Computational-experimental method for determining the height of steel fiber sedimentation in Ultra-High Performance Self-Compacting Concrete

. The uniform distribution of steel fibers in self-compacting Ultra-High Performance Steel Fiber Reinforced Concrete is one of the factors that determine the properties of the hardened material. In self-compacting concrete mixes characterized by low yield stress and plastic viscosity, steel fibers can settle under gravity, which can lead to a decrease in strength. In this study, a computational-experimental method is proposed for determining the height of steel fiber sedimentation, as well as the segregation index of a steel fiber reinforced concrete mixture. Mixtures were tested with a superplasticizer content of 1, 1.3, and 1.6% with a slump flow diameter of 288, 356, and 378 mm. According to the results of calculations performed according to the proposed method, it was found that the height of steel fiber sedimentation is 3.1, 5.9, and 22.8 mm in mixtures with a plasticizer content of 1, 1.3 and 1.6%. The calculated value of the segregation index of the steel fiber reinforced concrete is quite close to the experimental one with a correlation coefficient of 0.998. Based on the results obtained, a criterion was proposed for determining the maximum allowable height of fiber sedimentation, taking into account the actual height of the concreted structure, which can be used in the concrete mix design.


Introduction
Ultra-High Performance Fiber Reinforced Concrete (UHPFRC) is a relatively new structural material with improved physical and mechanical characteristics and durability.Compressive strength UHPFRC is in the range of 130…250 MPa [1][2][3].UHPFRC contains steel fiber in the amount of 1.5 ... 3% by volume.Steel fiber is introduced to increase the axial tensile strength up to 15...20 MPa, as well as to reduce the brittleness of the material [4,5].
In the vast majority of cases, freshly prepared UHPFRC is self-compacting [6][7][8].The use of self-compacting concrete mixtures has the following advantages: reduced labor intensity during casting, less air entrapped in the mixture [9], as well as the ability to control the orientation of the steel fiber in the desired direction due to the self-alignment of the fibers in the flow direction [10][11], which makes it possible to significantly increase the strength and fracture energy of the material.
A significant disadvantage of self-compacting concrete mixes is the risk of segregation of components under the action of gravity.Since there is no coarse aggregate in UHPFRC (maximum particle size is limited to 1 mm), there is a risk of settling only steel fibers due to the high density difference between steel and the surrounding concrete matrix.
To determine the criterion for the onset of steel fiber sedimentation, consider a sphere placed in a concrete mixture (Figure 1), which is subject to 3 forces: the Archimedes buoyancy force   , the resistance exerted by the concrete mixture due to the presence of yield stress   and gravity force   .•  0 , (2) Where: d -diameter of the particle, m;   -density of particle,   3  ⁄ ; g -free fall acceleration,   2  ⁄ ;   -density of fresh concrete surrounding the particle,   3  ⁄ ;  0 -yield stress of fresh concrete, Pa.From equation 2, we can obtain the value of the critical yield stress at which sedimentation of a particle will occur under the action of gravity: Taking as initial data the density values of steel and concrete mixture equal to 7800 и 2400   3  ⁄ , respectively, we can determine the critical value of the yield stress, below which the fiber will sediment.Particle diameter d is defined as the diameter of a sphere of equivalent volume ( = √1,5 •   2 •   3 ).For example, let's take the 2 most common fiber sizes: 13/0.2 and 30/1 mm.The equivalent diameter will be 0.92 and 3.56 mm, respectively.Substituting these values into equation 3, we obtain the values of the critical yield stress to 32 and 125 Pa for fibers 1 and 2, respectively.The value of the yield stress of most selfcompacting UHPC is in the range of 10...50 Pa [12-15], which is lower than the calculated values.This fact suggests that the steel fibers in the self-compacting UHPFRC will sediment under gravity, which can lead to deterioration in the strength characteristics of the hardened material.
The yield stress of the concrete mixture after the end of the mixing process increases with time, which is associated with the process of coagulation of small particles of cement and mineral fillers and the formation of a spatial structure [16].Since UHPC contains up to 60% by weight of finely ground components, this process proceeds more intensively compared to concretes of ordinary classes.If the dependence  0 = () is known, then it is possible to determine for how long the fiber will sediment in the concrete mixture.This process will proceed until the value of the yield stress reaches the critical value for a given fiber size, that is, the equality will be fulfilled:  0 =  0, .
The sedimentation rate of a fiber can be calculated from the Stokes equation, assuming that the fluid flow around the fiber is laminar: where () -function of the plastic viscosity change over time,  • .By integrating equation 4, it is possible to calculate to what depth the center of gravity of the i-th fiber will move from the moment the steel-fiber-concrete mixture is laid into the formwork: where   -the duration of sedimentation, which is defined as the amount of time required to establish equality  0 =  0, , min;  0, -is the coordinate of the center of gravity of the i-th fiber at the initial moment, m.
To solve equation 5, it is necessary to know the duration of sedimentation and the function  = (), which depend on a large number of factors and must be determined experimentally for each composition.
The purpose of this work was to establish the relationship  = () for compositions with different superplasticizer content, to calculate the depth of fiber sedimentation  , and the segregation index of the steel fiber reinforced concrete mixture   and to compare it with experimental data.
The procedure for carrying out calculations and experimental studies is given in the next section.

Materials and mix composition
Portland cement CEM I 42.5 N with a specific surface area of 345 m2kg⁄ and an activity of 47.5 MPa at the time of 28 days was used as a binder.Condensed silica fume (CSF) containing >85% amorphous silica was used as an active mineral additive.Limestone powder (LP) with a specific surface area of 280 m2kg⁄ was used as a mineral filler.The content of CSF and LP was constant and amounted to 15 and 20% by weight of the cement, respectively.As a fine aggregate, quartz sand of fractions 0.1-0.4 and 0.4-0.8 was used in a ratio of 30:70 by volume, which corresponds to the maximum particle packing density.Brass coated corrugated steel fiber 15 mm long and 0.3 mm in diameter (l/d=50) was used in the amount of 1.5% of the mixture volume.A superplasticizer based on polycarboxylate ethers Sika ViscoCrete 24 HE was used.
Table 1 shows the composition of concrete mixtures.To establish the dependences of the change of the rheological characteristics of concrete mixtures over time, 3 compositions without fiber were tested.The content of the superplasticizer was 1, 1.3 and 1.6% of the mass of cement, and the flow diameter on the Hagermann cone was 288, 356 and 378 mm respectively.To determine the segregation index of the steel fiber reinforced concrete mixture, compositions with the same content of the plasticizer were used, but with the addition of 1.5% steel fiber by volume.

Segregation index of the steel fiber reinforced concrete mixture
The segregation index of the steel fiber reinforced concrete mixture was determined according to GOST R 59535-2021.For this, the freshly prepared steel fiber reinforced concrete mixture was placed in a cylinder 300 mm high and 150 mm in diameter without additional compaction.After about 30 minutes, approximately half of the mixture is removed from the mold, and the height of the selected layer ℎ  is determined using a ruler.The height of the remaining layer is determined by the equation: where ℎ -the height of the cylinder in which the mixture is placed, mm Then each layer is thoroughly washed with water until the concrete mixture is completely removed.The fiber remaining after washing is dried to constant weight and weighed.
The segregation index of the steel fiber reinforced concrete mixture is determined by the following formula: where  , ,  , -the mass of steel fiber from the upper and lower layers, g For each formulation, the result is the arithmetic mean of two replicates.

Yield stress and plastic viscosity
The yield stress and plastic viscosity of liquids are determined on rheometers of various designs, which make it possible to obtain a "stress-shear rate" curve, from which the rheological characteristics are then obtained.The vast majority of construction laboratories do not have such measuring equipment due to its high cost and complexity in operation and maintenance.In order to use the results of the work in production, the yield stress and plastic viscosity were determined indirectly by measuring the flow time and the diameter of the flow of the mixture from the Hagermann cone [17].Test procedure is as follows: -the cone is mounted on a smooth Plexiglas plate moistened with a thin layer of water; -the cone is filled with a fresh concrete mixture in one step, the excess mixture is cut off with a knife or a ruler; -within 2-3 seconds the cone rises smoothly.At the same time, the stopwatch is turned on and the time required to obtain the diameter of the spread   = 200 mm is recorded; -after the mixture has stopped spreading, the diameter of the spread SF and the height of the resulting cake ℎ  are measured.
The yield stress of a mixture is determined by the following equation: where   -volume of the cone,  3 ; SF -slump flow of the fresh concrete mixture, m.The plastic viscosity of the mixture is determined by the equation: where ℎ  -cone height, m; The yield stress and plastic viscosity for each composition were determined immediately after mixing, as well as after 3 and 6 minutes after keeping in the cone.To do this, the freshly prepared mixture was placed in 3 cones, which then successively went up (Figure 3).According to the data obtained by the method of least squares, dependencies  0 = () and  = () were established to determine the height of sedimentation  , .

Calculated value of the segregation index of the steel fiber reinforced concrete mixture
By solving the integral equation 5, the height of sedimentation of the center of gravity of the fiber located at the top of the container ( 0, = 0) can be obtained.Assuming that each fiber settles independently of the others, each fiber will move to the calculated height along the vertical axis   .Thus, it is possible to determine the actual volume occupied by the steel fiber in the upper and lower parts of the container, having a height ℎ  and ℎ  , as well as a mass of steel fiber  ,, and  ,, (Figure 4):  The calculated value of the segregation index of the steel fiber reinforced concrete mixture  , is determined by the formula: 3 Results and discussion

Coefficient of segregation of steel fiber
The graphs of Figure 5 show the relationship between the experimentally determined segregation index and the slump flow diameter of the concrete mixture and the dosage of the superplasticizing additive.With an increase in the dosage of the superplasticizer from 1 to 1.6%, the flow diameter of the concrete mixture increased from 290 to 356 mm due to the adsorption of additive molecules on small particles of cement and mineral fillers, which prevents their coagulation and leads to the release of water contained inside the floccules from the smallest grains.All this leads to a decrease of the yield stress of the concrete mixture and an increase of the slump flow [18].An increase in the amount of additive leads to a decrease of the segregation index, which was 1.0, 0.77, and 0.23 with a plasticizer content of 1.0, 1.3, and 1.6% by weight of cement, respectively.With an increase in the amount of plasticizer, the yield stress of the mixture decreases and, as a result, more pronounced sedimentation of the fiber is observed.

Yield stress and plastic viscosity of fresh concrete mixture
The results of determining the yield stress and plastic viscosity of concrete mixtures calculated by equations 8, and 9 are presented in table 2. For further analysis, the obtained data were normalized concerning the initial value at t = 0 min.The graphs in Figure 5 show the change of the relative yield stress and plastic viscosity of concrete mixes with different plasticizer content over time.From the graph of Figure 5-a, it can be seen that the change of the relative yield stress of concrete mixtures is satisfactorily described by the equation  =  + , regardless of the amount of plasticizer, which was also found in [19].The most intense increase in shear stress is observed in the composition with 1% additive; as the content of the plasticizer increases, the intensity decreases, which is noted by a smaller slope angle between the approximating line and the abscissa axis.After 6 minutes, an increase in the ultimate shear stress of 1.76, 1.73, and 1.59 times relative to the initial value for compositions with 1, 1.3, and 1.6% superplasticizer was found.
From the graph in Figure 5-b, it can be seen that the relative plastic viscosity changes non-linearly.The experimental data were approximated by an exponential function  =   .It was found that the intensity of the change significantly depends on the amount of superplasticizer in the composition of the mixture.After 6 minutes, an increase in the plastic viscosity of the mixture was found to be 2.53, 2.24, and 1.37 times for compositions with 1, 1.3, and 1.6% superplasticizer.
Table 3 shows the values of the coefficients of empirical equations describing the change in the relative yield stress and plastic viscosity of the concrete mixture.

Prediction of sedimentation height and coefficient of sedimentation of steel fiber
To determine the sedimentation height of the center of gravity of the i-th fiber  , for each composition, the following expression was substituted in equation 5 instead of (): where  0 -the plastic viscosity of the mixture at time t = 0 min, Pa • s; b -an empirical coefficient (Table 2), 1  ⁄ .The general solution of the integral equation 5 is the following expression: Using the values of coefficient b from Table 2 and Equation 14, the depth of steel fiber sedimentation at various points in time was calculated, and the segregation index of the steel fiber concrete mixture  , was determined using Equations 10-12.The graphs in Figure 6 show the change in the depth of fiber sedimentation over time in mixtures with different contents of the superplasticizing additive, as well as a comparison of the actual and calculated segregation index.Figure 6-a shows that the active phase of fiber sedimentation occurs during the first 20…50 minutes, and the higher the superplasticizer dosage, the longer this period.Intensive settling of steel fibers in concrete mixtures occurs in the first 22, 28, and 50 minutes at a superplasticizer content of 1, 1.3, and 1.6%.During the hydration of Portland cement, CSH gel on the surface of the grains overlap the side chains of polycarboxylate molecules.If there is a reserve of non-adsorbed molecules in the water surrounding the particles, they will be adsorbed on the hydration products, which is visually expressed in greater preservation of the properties of the concrete mixture [20].At high dosages of the additive, a larger reserve of not yet adsorbed molecules remains in the pore space, as a result of which the viscosity of the mixture increases more slowly, allowing the fiber to move a greater distance under the influence of gravity.Figure 6-b shows a comparison between the calculated and experimental value of the segregation index.The results have good convergence.The correlation coefficient is 0.998.

Practical implementation
Consider a product molded horizontally (Figure 7).After casting the fiber reinforced concrete mixture, the fiber will sediment to a distance   .As a result, the part of the section located on the side of the formed surface will have a lower tensile strength compared to the rest, which can be critical if the product is used in a vertical position (for example, wall panels).Thus, in each specific case, the fiber sedimentation height should be limited to the value at which the ratio of the tensile strength of the upper and lower parts of the product will lie within the coefficient of variation (13.5%), which is expressed as follows: where  ,,1 ,  ,,2 -the tensile strength of the material in the upper and lower parts of the section, MPa.
The tensile strength of fiber reinforced concrete, other things being equal, depends on the volume content of steel fibers.Knowing the value   and height of the product   , we can determine the actual fiber content in the upper and lower half: , = ( where   -the volume content of fiber in the composition of the material, %. The coefficient K will be equal to the ratio of the actual volume content of the fiber in the upper and lower parts of the product: If К =   (equation 15), then formula 19 will look like this:  , = 0,036 •   (20) The resulting equations can be used in the selection of concrete composition.To do this, the critical sedimentation height of the fiber  , is first calculated using formula 20.Then several variants of the concrete matrix are tested and the coefficient b in equation 13 is determined.For each composition, the sedimentation height   is calculated using formula 14.Calculations can be made for different types of fibers.After that, the composition is selected that satisfies the requirements for workability and for which the condition   <  , is also fulfilled.

Conclusions
Based on the results obtained, the following conclusions can be drawn:  Increasing the content of the superplasticizer from 1 to 1.6% by weight of cement reduced the segregation index of the steel fiber reinforced concrete mixture from 1 to 0.23;  The intensity of change of the yield stress and plastic viscosity of mixtures with a high content of super plasticizing additives is less;  A computational-experimental method for determining the height of steel fiber sedimentation in self-compacting Ultra-High Performance Fiber Reinforced

Fig. 1 .
Fig. 1.Forces acting on the particle in the fresh concrete mixture.Sedimentation of a spherical particle occurs if the following inequality is fulfilled:  >   +

Fig. 3 .
Fig. 3. Determination of rheological properties of fresh concrete mixture over time.

Fig. 4 .
Fig. 4. Sketch for determination of the calculated value of the segregation index of the steel fiber reinforced concrete mixture.

Fig. 4 .
Fig. 4. a) Relationship between slump flow of fresh concrete mixture and superplasticizer dosage; b) relationship between fiber segregation coefficient and superplasticizer dosage.

Fig. 5 .
Fig. 5. a) Relative yield stress of fresh concrete mixture as a function of time; b) Relative plastic viscosity of fresh concrete mixture as a function of time.

Fig. 6 .
Fig. 6. a) Sedimentation of steel fiber as a function of time; b) Comparison between  , from eq. 10-12 and   from experiments.

Fig. 7 .
Fig. 7. Change in tensile strength due to the fiber segregation.

From formula 18 ,
the maximum allowable fiber sedimentation height can be expressed:

Table 2 .
Yield stress and plastic viscosity of UHPC depending on superplasticizer dosage and resting time.

Table 3 .
Values of empirical coefficients a and b.