Acta Polytechnica doi:10.14311/AP.2018.58.0355 Acta Polytechnica 58(6):355–364, 2018 © Czech Technical University in Prague, 2018 available online at http://ojs.cvut.cz/ojs/index.php/ap ULTRA-HIGH-PERFORMANCE FIBRE-REINFORCED CONCRETE UNDER HIGH-VELOCITY PROJECTILE IMPACT. PART II. APPLICABILITY OF PREDICTION MODELS Sebastjan Kravanjaa, Radoslav Sovjákb, ∗ a Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova cesta 2, Ljubljana 1000, Slovenia b Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: sovjak@fsv.cvut.cz Abstract. Semi-infinite targets of Ultra-High-Performance Fibre-Reinforced Concrete with various fibre volume fractions were subjected to the high-velocity projectile impact using in-service bullets. In this study, a variety of empirical and semi-analytical models for prediction of the depth of penetration and mass ejection were evaluated with respect to the experimental results. Models for the depth of penetration and spalling mass ejection were revisited and applied both with deformable and non- deformable projectiles parameters. The applicability of the prediction models was assessed through a statistical comparison of values from models with experimental results. The evaluation of the applicability was made through the newly proposed measure of a relative prediction accuracy for model selection and model estimation, which was verified with established statistical accuracy evaluations, such as accuracy ratio, logarithmic standard deviation and correlation coefficient. The best fit to the experimental readings was provided by newer semi-analytical models, which are incorporating additional concrete parameters beside compressive strength while the majority of older models failed to provide sufficient accuracy. Keywords: projectile impact; UHPFRC; prediction models; depth of penetration; mass ejection. 1. Introduction A set of results for Ultra-High-Performance Fibre- Reinforced Concrete (UHPFRC) with various fibre volume fractions under a high-velocity projectile im- pact was gathered for both rigid (non-deformable) and soft (deformable) projectile [1]. The UHPFRC was chosen due to its exceptional mechanical properties and impact resistance. In the framework of this study, UHPFRCs with unconfined compressive strengths over 110 MPa were reinforced with discrete steel fibres in five different volumetric fractions ranging from 0.125 % to 2.0 %, while additional control specimens without fibre inclusion (i.e., 0 %, plain UHPC) were tested as well. The rigid projectile was provided by a bullet with a full-metal jacket and mild-steel core (FMJ-MSC or MSC) while the soft projectile was provided by a bullet with a full-metal jacket and soft-lead core (FMJ-SLC or SLC). Both projectiles were chosen because they represent a world’s widespread intermediate cartridge for the military assault rifle AK-47. For semi-infinite targets investigated in this study, loaded with high-velocity projectile impact, a number of empirical and semi-analytical prediction models for predicting the penetration depth and mass ejec- tion were tested and evaluated through comparison to the experimental results. Predicting the effect of the projectile impact on cementitious composite is a very complex problem and although empirical formulas do exist, most accurate methods are semi-analytical mod- els and numerical simulations. Empirical formulas are based on the coefficients calibration and fitting of the empirical constants and do not contain any physical substance, while semi-analytical models are developed on the basis of a physical concept and then calibrated to experimental results. A large number of predictive models were used and its applicability for the case of this study was evaluated and discussed. 2. Theoretical background Throughout modern history, numerous empirical and semi-analytical prediction models have been developed and tested for the means of calculating the prediction of the penetration depth due to projectile impact on solid materials. The purpose was to verify if any of the existent penetration models fit the results gath- ered from the experimental part of this study. By this means, in total 42 penetration depth prediction models, which were publicly accessible, were evaluated and their results were compared with the results of the experimental analysis and between each other. The accuracy of the prediction models was evaluated with the use of value forecast model accuracy assessment∑ ln2 Q proposed by Tofallis [2], for which it was proven to be less biased than other known statistical 355 http://dx.doi.org/10.14311/AP.2018.58.0355 http://ojs.cvut.cz/ojs/index.php/ap Sebastjan Kravanja, Radoslav Sovják Acta Polytechnica assessments and can be calculated using ∑ ln2 Q = n∑ i=1 ln2 Qi, where Qi is named accuracy ratio and is calculated as Qi = Ft At , where Ft is predicted (forecast) value and At is the actual value of the compared quantity. In this case, the predicted value represented the result from pre- diction models (whether it was penetration depth or mass ejection) and the actual value represented the result from the experimental investigation. The model and experimental results were gathered for each of the six different fibre volumetric ratios (including the plain UHPC mixture), so the summation index i of the square of logarithmic values of accuracy ratios is assuming values from 1 to 6. Higher accuracy of the prediction models results in a lower value from the accuracy assessment equation; therefore, the most accurate models had the lowest accuracy estimation values. The efficiency of the newly proposed loga- rithmic assessment by Tofallis was verified by using much more established logarithmic standard deviation (LSD), which is calculated by the equation LSD = √∑ (s2/2 − ln2 Qi) n − 1 , where s2 is the sample variance of the accuracy ratio Qi. In this assessment, the lower value means lower error and higher accuracy of the prediction model. In addition, the statistical quantity of the correlation coefficient was calculated for each model with respect to the experimental results in order to evaluate the relative relation between these values [3]. In the vast majority of the cases of penetration mod- els, the unconfined compressive strength of a concrete target is a major parameter on the resistance side of the model. The major supposition in the develop- ment of prediction models was that the penetration depth and crater volume are in inverse correlation to the unconfined compressive strength of the con- crete. However, Kennedy [4] proposed, according to the one-dimensional theory of wave propagation, that significant reflected tensile wave also appears in the target with finite geometries. The tensile strength affects the spalling and scabbing part, and therefore cannot be neglected. This was appropriately incorpo- rated in the newly proposed semi-analytical model by Hwang et al. [5]. 2.1. Parameters. It is important to mention that the majority of the empirical and semi-analytical models with empirical factors were developed through the curve fitting and regression analysis on the basis of experimental re- sults, therefore establishing a strict application range Symbol Parameter Unit Impact parameters X Predicted penetration depth m vi Projectile impact velocity m/s D Diameter of the projectile m M Mass of the projectile kg CRH Calibre Radius Head – N∗ Nose shape factor – ρp Density of the projectile core material kg/m3 Ep Elastic modulus of the projectile core material Pa Yp Yield strength of the projectile core material Pa Concrete target parameters f′c Unconfined compressive strength Pa ρc Density kg/m3 ft Direct tensile strength Pa p Fibre volumetric fraction % sa Maximum diameter of coarse aggregate m h Target thickness m Table 1. Parameters used in prediction models. of validity [6]. The application range is a set of inter- vals of parameters, for which the models have been tested, proven or for which the empirical factors were calculated. In the majority of models, these param- eters are mass, diameter and impact velocity of the projectile and unconfined compressive strength of the concrete target. In penetration and mass ejection prediction models, a various set of parameters can be used. The param- eters are divided into two parts: impact parameters, which are considering the projectile physical proper- ties as well as impact velocity or impact kinetic energy, and target parameters, which are considering physical properties of the concrete target (Table 1). 3. The depth of penetration for non-deformable projectiles Five of the most known models and newest models, which were used in the study, are briefly described in this chapter. The original equation of the National Defence Research Committee (NDRC) was presented in 1946 and was based on the physical model of the impact process [7]. In this model, the force on the projectile is assumed to increase linearly to a pene- tration depth of x = 2d and further on remains at a constant value. Later, it has been shown that this model does not give an appropriate description of the penetration process, however, it was further modified to calculate total penetration depth with respect to the boundary condition of x = 2d. The penetration 356 vol. 58 no. 6/2018 Ultra-High-Performance Fibre-Reinforced Concrete. Part II. model consisted of impact parameters of projectile velocity, diameter and mass, and additional empirical factors of material dependence and the nose shape fac- tor. The latter two were later determined and a first modified NDRC formula was developed through the definition of the impact function G. In 1978, Kar [7] revised the NDRC formula to consider the type of the projectile material using a regression analysis. The type of the material was incorporated through the use of the ratio between Young’s elastic modulus of the deformable projectile material and the referenced elastic modulus of steel, and the modified impact function was proposed [8]. In 2012, Almusallam [9] proposed and further modified the NDRC equation to incorporate the effect of the fibre reinforcement in the concrete matrix. An exponential term was introduced to the impact function G while the general form of the NDRC equation remained unaltered. In 2015, the authors proposed the equation for calculating fibre empirical constants, based on fibre’s geometrical and mechanical properties [9, 10]. 4. Hwang et al. prediction model A new model for prediction of the penetration depth was presented by Hwang et al. [5]. The whole model is based on the principle of the energy conservation law, in which the authors are considering two main energies, which appear at the impact process: kinetic energy of the projectile EK and resistant energy of the concrete target ER. The latter is further divided into three resistant energies, which appear in the target of finite geometries: spalling ES, tunnelling ET and scabbing EC resistant energy. We have: EK = m 2 (v2i − v 2 r ), ER = ES + ET + EC. With respect to the energy conservation law, the ki- netic energy EK on the influence (impact) side of the problem is the same as the mobilized resistant energy ER of the concrete target. In the continuation, a brief summary of the energy model calculation is presented. The original model is entirely the work of the authors of this model [5] and is presented just with the intention of explaining its basic assumptions. 4.1. Spalling resistant energy The spalling energy is dissipated due to the reflected impact force (reflected tension wave) on the proximal face of the target. The dissipation of the energy emerges as an ejection of a concrete part, which is idealized with the truncated cone. The resistance force FS of the concrete cone is defined as FS = ftd ( tsbs tan θs + πd2 4 ) kskbs, where ftd is the concrete tensile strength increased by the strain rate according to the fib Model code 2010 [11]; ts is the allowable spalling depth; θs is the average failure cone angle; bs is the average perimeter of the concrete cone; ks is the size effect factor and kbs is the stress concentration factor. The proposed formula for the average perimeter of the concrete cone bs with a diameter of d + 2ts tan θs was corrected from bs = π(d + ts tan θs) to bs = π(d + 2ts tan θs) by the reason of the inferred error in the text, however, it was used only in the modified version of the model. Allowable spalling depth is the maximal depth of the ejected concrete cone. The model is proposing an estimation of the allowable spalling depth using four empirical factors, which take into consideration concrete target’s thickness, steel fibre volumetric ratio, concrete’s density and maximum size of the coarse aggregate. The average tangent value of an idealized cone angle was proposed by the authors of the model, which was experimentally acquired through larger se- ries of experimental data. For an ogive nose projectile, it was proposed to be 1.55, which corresponds to a cone failure angle of 57.17°. The expression of the resistance force FS is mul- tiplying the designed concrete tensile strength with truncated cone surface area (without bottom surface area on the face of the specimen). It is then multiplied by the size effect factor, which is determined by the concrete target’s thickness and stress concentration factor, which is an empirical factor proposed to a value 1.25 by authors. The non-conventional term for the cone surface area was not in a good agreement with experimental results of the crater cone area and it gave underestimated results and was, therefore, re- placed with the conventional geometrical equation for a lateral cone surface area Al: Al = π(r1 + r2) √ (r1 − r2)2 + t2s , where r1 = d + 2ts tan θs 2 , r2 = d 2 are the radii of the bottom and top circle surfaces, respectively, and ts is the originally proposed allowable spalling depth. The resistance force FS was then calculated by the modified expression FS = ( Al + πd2 4 ) ftdkskbs. The spalling resistant energy could then be determined by the equation ES = FS VSC ASP , where VSC is the volume of an idealized concrete cone and ASP is the projected area of the idealized concrete cone on the proximal face of the target. Both of these estimated quantities could also be compared with measured values. 357 Sebastjan Kravanja, Radoslav Sovják Acta Polytechnica 4.2. Tunnelling resistant energy After the spalling region, the projectile continues its way through the concrete material by a tight tun- nelling penetration. The projectile velocity is de- creased by the bond resistance between the projectile and concrete. The authors of the model suggest the following expression for the bond resistance Ft = πdttψτd, where tt is the allowable tunnelling depth, ψ is the nose shape factor (0.7 for ogive nosed) and τd is the bond strength, increased by a strain effect depending on the strain rate affecting the compressive strength according to the fib Model code 2010 [11]. The tun- nelling resistant energy could then be calculated with the equation ET = Ft ρpAp m. 4.3. Scabbing resistant energy The scabbing failure mode is, according to the authors, similar to the spalling failure mode, and therefore uses the same method for the resistant energy calculation. An allowable scabbing depth tc is assumed to be equal to the allowable spalling depth, while the average tangent value of an idealized failure cone angle on the scabbing part is proposed to be 2.0, regardless of the projectile nose shape. In addition, also in a part of the scabbing resistance energy calculation, the lateral area of the cone was replaced with the equation, which was presented in the spalling resistant energy section. The energy model with modified expressions for the lateral area of the concrete cone in both spalling and scabbing resistance energy section and corrected equation for the crater perimeter was labelled as Mod. Hwang et al. 4.4. Penetration depth calculation The total penetration depth (x) was calculated by an equation that is derived on the basis of the aforemen- tioned assumptions [5] for targets with semi-infinite geometries, where perforation does not occur, as x = mv2i 2ER,max (h − ts). Both the models, one with exact original formulas (Hwang et al.) and one with two modified expressions for the spalling and scabbing crater perimeters and failure cone lateral area (Mod. Hwang et al.) were compared with the data from the experimental study. 5. The depth of penetration for deformable projectiles An analytical algebraic formula for the penetration prediction of a deformable projectile impact into a deformable target material was proposed by Rubin and Yarin [12]. It is divided into two stages of pen- etration: first, a deformable penetration stage and second, a rigid penetration stage. In the first stage of the penetration P I, the projectile’s head deforms into a mushroom-like shape and its tail remains rigid – a projectile is eroding with its length decreasing while the penetration velocity is relatively constant. It is assumed by the authors that after the first stage, the remaining mushroom-like head and rigid tail continue to penetrate the target as a rigid body – the second stage P II begins. The total penetration x is calcu- lated by summarizing the penetration depths from both stages x = P I + P II. The first stage penetration depth P I can be calcu- lated with the use of the following formulas: P I = u vi − u Ler, Ler = (L0 − l0) ( 1 − exp −ρp(vi − u)2 2Yp ) , where u is the decreased velocity of the projectile due to the pressure at the target/projectile interface, calculated with the use of the projectile and target densities and assumed as constant, Ler is the length of the portion of the rigid tail of the projectile that has been eroded away, L0 is the initial projectile length and l0 is the length of the mushroom-like section. The model is derived for the penetration of the cylin- drical rod with an initial length L0. The analytical formula only considers cylinder-shaped blunt-nosed projectile. In this study, the head of the projectile was not blunt. However, since the resulting values of this analytical model are strongly influenced by the projectile length, an assumption of the equivalent length of the idealized cylindrical projectile was made and used in the equation. The equivalent length of the lead core was calculated by calculating the actual total volume of the core (by the cylinder and trun- cated cone) and then dividing it by the actual area of a cylindrical part of the deformable projectile core with a diameter of 6.32 mm, yielding an equivalent length L0 = 13.81 mm, which was further used in this model. Although the resulting values of the Rubin and Yarin analytical model are strongly influenced by the projectile length, the effect of the nose shape for a deformable projectile should also be noted. Walker and Anderson [13] reported that the conical nosed projectiles did not perform as well as the blunt-nosed projectiles when the target was sufficiently hard to cause a significant projectile erosion. It can be de- rived that the deformable ogive-nosed projectile used in this study, which undergone a significant erosion, possibly induces a smaller damage than a blunt-nosed projectile. Results from the analytical model should be, therefore, on the safe side. Again, it is important to mention that an equivalent length was established with an assumption that the major part of the crater- ing damage is governed by the length of the projectile. 358 vol. 58 no. 6/2018 Ultra-High-Performance Fibre-Reinforced Concrete. Part II. Figure 1. Comparison between experimental and representative prediction model results for DOP for non-deformable FMJ-MSC projectile impact on UHPFRC and linear regression line for experimental data. 6. Spalling mass ejection For the mass ejection prediction, two models were found accessible in public literature and assessed with corresponding input data. The model of Sier- akowski [14] is based on the physical principle of im- pulse. It is calculated by integrating resultant force with respect to time; however, in the case of an object of a constant mass (rigid projectile), the impulse can be expressed as a difference in momentum. The major supposition in this empirical model is that the volume of the impact crater is in inverse correlation with the square root of the unconfined compressive strength of the concrete. A prediction for the ejected mass from the front and the rear faces of fibre reinforced concrete slabs subjected to impact loads was proposed by Almusallam [15]. The mass defragmented from the front face was modelled by a combination of a crater and a tunnel in a manner similar to the one in the pre- diction model by Hwang et al., however, the shape of the front face crater at the proximal face was assumed to be elliptical and the shape of the crater at the pen- etration depth was assumed to be circular, whereas the transition was considered to be elliptical. The authors also proposed prediction equations for this quantity [16], which yielded a suitable usability of the prediction model in terms of designing of structures. The prediction model for equivalent crater diameter is incorporating the fibre effect directly through the use of the reinforcing index, whereas the prediction model for total mass ejection is incorporating this effect through the use of a penetration depth by Al- musallam et al. modified NDRC, equivalent crater diameter and additional tunnelling depth. FMJ-MSC FMJ-SLC Modulus of elasticity 210 44.3 Ep (GPa) Density ρp (kg/m3) 7850 10735 Yield strength Yp (MPa) 552 73.3 Table 2. Assessed input parameters regarding pro- jectile’s core material properties for consideration of deformability in prediction models. 7. Results for depth of penetration In total, 42 prediction models were tested for the prediction of the penetration depth. The projectile core diameter was used in the calculation in both the MSC and SLC cases. In the prediction models for the deformable projec- tile penetration, the projectile deformability is incor- porated through the use of ratios of the modulus of elasticity, density or yield strength of the projectile’s core and steel (Table 2). 7.1. Non-deformable FMJ-MSC projectile The results gathered through a calculation of pre- diction models of rigid projectile penetration gave values with a large dispersion. For better clarity, the prediction models, which take into account the fibre incorporation and/or high strain rate effect, were plot- ted (Figure 1). Additionally, original modified NDRC and ACE equations were added for their common and 359 Sebastjan Kravanja, Radoslav Sovják Acta Polytechnica Acc. Model Ref. ∑ ln2 Q Qi LSD ρFA scale 1 Mod. Hwang et al.* [5] 0.012 0.98 0.049 0.841 2 Haldar-Hamieh [8, 17] 0.057 1.1 0.105 0.82 3 Mod. Hughes* [8, 18] 0.078 1.03 0.124 0.983 4 Haldar & Miller [7, 19] 0.147 0.86 0.172 0.82 5 ACE [4, 8] 0.156 0.85 0.177 0.816 6 Almusallam et al. [15] 0.471 1.32 0.304 0.966 7 IRS [8, 20] 0.571 1.36 0.337 0.815 8 Whiffen [8, 21] 0.674 0.72 0.367 0.815 9 UKAEA [8, 22] 0.683 1.4 0.368 0.803 10 Amman & Whitney [4, 8] 0.692 0.71 0.372 0.803 11 Mod. Almusallam et al. [9] 0.708 1.41 0.375 0.82 12 Mod. NDRC [4, 8, 23] 0.712 1.41 0.376 0.803 13 Young/Sandia [24] 0.815 0.69 0.404 0.978 14 UMIST [8, 25, 26] 1.083 1.53 0.464 0.826 15 Hwang et al. [5] 1.4 1.62 0.528 0.902 16 Young [7] 1.584 1.67 0.56 0.978 17 Berezan [27] 1.844 1.74 0.601 -0.342 18 BRL [4, 8, 28] 1.989 1.78 0.629 0.821 19 Hughes (flexural) [8, 18] 2.076 1.7 0.507 0.951 20 Bergman [7, 29] 2.083 1.8 0.644 0.811 21 ConWep [30, 31] 2.32 1.86 0.679 0.803 22 Newton [32] 2.539 1.92 0.709 0.988 23 Zaidi et al. [33] 2.559 1.92 0.713 0.82 24 British formula [7, 34] 2.573 1.89 0.59 0.847 25 Tolch & Bushkovitch [7, 35] 3.461 2.14 0.823 -0.342 26 Mod. Petry I (k = 2.26 · 10−4) [4, 8, 36, 37] 4.006 2.26 0.885 -0.344 27 TBAA [7, 38] 4.782 2.44 0.975 0.827 28 Mod. Petry II (Kp = 0.01) [4, 8, 36, 37] 5.065 2.51 0.995 -0.344 29 Hughes (tension) [8, 18] 5.794 2.67 1.069 0.699 30 Forrestal et al. [7, 39, 40] 5.884 2.69 1.08 0.796 31 Mod. Forrestal et al. (Teland) [41] 6.264 2.78 1.115 0.809 32 Li & Chen [30, 42] 6.267 2.78 1.114 0.796 33 Mod. Forrestal et al. (Frew) [27] 6.726 2.88 1.154 0.785 34 Mod. Petry I (k = 3.39 · 10−4) [4, 8] 8.986 3.4 1.318 -0.344 35 Wen & Yang [43] 17.66 0.18 1.88 -0.036 36 Adeli & Amin – cubic [7, 28] 102.9 64.3 154.8 0.789 37 Adeli & Amin – quadratic [7, 28] – -19.8 – 0.789 38 Criepi [8, 44] – 0 – 0.795 *modified by authors of this study Table 3. Accuracy scale according to logarithmic accuracy assessment with accuracy ratio Qi, logarithmic standard deviation LSD and correlation coefficient values ρFA for FMJ-MSC impact. established use in the penetration prediction in the history and good agreement with experimental results according to other researchers. Modified Hughes equa- tion was added as a representative model since it takes into account tensile strength instead of the unconfined compressive strength. However, all of the tested models were assembled in an accuracy scale table according to their logarithmic accuracy value from the most accurate to the least accurate, resulting in 36 displayed models (Table 3). The accuracy ratio, LSD and correlation coefficient values were displayed as well as the control quantities. The modified version of the semi-analytical model, proposed by Hwang et al., was evaluated as a most accurate model in this case. However, the original version of this model turned out to be on the safe side and still displayed sufficient accuracy and is, therefore, preferred to use in engineering practice. The original modified NDRC equation overestimated the results, while the first proposed modification of this equation 360 vol. 58 no. 6/2018 Ultra-High-Performance Fibre-Reinforced Concrete. Part II. Figure 2. Comparison between experimental and representative prediction model results for DOP for FMJ-SLC projectile impact on UHPFRC and linear regression line for experimental data. by Almusallam et al. adjusted the results with the con- sideration of fibre volumetric fraction. The modified Hughes equation, where the high strain-rate effect on the tensile strength from the fib model code 2010 [11] was used, turned out to be relatively accurate, how- ever, it underestimated results in the cases of 1 % and 2 % fibre volumetric content. 7.2. Deformable FMJ-SLC projectile The only models, which take into account projectile de- formability, were displayed in this section. The model by Rubin & Yarin was tested with corresponding ma- terial parameters and effective length. In addition, the hydrodynamic limit value [12] was compared with prediction models (Figure 2). Only 10 models were appropriate for the deformable projectile penetration prediction, the other 32 models were developed for a rigid projectile penetration, and were, therefore, labelled as irrelevant (Table 4). The newest and most analytical model by Rubin & Yarin turned out to be the most accurate, while the hydrodynamic limit provides sufficiently accurate results for this kind of projectile. A model by Hwang et al. and its modified version did provide a good estimation of experimental results. The older mod- els turned out to be less accurate, while the Bernard model from 1977 did provide relatively accurate re- sults on the safe side. The use of Rubin & Yarin model to the UHPFRC targets yielded a very good agreement to experimental data; however, it is im- portant to note that the Rubin & Yarin model was originally developed for an eroding projectile pene- tration into metallic targets. The formula is limited to the case of long-rod penetration where both the projectile and the target experience significant plastic flow. In our case, plastic flow can be expected in the higher volumetric content of fibres in the UHPFRC mixture. Consistently, the prediction is slightly better within the higher volumetric content of fibres (1 % and 2 %) in the UHPFRC mixture than within lower fibre volumetric fractions. Due to the aforementioned reasons, the use of this model for future predictions on the UHPFRC targets should be cautious. 8. Results for spalling mass ejection Prediction models values were compared with exper- imental data, which were transformed from crater volumes values to mass ejection based on the concrete bulk density for each fibre volumetric ratio (Figure 3). Here it must be mentioned that for the evaluation of the model by Abbas et al., the whole projectile diameter was used since it gave a much more realistic description of an actual mass ejection than the core diameter. 8.1. Non-deformable FMJ-MSC projectile It is evident that newly prosed semi-analytical model by Abbas et al. provides much better correlation to the actual mass ejection than the older empiri- cal relation by Sierakowski (Figure 3). The latter is correct in the case of 0.125 % fibre volume fraction; however, this is probably just a coincidental occur- rence since the relation was developed for concretes with compressive strengths around 30 MPa. Further- more, it can be seen, that the Sierakowski relation is not following the decrement of the mass ejection values with an increment of fibre volumetric content; 361 Sebastjan Kravanja, Radoslav Sovják Acta Polytechnica Acc. Model Ref. ∑ ln2 Q Qi LSD ρFA scale 1 Rubin & Yarin [12, 41] 0.225 1.21 0.211 0.158 2 Hydrodynamic limit [41] 0.913 1.48 0.426 −0.079 3 Bernard (1977) [27] 1.022 1.51 0.448 0.266 4 Healey & Weissman [8, 45] 1.261 0.63 0.503 0.213 5 Mod. Hwang et al.* [5] 1.525 1.65 0.545 −0.438 6 Bernard (1978) [27] 1.660 1.69 0.570 0.266 7 Kar [8, 45, 46] 2.051 0.56 0.641 0.314 8 Bernard & Creighton [27] 2.931 2.01 0.755 0.268 9 Hwang et al. [5] 3.973 2.26 0.880 −0.183 10 Newton [32] 13.76 4.55 1.644 −0.078 *modified by authors of this study Table 4. Accuracy scale according to logarithmic accuracy estimation with accuracy ratio Qi, logarithmic standard deviation LSD and correlation coefficient values ρFA for FMJ-SLC impact. Figure 3. Comparison between experimental and prediction models values for mass ejection for FMJ-MSC. however, the model by Abbas et al. is approximat- ing the experimental values with a sufficient accuracy. The logarithmic accuracy assessment of the latter reached the value of 0.230, while the correlation coef- ficient was high: 0.90. 8.2. Deformable FMJ-SLC projectile In this case, the term for mass ejection of the cylin- drical tunnel in the model by Abbas et al. was not calculated, since it did not appear in the experimental work. The results are more similar than in the rigid pro- jectile case, since the Sierakowski relation was derived on the supposition of a constant mass, which is only true for the rigid projectile, while the model by Ab- bas et al. was developed on the basis of the modified NDRC equation, which was not corrected for the use of deformable projectile parameters (Figure 4). The log- arithmic accuracy assessment of the latter was 0.133, while the correlation coefficient was 0.74. 9. Conclusions A number of prediction models have been applied in the framework of this study and compared to the experimental readings. It can be deduced that newly proposed and more developed semi-analytical predic- tion models provide a better fit to the experimental data than older models. It was assessed that the most accurate model for a non-deformable projectile depth of penetration was a model by Hwang et al. with the modified expressions for the cone perimeter and lateral area, while for the deformable projectile, the first place was taken by the model by Rubin & Yarin. Furthermore, the newer model by Abbas et al. shows a very good agreement with the experimental data for the mass ejection prediction. 362 vol. 58 no. 6/2018 Ultra-High-Performance Fibre-Reinforced Concrete. Part II. Figure 4. Comparison between experimental and prediction models values for mass ejection for FMJ-SLC. Acknowledgements This work was supported by the Ministry of Interior of the Czech Republic [project No. VI20172020061]. 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J Struct Div 1978;104:809–16. 364 http://dx.doi.org/10.2172/562498 http://dx.doi.org/10.1142/8651 http://dx.doi.org/10.1016/0029-5493(85)90165-7 http://dx.doi.org/10.1016/j.ijimpeng.2016.04.013 http://dx.doi.org/10.3844/ajassp.2010.711.716 http://dx.doi.org/10.1016/0734-743X(94)80024-4 http://dx.doi.org/10.1016/0734-743X(95)00048-F http://dx.doi.org/10.1016/S0734-743X(02)00037-4 http://dx.doi.org/10.1016/j.ijimpeng.2013.11.008 http://dx.doi.org/10.1016/0029-5493(91)90121-W Acta Polytechnica 58(6):355–364, 2018 1 Introduction 2 Theoretical background 2.1 Parameters. 3 The depth of penetration for non-deformable projectiles 4 Hwang et al. prediction model 4.1 Spalling resistant energy 4.2 Tunnelling resistant energy 4.3 Scabbing resistant energy 4.4 Penetration depth calculation 5 The depth of penetration for deformable projectiles 6 Spalling mass ejection 7 Results for depth of penetration 7.1 Non-deformable FMJ-MSC projectile 7.2 Deformable FMJ-SLC projectile 8 Results for spalling mass ejection 8.1 Non-deformable FMJ-MSC projectile 8.2 Deformable FMJ-SLC projectile 9 Conclusions Acknowledgements References