatures of 300-400 °C is seen for tests at a hot state. Novak (2017) and Faiyadh (1989) which tested the tensile strength of concrete at hot state, reported an increase of T at temperatures of 350 and 400 °C, respectively. The lack of this (6) f t(T ) = T f t(a) (7) T = 1.514 × 10 −5 T 2 − 6.76 × 10 −3 T + +1.18 T ≤ 400 R 2 = 0.88  (Khalil, Kim, Abaeian, Bošnjak, Gao, Bastami) (Abaeian et al., 2018;Bastami et al., 2014;Bošnjak et al., 2019;Gao et al., 2012;Khalil, 2018;Kim & Lee, 2015), a descending trend for the residual tensile strength was reported. Based on this figure, it is seen that Eq. (7) provides the lowest value at temperatures range of 100-300 °C. At a temperature of 100 °C, the proposed model in this study has a slight difference from the experimental results of Khaliq and Waheed (Khaliq & Waheed, 2017). This comparison at a temperature of 200 °C, shows a minor deviation from the experimental results of Novak and Kohoutkova (Novák & Kohoutková, 2017). The experimental tensile strength reported by Rao et al. (Rao & Narayana, 2013) has a negligible deviation with the evaluated tensile strength at a temperature of 250 °C. The free water is evaporated at a temperature of 350 °C. Therefore, the proposed model based on the test at the hot state is almost consistent with the previously proposed model (Bažant & Chern, 1987; E.C. for S. (CEN) 1992; Gao et al., 2012;Chang et al., 2006;Zaici, 1998). Meanwhile, the comparison of the proposed model with the experimental result at the hot state shows that at a temperature of 600 °C, the evaluated and experimental tensile strength of Faiyadh (1989) is the same. These comparisons show that Eq. 7 estimates the overall trend of the normalized tensile strength of concrete with high accuracy.
The evaluated tensile strength using Eq. 6, and the deviation of the evaluated tensile strength from experimental results are presented in Table 5. It is seen that the average deviation of the evaluated and experimental data is 11.55%, which illustrates the high accuracy of this equation.   Page 10 of 16 Abdi Moghadam and Izadifard Int J Concr Struct Mater (2021) 15:47 illustrates the failure modes of fractured surfaces at temperatures of 450, 650, and 800 °C. It is seen that aggregates have a great role in the tensile strength behavior of concrete at temperatures below 500 °C. However, at temperatures above 500 °C, the expansion of cracks in the microstructure and incompatibility of the coefficient of thermal expansion breeds the detachment of aggregates. Owing to these justifications, the number of aggregates that contribute to load barring decreases. Furthermore, the red surface of fractured specimens at temperatures of 800 °C illustrates that tests were done in hot conditions. The comparison of test data of Table 5 and evaluated results from Eq. (6) are shown in Fig. 10. It is seen that with increasing the temperature up to 200 ℃, the tensile strength of concrete fell. Previous studies have shown that no chemical changes occur in the concrete microstructure in these temperature ranges (Kowalski, 2010). Therefore, the loss of tensile strength at these temperatures is connected to the evaporation of free water, which intensifies the internal pressure. At temperatures range of 200-400 ℃, the tensile strength increases. The hydration of un-hydrated cement owing to exposure to high temperature is attributed to this improvement (Xiao et al., 2018). An increase of surface forces between cement gel layers is another responsible factor for this improvement (Castillo, 1987). By increasing the temperature from 400 to 800 °C, chemical changes and degradation of microstructure intensifies, which results in a severe loss of tensile strength. Previous research of Abdi Moghadam and izadifard on the microstructure of mortars showed that  Page 11 of 16 Abdi Moghadam and Izadifard Int J Concr Struct Mater (2021) 15:47 the intensity of CSH peak is almost constant up to 400 °C and has dropped significantly at 800 °C. Furthermore, the calcium silicate peaks which are the product of C-S-H decomposition increased significantly at a temperature of 800 °C. Another product that was observed in XRD patterns of specimens that experienced 800 °C, is calcium silicate. It can be another reason for the sharp decrease in the mechanical properties of the mortar at this temperature. Other observations at 800 °C include a significant decomposition of calcium silicate hydrate peak and portlandite peak (Moghadam & Izadifard, 2020a). Based on this figure, it is seen that the deviation of the evaluated tensile strength from the experimental tensile strength is marginal, except at 200 ℃. At 200 ℃, owing to the simultaneous effect of the jack and vapor pressure resulting from the evaporation of the free water in capillary pores, a loud voice followed by the steam outlet was observed (Moghadam & Izadifard, 2020b). The average deviation of the predicted tensile strength from experimental results at temperatures of 28-800 ℃ is 11.55%, illustrating the high precious of Eq. 6.

The Tensile Strength of Steel Fiber Reinforced Concrete Exposed to High Temperature
Equation (8) illustrates that the tensile strength provided by steel fibers is considered as a function of volume fractions of steel fibers. In this equation, β T is a function of temperature considering the effect of volume fractions of steel fibers on the tensile strength of plain concrete: By calculating this coefficient at each temperature and applying a regression analysis, β T is calculated in the form of the following equation: 650°C 800°C Fig. 9 Failure modes of fractured surfaces.

Fig. 10
Variation of the tensile strength at high temperatures.
Page 12 of 16 Abdi Moghadam and Izadifard Int J Concr Struct Mater (2021) 15:47 Finally, the equation for the prediction of the tensile strength of steel fiber reinforced concrete is in the form of the following equation: where f t(FRC)(T ) is the tensile strength of steel fiber reinforced concrete at high temperature.
The experimental tensile strength with the corresponding (C.O.V.) and (St. Dev.) are shown in Table 7, where they are compared with the predicted tensile strength using Eq. (10). Based on this table, it is seen that the deviation of the predicted tensile strength from the experimental values for St25 and St50 is in the range of 2.95-14.17 and 0.35-11.5%, respectively. Fig. 11 represents changes in the tensile strength in the steel fiber-reinforced concrete at different temperatures, where they are compared with the predicted tensile strength using Eq. 10. This figure shows that the variation of the splitting tensile strength of the normal and the fiber reinforced concrete at various temperatures is almost similar. As could be observed, all specimens experienced a reduction up to 200 °C. At this temperature, the tensile strength of St25 and St50 is almost similar, and the inclusion of steel fibers improved the tensile strength of normal concrete. Interfacial adhesion and mechanical anchoring of steel fibers seem to be another reason for the increase of tensile strength. The inclusion of steel fibers allows heat to transfer easily inside the concrete, which reduces the thermal stress and the expansion of cracks. However, the rate of the tensile strength improvement is decreased at temperatures of 650 and (9) β T = −2. × 10 −7 T 3 + 6.14 × 10 −5 T 2 + 0.0062T + 1.28 T ≤ 400 R 2 = 0.91 β T = 4.21 × 10 −6 T 2 − 0.013T + 8.37 T > 400 R 2 = 0.80 (10) f t(FRC)(T ) = T f t(a) + β T V f 800 °C. This can be attributed to the effect of high temperature on the loss of bonding properties. Past studies have shown that the performance of the fibers does not change up to 500 °C and decreases at higher temperatures. The yield stress and ultimate stress of the steel fibers at 800 °C are 15% and 25% of ambient temperature, respectively (Abdallah et al., 2017). These changes in the steel fiber behavior cause the descending trend of the tensile strength at temperatures above 500 °C. Furthermore, the expansion of steel fibers creates radial cracks surrounded the steel fibers, which is another cause of the tensile strength loss. Table 7 shows that the average deviation of the predicted tensile strength from the experimental result of St25 and St50 is 7.68 and 5.58%, respectively. Fig. 11 presents a comparison between the predicted and experimental results. The tensile strength of steel fiber reinforced concrete has a descending trend at temperatures below 200 °C. With a further increase in temperature, the tensile strength improves and reaches a peak at 350 °C. Owing to the internal curing condition, which is the reason for strength gaining at temperatures of 400 °C, portlandite is consumed to produce higher content of CSH gels. The results of XRD illustrated that by increasing the temperature from 28 to 400 °C, the intensity of CaC O 3 increased from 12.2% to 19.8%, which justifies the decarburization of portlandite. However, at temperatures above 350 ℃, the overall trend of the tensile strength is descending. The oxidation and corrosion of steel fibers which result in a loss of cross section of steel fibers is another reason for the tensile strength loss. The Page 13 of 16 Abdi Moghadam and Izadifard Int J Concr Struct Mater (2021) 15:47 inclusion of 0.25 and 0.5% of steel fibers provide 58.48 and 80.29% improvement of the tensile strength at the tested temperatures on average. Meanwhile, the rate of increase of the tensile strength for a higher dosage of steel fibers is higher. This is due to the presence of a higher dosage of fibers at the tensile plane. Because aggregates detach from the cement matrix, the inclusion of fibers improves the tensile strength of plane concrete. Another reason for the improvement of the tensile strength owing to the inclusion of steel fibers can be attributed to the thermal conductivity of steel fibers. In specimens containing steel fibers, the heat is transferred easily which results in stress reduction. This issue reduced the crack development, and consequently the tensile strength improvement (Moghadam & Izadifard, 2020b).
To make Eq. (10) independent from the tensile strength of plain concrete at ambient temperature, and extend this equation for specimens with various compressive strength, Eq. (1) is substituted in Eq. (10): The difference between the predicted tensile strength and experimental values is shown in Table 8. This equation not only can predict the tensile strength of steel fiber reinforced concrete at high temperature without the tensile strength at ambient temperature but also this equation decreases the deviation of Eq. (10) at 200 °C. Based on this table, it is seen that the proposed equation can estimate the tensile strength of steel fibers reinforced concrete accurately. The average deviation of predicted tensile strength from the experimental results of St25 and St50 is 6.93% and 10.96%, respectively.

Conclusions
This study focused on the effects of high temperatures on the tensile strength of normal and steel fiber reinforced concrete. Based on the results presented in this paper, the following conclusions are drawn: -The overall trend of the tensile strength of plain and steel fiber reinforced concrete followed a similar trend at high temperatures. At temperatures below 200 °C, they experienced a sudden loss which can be attributed to the simultaneous effect of the jack and the evaporation of the free water. A review of the previous experimental results illustrated this loss did not occur for specimens that were tested after exposure to high temperatures. 200 °C onward, they recover their strength because of the hydration of un-hydrated cement owing to autoclave curing con-

Fig. 11
Comparison between the predicted tensile strength using Eq. 10 and experimental values.  Moghadam and Izadifard Int J Concr Struct Mater (2021) 15:47 dition. On the other hand, a further increase in temperature resulted in a descending trend in the tensile strength of the plain and steel fiber reinforced concrete.
-The incorporation of steel fiber improved the tensile strength of plain concrete under high-temperature exposure. The inclusion of 0.25 and 0.5% steel fibers improved the tensile strength of specimens at elevated temperatures averagely by 58.48% and 23.81%. These values are valid for concrete containing hooked end steel fibers. -The compressive strength has a great impact on the tensile strength of concrete, where an increase of compressive strength from 20.1 to 84.45 improves the tensile strength by 169.4% at ambient temperature. Using experimental data of this study and conducting regression analysis, an equation with high accuracy for the prediction of the tensile strength of normal concrete at ambient temperature is proposed. A comparison of the proposed equation in this study with the previous equations shows that this equation is most consistent with the previous equations in the compressive strength range of 40-60 MPa. Further verification of this equation by comparing the evaluated tensile strength with the previous experimental results reveals the average deviation of the predicted tensile strength from the experimental values is 7.53%, which describes the accuracy of the equation. This equation is developed to predict the tensile strength of steel fiber reinforced concrete at ambient temperature. The validity of this equation is verified by comparing the predicted tensile strength from previous experimental data having a wide range of volumes fractions (0.25-4%). The average deviation of the predicted tensile strength from the experimental results is 11%, which indicates high accuracy of the proposed equation. -To help a broader application of the steel fibers in structures with the risk of a fire accident, there is a need for an efficient model to predict the tensile strength under high-temperature exposure in the design process. Using experimental results of the tensile strength at temperatures of 28 to 800 °C, and conducting regression analysis, an equation that can predict the tensile strength of plain and steel fiber reinforced concrete is proposed. The comparison of the predicted tensile strength and experimental results of N, St25, and St50 shows a deviation of 11.55%, 6.93%, and 10.96%, respectively. However, the specification of steel fiber (shape, tensile strength, and the aspect ratio of the steel fiber) can greatly effect on the tensile strength improvement and the accurately of the proposed models. Mean-while, the heating regime including the heating rate and exposure times are other factor which can affect the results of this study. Further research on these parameter is warranted and would help researcher for a deep understand about the real behavior of the steel fiber reinforced concrete espouse to high temperature. However, the specification of steel fiber (shape, tensile strength, and the aspect ratio of the steel fiber) can affect the rate of improvement and the accuracy of the proposed models. Meanwhile, the heating regime including, the heating rate and the duration of exposure comprise other factors that can affect the results of this study. Further research on these parameters would help researchers for a deep understanding of the real behavior of the steel fiber reinforced concrete espouse to high temperature.