Reliability Assessment of Steel-Lined and Prestressed FRC Slabs against Projectile Impact

Abstract

A simulation-based probabilistic method is proposed for assessing the reliability of steel-lined and prestressed fiber-reinforced concrete (PFRC) slabs against the impact loads. The impact testing of several prestressed and non-prestressed FRC slabs of 800 × 800 × 100 mm in size was carried out. The experimental program involved projectile impact testing of prestressed and non-prestressed FRC slabs using a gas gun. Three parameters were varied for testing of slabs under the projectile impact, viz., a quantity of steel fibers in the concrete, steel lining on the backside of the slab, and the prestressing level. The probability-based reliability analysis of all the tested specimens was then performed to highlight the influence of steel lining, steel fibers, and prestress in enhancing the safety of RC (reinforced concrete) slabs against projectile impacts. Projectile impact velocity exceeding the slab’s ballistic limit was assumed to lead to the failure of the PFRC slab, i.e., the perforation failure of the slab. Study results indicate that when no steel fiber was present, slab reliability was low. Adding fibers to slabs increases slabs’ reliability significantly. Specimens with the highest steel fiber content (1.2%) showed the greatest increase in reliability. The steel lining on the back face makes the slab’s reliability almost doubled in comparison with those without lining. Additionally, steel lining makes the PFRC slabs as reliable as desired. It was further noticed that the prestressing helps in enhancing the slab safety against projectile impacts. Even a minimal amount of prestressing makes a noticeable improvement in the reliability of PFRC slabs.

1. Introduction

A wide range of infrastructure facilities utilize prestressed concrete structural members, such as nuclear containment facilities, skyscrapers, buried and above-ground water tanks, tunnels, retaining walls, traffic barriers, and girder bridges. Steel lining is often used on some of these prestressed or non-prestressed structures, such as the outer containment of a nuclear power plant. Prestressed concrete structures are widely used because of their excellent shielding properties and safety against radioactive leaks and gas leaks. In addition, they are easy to maintain, strong, and ductile. An explosion, a projectile strike, an aircraft crash, or a vehicle collision may cause damage to these structures. Therefore, in addition to being safe under normal loading conditions, these structures must be safe during accidental overloading as well. The reliability of these structures under such complex dynamic loading conditions is often unknown in many of these cases. Recent studies have found that fibers are effective at improving the reliability of concrete against impact loading [1,2,3]. However, no study has yet investigated the reliability of prestressed fiber-added concrete (or prestressed fiber-reinforced concrete, PFRC) with steel lining against impact loading. Prestressed fiber-reinforced concrete (PFRC) has the benefits of both fibers and prestressing. The availability of fibers and prestressing increase concrete’s tensile strength, resulting in higher flexural strength, shear capacity, and impact resistance for concrete members.

Precast concrete (PC) and prestressed fiber-reinforced concrete (PFRC) have been studied much less than RC and FRC. In a study by Yi et al. [4], nine slabs of reinforced concrete, post-tensioned concrete without reinforcement bars, and post-tensioned concrete with reinforcement bars were tested. They found that post-tensioned concrete with rebars had a greater impact resistance than concrete without rebars or reinforced concrete.

Experimentally and numerically, Kristoffersen et al. [5] examined the perforation resistance of plain concrete slabs that were impacted by steel projectiles of 20 mm diameter. They concluded that (i) the impact resistance of thin slabs is governed by tensile strength, and (ii) the ballistic limit of the slabs increases almost linearly with concrete strength. The punching strength of concrete slabs was studied by Orbovic et al. [6] using a 47 kg hard projectile and longitudinal prestressing. Concrete had a compressive strength of 45 to 52 MPa. While permanent deflection was lower for prestressed concrete slabs without transverse reinforcement, the scabbed area was found to be more than RC elements. Residual exit velocity and scabbed area were significantly influenced by transverse reinforcement. Rosenberg et al. [7] validated their earlier published model for concrete slab perforation by rigid projectiles. Ballistic limits and residual velocities obtained by the model agreed well with several sets of concrete slab perforation data.

Wang et al. [8] studied the prestressed concrete (PC) slabs subjected to hard missile impacts. They found that prestressing and reinforcement ratio (due to prestressing bars) are the main factors affecting the ballistic resistance of PC slabs. They further observed that concrete confinement also increases the ballistic resistance of PC slabs.

In situ prestressed concrete sleepers were studied by Kaewunruen and Remennikov [9]. Precast concrete sleepers were investigated for failure mechanisms, impact energy absorption, flexural toughness, crack propagation, and energy transfer mechanisms. In the study, it was found that split cracks are the prime cause of prestressed concrete sleeper failure.

In a study, Rajput and Iqbal [10] examined the ballistic performance of 45 cm × 45 cm slabs of plain concrete, reinforced concrete, and prestressed concrete using steel projectiles of 1 kg weight. The slabs ranged in thickness from 60 mm to 100 mm. Concrete having a compressive strength of 48 MPa was prestressed with 10% of its compressive strength. The experiments measured damage volume, spalling, scabbing, and ballistic resistance at impact speeds ranging from 65 to 220 m/s. Of the three concretes examined, prestressed concrete showed the best ballistic performance. Additionally, they observed that prestressing effects are more noticeable as target thickness increases.

Using a 234 kg steel hammer for free fall, Kumar et al. examined 12 samples of prestressed and reinforced concrete slabs (80 cm × 80 cm × 10 cm) of 60 MPa compressive strength. There were two levels of prestressing: 10% and 20%. They found that when moving from reinforced concrete to 10% prestressing, peak displacement was reduced more than when moving from 10% to 20% prestressing. Liu et al. [11] reviewed the state-of-the-art research on thick ultra-high-performance concrete (UHPC) targets subjected to high-velocity projectile impacts. The researchers also evaluated existing empirical and semi-empirical formulas that can be used to predict the depth of penetration in conventional concrete and assessed their suitability for UHPC.

Post-tensioned slabs under impact loads were studied by Al Rawi et al. [12]. Concrete slabs were post-tensioned to 3% of their strength. A 605 kg impactor was dropped on the slabs from a height of 20 m. There was no difference between RC and post-tensioned slabs in terms of maximum deflection, but post-tensioned slabs had 35.2% more impact force, and RC slabs had 14.9% higher maximum deflection.

In the literature, there are few studies assessing the reliability of structures that have been subjected to aircraft, missiles, and projectile impact loads. Siddiqui et al. [13] presented a method for assessing concrete targets’ reliability to rigid missile impacts at specific depths. An expression for missile velocity at any depth in the soil was derived based on penetration depth. Following this, the First Order Reliability Method was employed for assessing concrete target reliability. Sensitivity analysis also revealed results of practical interest. The steel plates shielded with RC were tested against hard projectile strikes using a simple experiment designed by Siddiqui et al. [14]. A Monte Carlo Simulation (MCS) technique was used to estimate the reliability of RC-shielded steel plates for different impact velocities, and the outputs were correlated with failure scenarios. Furthermore, they extended their work [15] to investigate how well a containment structure having double walls holds up against an impact from a rigid projectile. A correlation was made between a containment’s ballistic limit and its reliability indices and probability of failure. The reliability of containment structures was also examined through a number of parametric studies.

Considering the uncertainties associated with material, geometric, and impact parameters, Siddiqui et al. [16] devised a probabilistic method for estimating HFRC slab reliability against hemispherical nose projectile impacts. Additionally, hybrid fibers were investigated for their effect on the reliability of RC slabs against projectile loads. A projectile impacting the HFRC slab above its ballistic limit caused the slab to perforate, resulting in failure. Various impact velocities for a given projectile were used to assess reliability for different impact loads. By analyzing the reliability of the HFRC slabs, they determined their safety level against projectile impact.

The above literature review examines the various studies on RC/FRC/PC members’ impact behavior. Although the impact response of concrete targets has been studied in sufficient detail, reliability aspects are not covered, which is the focus of this study.

2. Research Significance

The above review shows that while some researchers have studied prestressed concrete for low-velocity impacts, there were no studies on the safety or reliability of prestressed fiber-reinforced concrete (PFRC) members and PFRCs with steel linings under impact loads. This paper proposes a simulation-based probabilistic method for assessing the reliability index of prestressed and steel-lined PFRC slabs against projectile loads, taking into account the uncertainties involved in various parameters related to geometry, material, and impact. It is expected that the study would be useful for practicing engineers and researchers in understanding the sensitivity of various parameters influencing the impact response and reliability of PFRC members with or without steel linings.

3. Formulation for Reliability Analysis

The main objective of reliability assessment is to estimate the probability that a structure or structural member will not violate or exceed the specified limit state during its useful life. Limit states are said to be violated when prestressed, and steel-lined FRC slabs have ballistic limits lower than impacting projectile velocities. Alternatively, a slab is said to fail if the projectile impact velocity is more than its ballistic limit. MCS is used to assess the reliability index and probability of failure for PFRC slabs. Limit state functions are used to represent failure modes or types in reliability analyses of structures or structural components. The value of the limit state function can be zero, positive, or negative. Failures of structures or structural components are represented by zero or negative values of the function. A positive function value, on the other hand, indicates that a structural component or structure is reliable. Therefore, we can express failure probability as follows:

$$P_f=P\left[g\left(\underset{\_}{x}\right)\le 0\right]$$

(1)

Here, $P_f$ denotes the failure probability, $g\left(\underset{\_}{x}\right)$ represents the limit state function, and $\underset{\_}{x}$ represents the vector of those variables that are random. Note, that due to the large concentration of stress and stress waves in the surrounding areas of the projectile impact, failure related to projectile impact was a local failure.

3.1. Limit State Function

A normal strike of a hemispherical nose projectile was assumed on prestressed and steel-lined FRC slabs for deriving the desired limit state function. Slab failure was assumed if a given projectile’s impact velocity exceeded the ballistic limit of the slab. We can write the limit state function as follows, using $V_0$ and $V_p$ as impact and ballistic velocity, respectively:

$$g\left(\underset{\_}{x}\right)=V_p-V_0$$

(2)

If the projectile’s impact velocity exceeds the PFRC slab’s ballistic limit, the slab is likely to be perforated (i.e., penetrated). For the current study, the ballistic limit expression proposed in Equations (3)–(6) were used for prestressed and steel-lined FRC targets.

$$V_p=V_a \mathrm{for} V_a\le 70 \mathrm{m}/\mathrm{s}$$

(3)

$$\mathrm{and} V_p=V_a\left[1+{\left(\frac{V_a}{500}\right)}^2\right] \mathrm{for} V_a>70 \mathrm{m}/\mathrm{s}$$

(4)

$$\mathrm{where}, V_a=3.1 {\rho }_c{}^{0.14} f_{teff}{}^{0.39}{\left(\frac{d{H}^2}{M}\right)}^{0.4}{\left(r+0.3\right)}^{0.5}\left[1.2-0.6\left(\frac{c_r}{H}\right)\right]Exp\left(0.5{\left(0.5RI\right)}^5\right)$$

(5)

$$\mathrm{and} r=A_{st }\%+\left(\frac{100\times T_l}{H}\right)$$

(6)

in which $M$ denotes the mass of the projectile (kg), $H$ represents the concrete target thickness (mm), $c_r$ denotes the rebar spacing (mm), $r$ is the percentage of steel reinforcement (rebars + steel lining), $d$ is the projectile diameter, $T_l$ is the thickness of steel lining (mm), $RI$ is the reinforcing index, ${\rho }_c$ is the concrete density (kg/m³), and $f_{teff}$ is the effective tensile strength of concrete.

The effective tensile strength of concrete $\left(f_{teff}\right)$ is calculated by incorporating the effect of prestressing. Thus, it is calculated using the following:

$$f_{teff}=f_t+f_p$$

(7)

where, $f_t$ denotes the concrete’s tensile strength (N/m²),$f_p$ is prestress in concrete (N/m²).

The substitution of the expression of $V_p$ in Equation (2), makes the limit state a function of the variables ${\rho }_c,f_{teff},H,r, c_r,M, d,RI,$ and $V_0$. Due to various factors, these variables are subject to significant uncertainties. As a result, these variables are assumed random in the present study. The variables discussed above can be arrayed as follows:

$$\underset{\_}{x}=\left(V_0, {\rho }_c,f_{teff},H,r, c_r,M, d,RI\right)$$

(8)

The reliability and failure probability of PFRC slabs against projectile impacts are estimated using the derived limit state function. For this purpose, the MCS technique [17] was used.

3.2. Steps Followed for Reliability Analysis

During MCS-based reliability analysis, the following steps were followed:

• Choose the probability distributions of all random variables, input their nominal values, and specify their statistical properties (e.g., Bias factors and COV).
• Calculate probability distribution parameters.
• Perform MCS using N simulations (e.g., $N$ = 500,000).
• For all random variables, generate random values based on their probability distributions in each simulation cycle.
• Feed the generated random values of all the variables (i.e., random variables) into the function (Equation (2)).
• For $N$ simulations, repeat steps 4 and 5. Determine the number of simulations with a negative limit state function; say it is $N_f$.
• Calculate $P_f$ and the reliability index $\beta$ by:

  $$P_f=N_f/N$$

  (9)

  $$\mathrm{and} \beta =-{\mathsf{\Phi }}^{-1}\left(P_f\right)$$

  (10)

  where, ${\mathsf{\Phi }}^{-1}\left(P_f\right)$ is the inverse of standard normal CDF (cumulative distribution function).
• Use the following equation to check the convergence of MCS employing the coefficient of variation of the estimated failure probability:

  $$\mathrm{COV}\left(P_f\right)\cong \frac{\sqrt{\frac{\left(1-P_f\right)P_f}{N}}}{P_f}$$

  (11)

A low coefficient of variation indicates a more accurate estimation of failure probability. If the $\mathrm{COV}\left(P_f\right)$ is less than 5%, that number of simulations is considered sufficient for real-life calculations.

4. Experimental Program

The impact testing of several prestressed and non-prestressed FRC slabs of 800 × 800 × 100 mm in size (Figure 1) was carried out, and the results of the tests were used for the probabilistic analysis. The experimental program involved impact testing of prestressed and non-prestressed FRC slabs using a gas gun (Figure 2). Three parameters were varied for testing of slabs under the projectile impact, viz., a quantity of steel fibers in the concrete, steel lining on the backside of the slab, and the prestressing level. All test specimens were reinforced with ϕ8@100 mm both ways provided on both faces of the slabs.

There were twelve groups of slabs. The groups were different in (i) volume fraction of steel fibers (0%, 0.6%, and 1.2%), (ii) steel lining (no lining and steel lining at the back face), and (iii) prestressing (no prestressing and prestressing). Three slabs of each type were prepared for testing (except in one group where only two slabs were cast for testing), each at a different projectile strike velocity. Thus, a total of 35 slabs (

Table 1. Details of the test specimens employed for reliability analysis.

Specimen Number	Specimen ID	Fiber (%)	Prestressed	Steel Lining on the Back Face
1	RE-NL-F0-N-1	0%	No	No
2	RE-NL-F0-N-2	0%	No	No
3	RE-NL-F0-N-3	0%	No	No
4	RE-SL-F0-N-1	0%	No	Yes
5	RE-SL-F0-N-2	0%	No	Yes
6	RE-SL-F0-N-3	0%	No	Yes
7	RE-NL-F1-N-1	0.6%	No	No
8	RE-NL-F1-N-2	0.6%	No	No
9	RE-NL-F1-N-3	0.6%	No	No
10	RE-SL-F1-N-1	0.6%	No	Yes
11	RE-SL-F1-N-2	0.6%	No	Yes
12	RE-SL-F1-N-3	0.6%	No	Yes
13	RE-NL-F2-N-1	1.2%	No	No
14	RE-NL-F2-N-2	1.2%	No	No
15	RE-NL-F2-N-3	1.2%	No	No
16	RE-SL-F2-N-1	1.2%	No	Yes
17	RE-SL-F2-N-2	1.2%	No	Yes
18	RE-SL-F2-N-3	1.2%	No	Yes
19	RE-NL-F0-P-1	0%	Yes	No
20	RE-NL-F0-P-2	0%	Yes	No
21	RE-SL-F0-P-1	0%	Yes	Yes
22	RE-SL-F0-P-2	0%	Yes	Yes
23	RE-SL-F0-P-3	0%	Yes	Yes
24	RE-NL-F1-P-1	0.6%	Yes	No
25	RE-NL-F1-P-2	0.6%	Yes	No
26	RE-NL-F1-P-3	0.6%	Yes	No
27	RE-SL-F1-P-1	0.6%	Yes	Yes
28	RE-SL-F1-P-2	0.6%	Yes	Yes
29	RE-SL-F1-P-3	0.6%	Yes	Yes
30	RE-NL-F2-P-1	1.2%	Yes	No
31	RE-NL-F2-P-2	1.2%	Yes	No
32	RE-NL-F2-P-3	1.2%	Yes	No
33	RE-SL-F2-P-1	1.2%	Yes	Yes
34	RE-SL-F2-P-2	1.2%	Yes	Yes
35	RE-SL-F2-P-3	1.2%	Yes	Yes

) were tested. The prestressing was conducted in both directions using a post-tensioning system, which employed high-strength prestressing rods threaded and anchored at the ends using high-strength anchor nuts. The PVC ducts were embedded in concrete before casting for passing prestressing rods. Thick steel blocks were used for the distribution of prestressing force at the end anchorages. The 1.8-mm thick steel lining (provided at the back face) was anchored to concrete using shear studs of ϕ8 steel rebars, which were welded to the steel lining (Figure 3). Additional details about the experimental program, data analysis, and damage mode can be found in Abbas et al. [18].

5. Data for Reliability Analysis

Prestressed and steel-lined fiber-reinforced concrete (PFRC) slabs are subject to substantial uncertainties, so we identified those variables that contain substantial uncertainties and considered them as random, as shown in

Table 2. Random variables and statistical data.

Random Variable	Nominal	Bias Factor	COV (Coefficient of Variation)	Probability Distribution	Reference
PFRC slab
$\mathrm{Density} \mathrm{of} \mathrm{concrete}, {\rho }_c$(kg/ m3)	2500	1.05	0.10	Lognormal	[19]
$\mathrm{Effective} \mathrm{tensile} \mathrm{strength} \mathrm{of} \mathrm{concrete}, f_{teff}$ (MPa)	Variable	1.10	0.10	Lognormal	Assumed
Thickness of slab, $H$ (mm)	100	1.00	0.05	Normal	[19]
Reinforcement ratio, $r$ (%)	Estimated $	1.10	0.10	Normal	Assumed
Steel rebar spacing, $c_r$ (mm)	100	0.90	0.05	Lognormal	[16]
Reinforcing index. $RI$	Estimated $	1.10	0.12	Lognormal	[16]
Hemispherical nose projectile
Projectile mass, $M$ (kg)	0.80	1.10	0.05	Lognormal	[20]
Projectile diameter, $d$ (mm)	40.0	1.05	0.05	Normal	[20]
Projectile strike velocity, $V_0$ (m/s)	Variable	0.90	0.10	Extreme Type I	[16]

^$ Nominal value was estimated using the appropriate equation.

. The bias factor in this table is calculated by dividing the mean value of the random variable by its nominal value (which is fixed in a deterministic sense). The nominal value and the mean value are the same when this factor is 1. Bias factors are greater than one for resistance-related variables. For variables related to loading, this factor is less than 1.0.

Reliability analysis of PFRC slab specimens also requires probability distributions of the expected extreme impact load, measured in terms of impact velocity. In this study, projectile velocity was described by the Extreme Type I distribution (Table 2). This distribution’s probability density function (PDF) and CDF are given by Nowak and Collins [17]:

$$\mathrm{PDF}: f\left(x\right)=\alpha \exp \left\{-e^{-\alpha \left(x-u\right)}\right\}\exp \left\{-\alpha \left(x-u\right)\right\}$$

(12)

$$\mathrm{CDF}: F\left(x\right)=\exp \left\{-e^{-\alpha \left(x-u\right)}\right\} \mathrm{for} -\infty \le x\le \infty$$

(13)

Here, $\alpha$ and $u$ represent distribution parameters. In order to estimate distribution parameters, Nowak and Collins [17] proposed the following formulas:

$$\alpha \approx \frac{1.282}{{\sigma }_x}$$

(14)

$$u\approx {\mu }_x-0.45{\sigma }_x$$

(15)

in this equation, ${\mu }_x$ represents the mean and ${\sigma }_x$ represents the standard deviation. References for the probability distributions and coefficients of variation are given in Table 2.

It is critical to know the required number of simulations when performing a reliability analysis with MCS. An example of how $COV\left(P_f\right)$ varies with the simulation number shown in Figure 4. This slab specimen (RE-SL-F2-P1) was impacted by the projectile at a nominal strike velocity of 185 m/s. Figure 4 shows that $COV\left(P_f\right)$ decreases with increasing simulation cycles (N). For the present specimen, N greater than 1,000,000 (1 million) gives an adequate $P_f$(i.e., $COV\left(P_f\right)$ smaller than 5%). In the current study, the MCS was performed with enough simulations for $COV\left(P_f\right)$ to be less than 5%.

6. Discussion of Results

The last two columns of Table 2 show the probability of failure ($P_f$) and reliability indices ($\beta$) of PFRC slab specimens determined by the MCS technique and the output of the analysis are given in

Table 3. Output of the MCS-based reliability analysis.

Specimen Number	Specimen ID	${\mathit{V}}_{\mathit{P}}$ (m/s)	${\mathit{V}}_0$ (m/s)	${\mathit{V}}_0$$/{\mathit{V}}_{\mathit{P}}$	Damage	${\mathit{P}}_{\mathit{f}}$	β
1	RE-NL-F0-N-1	118.7	116.1	1.0	NP	6.11 × 10−2	1.846
2	RE-NL-F0-N-2	118.7	147.5	1.2	P	3.58 × 10−1	0.363
3	RE-NL-F0-N-3	118.7	136.2	1.1	P	2.19 × 10−1	0.774
4	RE-SL-F0-N-1	182.5	136.2	0.7	NP	4.17 × 10−3	2.638
5	RE-SL-F0-N-2	182.5	158.5	0.9	NP	2.08 × 10−2	2.037
6	RE-SL-F0-N-3	182.5	177.1	1.0	NP	6.05 × 10−2	1.551
7	RE-NL-F1-N-1	153.6	116.1	0.8	NP	3.86 × 10−3	2.664
8	RE-NL-F1-N-2	153.6	143.8	0.9	NP	4.17 × 10−2	1.742
9	RE-NL-F1-N-3	153.6	136.2	0.9	NP	2.29 × 10−2	1.997
10	RE-SL-F1-N-1	242.8	158.5	0.7	NP	6.85 × 10−4	3.201
11	RE-SL-F1-N-2	242.8	200.0	0.8	NP	1.42 × 10−2	2.192
12	RE-SL-F1-N-3	242.8	218.0	0.9	NP	3.20 × 10−2	1.852
13	RE-NL-F2-N-1	165.9	136.2	0.8	NP	8.45 × 10−3	2.389
14	RE-NL-F2-N-2	165.9	158.5	1.0	NP	4.38 × 10−2	1.709
15	RE-NL-F2-N-3	165.9	169.5	1.0	NP	8.19 × 10−2	1.392
16	RE-SL-F2-N-1	264.8	185.0	0.7	NP	1.83 × 10−3	2.906
17	RE-SL-F2-N-2	264.8	218.0	0.8	NP	1.26 × 10−2	2.238
18	RE-SL-F2-N-3	264.8	136.2	0.5	NP	8.00 × 10−6	4.314
19	RE-NL-F0-P-1	127.7	136.2	1.1	P	1.28 × 10−1	1.134
20	RE-NL-F0-P-2	127.7	125.0	1.0	NP	6.20 × 10−2	1.539
21	RE-SL-F0-P-1	197.7	218.0	1.1	P	1.61 × 10−1	0.992
22	RE-SL-F0-P-2	197.7	190.0	1.0	NP	5.65 × 10−2	1.585
23	RE-SL-F0-P-3	197.7	136.2	0.7	NP	1.26 × 10−3	3.022
24	RE-NL-F1-P-1	160.6	158.5	1.0	NP	6.71 × 10−2	1.498
25	RE-NL-F1-P-2	160.6	178.8	1.1	P	1.72 × 10−1	0.947
26	RE-NL-F1-P-3	160.6	136.2	0.8	NP	1.48 × 10−2	2.175
27	RE-SL-F1-P-1	255.3	218.0	0.9	NP	2.02 × 10−2	2.051
28	RE-SL-F1-P-2	255.3	200.0	0.8	NP	7.96 × 10−3	2.411
29	RE-SL-F1-P-3	255.3	136.2	0.5	NP	2.60 × 10−5	4.046
30	RE-NL-F2-P-1	173.0	185.0	1.1	P	1.16 × 10−1	1.193
31	RE-NL-F2-P-2	173.0	136.0	0.8	NP	1.53 × 10−2	2.555
32	RE-NL-F2-P-3	173.0	192.3	1.1	P	1.56 × 10−1	1.013
33	RE-SL-F2-P-1	277.7	185.0	0.7	NP	9.02 × 10−4	3.121
34	RE-SL-F2-P-2	277.7	225.0	0.8	NP	1.04 × 10−2	2.310
35	RE-SL-F2-P-3	277.7	136.2	0.5	NP	6.00 × 10−6	4.378

$V_p$: Ballistic limit; $V_0$: Impact velocity; P: Perforated (failed); NP: Not Perforated (Not failed).

. The nominal strike velocities ($V_0$) shown in this table are the same as those used in the experiment. The PFRC slab specimens of 800 × 800 × 100 mm were tested for impact resistance against steel projectiles with hemispherical nose shapes.

Table 3 shows the $P_f$ and $\beta$ for PFRC slab specimens. There is a clear correlation between the projectile strike velocity and the $P_f$ of the PFRC specimens. The $P_f$ of the PFRC specimens increases with increasing projectile impact velocity. This is a trend that is expected. In Table 3, specimens with $\beta$ values above 1.0 have a sufficiently low failure probability. Due to their combined uncertainty, the parameters affecting the ballistic limit velocity of the PFRC specimens do not bring it below the strike velocity. A $P_f$ of 0.04 for RE-NL-F1-N-2, for example, indicates that among 100 specimens subjected to the same impact velocity, only four would fall below the ballistic limit, i.e., only four specimens out of 100 would be perforated, and 96 would not be perforated. In other words, there is a very small probability that this specimen would be perforated when impacted by a projectile with a nominal strike velocity of 143.8 m/s.

It is imperative to note that when the striking velocity exceeds the ballistic limit, failure occurs deterministically. However, when viewed probabilistically, it means that failure is more likely. In Table 3, all specimens with a ratio greater than one support this claim. In the case of specimens RE-NL-F0-N-2, RE-NL-F0-N-3, RE-NL-F0-P-1, RE-SL-F0-P-1, RE-NL-F1-P-2, RE-NL-F2-P-1, and RE-NL-F2-P-3, failure probabilities are substantially high ($\beta$ are less than 3.0). Generally, the reliability index of a structure or structural component should exceed 3.0 to consider it to be safe enough [13,14,15,16,17,21]. The probabilistic analysis of the specimens shows that only six specimens (RE-SL-F1-N-1, RE-SL-F2-N-3, RE-SL-F0-P-3, RE-SL-F1-P-3, RE-SL-F2-P-1, and RE-SL-F2-P-3) are as reliable as desired for the considered strike velocities. All other specimens have reliability indices less than 3.0.

Figure 5 plots the reliability index against the $V_0/V_P$ ratio in order to determine the impact velocity at which slab specimens can be considered reliable. All specimens were plotted on the graph. Figure 5 shows that reliability decreases sharply with increasing impact velocity. The reliability index is around 4 when the projectile strikes PFRC slabs at half of their ballistic limit (~0.5$V_P$). However, if the impact velocity approaches 1.2 times the slab’s ballistic limit (~1.2$V_P$), the reliability index reduces to around 0.75, indicating less reliability (i.e., high failure probability). When the ratio $V_0$/$V_P$ is 0.7 or less, the reliability is three or above. For PFRC slabs to achieve at least a 3.0 reliability index, their ballistic limit must be designed for $V_0$/0.7~1.4$V_0$.

Figure 6 illustrates the variation in the reliability of PFRC slabs with the proportion of steel fibers subjected to the projectile strike velocity equal to the ballistic limit of that slab which is without fibers, prestress, or steel lining (118.7 m/s), i.e., RE-NL-F0-N. Without fiber, the slab’s reliability was substantially less, as shown in the figure. The reliability of slabs improved considerably after fibers were added. For specimens with the highest amounts of steel fibers (1.2%), the maximum increase in reliability was observed. The second-highest reliability belongs to the specimen with the second-highest percentage of steel fibers. This figure also shows that an FRC slab without prestressing and steel lining, subjected to the projectile strike velocity of 118.7 m/s, should have a steel fiber proportion of about 1.2% to attain the desired reliability index of 3.0. The effect of prestressing is also obvious from this figure that it makes a noticeable improvement in reliability.

Figure 7 shows how steel lining improves the safety of RC slabs. This figure shows that steel lining can improve the reliability of both prestressed and non-prestressed slabs substantially. When there was no steel lining (and no steel fibers as well), the reliability index of both prestressed and non-prestressed slabs was far below the desired reliability index of 3.0 when impacted by the projectile of 118.7 m/s impact velocity. With the use of 1.8 mm thick steel lining, the reliability improves to 3.33 and 3.68 for non-prestressed and prestressed slabs, respectively. Thus, the steel lining on the back face makes the slab reliability almost double when the slab has no lining. Additionally, steel makes them as reliable as desired. The effect of prestressing is again obvious from this figure that it makes a noticeable improvement in reliability.

7. Conclusions

Based on the findings of the present reliability study, the following conclusions can be drawn:

• When striking velocity exceeds the ballistic limit, failure is inevitable, but from a probabilistic perspective, it simply means that the probability of failure is high.
• Out of 35 specimens, only six specimens are as reliable as desired at the considered impact velocities, and all the rest have reliability indices below 3.0.
• Increasing impact velocity sharply decreases reliability. The reliability index is around 4 when the projectile strikes PFRC slabs at half of their ballistic limit (~$0.5V_P)$. However, if the impact velocity ($V_0)$ approaches 1.2 times the slab’s ballistic limit ($~1.2V_P$), the reliability index reduces to around 0.75, indicating very low reliability or a high failure probability. A $V_0/V_P$ ratio of 0.7 or lesser results in a reliability of three and above.
• As a design guideline, PFRC slabs should have a ballistic limit of around $V_0$ /0.7~1.4$V_0$ to have the slab’s desired reliability index of at least 3.0.
• Without fiber, the slab’s reliability was substantially low. The reliability of slabs improved considerably after fibers were added. The maximum increase in reliability was observed for specimens with the highest amounts of steel fibers (1.2%). Second-highest reliability belongs to the specimen with the second-highest percentage of steel fibers.
• An FRC slab without prestressing and steel lining, subjected to a projectile strike velocity of 118.7 m/s, should have a steel fiber proportion of about 1.2% to attain the desired reliability index of 3.0.
• The steel lining on the back face makes the slab’s reliability almost double when there is no lining. Additionally, steel lining makes the PFRC slabs as reliable as desired.
• A prestressed slab has a higher safety level. Even a minimal amount of prestressing makes a noticeable improvement in the reliability of PFRC slabs.