1 Introduction

Frost durability is a key characteristic of concrete in severe environments. It is the duty of engineers to improve the durability and prolong the service life of concrete in cold regions. Powers stated that the hydraulic pressure theory accounts for frost damage in concrete [1]. In this regard, when water freezes, a tensile force is exerted and internal cracks are induced in the body of the concrete. During thawing, the water moves through the cracks and expands them; it remains present there to cause more damage when freezing occurs again. This theory was accepted by many researchers only in very saturated conditions; Even Powers replaced it with the theory of osmotic pressure [2]. Despite the fact that frost mechanisms in concrete may be fully understood by engineers, frost action is still the main reason for the deterioration of concrete structures.

It is generally accepted that air-entrained agents can relieve internal pressure generated by volume expansion. Therefore, the use of entrained air with a size of 50–200 µm and a spacing factor of less than 0.2 mm (0.008 in) protects concrete from inner destruction [3]. Air entraining also adversely affects strength properties of concrete; therefore, contractors in cold regions prefer to produce non air-entrained concrete. That’s why some researchers tried to use various types of fibers as a replacement for air-entrained agents. Their main findings are summarized in Table 1. Moreover, testing and assessing the freeze–thaw resistance of concrete with and without an air-entrained agent is complicated and time consuming. For this reason, finding proper design, production, and evaluation techniques which guarantee the frost resistance of concrete is still a vital necessity.

Table 1 Main findings of researches about fiber application in frost resistance of concrete
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In the present study, fibers were used as an alternative for AE agents, because the inclusion of fibers in concrete, mortar, and cement paste enhances many engineering properties, such as flexural strength, fracture toughness, thermal shock strength, and resistance under impact loadings [15–21].

Furthermore, a faster method based on the artificial neural network (ANN) was also used for the gathered investigated data. This technique has been recently used by researchers in various structural material characterization and modeling studies. The use of back-propagation neural network (BPNN) for modeling the behavior of conventional materials such as concrete in the state of plane stress under monotonic biaxial loading is described [22]. Some researchers demonstrate the applicability of NNs to composite material characterization. They use the BP algorithm to predict the thermal and mechanical properties of composite materials [23–25].

Subsequently, the present research may answer questions regarding which types of fibers, steel or polypropylene may increase the frost durability of concrete and whether the fibers can behave as an AE agent for releasing exerted stress. Moreover, the efficacy of the results was examined using the ANN technique.

2 Materials and methods

2.1 Mix proportions

The maximum size of the aggregate used in this study was limited to 19 mm. The physical properties of the aggregates are provided in Table 2. A water-cement ratio of 0.465 was used. Ordinary Portland cement (type I) was obtained from the Hekmatan factory in Hamedan, Iran. A high range water-reducing agent with the commercial name of Glenium 110 P was used to adjust the workability of the concrete, and an air-entrained agent of Micro-Air 100IR was used to adjust the air content of concrete. Both were obtained from the BASF Company. Polypropylene and hooked-end steel fibers were employed in this work. The geometry and the properties of the fibers are provided in Fig. 1 and Table 3, respectively.

Table 2 Properties of aggregates
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Fig. 1
figure 1

Geometry and apparent shape of a steel, b polypropylene fibers

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Table 3 Properties of fibers
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Tests were performed on 11 different mix designs. Table 4 presents the composition and fresh concrete properties of the mixes. The reference and air-entrained specimens are labeled as A0 and AE, respectively. The steel fiber concrete specimens with 0.2–1.0% fiber are shown as S1 to S5. Similarly, the polypropylene fiber concrete specimens with 0.2–0.5% by volume fractions are shown as P1 to P4, respectively. The results of the slump test revealed that, by properly selecting the super-plasticizer, it was possible to make a workable fiber-reinforced mixture.

Table 4 Mix proportions of the concrete mixtures
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Each type of freshly mixed concrete was cast into cubic (100 mm), cylindrical (100 × 200 mm), and prismatic (75 × 102 × 406 mm) shapes for compressive, splitting tensile, and strain measurement tests, respectively.

2.2 Artificial neural network

Artificial neural networks (ANNs) as information-processing systems have certain performance characteristics in common with biological NNs which consist of a large number of simple processing elements called neurons or nodes. The ANN is characterized by the following three functions: (1) the pattern of connections between the neurons, (2) the method by which the weight of connections is determined which is called training or learning algorithm, and (3) the activation function. The feedforward NNs trained with the back-propagation (BP) learning algorithm are the most widely used NNs [26, 27]. Determining the best values of all the weights is called training the ANN [28, 29].

The Levenberg–Marquardt (LM) algorithm, a standard second-order nonlinear least-squares technique based on the back-propagation process, was used to train the ANN models [30, 31]. The hyperbolic tangent function was also used as the activation function to determine the relationship between input and output of a node and a network.

In the present work, because of the behavior of hyperbolic tangent function, data was normalized in the 0.2–0.8 interval. Data scaling is important as a preprocessing stage to ensure that all variables receive equal attention during training. The output variables have to be scaled to be commensurate with the limits of the transfer functions used in the output layer [32]. Scaling of the data was done individually using the following equation [33]:

$$x_{\text{nor}} = a\left( {\frac{{x_{i} - x_{\hbox{min} } }}{{x_{\hbox{max} } - x_{\hbox{min} } }}} \right) + b$$
(1)

where x nor, x min, and x max are normalized, minimum, and maximum amount of data, respectively, and a and b are 0.2 and 0.8, respectively.

For any given ANN and set of connection weight values and training set, there exists an overall root mean squared (RMS) error of prediction. Training was finished when the network found the lowest RMSE or when the RMSE had not changed during a number of iterations.

Once the network is trained, the neural network may then be tested with unknown data, which has not been seen by the network [34, 35]. The ANN modeling was implemented within the MATLAB neural networks toolbox.

3 Testing procedures

Deterioration due to freeze-thaw cycles is assessed by various criteria, such as relative dynamic modulus of elasticity and length change (ASTM C666 [36]), mass loss (ASTM C672 [37]) and compressive strength (GOST 10060.0-95 [38], GOST 10060.1-95 [39]). In the present study, compressive and splitting tensile strength, and longitudinal strain as an index for length change are chosen.

Moreover, slow freeze–thaw cycles have been used by some researchers for better coincidence with real conditions [40–42]. In this research, a modified and more field representative testing procedure that involves slow freeze–thaw cycling is also used.

Compressive and tensile strength tests were carried out on the standard cured specimens at age 28 days. Furthermore, 60 slow freeze–thaw cycles were applied to 14-day cured specimens. The compressive strength, tensile strength, and longitudinal strain of the exposed specimens were measured every 20 freeze–thaw cycles. A system was designed to apply two freeze–thaw cycles (between −15 and +5 °C) every 24 h. The freezing and thawing periods were lengthened 8 and 4 h, respectively.

4 Experimental results

4.1 Compressive strength

The effects of the freeze–thaw cycles on compressive strength are summarized in Table 5. As shown, the compressive strength of the samples was decreased as the number of freeze–thaw cycles increased. This is due to the growth and propagation of cracks in the body of the concrete specimens [43–46].

Table 5 Compressive strength of concrete mixtures
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As the compressive strengths are relatively close in various cycles, an error analysis is performed based on error propagation. Uncertainty of residual strength is calculated by multiplying the relative error and residual strength [47]:

$${\text{Uncertainty }}\left( \% \right) = 100\left( {\frac{{f_{c}^{0} }}{{f_{c}^{60} }}} \right)\sqrt {\left( {\frac{{e_{f} }}{{f_{c}^{60} }}} \right)^{2} + \left( {\frac{{e_{f} }}{{f_{c}^{0} }}} \right)^{2} } ,$$
(2)

where \(f_{c}^{0}\) and \(f_{c}^{60}\) are compressive strengths at 0 and 60 cycles, respectively. e f is strength uncertainty, which equals to accuracy of device or 0.01 MPa. The maximum value of residual strength uncertainty is very slight and acceptable value of 0.045%.

The maximum decrease of 13.2% belonged to the reference specimens P0S0A0. As shown, the increase in cycles had no negative impact on compressive strength of the air-entrained specimens. By addition of maximum amount of steel and PP fibers, the slight 1.54 and 3.42% decrease in compressive strength are achieved, respectively. The results also showed that the higher fiber volume fractions in the specimens led to a lower decrease of compressive strength.

Figures 2 and 3 illustrate the comparison of primary and residual compressive strength after 60 freeze–thaw cycles. The results show that the frost resistance of the steel fiber specimens (with 1% fiber) was similar to that for air-entrained (P0S0AE) specimens. Minimum fiber volumes of 0.4 and 0.5% lead to a residual strength more than 95% in the polypropylene and steel fiber specimens, respectively.

Fig. 2
figure 2

Compressive strength of samples with steel fibers

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Fig. 3
figure 3

Compressive strength of samples with PP fibers

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The fibrous concrete is higher in price than AE-concrete, however, the latter is not applicable easily in special concrete structures such as protective structures which should suffer both impact loading and freeze–thaw cycles simultaneously. This is the reason that the total cost of the fibrous concrete may be accepted by consumers.

4.2 Splitting tensile strength

The results of tensile strength tests for frost-attacked specimens are given in Table 6. An error analysis is done in a similar manner mentioned for compressive strength values. It can be seen that, by increasing the number of freeze–thaw cycles to 60, a considerable decrease of 37. 9% in tensile strength was observed in the referenced specimen; however, by addition of maximum amount of steel and PP fibers, the 5.35 and 21.74% decrease in compressive strength are achieved, respectively. This considerable difference suggests superiority of steel fibers to PP ones from tensile strength point of view. Increasing the fiber volume fraction led to less deterioration. Furthermore, the freeze–thaw cycles had no impact on the tensile strength of the AE or the high volume steel fiber specimens. These results are similar to those that were obtained for the compressive strength tests. Figures 4 and 5 depict tensile strength and residual tensile strength after 60 freeze–thaw cycles

Table 6 Splitting tensile strength of concrete mixtures
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Fig. 4
figure 4

Splitting tensile strength of samples with steel fibers

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Fig. 5
figure 5

Splitting tensile strength of samples with PP fibers

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4.3 Longitudinal strain results

The measured longitudinal strains for the control, AE, and fiber-reinforced specimens are given in Table 7. It can be seen that the strains increased after freeze–thaw cycles; the maximum strain of 0.14% belonged to the reference specimen. A higher fiber volume fraction in the samples resulted in a lower increase in longitudinal strain. Figures 6 and 7 also show how strains of various mixes rose with freeze–thaw cycles. Similar to previous results of strength measurements, the positive effect of fiber volume fraction and the superiority of steel fiber compared with polypropylene fiber are obvious.

Table 7 Longitudinal strain of concrete mixtures
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Fig. 6
figure 6

Longitudinal strain of samples with steel fiber in various cycles

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Fig. 7
figure 7

Longitudinal strain of samples with PP fiber in various cycles

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Figure 8 depicts the splitting tensile strength versus the longitudinal strain after 60 freeze–thaw cycles. It can be concluded that the decrease in strain agrees with the increase in longitudinal strain at high correlation coefficients of R 2 equaling 0.99 and 0.97 for steel and PP fiber, respectively.

Fig. 8
figure 8

Splitting tensile strength of fiber samples vs. final strain

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5 Modeling results

In this study, the problem was proposed to the network models by means of four input and three output parameters. The parameters of air content, number of cycles, and percentage of steel and polypropylene fibers were selected as input variables. The output variables were compressive strength, tensile strength, and longitudinal strains.

The ANN model of compressive strength of samples is defined by the following expression:

$$f_{\text{c}}^{\text{nor}} = \left[ {\begin{array}{*{20}c} { - 1.8555} \\ { - 0.8882} \\ {0.9024} \\ \end{array} } \right]^{T} \tanh \left( {\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 0.3801} & { - 1.4636} \\ \end{array} } & {\begin{array}{*{20}c} {2.2933} & {0.1461} \\ \end{array} } \\ {\begin{array}{*{20}c} { - 0.6255} & { - 1.0427} \\ {0.1177} & {0.2845} \\ \end{array} } & {\begin{array}{*{20}c} { - 0.0613} & {0.1904} \\ {0.1147} & {0.8956} \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} P \\ S \\ \end{array} } \\ A \\ C \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.1102} \\ {0.2602} \\ \end{array} } \\ { - 2.7691} \\ \end{array} } \right]} \right) + 0.50374 ,$$
(3)

where P, S, A, and C are volume fraction percentages of polypropylene and steel fibers, air percent, and number of cycles, respectively. Since the model gives normalized compressive strength (\(f_{c}^{nor}\)) in the [0.2, 0.8] interval corresponding the [31.72, 44.89] interval of ordinary compressive strength, the following linear relationship converts it into an ordinary compressive strength value (f c), which was obtained based on linear interpolation:

$$f_{c} = 21.95 f_{c}^{nor} + 27.33$$
(4)

Similarly, the ANN model of splitting tensile strength and longitudinal strain of samples are defined by the following expressions:

$$f_{t}^{nor} = \left[ {\begin{array}{*{20}c} { - 0.6365} \\ {1.3358} \\ {0.9043} \\ \end{array} } \right]^{T} \tanh \left( {\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.2084} & { - 2.4364} \\ \end{array} } & {\begin{array}{*{20}c} {0.5829} & {0.1116} \\ \end{array} } \\ {\begin{array}{*{20}c} { - 0.0085} & {1.4231} \\ {0.4832} & { - 0.5524} \\ \end{array} } & {\begin{array}{*{20}c} {0.4952} & { - 0.1349} \\ { - 1.3084} & { - 0.4432} \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} P \\ S \\ \end{array} } \\ A \\ C \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {1.8155} \\ {2.0848} \\ \end{array} } \\ { - 1.5472} \\ \end{array} } \right]} \right) + 0.1063$$
(5)

Linear interpolation corresponding to [0.2, 0.8] and [2, 4.11] intervals leads to:

$$f_{t} = 3.52 f_{t}^{nor} + 1.296$$
(6)
$$\varepsilon^{nor} = \left[ {\begin{array}{*{20}c} { - 1.2084} \\ { - 1.4} \\ { - 1.2295} \\ \end{array} } \right]^{T} \tanh \left( {\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1.7029} & { - 0.6266} \\ \end{array} } & {\begin{array}{*{20}c} {1.1231} & {0.2233} \\ \end{array} } \\ {\begin{array}{*{20}c} {0.8454} & {0.9907} \\ {0.0899} & {0.1871} \\ \end{array} } & {\begin{array}{*{20}c} {0.0549} & { - 0.5268} \\ { - 0.5584} & { - 1.3015} \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} P \\ S \\ \end{array} } \\ A \\ C \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 1.6411} \\ {1.7593} \\ \end{array} } \\ { - 1.623} \\ \end{array} } \right]} \right) - 1.6018$$
(7)

Similarly, Linear interpolation corresponding to [0.2, 0.8] and [0, 0.14] intervals leads to:

$$\varepsilon = 0.233\varepsilon^{nor} - 0.0467,$$
(8)

where f t and ε are splitting tensile strength and longitudinal strain, respectively. The performance of training and test sets are shown in Figs. 9, 10 and 11. It can be seen that the ANN model predicted the compressive strength, splitting tensile strength, and longitudinal strain with an R 2 of 0.98, 0.99, and 0.99, respectively.

Fig. 9
figure 9

Scatter plot of the observed versus predicted compressive strength of samples

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Fig. 10
figure 10

Scatter plot of the observed versus predicted splitting tensile strength of samples

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Fig. 11
figure 11

Scatter plot of the observed versus predicted longitudinal strain of samples

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6 Conclusions

  1. 1.

    The effect of fiber volume on the frost resistance of polypropylene and steel fiber specimens exposed to 60 slow freeze–thaw cycles was examined. The results were compared with the air-entrained specimen. The following conclusions can be drawn from the results.

  2. 2.

    Inclusion of fibers into the concrete specimens enhanced both the compressive and splitting strength of the specimens compared to those without fibers. The maximum strength of 47.5 MPa belonged to the 1% steel fiber specimens.

  3. 3.

    After 60 freeze–thaw cycles, the compressive strength of the specimens decreased 1.54 and 3.42% by addition of maximum amount of steel and PP fibers, respectively. The tensile strength of the specimens also decreased 5.35 and 21.74% by addition of maximum amount of steel and PP fibers, respectively, but the strain of the specimens increased. As expected, the air-entrained sample attained the best resistance against freeze–thaw cycles.

  4. 4.

    Introducing fibers at a high fraction volume, in particular steel fibers, into the concrete mixtures prevented frost deterioration in the specimens. Minimum fiber volumes of 0.4 and 0.5% were required for frost resistance in the polypropylene and steel fiber specimens, respectively.

  5. 5.

    The frost resistance of the steel fiber specimen (1% fiber volume) was similar to that of the air-entrained ones.

  6. 6.

    From tensile strength point of view, Steel fiber is far more superior to PP, but steel fiber is only slightly better in compressive strength and strain changes of specimens.

  7. 7.

    The best correlation was obtained between decrease in splitting tensile strength and increase in strain of frost-exposed samples; this revealed that strain increase may be considered as a proper index for frost actions.

  8. 8.

    The ANN was found to be capable of generalizing between the input variables and the output of compressive and tensile strength and strain with good reasonable predictions.

  9. 9.

    From the viewpoints of strength and durability, steel fibers can improve frost durability without sacrificing strength. These important results can relieve engineers’ anxiety about using air-entrained agents in concrete. Furthermore, AE-concrete is not applicable easily in special concrete structures such as protective structures, which should suffer both impact loading and freeze–thaw cycles simultaneously. This is the reason that the total cost of the fibrous concrete may be accepted by consumers.