The impacts of contact explosions on ultra-high performance reinforced concrete slabs: experimental study and dimensional analysis

Abstract

Ultra-high performance concrete (UHPC) is becoming a prevailing construction material in protective engineering. However, an insufficient research basis causes difficulty in the anti-explosion structural design of UHPC. To investigate the blast resistance of the UHPC slab (UHPCS), contact explosion tests were carried out. UHPCS exhibited superior blast resistance and a lower threshold range of failure modes than the normal reinforced concrete slab (NRCS). Compared with the NRCS, the UHPCS reached a lower damage level under the same scaled distance and had smaller craters and spalls, fewer cracks, and an absence of cross-shaped cracks. For NRCS, failure modes of medium spall, severe spall, and perforation were reached under charges of 1.0, 1.6, and 2.2 kg. In contrast, UHPCS reached medium spall, severe spall, and perforation under charges of 1.6, 3.3, and 5.0 kg. The scaled slab thicknesses (T/W^1/3) of the medium spall, severe spall, and perforation of the NRCSs were 1.17, 1.27, and 1.32 times larger than those of the UHPCSs, respectively. The formulae for predicting the spall and perforation thresholds of the UHPCSs were proposed. The reduction factors were used to evaluate the mitigation effect of the blast damage on the UHPC compared to that on the NRC. The reduction factors of the crater diameter () and the spall diameter () were determined to be 0.507 and 0.721, respectively. For quantitative analysis of the anti-explosion performance of UHPC, a simple estimation method for predicting the dimensions of the crater and spall of the UHPCS was proposed and verified based on the dimensional analysis method.

Introduction

In recent years, with terrorism events and accidental explosive mishaps progressively increasing, public safety and property protection have become prominent issues for engineering design and construction¹. Concrete is one of the most commonly used construction materials in civil engineering. Therefore, the blast resistance performance of concrete members plays an important role in the design of civil and protective structures, which has become a popular issue in engineering research^2–4.

When a concrete slab is subjected to an intensive blast load, different portions of the slab experience different damage patterns. On the front surface of a concrete slab facing the blast load, the concrete is compressed and sheared under high compressive pressure, and debris is generated and ejected at high speeds. On the free rear surface of the concrete slab, the compressional waves turn into tensile waves. When the tensile stress waves are reflected from the free surface of at rear surface into the concrete and interact with the attenuated compression wave to produce local tensile stresses that exceeds the dynamic tensile fracture strength of the material⁵, a spall is formed and ejected. It poses a considerable threat to personnel and facilities⁶.

In the past few decades, spall damage of concrete under blast loads has been studied by researchers, and many substantial results have been obtained. In the 1970s, Kot et al.^7,8 proposed a theoretical prediction method for concrete wall spall damage. However, this method had limitations in terms of the explosion source and was mainly suitable for light and medium bombs. In the 1980s, Weerheijm et al.⁹ proposed a method for calculating the critical thickness to prevent a complete breach of a concrete slab. They found that the compressive strength and the boundary conditions of concrete slabs under contact explosion did not affect the critical slab thickness. McVay¹⁰ summarized a series of concrete spall tests from different sources and subsequently conducted a parametric study on the concrete spall, including the scale distance, explosive weight, wall thickness, concrete strength, concrete admixture, and reinforcing bar spacing. Based on the test results, an empirical method for determining the crack initiation of concrete spalls induced by tensile waves was derived. However, the wave divergence was the only considered variation in the stress wave propagation. The attenuation and dispersion effects of the stress wave were disregarded. In the 1990s, Lonnqvist et al.¹¹ carried out a dimensional analysis on the test results of contact explosions. They established a model to predict the fracture diameter of a concrete slab, and parameters such as the explosive mass, explosive density, detonation specific energy, concrete strength, and concrete density were included. At the beginning of this century, Morishita et al.¹² proposed an explosion collapse coefficient for concrete slabs. In 2008, the United States Department of Defense proposed the Uniform Facility Standard (UFC) 3-340-02¹³ for predicting the damage of concrete members under contact and close-in explosion. Spall and perforation threshold curves, used as the approximate upper bounds of the spall and perforation data points, were proposed. Wang et al.^14,15 proposed an empirical formula to correct the proportional relationship between the model and prototype.

For a long time, the brittleness and low tensile strength of concrete have caused difficulty in the improvement of the spall resistance of a concrete structure. To overcome this limitation, many efforts have been made. Innovation on reinforcement is a common and popular method. Basalt fiber reinforced polymer (BFRP) bars^16,17 and glass fiber reinforced polymer (GFRP) bars¹⁸ had a good improvement on the anti-explosion performance of concrete slabs and beams. The reinforcement form of adding studs to the steel bar has also achieved good results in enhancing the anti-explosion ability of the concrete slab¹⁹. Adding an absorbing layer on the rear surface of a concrete slab was another commonly used method to promote the anti-explosion performance. The polyisocyanate-oxazodone (PODZ), glass fiber textiles²⁰, aramid fiber textiles²¹ and carbon fiber textiles²² were used as absorbing layers. The added layer could mitigate and reduce the tensile wave reflected from the rear surface, and increase the tensile damage area, which effectively decreased the structural damage.

In addition to strengthening methods for concrete structures, developing the concrete material itself is also a popular approach. Many approaches, such as varying the composition, adding fibers, adjusting the mix ratio, and changing the curing conditions, have been used to develop new types of concrete for different purposes. By adding high-strength fiber, fiber-reinforced concrete (FRC) can provide superior mechanical performance compared with normal concrete. These high-strength fibers generate a bridging action to promote the joint forces among the skeletons, which improves the macroscopic tensile strength of the concrete²³. Ultra-high performance concrete (UHPC) has been the most widely applied FRC in recent years. By adding a certain percentage of steel fibers, UHPC exhibits high strength, high performance, and high durability^24,25. Its quasi-static and dynamic mechanical properties, such as strength, toughness, and impact resistance, are significantly greater than those of traditional commercial concrete^26–28. As a new type of building material with excellent mechanical properties, the anti-explosion performance of UHPC has also been studied. These studies have focused mainly on the anti-explosion performance of UHPC columns and the close-in explosion of UHPC thin slabs. Roller et al.²⁹ designed UHPC as a reinforcing layer for reinforced concrete columns. Contact explosion tests were carried out to investigate the residual bearing capacity of the columns. Their results showed that UHPC effectively improved the residual bearing capacity of the members. Astarioglu and Krauthammer³⁰ used a numerical simulation method to analyze the response of normal reinforced concrete (NRC) and UHPC columns under a blast load. Their results showed that UHPC columns could withstand four times the pulse required to cause damage to NRC columns. Ellis et al.³¹ conducted an impact test of a 51 mm-thick UHPC unidirectional plate. The dimensional geometry, packing, and fiber volume fraction were found to be the keys to improving the blast resistance performance. Wu et al.³² conducted a series of close-in explosion tests on thin plates with a thickness of 100 mm. Their results showed that the damage degrees of UHPC slabs were lower than those of NRC slabs, which indicated that UHPC was more effective in anti-blast design. Li et al.^33,34 conducted contact explosion tests on five steel bar-reinforced UHPC slabs. The influences of the reinforcement spacing on the compression crater and spall area were quantitatively analyzed. However, these existing studies do not sufficiently elucidate the anti-explosion characteristics of UHPC.

To investigate the blast resistance and damage pattern of UHPC slabs, contact explosion tests were performed on ten thick slabs with a 300 mm thickness, and there were five slabs each for UHPC and NRC. The typical damage areas, such as craters and spalls, were quantitatively analyzed and compared. The superiority of the anti-explosion performance of UHPC was discussed. Formulae for predicting the crater volume and spall diameter of the thick UHPC slabs were proposed through a dimensional analysis method. The precision and feasibility of the existing empirical methods for predicting the damage of thick UHPC slabs were also discussed.

Design of the experiments

Slab specifications

Two types of slabs, i.e., a normal reinforced concrete slab (NRCS) and an ultra-high performance concrete slab (UHPCS), were tested. All slabs had the same dimensions. The length, width, and thickness were 1500, 1500, and 300 mm, respectively.

The concrete used for producing the NRCS was the same batch of commercial concrete with a strength grade of C40. The 28-d curing strength of the concrete was 42.5 MPa. The concrete composition is listed in Table 1. Three layers of reinforcing bars were laid inside the NRCS, as shown in Fig. 1. The spacing distance between each reinforcing layer was 130 mm. Seven bars were set in both the latitudinal and longitudinal directions, and the spacing distance was 240 mm. Distribution bars were set at every intersection of the reinforcing bars to connect and fix each layer. The rebar was HPB 235 with a diameter of 8 mm. The yield strength was 235 MPa, and the elastic modulus was 210 GPa. A 30 mm-thick concrete protective layer was reserved on the side of the slab, and a 20 mm-thick concrete protective layer was reserved on the frontal surface and rear surface.

Table 1.

Mix proportions of the concrete (unit: kg/m³).

	Cement	Fly ash	Silica fume	Coarse aggregate	Fine aggregate	Superplasticizer	Water	Steel fiber
C40	320 (P·O42.5)	60	18	1165	669	7	145	--
UHPC	850 (P·O52.5)	--	200	--	1000	40	150	79

Fig. 1.

Geometry of the NRCS (unit: mm).

The 28-d curing strength of the UHPC was 129.3 MPa. The splitting tensile strength was 6.97 MPa. The composition of the UHPC is listed in Table 1. Silica fume (particle size of 0.10–0.26 μm, density of 2.1 g/cm³, specific surface area of 20,500 m²/kg) was from Elkem company. Steel fibers were used as micro reinforcement with a volume dosage of 1%. The fiber length, diameter, and tensile strength were 12 mm, 0.2 mm, and 2850 MPa, respectively. Considering the enhancement effect of the steel fibers on joining the aggregates, the confinement effect of the rebar can be neglected. There was no reinforcement set in the UHPCS.

TNT explosives

TNT explosives, which are standard high explosives with chemical safety, were used in the tests. The detonation heat density of the TNT was 4521 kJ/kg. The density of the TNT was 1.65 g/cm³. Five charges were designed, i.e., 1.0, 1.6, 2.2, 3.3, and 5.0 kg, for each type of slab. The TNT shape was cylindrical with a length-to-diameter ratio of 1. The TNT was placed on the top center of the slab, as shown in Fig. 2. An electrical detonator was placed at the top of the TNT. The explosive in the detonator was Hexogen (RDX). Each detonator contained 0.4 g to 0.6 g of RDX. The equivalent TNT mass of the detonator was less than 1 g, this mass was three orders of magnitude smaller than the weight of the explosive used in the test. Thus, the influence of the mass of the detonator could be reasonably neglected.

Fig. 2.

Layout of the TNT explosive and the detonator.

Experimental setup

Previous studies^35,36 have shown that the localized response and damage caused by contact explosion are independent of the boundary conditions. To facilitate measuring the spall effect on the rear surface of the slab, simply supported boundary conditions were adopted. The support system consisted of four concrete piers and one square steel frame, as shown in Fig. 3. The four concrete piers with a strength grade of C40 acted as support foundations. Their height and planeness were carefully calibrated to maintain the same heights for the top faces. The steel frame adopts Q345 and the yield strength is 345 MPa.The four corners of the square steel frame were placed on the top faces of the piers. The slab was then placed on a square steel frame to form simple support. The outer side length of the square steel frame was 1500 mm, this value was the same as the side length of the slab. The assembly of the slab was carefully performed to ensure that the slab edges were flush with the outer edges of the square steel frame. The inner side length of the square steel frame was 1340 mm. This allowed most of the rear surface of the slab to be exposed to air.

Fig. 3.

Experimental setup and support system (unit: mm).

Test program

Two types of slabs, the NRCS and the UHPCS, were planned for the contact explosion tests. Five slabs were made for each type of slab. Five charges of TNT, i.e., 1.0, 1.6, 2.2, 3.3, and 5.0 kg, were tested for the five slabs, as shown in Table 2.

Table 2.

Test scheme.

TNT charge weight(kg)	Concrete type
	NRC	UHPC
1.0	NRCS-1	UHPCS-1
1.6	NRCS-2	UHPCS-2
2.2	NRCS-3	UHPCS-3
3.3	NRCS-4	UHPCS-4
5.0	NRCS-5	UHPCS-5

Results and discussion

Failure mode and results

Failure mode classification

According to the damage-severity degree¹⁰ of the concrete slabs under contact explosion, the failure modes are typically divided into five categories.

1. Crater mode, as shown in Fig. 4(a):

   • When the charge weight is relatively low, the explosion can only cause damage on the frontal surface of the slab. The damaged area is usually a cone-shaped crater. No damage is generated on the rear surface. This situation is defined as the crater mode.

• 2.Threshold spall mode, as shown in Fig. 4(b):
  • As the magnitude of the blast wave induced by the explosion of TNT increases, the peak tensile wave reflected from the free rear surface of the slab reaches the tensile strength of the concrete, and cracks begin to be generated at the rear surface of the slab. A slight bulge with a gentle slope may also be generated as the cracks propagate. The boundary of the bulge cannot be clearly distinguished by the naked eye. This failure mode is defined as the threshold spall mode. The charge required for the threshold spall mode is called the threshold spall charge.

• 3.Medium spall mode, as shown in Fig. 4(c):
  • As the magnitude of the explosive load increases with further increase in the charge weight, the failure mode of the slab will reach the medium spall mode.
  • In the medium spall mode, the reflected tensile wave causes a large bulge with concentrated cracks and fractures on the rear surface of the slab. A clear boundary of the bulge can be identified. Fewer cracks outside the bulge boundary can be observed than inside the bulge. Although severe damage has been generated inside the bulge, the residual strength of the concrete or the connecting force of the cement matrix is still sufficient to prevent the entire bulge from peeling off from the slab. However, at the center of the bulge, a local spall usually occurs due to the peak value of the reflected tensile wave, as shown in Fig. 4(c1). Therefore, in the medium spall mode, in addition to the crater in the frontal surface, a local spall area can also be detected and measured.
  • For concrete reinforced with fibers, the medium spall mode behavior may be different. For example, a large number of steel fibers in UHPC provide a bridging action for the cement matrix. The extra tensile force provided by the bridging action prevents the local spall from peeling off. The entire bulge with a distinct boundary can be observed at the rear surface of the slab, as shown in Fig. 4(c2).

• 4.Severe spall mode, as shown in Fig. 4(d):
  • As the explosive charge continuously increases, the reflected tensile wave at the rear surface of the slab not only exceeds the tensile strength of the concrete, but also has residual tensile energy that can overcome the residual connecting forces between the bulge and slab, and the entire bulge area on the rear surface detaches. The detached portion is defined as a spall. The corresponding failure mode is defined as the severe spall mode. The typical failure zone of this mode includes the front crater and rear spall.

• 5.Perforation mode, as shown in Fig. 4(e):
  • As the explosive charge exceeds that needed for the severe spall mode, the dimensions of the carter and spall increase. The perforation mode is reached upon connection of the crater and the spall.

Fig. 4.

Typical failure modes of concrete slabs under contact explosion.

Measurement methods and results

The failure modes and measurement results of the NRCSs and UHPCSs are shown in Table 3. The symbols Φ and D represent the diameter and depth of the damaged areas, respectively. The subscripts C and S represent the crater and spall, respectively. As shown in Fig. 5(a), the depths of the crater and spall are denoted as D_C and D_S, respectively. The diameters of the crater and spall are denoted as Φ_C and Φ_S, respectively. The depth was determined by measuring the distance from the outer surface to the deepest point. Due to the irregular shapes of the crater and spall area, an averaging method was used to calculate Φ_C and Φ_S. As shown in Fig. 5(b), four data points along the longitudinal, latitudinal, and two diagonal directions, denoted as Φ_Ca(Φ_Sa), Φ_Cb(Φ_Sb), Φ_Cc(Φ_Sc), and Φ_Cd(Φ_Sd), respectively, were measured. Then, Φ_C(Φ_S) was obtained by averaging the values of Φ_Ca(Φ_Sa), Φ_Cb(Φ_Sb), Φ_Cc(Φ_Sc), and Φ_Cd(Φ_Sd).

Table 3.

Failure modes and dimensions of the slabs. (unit: cm)

Slab no.	W (kg)	Failure mode	ΦC	ΦCa	ΦCb	ΦCc	ΦCd	DC	ΦS	ΦSa	ΦSb	ΦSc	ΦSd	DS
NRCS-1	1.0	Medium spall	44.9	45.4	41.6	45.9	46.6	7.3	76.5	81.6	71.4	80.9	72.1	6.8*
NRCS-2	1.6	Severe spall	66.6	59.5	65.3	66.0	75.5	8.5	77.8	77.8	72.0	81.5	80.0	11.3
NRCS-3	2.2	Perforation	75.0	72.0	86.0	73.0	69.0	8.0	99.0	94.0	104.0	97.0	101.0	22.0
NRCS-4	3.3	Perforation	70.0	66.0	75.0	79.0	60.0	11.0	104.5	104.0	105.0	101.0	107.5	19.0
NRCS-5	5.0	Perforation	108.0	109.0	101.0	120.0	102.0	14.0	109	107.0	105.0	108.0	116.0	16.0
UHPCS-1	1.0	Threshold spall	25.3	24.0	27.0	24.0	26.0	5.0	——	——	——	——	——	1.6*
UHPCS-2	1.6	Medium spall	34.3	31.0	37.0	30.0	39.0	6.5	57.8	55.0	60.0	58.0	58.0	6.5*
UHPCS-3	2.2	Medium spall	33.3	32.0	35.0	32.0	34.0	7.5	76.3	70.0	80.0	73.0	82.0	12.0*
UHPCS-4	3.3	Severe spall	34.8	35.0	40.0	30.0	34.0	7.0	66.8	68.0	68.0	58.0	73.0	17.0
UHPCS-5	5.0	Perforation	56.3	54.0	56.0	56.0	59.0	7.0	80.0	77.0	79.0	84.0	80.0	23.0

Fig. 5.

(a) Dimensions of the crater and spall. (b) Measurements of the diameters of the crater and spall.

Note that because the bulges in the threshold spall and medium spall modes were not detached from the slab, the spall depth could not be directly measured. The D_S data points for NRCS-1, UHPCS-1, UHPCS-2, and UHPCS-3 were the heights of the bulges from the peak to the rear surface, which are marked with * in Table 3.

In addition, to better compare and analyze the anti-explosion performance of the two types of concrete slabs, we further processed the test data according to Reference¹⁰, and the parameters of the slab thickness (T), the standoff distance (R), and the equivalent mass of TNT (W) were considered the key variables. The scaled slab thickness(T/W^1/3), the scaled distance(R/W^1/3) and the corresponding failure mode are listed in Table 4.

Table 4.

Parameters of the tested slabs.

Slab no.	W(kg)	T(m)	R(m)	T/W1/3(m/kg1/3)	R/W1/3(m/kg1/3)	Failure mode
NRCS-1	1.0	0.3	0.392	0.300	0.392	Medium spall
NRCS-2	1.6	0.3	0.407	0.256	0.348	Severe spall
NRCS-3	2.2	0.3	0.419	0.231	0.322	Perforation
NRCS-4	3.3	0.3	0.437	0.202	0.293	Perforation
NRCS-5	5.0	0.3	0.457	0.175	0.267	Perforation
UHPCS-1	1.0	0.3	0.392	0.300	0.392	Threshold spall
UHPCS-2	1.6	0.3	0.407	0.256	0.348	Medium spall
UHPCS-3	2.2	0.3	0.419	0.231	0.322	Medium spall
UHPCS-4	3.3	0.3	0.437	0.202	0.293	Severe spall
UHPCS-5	5.0	0.3	0.457	0.175	0.267	Perforation

Failure characteristics

The 1.0 kg-TNT explosion

The failure modes of the slabs under contact explosion of 1.0 kg TNT are shown in Fig. 6.

Fig. 6.

Failure characteristics of NRCS-1 and UHPCS-1.

NRCS-1 reached the medium spall mode. On the frontal surface, a compression crater with a diameter of 44.9 cm and a depth of 7.3 cm was generated. Numerous radial and circumferential cracks were observed outside the crater. A cross-shaped crack was observed and extended along the longitudinal and latitudinal directions of the central reinforcements. On the rear surface, a circular bulge with a diameter of 76.5 cm was generated. In the center of the bulge, an irregular spall area with a diameter of 15 cm and a depth of 6.8 cm, was observed. Many radial cracks were concentrated between the margin of the bulge and the margin of the spall. These results indicated severe damage in this area. A cross-shaped crack that extended along the longitudinal and latitudinal directions of the central reinforcements was also observed on the rear surface. On each of the four side surfaces, a crack was generated along the centerline. These four cracks were connected with the cross-shaped crack on the rear surface. Therefore, it is reasonable to predict that fault surfaces might exist along the centerlines of the slab.

UHPCS-1 reached the threshold spall mode. Compared to NRCS-1, UHPCS-1 exhibited a much lower degree of damage. On the frontal surface, a circular crater was generated. The diameter and depth of the crater were 25.3 cm and 5.0 cm, respectively, which were 43.7% and 31.5% smaller than those of NRCS-1. Only one short radial crack and one partial circumferential crack were observed outside the crater. On the rear surface, although no spall was present, a bulge with a height of 1.6 cm was observed. The slope of the bulge was relatively low, which caused difficulty to clearly detect the bulge boundary. In the center of the bulge, a dense crack zone with a diameter of 30 mm was observed, this result indicated some inner damage. On the four side surfaces of the slab, only two fine cracks were observed on side surface B.

The 1.6 kg-TNT explosion

The failure modes of the slabs under contact explosion of 1.6 kg TNT are shown in Fig. 7.

Fig. 7.

Failure characteristics of NRCS-2 and UHPCS-2.

NRCS-2 reached the severe spall mode. The frontal surface was cratered with a diameter of 66.6 cm and a depth of 8.5 cm. Outside the crater, radial and circumferential cracks were observed. Cross-shaped cracks along the longitudinal and latitudinal directions of the central reinforcements were also observed. The central reinforcement in the latitudinal direction was fractured. On the rear surface, a spall with a diameter of 77.8 cm and a depth of 11.3 cm was generated through the detachment of the entire bulge. Cross-shaped cracks grew along the longitudinal and latitudinal directions of the central reinforcements. The cross-shaped cracks propagated through the entire rear surface. Many radial cracks were observed outside the spall. The central reinforcements were fractured in the middle. On each side surface, a crack propagating through the entire surface along the centerline was observed.

UHPCS-2 reached the medium spall mode. On the frontal surface, a crater with a diameter of 34.3 cm and a depth of 6.5 cm was observed. Compared with NRCS-2, the diameter and depth of the crater were reduced by 48.5% and 23.5%, respectively. Circumferential cracks were observed around the crater. On the rear surface, an evident circular bulge was produced. Although the boundary of the bulge was clear, the bulge was still attached to the slab, and spall did not occur. The diameter of the bulge was 57.8 cm, and its height was 6.5 cm. A concentration phenomenon of the radial and circumferential cracks was observed inside the bulge. Outside the bulge, relatively sparsely distributed radial and circumferential cracks were observed. On the side surfaces, four to five cracks were observed. The widths of the cracks were much finer than those of NRCS-2.

The 2.2 kg-TNT explosion

The failure modes of the slabs under contact explosion of 2.2 kg TNT are shown in Fig. 8.

Fig. 8.

Failure characteristics of NRCS-3 and UHPCS-3.

NRCS-3 reached the perforation mode. On the frontal surface, a crater with a diameter of 75.0 cm and a depth of 8.0 cm was generated. The steel bars were fractured at the center of the frontal surface. Cross-shaped cracks formed outside the crater. In addition, numerous radial and circumferential cracks with widths close to those of the cross-shaped cracks were generated. On the rear surface, a large spall with a diameter of 99.0 cm was formed. A large number of wide cracks propagated radially outside the spall. Under a strong blast load, the reinforcements were pulled out and bent, and fractures also occurred. On the side surfaces of the slab, one to three cracks with a considerably wide width were generated.

UHPCS-3 reached the medium spall mode. A crater was generated on the frontal surface. The diameter and depth of the crater were 33.3 cm and 7.5 cm, which were 55.6% and 6.3% smaller than those of NRCS-3, respectively. Some fine circumferential cracks were observed outside the crater. On the rear surface, a bulge with a clear boundary and a certain height was observed. Spalling of the bulge was prevented by the bridging action of the steel fibers. The diameter of the bulge was 76.3 cm, which was 22.9% smaller than that of NRCS-3. The height of the bulge was 12 cm. This observation indicated severe damage inside the bulge. Along the centerline of the bulge, a wide crack was formed and separated the bulge into several parts. Parts of the bulge exhibited valgus tendencies as they grew with the wide crack propagation. A large number of radial and circumferential cracks were concentrated inside the bulge. Outside the bulge, several intermittent circumferential cracks were observed. On each side surface, four to five cracks were observed.

The 3.3 kg-TNT explosion

The failure modes of the slabs under contact explosion of 3.3 kg TNT are shown in Fig. 9.

Fig. 9.

Failure characteristics of NRCS-4 and UHPCS-4.

NRCS-4 reached the perforation mode. Although NRCS-4 exhibited the same failure mode as NRCS-3, NRCS-4 exhibited much more severe damage than NRCS-3. On the frontal surface, a crater with a diameter of 70.0 cm and a depth of 11.0 cm was observed. Numerous radial and circumferential cracks were generated outside the crater. Cross-shaped cracks along the midline direction of the concrete slab were also observed. The edges of the cross-shaped cracks were connected to the cracks on the four side surfaces. The steel bars at the center of the slab were fractured. On the rear surface, a spall with a diameter of 104.5 cm and a depth of 19.0 cm was generated. Outside the spall, radial cracks propagating to the edges of the surface were generated. The steel rebars in the bottom and middle layers were severely pulled out and bent, and fractures were even observed at the center. On each side surface, several vertical cracks were observed. In the center, approximately one to three wide cracks penetrating through the surface were generated. Their edges were connected with the radial cracks on the frontal and rear surfaces.

UHPC-4 reached the severe spall mode. It exhibited much better blast resistance than NRCS-4. On the frontal surface, a crater with a diameter and a depth of 38.4 cm and 7.0 cm, respectively, was generated, which were 45.1% and 36.4% smaller than those of NRCS-4. Some clear circumferential cracks were observed outside the crater, but the widths of the cracks were much narrower than those of NRCS-4. On the rear surface, a spall with a diameter of 66.8 cm and a depth of 17.0 cm was observed, which were 36.1% and 10.5% smaller than those of NRCS-4. The spall indicated that the tensile wave reflected from the rear surface exceeded the tensile strength of UHPC subjected to TNT with a charge weight of 3.3 kg. Outside the spall, some radial and circumferential cracks were observed. On each of the four side surfaces, some short cracks were observed in the center.

The 5.0 kg-TNT explosion

The failure modes of the slabs under contact explosion of 5.0 kg TNT are shown in Fig. 10.

Fig. 10.

Failure characteristics of NRCS-5 and UHPCS-5.

A more severe perforation level was reached in NRCS-5 than in NRCS-3 and NRCS-4. On the frontal surface, a crater with a diameter of 108.0 cm and a depth of 14.0 cm was observed. A large number of wide radial and circumferential cracks were observed outside the crater. Among them, the widths of the cracks in the midline direction and diagonal direction were the largest. On the rear surface, a spall with a diameter of 109.0 cm and a depth of 16.0 cm was observed. Outside the spall, the damage severity of the radial and circumferential cracks was approximately the same as that on the frontal surface. In both the frontal and rear surfaces, pulling out, bending, and fracturing of the steel bar reinforcements were also observed. On each side surface, several vertical cracks were observed. In the center of the side surfaces, a very wide crack with a width of ~ 5 cm penetrated through the entire surface, indicating that the concrete was split by this level of blast load. The edges of the vertical cracks were connected to radial cracks on the frontal and rear surfaces. The severe failure characteristics indicated that the concrete matrix of the entire slab almost reached the verge of fracture and disintegration, and separation could potentially be observed as the charge increased.

Under the contact explosion generated by 5 kg of TNT, UHPCS-5 also reached the perforation mode. On the frontal surface, a crater was generated. The diameter and depth of the crater were 56.3 cm and 7.0 cm, respectively, which were 47.9% and 50.0% smaller than those of NRCS-5. Outside the crater, many wide circumferential cracks with a certain width were observed. On the rear surface, a conical spall with a diameter and a depth of 80.0 cm and 23.0 cm was generated. The diameter of the spall was 26.6% smaller than that of NRCS-5. Outside the spall, distinct radial and circumferential cracks with certain widths were also observed. In the center of each side surface, one wide crack penetrating through the surface was generated. The crack in side surface D had the largest width. In addition, multiple finer cracks could be found on all four side surfaces.

Comparison of the results

The Φ_C and D_C values are shown in Fig. 11(a). The solid black points and the hollow black points denote the Φ_C and D_C of the NRCSs, respectively. The solid red points and the hollow red points denote the Φ_C and D_C of the UHPCSs, respectively. The lines outline the tendencies of Φ_C and D_C with varying scaled distance (R/W^1/3). As R/W^1/3 decreases, the Φ_C and D_C of the two types of concrete increase. The slope of the NRCS is much sharper than that of the UHPCS. Although the UHPCS is perforated under the minimum R/W^1/3 of 0.267, the dimensions (Φ_C of 56.3 cm, D_C of 7.0 cm) of the UHPCS crater are similar to those (Φ_C of 44.9 cm, D_C of 7.3 cm) of the NRCS crater under the maximum R/W^1/3 of 0.392. Considering that the crater size is mainly determined by the compressive and shear performance of the material, the high compressive strength of UHPC is presumed to be beneficial for reducing the crater size.

Fig. 11.

Dimensions of the craters.

Under the same R/W^1/3, a much larger crater is generated in both diameter and depth in the NRCS than in the UHPCS. The reduction factors for the Φ_C and D_C of UHPC are defined as follows:

1

where Φ_CUHPC is the Φ_C of the UHPCS, and Φ_CNRC is the Φ_C of the NRCS. D_CUHPC is the D_C of the UHPCS, and D_CNRC is the D_C of the NRCS. The calculated values of and are shown in Fig. 11(b). As R/W^1/3 decreases from 0.392 to 0.267, initially decreases and then increases with slight variation, and the maximum, minimum, and average values are 0.562, 0.443, and 0.507, respectively. These results indicate that is insensitive to R/W^1/3, and that Φ_CUHPC is approximately half of Φ_CNRC as R/W^1/3 changes. In contrast, exhibits a certain sensitivity to R/W^1/3. As R/W^1/3 decreases from 0.392 to 0.267, initially increases and then decreases with significant variation, and the maximum, minimum, and average values are 0.938, 0.500, and 0.705, respectively.

The Φ_S and D_S are compared in Fig. 12(a). As R/W^1/3 decreases, the Φ_S and D_S of the UHPCSs increase. The Φ_S of the NRCSs increases as R/W^1/3 decreases. In contrast, the D_S of the NRCS initially increases as R/W^1/3 decreases. D_S reaches a maximum at an R/W^1/3 of 0.322 for NRCS-3. After the maximum, the D_S of the NRCSs decreases as R/W^1/3 decreases. This phenomenon can be explained by the aggravation of the perforation level as R/W^1/3 decreases for the slabs with a certain thickness. When R/W^1/3 is lower than 0.322, the slab is clearly perforated, and the sum of D_C and D_S equals the slab thickness of 30 cm. Because the blast load is directly loaded on the frontal surface of the slab, the crater generation is the primary path for blast energy consumption. Thus, as the blast load increases with a decrease in R/W^1/3 from 0.322, the D_C clearly increases, which results in a decrease in D_S. Moreover, the increases in the blast load can still aggravate the spall, which is expressed as an increase in Φ_S, as shown in Fig. 12(a). Thus, a similar phenomenon can be reasonably deduced for the UHPCSs when R/W^1/3 continuously decreases from 0.267. As shown in Fig. 12(b), NRCS-3 and UHPCS-5 reach perforation under R/W^1/3 values of 0.322 and 0.267, respectively. The D_S of UHPCS-5 is 23.0 cm, and the D_S of NRCS-3 is 22.0 cm. These two types of slabs have similar D_S values when perforation is reached only for the two types of slabs with the same thickness. However, the Φ_S of UHPCS-5 is 80 cm, and the Φ_S of NRC-3 is 99.0 cm. The former is 80.8% of the latter. This demonstrated that UHPC can mitigate spall damage under the perforation mode. It is reasonable to presume that the higher tensile strength of UHPC the mixing of steel fibers plays a positive effect.

Fig. 12.

Dimensions of the spalls.

Under the same R/W^1/3, a much larger spall in both diameter and depth is generated in the NRCS than in the UHPCS. The reduction factors for the Φ_S and D_S of UHPC are defined as follows:

2

where Φ_SUHPC is the Φ_S of the UHPCS and Φ_SNRC is the Φ_S of the NRCS. D_SUHPC is the D_S of the UHPCS, and D_SNRC is the D_S of the NRCS. The calculated values of and are shown in Fig. 12(b). The maximum and minimum values of are 0.770, and 0.639, respectively. As R/W^1/3 decreases from 0.392 to 0.267, changes with slight fluctuation around an average value of 0.721. This result indicates that UHPCS exhibits a consistently better anti-collapse performance than NRCS, potentially due to the enhancement effect of the steel fibers. Steel fibers³⁷ can effectively improve the spalling strength of UHPC, reduce damage to the matrix, and prevent the formation and propagation of cracks. Only two valid data points of are present; the maximum and minimum values of are 1.438, and 0.895, respectively.

Under the same charge weight, the dimensions of the crater and spall of the UHPCS are much smaller than those of the NRCS; these results indicate that the damage degree of the UHPCS is much slighter than that of the NRCS. Similar conclusions have also been drawn based on the near-field explosion tests of 10 cm slabs conducted by Wu et al.³² and the contact explosion tests of 10 cm slabs conducted by Li et al.³³.

Validation of the failure mode predictions

Under contact explosion, many unknown parameters and uncertainties, such as the influence of the geometric shape of the charge on the explosion³⁸, the propagation and attenuation rate of the stress wave in the concrete, the dispersion effect of the wave, and the tensile and compressive strength of the concrete material under a high strain rate, cause difficulty for the use of a complete analytical method to deduce the stress state and failure mode of the concrete slab. To evaluate the damage degree of concrete members under explosion loads, many empirical methods have been proposed.

By analyzing more than three hundred experimental data points, McVay¹⁰ proposed an empirical formula to predict the local damage of the concrete slabs under chemical explosion. The failure modes are differentiated by the values of the scaled slab thickness (T/W^1/3) and the scaled distance (R/W^1/3). For each slab tested in this study, T/W^1/3 and R/W^1/3 were calculated and summarized in Table 4. The T/W^1/3 values of the NRCSs reaching medium spall, severe spall, and perforation are 1.17, 1.27, and 1.32 times the T/W^1/3 values of the UHPCSs, respectively. The R/W^1/3 of NRCSs reaching medium spall, severe spall, and perforation are 1.13, 1.19, and 1.21 times the R/W^1/3 values of the UHPCSs, respectively.

McVay’s formula and the experimental data¹⁰ are replotted in Fig. 13. The upper line is used to differentiate the no damage mode and the spall mode, and the lower line differentiates the spall mode and the perforation mode. Ten experimental results from the present study are also plotted in Fig. 13 for comparison. All ten slabs are located in the no damage zone, indicating an underestimation of McVay’s method for the contact explosion tests. The main reason for the inaccurate evaluation with McVay’s formula can be attributed to the experimental data used. McVay’s formula was proposed mainly based on the data of close-in explosion tests, and the data from the different failure modes practically fall into the corresponding area distinguished by the thresholds, as shown in Fig. 13. However, the energy action mechanism of a close-in explosion on a concrete slab is far different from that of a contact explosion. In the case of a contact explosion, the blast energy can directly load on the concrete slab without consumption. For a close-in explosion⁷, the certain layer of air between the explosive and the slab consumes a large amount of explosive energy as the blast wave propagates and attenuates in the air. Another reason for the deviation of the prediction results of the UHPCS is the strength grade. The recommended application range of McVay’s formula is normal concrete with a compressive strength of 17 ~ 48 MPa. This formula is not applicable to concrete with a higher strength, and additives such as steel fibers are not considered. The UHPC used in this study has a compressive strength of 129.3 MPa with a 1% dosage of steel fibers, which is far beyond the scope of the recommended conditions. This causes the unsatisfactory prediction results, as shown in Fig. 13. In protective engineering, the underestimation of the failure mode will lead to a reduction in the resistance design index, which will cause the engineering structure to face irreparable consequences. Therefore, it should be careful as much as possible to use McVay’s formula to evaluate the failure mode of concrete slabs under contact explosion.

Fig. 13.

Validation of the empirical predictions of McVay’s formula¹⁰.

Combined with experimental results, Morishita et al.¹² proposed damage prediction formulae for concrete slabs under contact explosion based on McVay’s formula, which are defined as follows (the unit is m/kg^1/3):

3

4

5

The experimental results and the predictions calculated from Morishita’s formulae are compared in Fig. 14. Figure 14 is divided into three windows: the left window shows the predictions of Morishita’s formulae, the middle window shows the NRCS results, and the right window shows the UHPCS results. In each window, the spall and perforation thresholds are plotted by the blue lines and the red lines, respectively. Although some experimental results (NRCS-1, NRCS-2, NRCS-5, UHPCS-2, UHPCS-3, UHPCS-4, and UHPCS-5) are effectively predicted by the formulae, the spall and perforation thresholds of the proposed formulae exhibit distinct differences from the threshold values measured in our study. The spall threshold proposed by Morishita’s formulae is 0.36. The spall threshold of the NRCS is 0.3 ~ 0.323 (the upper limit is obtained from our previous experiments¹⁶, and the lower limit is determined by NRCS-1); this result is lower than 0.36. The spall threshold of the UHPCS exhibits a further lower value of ~ 0.3, which is 0 ~ 7.1% lower than that of the NRCS. The perforation threshold proposed by Morishita’s formulae is 0.20. The perforation threshold of the NRCS shows a higher value of 0.242 ~ 0.256. The perforation threshold of the UHPCS is 0.175 ~ 0.202, which is 16.5 ~ 31.6% less than that of the NRCS. The perforation threshold of the UHPCS showed good agreement with the value of 0.2 proposed by Morishita’s formulae. However, this agreement seems to be just a coincidence, because Morishita’s formulae are proposed based on the experimental data of normal concrete, and no UHPC data are considered.

Fig. 14.

Validation of the empirical predictions of Morishita’s formulae.

To verify the thresholds for the UHPCSs proposed in this study, the data of UHPCSs under contact explosion from the literature³⁴ and literature³⁹ are plotted in the right window of Fig. 14 as the solid blue points and hollow blue triangles, respectively. According to the experimental results in the literature³⁴, UHPCSs reach perforation mode at T/W^1/3 values of 0.12, 0.1, and 0.15. As shown in Fig. 14, points 0.12, 0.1, and 0.15 are accurately located in the perforation mode area, which verifies the validity of the perforation threshold for predicting the perforation mode. The UHPCS reaches the crater mode at a T/W^1/3 of 0.258 in the literature³⁴. However, this data point is located in the spall mode area, which represents a prediction of a severe failure level. The deviation in the prediction of the failure mode may be induced by the size effect⁴⁰. In an equivalent explosion experiment, the damage degrees of small-size slabs under small equivalent explosions were slightly less than those of large-sized concrete slabs. The thickness of the slabs in the literature³⁴ was 100 ~ 150 mm, which was 1/3 ~ 1/2 the thickness of the 300 mm-thick slabs in the present study. According to the experimental results in the literature³⁹, the UHPCSs reach the spall mode at T/W^1/3 values of 0.295, 0.277, 0.261, 0.242, 0.216, 0.207, and 0.188. As shown in Fig. 14, the data points are accurately located in the spall mode area, which is bounded by the upper limit of the spall threshold and the lower limit of the perforation threshold. The data point 0.295 is close to the spall threshold in Fig. 14. Moreover, the failure characteristics at a T/W^1/3 of 0.295 approach the spall threshold: a crater with a diameter of 45 cm is generated on the frontal surface, and a tiny spall with various radical cracks is generated on the rear surface³⁹. The data point 0.188 is close to the perforation threshold in Fig. 14. Moreover, the failure characteristics at a T/W^1/3 of 0.188 reach the severe spall mode, approaching the perforation threshold: the crater size is significantly enlarged, with a depth of 12.3 cm. A spall with a diameter of 95 cm and a depth of 20.4 cm is generated. Numerous cracks propagate around the spall, and some even extend to the four sides of the slab³⁹.

Based on the experimental data of contact explosion, the spalling damage coefficient K_z was established to determine the failure mode of a concrete slab^18,36, as shown in Eq. (6). Different from Morishita’s formulae, the scaled distance R/W^1/3 is considered in these formulae. A smaller K_z correlates to a more severe damage mode of the concrete slab.

6

where T is the concrete slab thickness, in meters; e is the charge center height, in meters; and W is the TNT mass, in kilograms. For clarity, the threshold value of K_z distinguishing the crater mode from the spall mode is deted as K_zcs; the threshold value of K_z distinguishing the spall mode from the perforation mode is denoted as K_zsp.

The experimental results and the formula predictions are compared in Fig. 15, and great differences can be observed. The boundary of the spall mode exhibits a relatively wide range, which leads to all ten data points falling into the spall mode area. The test results show that the K_zcs values for the NRCS and UHPCS are 0.392 ~ 0.415 (the upper limit is obtained from our previous experiments¹⁶, and the lower limit is determined by NRCS-1) and ~ 0.392, respectively. In contrast, the K_zcs of 0.8 provided by the formula is too high. The approximately 0 ~ 5.5% lower value of K_zcs of the UHPCS than that of the NRCS demonstrates the better blast resistance of the former. The test results show that the K_zsp values for the NRCS and UHPCS are 0.322 ~ 0.348 and 0.267 ~ 0.293, respectively. The K_zsp of 0.25 provided by the formula is too low. The K_zsp of the UHPCS is 9.0 ~ 23.3% lower than that of the NRCS.

Fig. 15.

Validation of the empirical predictions of the spalling damage coefficient K_z.

The experimental data in the literature³⁴ and literature³⁹ are also used to verify the proposed threshold values of K_z, as shown in the right window of Fig. 15. The data from the literature³⁴ and literature³⁹ are plotted as solid blue points and hollow blue triangles, respectively. According to the experimental results in the literature³⁴, only one point at a K_z of 0.302 indicates an inaccurate prediction, and satisfactory prediction results are provided for the other three points, in which the prediction of the perforation mode is consistent with the experimental results. Similar to the discussion about Morishita’s formulae, the tested failure mode at a K_z of 0.302 is the crater mode, which is slighter than the predicted spall mode. This difference may also be induced by the size effect. According to the experimental results in the literature³⁹, the UHPCSs reach the spall mode at K_z of 0.331, 0.321, 0.302, and 0.28, which is consistent with the experimental results. However, deviations are found at three points when K_z equals 0.257, 0.246, and 0.229. The slabs reach the spall mode in the experiments, but the prediction provides an incorrect result and shows the perforation mode.

From the prediction results, Morishita’s formulae exhibit greater accuracy than the spalling damage coefficient K_z in predicting the failure mode of UHPCSs under contact explosion. The charge center height e is included in the formulae K_z, and is not considered in Morishita’s formulae. The reason for the inaccurate prediction may be caused by the shape of the TNT charge, detonation location, etc. A further discussion of the underlying reasons needs be carried out on the basis of adequate experimental and simulation results in the future.

The variables considered by the three types of formulae include the scaled slab thickness (T/W^1/3) and the scaled distance (R/W ^1/3), however, variables related to the material of the concrete slab, such as the compressive strength, and tensile strength, are not considered. These formulae were proposed relatively early, and the test results used were mainly obtained from normal concrete with relatively low strength. The range of the material strength was not large enough to be considered a variable. However, as discussed above, for two types of concrete slabs with the same dimensions loaded by the same charge and form of TNT, the three formulae cannot distinguish the difference in the failure mode. The significant difference in the blast resistance of the two types of concrete slabs cannot be disregarded. The effect induced by the substantial increase in the material strength needs to be considered. To provide a reliable prediction of the blast failure modes of UHPC, further modification of these empirical formulae is necessary.

Dimensional analysis of the crater and spall

Explosion is a complicated phenomenon. Energy transformation, accompanied by high-pressure generation, high-temperature release, and shock wave propagation, occurs in a transient period and limited space. The dimensional analysis method⁴¹ has been successfully used to describe explosion scenarios in several fields, such as air explosion, explosive forming, explosive welding, and excavation blasting for tunnels. To facilitate resolving the contact explosion problems confronted in protective engineering, the dimensional analysis method is used to quantitatively describe the dimensions of the crater and the spall of the two types of concrete slabs.

For contact explosions, the main parameters that influence the crater volume V_C can be divided into two parts. The first part includes the parameters related to the explosive charge, and these are the charge weight W, the explosive density _e, the chemical energy released per unit mass of explosive E_e, and the dilatation index of detonation products _e. The second part includes the parameters related to the target medium, and these are the slab thickness T, the material density , the elastic modulus E, the Poisson’s ratio , and the compressive strength Y_C. All the parameters are presented in Table 5.

Table 5.

Parameters considered for dimensional analysis.

Parameter type	Unit	Dimension
Charge parameters
Charge weight W	kg	M
Explosive density e	kg/m3	ML− 3
Chemical energy Ee,	J/kg	L2T− 2
Dilatation index e.	—	—
Slab parameters
Slab thickness T	m	L
Spall diameter ΦS	m	L
Crater volume VC	m3	L3
Material density	kg/m3	ML− 3
Elastic modulus E	MPa	ML− 1T− 2
Poisson’s ratio	—	—
Compressive(Tensile) strength YC(Yt)	MPa	ML− 1T− 2

Then, the V_C can be expressed as the following function:

7

Setting , , and as a unit system, Eq. (7) can be expressed as follows:

8

Considering that the same type of explosive is used, that UHPC and NRC have similar densities, and that the influence of elastic deformation can be ignored in the case of large deformation; thus, Eq. (8) can be simplified as follows:

9

Taking the log of both sides of Eq. (9) produces the following equation:

10

where, , , and c is a constant. Combining with test data, the fitting formula is obtained as follows:

11

Specifically, the following equation can be used:

12

The fitted formulae and the experimental results are shown in Fig. 16. The lower amplitude of the red line of the UHPCS than the black line of the NRCS demonstrates the smaller crater size of the UHPCS under the same . Figure 17 shows the comparison between the experimental data in the literature and the prediction curve. As shown in the figure, the predicted results are in good agreement with the experimental data. Therefore, the dimensional analysis method in this study can be used to effectively predict the crater volume of thick concrete slab under contact explosion.

Fig. 16.

Dimensional analysis of the crater.

Fig. 17.

Comparisons of experimental data and dimensional analysis of (a) Experiment 1 (b) Experiment 2 (c) Experiment 3.

For the spall on the rear surface of the slabs, the diameter of the spall Φ_S is analyzed since only two available data points for D_SUHPC can be used. Similar to the crater, the parameters that need to be considered include W, _e, E_e, _e, T, , E, , and Y_t. Note that the tensile strength Y_t of the concrete is used instead of the compressive strength Y_C because the spall damage is mainly determined by Y_t. All parameters are presented in Table 5.

Φ_S can be expressed as the following function:

13

Setting , , and as a unit system, Eq. (13) can be expressed as follows:

14

Ignoring the unchanged variables related to the explosive, the concrete density, and the elastic deformation, Eq. (14) is then simplified as follows:

15

Taking the log of both sides of Eq. (15) produces the following equation:

16

where , according to the calculations in literature⁴², . As shown in Fig. 18, combined with the test data, the fitting formula is obtained as follows:

17

Fig. 18.

Dimensional analysis of the spall.

Specifically, the following equation can be used:

18

Figure 19 shows the comparison between the experimental data in the literature and the prediction curves. As shown in the figure, the predicted results are in good agreement with the experimental data. Therefore, the dimensional analysis method in this study can be used to effectively predict the spall diameter of thick concrete slab under contact explosion.

Fig. 19.

Comparisons of the experimental data and dimensional analysis of (a) Experiment 1 (b) Experiment 2.

Conclusions

The blast resistance of UHPCS and NRCS under contact explosion was compared and analyzed through experiments. The failure modes were crater, spall, and perforation. The feasibility of the empirical formulae for predicting the failure mode of the two types of concrete was discussed. The following conclusions can be drawn:

(1) UHPCS exhibits superior blast resistance compared with NRCS. The ubiquity of cross-shaped cracks in all the failure modes of the NRCS indicates the drastic integral damage effect induced by the blast load. In contrast, cross-shaped cracks were not observed for all the failure modes of the UHPCS. The mix of steel fibers in the UHPCS effectively inhibits the generation of cracks through the central lines of the front, rear, and side surfaces. The fewer quantity of cracks in the UHPCS exhibited a shorter width and length than those of the NRCS. Different from the prevailing circumferential cracks of the NRCS, circumferential cracks with a certain length and width were not generated until the UHPCS reached the perforation mode. The scaled slab thicknesses (T/W^1/3) of the medium spall, severe spall, and perforation of the NRCSs were 1.17, 1.27, and 1.32 times greater than those of the UHPCSs, respectively.

(2) The reduction factors exhibit constant values, and these values are independent of T/W^1/3. The reduction factors might be determined by the properties of the UHPC and NRC. , and are 0.507 and 0.721, respectively.

(3) For predicting the spall and perforation of concrete slabs under contact explosion, the Morishita’s formula with modified threshold has the best applicability. The appropriate values for the spall and perforation thresholds of UHPC to modify Morishita’s formulae are ~ 0.3 and 0.175 ~ 0.202. The prediction accuracy of the spalling damage coefficient K_z is not as satisfactory as that of Morishita’s formulae. The use of McVay’s formula should be cautious.

(4) Dimensional analysis shows that V_C and Φ_S can be expressed as a function of T/W^1/3, Y_c/Y_t, and T. Analytical equations for V_C and Φ_S are also proposed and validated.

Future work

We will conduct in-depth research on the following aspects in the future.

1. Study on the anti-explosion performance of UHPC reinforced with different fibers. With the development of UHPC, more and more different kinds of fibers have been applied to UHPC and show good mechanical properties. Research on the anti-explosion performance of UHPC reinforced with different fibers will also become an important part of its engineering application.

2. Application of numerical simulation in the study of explosion resistance of UHPC. Explosion-related tests are difficult and costly to perform, and it is difficult to carry out very comprehensive experimental research. Thus, numerical simulation will likely be the development direction of UHPC anti-explosion performance research.

3. Study on the anti-explosion performance of UHPC under close-in explosion. To understand the anti-explosion performance of UHPC under different working conditions and comprehensively deal with the potential danger in extreme environments. The anti-explosion performance of UHPC under close-in explosion is an important part of its anti-explosion performance.

Author contributions

W.Z.: Conceptualization, methodology, investigation, writing—original draft preparation; J.M.: methodology, validation, investigation; X.Y.:methodology, investigation, data curation; B.Z.: validation, investigation, supervision; L.W.:methodology, data curation; F.H.:investigation; D.L.: formal analysis; M.C.:data curation.All authors reviewed the manuscript.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.