1 Introduction

Fiber reinforced concrete (FRC) has been studied over the past three decades in terms of the improved crack control. This is of further importance when fibers are incorporated in an inherently brittle material such as high strength concrete (HSC), the use of which has increased progressively not only due to the higher load carrying capacity but also due to the improvement in durability and service life. The higher load carrying capacity of HSC is normally accompanied by more brittle behavior, which can be compensated in a rational manner through the incorporation of fibers. Many studies on the degradation of concrete when it is exposed to high temperatures have been reported. Concrete structures may be exposed to high temperatures, by accidental causes or by the characteristics of the structural application. As a consequence, concrete undergoes changes that may result, in many cases, in extensive cracking. In this sense, it is interesting to study the contribution of fibers to control crack formation and propagation [10].

The mechanical properties of fiber reinforced concrete (FRC) after high temperatures have received considerable attention in recent years [8, 11, 17, 19]. When PP fibers are utilized to control fresh and hardened properties of cement-based materials at ambient temperature, it has been found that PP fibers can decrease the plastic shrinkage [1], and they also have a minor effect on the compressive and flexural strengths. The effect on strength, in fact, has been reported to be contradictory [1, 2]. Therefore, the beneficial effect of avoiding or reducing explosive spalling raises the question of how much PP fibers will affect the residual mechanical behavior of high performance concrete (HPC) exposed to elevated temperatures. The investigation on cement paste by Komonen and Penttala [12] have indicated that inclusion of PP fibers produces a finer residual capillary pore structure, decreases residual compressive strength and improves residual flexural strength when temperature ranged from 150°C to 440°C, whereas the residual flexural strength decreases considerably when temperature rises beyond 440°C to 520°C. Furthermore, Poon et al. [17] have concluded that inclusion of PP fibers results in a quicker loss of the compressive strength and toughness of concrete (besides Portland cement, cement both with and without metakaolin or silica fume were included in their research) after exposure to elevated temperature (up to 800°C). However, they also have found that the residual compressive strength of HPC with ordinary Portland cement containing PP fibers (0.22% by volume) increases 4.6% after exposure to 600°C, while it decreases 3.2% after exposure to 800°C, compared with that for HPC without PP fibers. From their investigation, it may be deduced that the effects of PP fibers on the residual mechanical strength of HPCs after exposure to elevated temperatures still need to be further studied [21].

Structural fire safety is one of the primary considerations in the design of high-rise buildings and infrastructures, where concrete is often the material of choice for structural members. At present, the fire-resistance of reinforced concrete (RC) members is generally established using prescriptive approaches that are based on either the standard fire-resistance tests or empirical methods of calculation. Although these approaches have drawbacks, there have been no significant failures of concrete structures or members made of either high-strength PPFRC exposed to high temperatures when designed in accordance with current codes, there is an increased focus on the use of numerical methods for evaluating the fire performance of structural members. Because this depends on the properties of the constituent materials, knowledge of the elevated-temperature properties of concrete is critical for fire-resistance assessment under performance-based codes [3].

The properties that are known to control PPFRC behavior at elevated temperatures are compressive strength, tensile strength, peak strain (i.e. strain at peak stress), modulus of elasticity, flexural tensile strength (modulus of rupture), and others that are non-linear functions of temperature. Many compressive and tensile constitutive models for concrete at normal temperature are available. The constitutive laws of concrete materials under fire conditions are complicated and current knowledge of thermal properties is based on the outcome of limited experimental tests of material properties. There are only limited test data for some high-temperature properties of PPFRC and there are considerable variations and discrepancies in the high-temperature test data for other properties of PPFRC. This paper proposes reliable constitutive relationships for high strength PPFRC for fire-resistance predictions of RC members.

2 Research Significance

Although computational methods and techniques for evaluating the fire performance of structural members of buildings have been developed in recent years, research related to supplying input information (material properties) into these computational methods has not developed enough. There is an urgent need to establish constitutive relationships for modeling the fire response of PPFRC members because the use of PPFRC has considerably developed during the last years. The objectives of this study are: a) proposing new mechanical properties relationships for PPFRC mixtures at elevated temperature (i.e. compressive and tensile strengths, modulus of elasticity, modulus of rupture, and peak strain at maximum compressive strength), b) proposing new compressive and tensile stress–strain relationships for PPFRC at elevated temperatures.

3 Database of Experimental Results

An experimental results database from various published investigations is an effective tool for studying the applicability of the various high temperature behaviors for PPFRC. To apply the models to a particular concrete mixture accurately, it is necessary to use only investigations that are sufficiently consistent with the applied testing methodology. The PPFRC experimental results included in the database were gathered mainly from papers presented at various published articles. The database includes information regarding the composition of the mixtures, fresh properties of PPFRC, testing methodology, and conditions. Mechanical properties at high temperatures have not been investigated as much as the other aspects of PPFRC.

Tables 1 and 2 include general information about the experimental tests, such as fiber content (V f ), type of fiber, aggregate and cement type, temperatures, rate of heating, and specimens type. Table 1 shows that general type of cement that is used in the most of researches is Ordinary Portland Cement (OPC). Also, PP fiber content (V f ) is varied between 0.11% and 0.6%. Moreover, most of fine aggregate that are used in the database is natural river sand and type of coarse aggregate is varied. Table 2 indicates different type of PPFRC specimens are used for thermal behavior analysis. Also, heating rate range in the experimental results database is between 0.5°C/min to 10°C/min. The temperature that is used for different research is varied between 20°C to 900°C.

Table 1 Experimental Results Database Properties
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Table 2 Continued Experimental Results Database Properties
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4 Compressive Strength of PPFRC at Elevated Temperatures

The residual compressive behavior of concrete has been under investigation since the early 1960s (see the contributions by Zoldners, Dougill, Harmathy, Crook, Kasami et al., Schneider and Diederiches, all quoted in [18]. Attention has been focused mostly on the compressive strength (the strength at room temperature after a specimen has been heated to a test temperature and subsequently cooled) as such, on the residual strain and on strength recovery with time [9]. In this contribution the efforts are focused on producing compressive strength relationships for PPFRC of different strength classes which incorporate different PP fiber content, in order to investigate their performance after exposure at gradually up scaled temperature.

In this study, the relationships proposed for the compressive strength of PPFRC by considering PP fiber various volume fractions at elevated temperature are based on regression analyses on existing experimental data with the results expressed as Equations (1) to (2). The main aim of regression analyses is considering the variation experimental compressive strength of PPFRC behaviors at different elevated temperatures and developing the rational and simple relationships that can fit well with experimental data.

The nonlinear regression understating and interpreting are described as follow. Similar to linear regression, the goal of nonlinear regression is to determine the best-fit parameters for a model by minimizing a chosen merit function. Where nonlinear regression differs is that the model has a nonlinear dependence on the unknown parameters, and the process of merit function minimization is an iterative approach. The process is to start with some initial estimates and incorporates algorithms to improve the estimates iteratively. The new estimates then become a starting point for the next iteration. These iterations continue until the merit function effectively stops decreasing.

The compressive strength versus fiber volume fraction prediction is proposed as Eq. (1). Also, the PPFRC compressive strength at elevated temperatures based on the experimental results are captured as shown in Equation (2). The Equation (1) should use in the Equation (2) to appropriate prediction of PPFRC compressive strength at different high temperatures and fiber content. These proposed relationships are compared separately with test results, as shown in Figures 1 to 2.

$$ f_{cf}^{\prime } = f_{c}^{\prime } - 46.36\,\,V_{f} $$
(1)
$$ f_{cT}^{\prime } = f_{cf}^{\prime } \,\left\{ {\begin{array}{*{20}c} {1.0} \hfill & {20^{^\circ } {\text C}} \hfill \\ {1.0237 - 0.00105\,\,T + 1 \times 10^{ - 7} \,T^{2} } \hfill & {100^{^\circ }{\text C} \le T \le 800^{^\circ }{\text C}} \hfill \\ \end{array} } \right\} $$
(2)

where \( f_{c}^{\prime } \) is the compressive strength without fiber, \( f_{c}^{\prime } \) is the compressive strength of fiber reinforced concrete. Figure 1 makes an evaluation between proposed relationship for compressive strength against PP fiber volume fraction. Experimental results show that compressive strength will be decreased by increasing PP fiber content. Comparison of compressive strength of concrete with different fiber content shows that compressive strength will decreases 10.03%, 20%, 30.1%, and 40.1% by adding 0.2%, 0.4%, 0.6%, and 0.8% fiber to the concrete, respectively. Figure 2 creates comparison between proposed relationship for compressive strength of PPFRC at different temperatures against published unstressed experimental test results (unstressed tests: the specimen is heated, without preload, at a constant rate to the target temperature, which is maintained until a thermal steady state is achieved) [7, 8, 11, 14, 16, 20, 21]. The experimental results indicate that compressive strength decreases up to 38.30% at 400°C temperature and it reduces to 75.23% at 800°C temperature. The proposed relationships fit the experimental results well in comparison with others.

Figure 1
figure 1

Effect of fiber volume on compressive strength

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Figure 2
figure 2

Comparison between compressive strength proposed relationship of PPFRC with experimental test results

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5 Tensile Strength of PPFRC at Elevated Temperatures

Very little attention has been paid to concrete behaviour in tension, either direct or indirect (in bending or splitting) at high temperatures. Before the mid-1980s [18], the studies in this area are limited and a few of them are still unpublished. Furthermore, another reason to investigate concrete properties in tension is spalling of the material [9]. In this study, the relationships proposed for the tensile strength of PPFRC by including fiber content at elevated temperature are based on regression analyses on existing experimental data with the results expressed as Equations (3) to (4). The main aim of regression analyses is considering the changeable experimental tensile strength of PPFRC behaviors at different elevated temperatures and developing the rational and simple relationships that are in good correlation with test results. The tensile strength versus fiber volume fraction prediction is proposed as Equation (3). The Equation (3) should use in the Equation (4) to suitable prediction of PPFRC tensile strength at different high temperatures and fiber content.

$$ f_{ctf} = f_{ct} + 0.626\,\,V_{f} $$
(3)
$$ f_{ctT} = f_{ctf} \;\left\{ {\begin{array}{*{20}c} {1.0} \hfill & {20^{^\circ } {\text C}} \hfill \\ {1.0237 - 0.00107\,\,T + 1 \times 10^{ - 7} \,T^{2} } \hfill & {100^{^\circ } {\text C} \le T \le 800^{^\circ } {\text C}} \hfill \\ \end{array} } \right\} $$
(4)

where \( f_{ct} \) is the tensile strength without fiber, \( f_{ctf} \) is the tensile strength of fiber reinforced concrete. Figure 3 creates an evaluation between proposed relationship for tensile strength against steel fiber volume fraction. Experimental results show that tensile strength will be increase by increasing steel fiber content. Comparison of tensile strength of concrete with different fiber content shows that tensile strength will increases 2.18%, 4.26%, 6.26%, and 8.18% by adding 0.2%, 0.4%, 0.6%, and 0.8% fiber to the concrete, respectively. Figure 4 makes comparison between PPFRC tensile strength proposed relationship at different temperatures against published unstressed experimental test results [7, 8, 15, 20]. The experimental results indicate that tensile strength decreases up to 38.83% at 400°C temperature and it reduces to 76.83% at 800°C temperature. The proposed relationships for tensile strength of PPFRC against the unstressed experimental results are shown that the results provide a reasonable fit to the available experimental data.

Figure 3
figure 3

Effect of fiber volume on tensile strength

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Figure 4
figure 4

Comparison between tensile strength proposed relationship of PPFRC with experimental test results

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6 Modulus of Elasticity of PPFRC at Elevated Temperatures

The elastic modulus of concrete could be affected primarily by the same factors that influence its compressive strength [13]. The modulus of elasticity versus fiber volume fraction prediction is proposed as Equation (5). Moreover, a relationship is proposed to evaluate the elasticity modulus of PPFRC at elevated temperatures using regression analyses conducted on experimental data and is expressed as Equation (6). The regression analyses is considering the changeable experimental elastic modulus of PPFRC behaviors at different elevated temperatures and developing the rational and simple relationships that can fits well with experimental data. The Equation (5) should use in the Equation (6) to proper prediction of PPFRC modulus of elasticity at different high temperatures and fiber content.

$$ E_{cf} = E{}_{c} - 31.177\,\,V_{f} $$
(5)
$$ E_{cT} = E_{cf} \;\left\{ {\begin{array}{*{20}c} {1.0} \hfill & {20^\circ {\text C}} \hfill \\ {1.01 - 0.0013\;T + 10^{ - 7} \,T^{2} } \hfill & {100^\circ {\text C} \le T \le 800^\circ {\text C}} \hfill \\ \end{array} } \right\} $$
(6)

where \( E_{c} \) is the modulus of elasticity without fiber, \( E_{cf} \) is the modulus of elasticity of fiber reinforced concrete. Figure 5 makes comparison between proposed relationship for modulus of elasticity against PP fiber volume fraction. Experimental results show that modulus of elasticity will be slightly increase by increasing PP fiber content. Comparison of modulus of elasticity of concrete with different fiber content shows that modulus of elasticity will increases 13.67%, 27.35%, and 41.03% by adding 0.2%, 0.4%, and 0.6% fiber to the concrete, respectively. Figure 6 makes comparison between PPFRC modulus of elasticity proposed relationship at different temperatures against published unstressed experimental test results [14, 16, 20]. The experimental results indicate that modulus of elasticity decreases up to 49.4% at 400°C temperature and it reduces to 96.6% at 800°C temperature. The proposed relationship is in agreement with the experimental test results.

Figure 5
figure 5

Effect of fiber volume on modulus of elasticity

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Figure 6
figure 6

Comparison between modulus of elasticity proposed relationship of PPFRC with experimental test results

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7 Modulus of Rupture of PPFRC at Elevated Temperatures

There are studies on flexural tensile strength (modulus of rupture). For example, Lau and Anson (2006) carried out both compression and flexural tests on both plain concrete and 1% PPFRC under high temperatures ranging between 105°C and 1,200°C. Their results indicated a decrease in both compressive and flexural strength for both the plain and PPFRC. However, PPFRC was able to resist high temperatures much better than plain concrete—as seen by a much higher residual strength at all temperature levels. Poon et al. [17] experimented with compression to determine strength and toughness of PPFRC and polypropylene FRC, at temperatures between 200°C and 800°C. At temperatures lower than 200°C, the compressive strength of both plain and FRC remained unchanged. The strength was found to decrease linearly as the temperature increased above 200°C. As for compression toughness, PPFRC was found to maintain its energy absorption better than plain concrete even at the highest temperatures. The modulus of rupture versus fiber volume fraction prediction is proposed as Equation (7). Also, a relationship is suggested to calculate the modulus of rupture of PPFRC at elevated temperatures using regression investigates conducted on experimental data and is expressed as Equation (8). The regression analyses is considering the variable experimental modulus of rupture of PPFRC behaviors at different elevated temperatures and developing the rational and simple relationships that can fits well with experimental data. The Equation (7) should use in the Equation (8) to proper calculation of PPFRC modulus of rupture at different high temperatures and fiber content.

$$ f_{crf} = f_{cr} - 1.726\,\,V_{f} $$
(7)
$$ f_{crT} = f_{crf} \;\left\{ {\begin{array}{*{20}c} {1.0} \hfill & {20^\circ {\text C}} \hfill \\ {1.1 - 0.0019\;T + 8 \times 10^{ - 7} \,T^{2} } \hfill & {100^\circ {\text C} \le T \le 900^\circ {\text C}} \hfill \\ \end{array} } \right\} $$
(8)

where \( f_{cr} \) is the modulus of rupture without fiber, \( f_{crf} \) is the modulus of rupture of fiber reinforced concrete. Figure 7 makes comparison between proposed relationship for modulus of rupture against PP fiber volume fraction. Experimental results show that modulus of rupture will be decreased by increasing PP fiber content. Comparison of modulus of rupture of concrete with different fiber content shows that modulus of rupture will increases 5.22% and 10.45% by adding 0.2% and 0.4% fiber to the concrete, respectively. Figure 8 makes comparison between PPFRC modulus of rupture proposed relationship at different temperatures against published unstressed experimental test results [16, 21]. The experimental results indicate that modulus of rupture decreases up to 53.2% at 400°C and 96.2% at 900°C temperature. The proposed relationship is in agreement with the experimental test results.

Figure 7
figure 7

Effect of fiber volume on modulus of rupture

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Figure 8
figure 8

Comparison between modulus of rupture proposed relationship of PPFRC with experimental test results

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8 Strain at Peak Stress (Peak Strain) at Elevated Temperatures

In this study, the relationships proposed for the peak strain of PPFRC at elevated temperature are based on regression analyses on existing experimental data with the results expressed as Equations 9 and 10. The Equation (9) should use in the Equation (10) to suitable calculation of PPFRC peak strain at different high temperatures and fiber content.

$$ \varepsilon_{cf} = 0.0088 - 0.0075\;V_{f} $$
(9)
$$ \varepsilon_{cT} = \varepsilon_{cf} \;\left\{ {\begin{array}{*{20}c} {1.0} \hfill & {20^\circ {\text C}} \hfill \\ {1.0037 - 0.0001\;T} \hfill & {100^\circ {\text C} \le T \le 500^\circ {\text C}} \hfill \\ {1.0266 - 0.0014\;T + 2.2 \times 10^{ - 6} \,T^{2} } \hfill & {600^\circ {\text C} \le T \le 800^\circ {\text C}} \hfill \\ \end{array} } \right\} $$
(10)

where \( \varepsilon_{cf} \) is the peak strain of fiber reinforced concrete. Figure 9 makes comparison between proposed relationship for peak strain against PP fiber volume fraction. Experimental results show that peak strain will be decrease by increasing PP fiber content. Comparison of peak strain of concrete with different fiber content shows that peak strain will increases 8.52%, 17.04%, and 21.30% by adding 0.1%, 0.2%, and 0.25% fiber to the concrete, respectively. Figure 10 makes comparison between PPFRC peak strain proposed relationship at different temperatures against published unstressed experimental test results [14, 17]. The experimental results indicate that peak strain decreases up to 4.65% at 400°C temperature and it rose to 31.46% at 800°C temperature. The proposed relationship is in agreement with the experimental test results.

Figure 9
figure 9

Effect of fiber volume on peak strain

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Figure 10
figure 10

Comparison between peak strain proposed relationship of PPFRC with experimental test results

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9 Compressive Stress–Strain Relationship for PPFRC at Elevated Temperatures

In the structural design of heated concrete, the entire stress–strain curve, often in idealized form, must be considered as a function of temperature. In this study, a compressive stress–strain relationship for PPFRC at elevated temperatures that is based on modified Authors’ [5, 6] model and is developed by using proposed compressive strength [Equations (1) to (2)], elastic modulus [Equations (5) to (6)], and peak strain [Equations (9) to (10)] relationships.

$$ \frac{{\sigma_{c} }}{{f_{c}^{\prime } }} = \frac{{n\,\left( {\frac{{\varepsilon_{c} }}{{\varepsilon_{c}^{'} }}} \right)}}{{n - 1 + \left( {\frac{{\varepsilon_{c} }}{{\varepsilon_{c}^{'} }}} \right)^{n} }} $$
(11)
$$ n = n_{1} = \left[ {1.02 - 1.17\left( {E_{\sec } /E_{c} } \right)} \right]^{ - 0.74} \quad {\text{if}}\;\varepsilon_{c} \le \varepsilon_{c}^{'} $$
(12)
$$ n = n_{2} = n_{1} + \left( {\lambda + 28 \times \mu } \right)\quad {\text{if}}\;\varepsilon_{c} \ge \varepsilon_{c}^{'} $$
(13)
$$ \lambda = \left( {135.16 - 0.1744f_{{_{c} }}^{\prime } } \right)^{ - 0.46} $$
(14)
$$ \mu = 0.83\exp \,\left( { - 911/f_{c}^{\prime } } \right) $$
(15)
$$ E_{\sec } = f_{c}^{\prime } /\varepsilon_{c}^{\prime } $$
(16)
$$ \varepsilon_{c}^{\prime } = \left( {\frac{{f_{c}^{\prime } }}{{E_{c} }}} \right)\,\left( {\frac{\psi }{\psi - 1}} \right) $$
(17)
$$ \psi = \frac{{f_{c}^{\prime } }}{17} + 0.8 $$
(18)

where σ cf is fibre reinforced concrete stress, \( f_{cf}^{\prime } \) is maximum compressive strength of fibre reinforced concrete, n is material parameter that depends on the shape of the stress–strain curve, ε cf is fibre reinforced concrete strain, \( \varepsilon_{cf}^{\prime } \) is strain corresponding with the maximum stress \( f_{cf}^{\prime } \), n 1 is modified material parameter at the ascending branch, n 2 is modified material parameter at the descending branch, E cf is modulus of elasticity of fibre reinforced concrete, E sec is secant modulus of elasticity, n 1 is modified material parameter at the ascending branch, n 2 is modified material parameter at the descending branch, and λ, μ are coefficients of linear equation.

To account for transient creep effects, [4] considered that the total strain is composed of separate strain components. The thermal strain is a function of the temperature, and thus can be separated easily from the total strain. To calculate the transient creep strain, an assumption has to be made for the corresponding stress. This leads to an iterative solution. For more information regarding to thermal strain and/or transient strain and/or creep refer to Aslani [4].

Figure 11 shows comparisons between the proposed relationship for PPFRC against the unstressed experimental results at 20°C and 200°C reported by Noumowe [14]. Figures 12, 13, 14 show comparisons between the proposed relationship for normal and silica fume (type I and II) PPFRC against the unstressed experimental results at 20°C, 600°C, and 800°C reported by Poon et al. [17]. The proposed compressive stress–strain relationship fits the experimental results at elevated temperature well.

Figure 11
figure 11

Comparisons between proposed compressive stress–strain relationship for PPFRC against the experimental results Noumowe [14] at 20°C and 200°C

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Figure 12
figure 12

Comparisons between proposed compressive stress–strain relationship for normal concrete against the experimental results Poon et al. [17] at 20°C, 600°C, and 800°C

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Figure 13
figure 13

Comparisons between proposed compressive stress–strain relationship for silica fume type I PPFRC against the experimental results Poon et al. [17] at 20°C, 600°C, and 800°C

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Figure 14
figure 14

Comparisons between proposed compressive stress–strain relationship for silica fume type II PPFRC against the experimental results Poon et al. [17] at 20°C, 600°C, and 800°C

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Proposed models have following limitations: (1) these models did not include confinement effects, (2) they are applicable in the various range of compressive strength because normalized compressive strength is used, (3) PP fibre content is varied between 0.11% and 0.6%, (4) Common fine aggregate that is used in the database is natural river sand and type of coarse aggregate is varied, (5) heating rate range is between 0.5°C/min and 10°C/min, and (6) temperature that is used for different research is varied between 20°C and 900°C.

10 Conclusions

The following conclusions can be drawn from this study:

1. The proposed compressive stress–strain relationship of PPFRC at elevated temperature is based on authors’ model with some modifications and is developed by using the proposed compressive strength, elastic modulus and peak strain relationships that is in good agreement with the experimental test results for PPFRC at different temperatures.

2. The proposed compressive stress–strain relationship is simple and reliable for modeling the compressive behavior of PPFRC at elevated temperatures. Also, using these relationships in the finite element method (FEM) is more simple and suitable.

3. The proposed relationships for the compressive and tensile strength, elasticity modulus, modulus of rupture, and peak strain of PPFRC with different content of fiber at elevated temperature are in good reasonable agreement with the experimental results. Also, the relationships for above mechanical properties are proposed that can calculate these properties related to the fiber content.

4. The paper stressed the fact that additional tests at different temperatures are needed to investigate the role of initial compressive and tensile stresses on the PPFRC compressive strength, strain at peak stress, modulus of elasticity, free thermal strain, load induced thermal strain, creep strain, transient strain, and fire spalling.