Abstract

The explosion in fireworks production facilities has its particularities. On the one hand, due to the enclosed structure of the fireworks production facility, the explosion of explosive material inside the structure is considered a confined explosion. On the other hand, the explosive material (detonators, etc.) are stored in two locations within the fireworks facility, resulting in two explosion sources.

There is limited research on the explosion loads and structural responses caused by dual explosion sources within enclosed spaces, both domestically and internationally. This paper focuses on studying the explosion loads and structural responses of dual explosion sources within fireworks facilities. It proposes an explosion resistance evaluation method that can be applied in engineering practice for explosive actuated power devices. The main research contents are as follows:

1.   Through explicit dynamic analysis software LS-DYNA, the explosion of dual sources in free air is simulated, and the propagation characteristics of the explosion shock waves from the dual sources are analyzed. The conclusion is that when dual explosion sources explode, there is an interaction between the two explosion shock waves, similar to the Mach reflection mechanism, resulting in an 89.7% enhancement of the explosion shock waves. As the distance between the two sources increases, the Mach reflection region remains essentially unchanged, while the enhancement effect of the Mach reflection weakens accordingly.

2.   The explosion process of dual explosion sources within fireworks facilities is simulated using explicit dynamic analysis software LS-DYNA. The distribution characteristics of the explosion loads on each wall are as follows: the distribution is extremely uneven along the walls of the fireworks facility, with two peak regions located in the collision zone of the two explosion shock waves and the corners of the walls.

3.   The existing methods for determining confined explosion loads in regulations have two shortcomings: 1) They assume a uniform distribution of explosion loads along the walls; 2) They only provide determination methods for explosion loads from a single source, which are not applicable to determining explosion loads from dual internal sources. Considering these factors, the walls of the fireworks facility are divided into zones, the explosion load enhancement coefficient λ is introduced, and a multi-point integration method is used to determine the enhancement coefficients in each zone on each wall. A method for determining explosion loads from dual internal sources is proposed.

4.   Based on the proposed method for determining explosion loads from dual internal sources, the dynamic response of the fireworks facility structure under the influence of dual explosion sources is analyzed, and the evaluation method for explosion resistance of the fireworks facility is improved. Under the action of 7.0 kg equivalent TNT, the structure’s walls undergo varying degrees of bending deformation. The rear wall experiences the most severe deformation, with the maximum displacement occurring at the mid-span of each wall. The maximum displacement of the rear wall is 76.7 mm, with a residual displacement of 27.6 mm, indicating mild damage to the structure. Due to the enhanced effect of Mach reflection, the side walls also experience significant deformation in their Mach reflection regions and mid-span regions.

Keywords: fireworks facility mass explosion hazard, confined explosion, dual explosion sources, explosion loads, rapid assessment.

Chapter 1: Introduction

1.1 Research Background and Significance

Explosion disasters occur due to various reasons, such as natural disasters caused by severe weather conditions or earthquakes, accidental explosions in social production activities via safety fuses, and man-made disasters like terrorist attacks or even wars. Regardless of the causes, explosion disasters result in significant casualties and economic losses, and to some extent, they cause social panic. Due to the unpredictability of explosion disasters, blast-resistant designs are not sufficiently robust. The instantaneous nature of explosions makes it difficult to control their impact, and the destructive damage caused by explosions renders structures unusable. These factors have led to an increasing focus on explosion disasters by humanity.

Throughout international history, mass explosion disasters have frequently occurred worldwide. Typical explosive events lead to enormous economic losses and disrupt normal social production and life. With the rapid improvement in the social production levels of various countries, the rapid expansion of manufacturing scales (explosives manufacturing building), and the increasing severity of conflicts between countries, explosion incidents have become more frequent internationally.

On April 19, 1995, a shocking car bomb terrorist attack occurred in Oklahoma City, USA, resulting in an explosion at the Alfred P. Murrah Federal Building, causing structural damage and collapse within a radius of several hundred meters (Figure 1-1). Nearly 200 people were killed, and thousands were injured, causing an economic loss of up to $650 million. Three years later, simultaneous car bomb terrorist attacks targeted the United States embassies in Dar es Salaam, the capital of Tanzania, and Nairobi, the capital of Kenya (Figure 1-2).

The death toll reached 224 people, with over 4,500 injured. The most significant explosive material event in international history is the “9/11 terrorist attacks.” On the morning of September 11, 2001, terrorists hijacked two civilian aircraft, which crashed into the Word Trade Center Building 1 and Word Trade Center Building 2 in New York City. The two buildings were severely impacted, resulting in explosions and fires (Figure 1-3). The death toll reached 2,996 people, with 6,000 injured, and the economic loss amounted to $200 billion. In the Middle East, suicide bombings occur frequently due to ongoing conflicts. On April 25, 2016, a car bomb attack took place in the capital city of Baghdad, Iraq (Figure 1-4), resulting in the deaths of 7 people and injuring 30 others.

Image 1-1: Oklahoma City Federal Building, 1995

Image 1-2: Nairobi Embassy, 1998

Image 1-3: World Trade Center, United States, 2001

Image 1-4: Baghdad Community, Capital of Iraq, 2016

Apart from accidental explosions caused by terrorist attacks, accidental explosions occur occasionally in human production and daily life. As a characteristic industry in China, the fireworks manufacturing industry has developed rapidly, and its production scale has expanded. In the Liuyang area of Hunan Province, a large-scale fireworks production and manufacturing park has been formed. However, accidental explosions also frequently occur in producing and manufacturing fireworks.

On February 5, 2018, a fireworks and firecracker warehouse area in Shangli County, Pingxiang City, Jiangxi Province, experienced a combustion explosion, accompanied by a massive mushroom cloud, and sustained explosions for a while, resulting in one person missing. On August 16, 2010, a fireworks and firecracker factory in Yichun City, Heilongjiang Province, exploded, causing extensive fires and resulting in 34 deaths, 153 injuries, and an economic loss of 320 million yuan. On January 20, 2016, an accidental explosion occurred in a fireworks factory in Guangfeng District, Shangrao City, Jiangxi Province, triggering a fire and causing strong vibrations throughout the city, resulting in 3 deaths and 53 injuries.

In conclusion, various explosion disasters caused by different factors have been increasing domestically and internationally, causing significant dynamic damage to various structures. The phenomenon of progressive collapse of high-rise buildings under blast loads, the structural collapse mechanism of large-scale space-reinforced concrete structures, and the structural dynamic damage caused by internal explosions in small enclosed reinforced concrete structures have become key research directions in engineering studies both domestically and internationally. A systematic and comprehensive study of the objective laws governing the propagation of blast loads and the blast resistance of building structures holds great significance.

• The particularities of fireworks production facility explosions
• The confined explosion within the enclosed structure
• Dual explosion sources within fireworks facilities
• Limited research on explosion loads and structural responses
• Focus on studying explosion loads and structural responses
• Simulation of dual explosion sources in free air
• Analysis of explosion shock wave propagation characteristics
• Interaction between the two explosion shock waves
• Mach reflection mechanism and its enhancement effect
• Distribution characteristics of explosion loads within fireworks facilities
• Uneven distribution along the walls
• Peak regions in collision zone and corners of walls
• Shortcomings of existing methods for determining confined explosion loads
• Introduction of explosion load enhancement coefficient λ
• Multi-point integration method for determining enhancement coefficients
• The proposed method for determining explosion loads from dual internal sources
• Analysis of the dynamic response of fireworks facility structure
• Evaluation method for explosion resistance improvement
• Bending deformation and displacement of structure walls
• Severe deformation in the rear wall
• Maximum displacement and residual displacement
• Mild damage to the structure
• Significant deformation in Mach reflection and mid-span regions of side walls

1.2 Research Status of Domestic and Foreign Explosion Loads

In response to the diverse types of explosion disasters in various parts of the world in recent years, scholars, both domestically and internationally, have conducted systematic research. In the research direction of the propagation law of explosion loads, scholars from both domestic and foreign origins have conducted extensive research using field experiments and numerical simulations. Stoner and Bleakney [1] studied the influence of explosive material shape on pressure-time curves.

They measured the peak pressures of spherical, cylindrical, and cubic explosive material through experiments, proposing a semi-empirical formula for calculating peak pressures. Phillips [2] compared the different peak pressures produced by end initiation (initiating explosives) and center initiation of cylindrical explosive material, finding that end initiation resulted in higher peak pressures than center initiation (initiating explosives). Hu and Wu [3-4] studied the effects of different explosive material shapes, placement orientations, etc., on pressure distribution on structural surfaces. Victor [5] pointed out significant differences in numerical values and directional characteristics of explosion parameters between non-spherical commercial explosives and spherical such explosives material with the same equivalent charge.

Knock and Davies [6] proposed a semi-empirical formula for calculating incident peak pressures based on field experiments with cylindrical PE4 explosive material. Hao [7-8] and others conducted parameterized analyses of ground explosion loads using numerical simulations and proposed semi-empirical formulas for calculating explosion pressures using rigid walls. Alex and Rose [9-10] conducted numerical simulations using Air3D to study the propagation characteristics of explosion shockwaves in complex structures of urban building clusters.

They investigated the influence of adjacent building structures on explosion loads. Shi Yanchao [11] employed AUTO-DYN for numerical simulations to study the influence of different grid sizes on various parameters of explosion loads and proposed a correction method to eliminate the grid size effect. Lu Hongqin, Li Weiping, Wang Keqiang [12-14], and others compared the numerical simulation results of explosion loads using LS-DYNA with empirical formulas and experimental results and found that the results were generally consistent, indicating a certain degree of reliability in the numerical simulation of explosion loads using LS-DYNA software.

Hu Ye [15] conducted a comparison of the effects of end initiation and middle initiation on explosion load responses of cylindrical explosive material and verified the results using numerical simulations and field experiments. The objective laws of explosion loads for cylindrical explosive material with end initiation were analyzed, and a simplified model for the explosion loads of nearly detonated cylindrical explosive material with end initiation was proposed.

Scholars domestically and internationally have conducted extensive studies in researching dynamic damage and blast resistance of building structures. Mosalam [16-17] conducted a numerical simulation analysis on the dynamic damage of CFRP-reinforced concrete slab structures under explosive action, verified through experiments.

Dakhakhni [18] analyzed the dynamic damage of bi-directionally reinforced concrete slabs under explosive action using single-degree-of-freedom analysis. Shi [19] used LS-DYNA to perform parametric analysis by propellant actuated power devices on reinforced concrete columns and proposed coefficients such as increasing the moment of inertia and concrete axial strength to enhance the blast resistance of reinforced concrete columns. Ngo, Namet [20-22] studied the blast resistance of high-strength concrete slab structures under explosive material loads.

Luccioni [23] employed AUTODYN software to investigate the collapse mechanism of reinforced concrete building structures under explosive action. Hsin [24] conducted a reliability analysis on the direct shear failure and bending failure of reinforced concrete slab structures under explosive action. Zhou [25] studied the dynamic damage of a medium-scale reinforced concrete model under contact explosion. Yang Kezhi [26] researched the dynamic response of structures subjected to explosive material loads in enclosed containers and conducted on-site tests for verification.

Tian Li [27] studied the continuous collapse mechanism of basement structures through numerical simulation, compared the dynamic damage under explosive material loads between aboveground and underground structures, and proposed feasible suggestions for structural protection. Zhai Hongbo [28] conducted on-site tests on a 1:8 scaled ship compartment model structure with the dual-point synchronous explosion, comparing the differences between single-point and dual-point synchronous explosions. Hu Hongwei [29] studied the objective laws of the overlapped area of ground explosion shock waves under multiple-point explosions. Liu Sai [30] used LS-DYNA to study the propagation characteristics of explosion shock waves under indoor explosion conditions and proposed a simplified load model for indoor explosions. Parametric analysis was conducted on steel columns using numerical simulation.

From the current domestic and international research status, it can be observed that scholars’ research focuses mainly on the explosion load of single-point detonation or single explosive source, as well as the influence of the form and shape of explosive material on the explosion load. Limited research on the explosion load of dual-point detonation or dual explosive sources exists. Due to the interaction between the two explosive sources, the explosion load of dual explosive sources experiences complex reflection phenomena, significantly enhancing the intensity of the explosion shockwave.

This results in more severe dynamic damage to structures in the vicinity of the dual explosive sources in blast holes compared to explosions caused by a single explosive source. In existing research, both domestically and internationally, the focus is mainly on outdoor explosions, and there is limited research on indoor explosions. Furthermore, there is a lack of practical methods for evaluating the blast resistance of structures subjected to indoor explosions caused by dual explosive sources.

1.3 Main Research Content of this Paper

Based on the evaluation of the blast resistance of a fireworks production facility in Liuyang City, Hunan Province, this paper researches two aspects: the explosion load of dual explosive combustible materials sources and the dynamic response of the structure of the fireworks facility under the action of internal dual explosive sources. The main research content is as follows:

(1) Systematically study the propagation characteristics of the explosion shockwave of dual explosive sources and use numerical simulation analysis methods to obtain pressure-time curves at different distances during the free-air explosion of dual explosive sources. Investigate the influence of the interaction between the two explosive shockwaves on the explosion load.

(2) Numerical simulations were conducted to study the constrained explosion of dual explosion sources in fireworks factory buildings. The propagation characteristics of shock waves were investigated when the dual explosion sources occurred within the constrained structure. The reflected pressure (Pr) and reflected impulse (ir) on each wall of the constrained structure were compared, and the load distribution on the walls of the fireworks production facility under the influence of internal dual explosion sources was analyzed. Considering the non-uniform distribution of explosion loads from the internal dual explosion sources, the walls were divided into zones, and an enhancement coefficient (λ) was introduced. A novel method for determining the explosion load under the influence of internal dual explosion sources was proposed.

(3) Based on the proposed method for determining the explosion load, an improved evaluation method for the blast resistance of the fireworks production facility with internal dual explosion sources was developed. This evaluation method can be applied in practical engineering. According to the structural zoning, the explosion load of the fireworks production facility under the influence of a single internal explosion source was determined based on specifications.

This load was multiplied by the corresponding Mach reflection enhancement coefficient in the respective zone, and the explosion load on the structural walls was directly applied to the corresponding zones. The dynamic response of the structure under the action of internal dual explosion sources was studied, enabling the rapid assessment of dynamic damage to building structures in practical engineering.

Explore the unique characteristics of explosions in fireworks production facilities and their impact on structural integrity. This research paper investigates the explosion loads and structural responses caused by dual explosion sources within enclosed spaces. LS-DYNA software analyzes the propagation characteristics of explosion shock waves and the distribution of explosion loads. A novel method for determining explosion loads from dual internal sources is proposed, leading to an improved evaluation of explosion resistance in fireworks facilities. Gain insights into the dynamic response and deformation patterns of facility structures under the influence of dual explosion sources.

Chapter 2: Propagation Laws of Blast Waves from Dual Explosions

2.1 Introduction

An explosion is a sudden physical or chemical change in the state of matter, accompanied by the rapid release of motion and energy [31]. Explosions can take various forms and can be classified into two types: physical explosions and chemical explosions, depending on their causes. This paper primarily focuses on chemical explosions, where chemical explosive material (combustible materials) rapidly transform into gaseous detonation products in an instantaneous time frame.

These gaseous detonation products rapidly expand outward, compressing the surrounding exposures gas and forming a compressed air layer known as a shock wave. The gaseous detonation products cease propagation after a certain period of time, while the explosion shock wave continues to propagate outward, gradually attenuating over time.

An explosion wave occurs when the products generated by the explosion transform into high-temperature, high-pressure gas and create a high-pressure wavefront that radiates outward into the surrounding exposures atmosphere, powered by the high-temperature gas [32]. This disturbance wavefront is called the explosion shock wave, which instantly elevates the ambient atmospheric pressure to the peak incident pressure and propagates outward from the detonation point at supersonic speeds. As the distance increases, the propagation rate of the shock wave gradually decreases, and the air pressure gradually decreases.

2.1.1 Unconfined and Confined Explosion Loads

Explosion loads acting on structures can be classified into unconfined explosion loads and confined explosion loads based on the constraints of the explosion. Unconfined explosion loads include free air burst explosions, air burst explosions, and surface burst explosions. Confined explosion loads can be further categorized as fully vented, partially confined, and fully confined explosions [33-34].

2.1.1.1 Free Air Explosion

An explosion occurs in free air when the source is adjacent to or above a structure. The initial explosion shockwave does not increase due to any reflections during propagation, and it subsequently acts on the structure, resulting in the explosion load known as the free air explosion load. As an incident wave, the explosion shockwave radiates outward from the explosion source.

When it collides with the structure, the explosion shockwave is influenced by the structure, leading to reflection and amplification. As shown in Figure 2-1, when the propagation path of the explosion shockwave is perpendicular to the structure’s surface and collides with it, the resulting shockwave pressure and impulse reach their maximum values. When the propagation path of the shockwave is parallel to the structure’s surface, the resulting shockwave pressure and impulse are considered to be at their minimum values.

Figure 2-1: Propagation of Free Air Explosion Shock Wave

Figure 2-2 shows the typical pressure-time curve (from maximum to minimum) of the explosion shock wave generated by a free-air explosion. After a certain time, tA, the shock wave reaches the target point, where the pressure instantaneously rises to Ps0. After passing through the target point, the pressure gradually decreases from the peak value Ps0 and returns to atmospheric pressure. As the shock wave continues to propagate, the pressure at the target point continues to decrease, creating negative pressure.

Once the pressure at the target point drops to atmospheric pressure, it further decreases to the negative pressure peak Ps0-. After the shock wave moves away from the target point, the negative pressure begins to recover, and the pressure at the target point gradually returns to atmospheric pressure. The duration of positive-phase pressure is denoted as t0, and the duration of negative-phase pressure is denoted as t0-.

During the propagation of the shock wave, when it collides with a structural surface, it will experience positive and negative reflections. After the reflection, the peak pressure and impulse of the shock wave will increase due to the coupling between the incident wave and the reflected wave. The pressure-time curve of the reflected pressure generated by the explosion shock wave is consistent with the incident pressure. The peak pressure Pr of the reflected shock wave is multiple times higher than the peak pressure Ps0 of the incident shock wave.

Figure 2-2: Typical Pressure-Time Curve of a Single Source Explosion Shock Wave

2.1.1.2 Air Blast Load

The air blast load occurs near the ground, with a certain distance between the source and the structure. The initial shock wave generated by the source expands outward. Before reaching the structure’s surface, the shock wave collides with the ground, producing a reflected shock wave. The reflected shock wave continues propagating along the ground and couples with the initial shock wave to form a Mach wave. This coupling interaction between the shock waves results in an enhanced effect.

Figure 2-3: Propagation of Air Blast Shock Wave

As shown in Figure 2-3, the explosion shock wave generated by the explosion source collides with the ground, producing a reflected shock wave. After coupling with the incident shock wave generated by the explosion source, it forms a Mach wave [33-34]. The incident shock wave, ground reflected shock wave, and Mach wave create a three-wave coupling point, and the changing path of the three-wave coupling point is shown in Figure 2-3. 

Some shock waves exceed the height of the three-wave coupling point (the height of the Mach wave), and the peak pressure of these shock waves is smaller than that of the Mach wave. Therefore, for design purposes, the portion of the shock wave that exceeds the height of the Mach wave is ignored. The shock wave can be regarded as a parallel wavefront within the height range of the Mach wave’s wavefront. The distance from the coupling point of the initial shock wave, ground reflected wave, and Mach wave to the ground is called the Mach wave’s wavefront height, also known as the Mach stem. The Mach wave’s wavefront height gradually increases outward from the explosion source.

If the height of the three-wave coupling point does not exceed the height of the structure, it is considered a uniform Mach wave within the height below the three-wave coupling point. Within the height range above the three-wave coupling point, it is considered the coupling of the incident wave and the reflected wave, with a magnitude lower than that of the Mach wave. In general design conditions, the distance between the explosion source and the structure is sufficiently large, and the height of the three-wave coupling point is much higher than the height of the structure.

2.1.1.3 Ground Explosion Load

When an explosion occurs on the ground or very close to the ground, the resulting explosion load is called a ground explosion load. After the initial shock wave generated by the explosion source collides with the ground, it enhances and forms a reflected shock wave. Unlike air blast loads, ground explosion loads occur on the ground, and the reflected and incident waves couple together at the detonation point, creating a new shock wave similar to the Mach wave in air blast loads. It enhances the initial shock wave and uniformly propagates outward, but its waveform essentially appears as a hemispherical shape.

2.1.1.4 Constrained Blast Load

A constrained blast refers to an explosion that occurs within a confined space. When the explosion occurs inside a structure, the initial peak pressure of the shockwave increases sharply due to multiple reflections of the blast shockwave within the structure.

Additionally, depending on the different constraints imposed by the confined space, the constrained blast shockwave will be influenced to varying degrees by high temperatures and the accumulation of gas products generated by the explosion’s accompanying chemical explosive reaction within the structure. It leads to varying increases in the peak pressure and duration of the blast shockwave. The combined effects of these pressures can cause complete internal destruction of the structure (chemical compound), requiring protective measures to be incorporated during the design stage to counteract this internal pressure. A typical protective measure is venting, which reduces the peak pressure’s magnitude and the blast shockwave’s duration through controlled release.

Constrained blast loads can be categorized based on the type of confinement within the structure: fully vented explosion, partially confined explosion, and fully confined explosion.

Fully vented explosion: When an explosion occurs in a structure adjacent to one or more free faces open to the atmosphere or within such a structure, this type of blast load is called a fully vented explosion. In a fully vented explosion, the initial wave of the blast shockwave is intensified upon collision with the rigid face of the structure, and the blast products are ejected into the atmosphere, forming a vibration shockwave that propagates from the structure’s free faces into the atmosphere.

Partial Constraint Blast Load: When an explosion occurs inside a structure with a finite-sized opening or a fragile surface, this type of blast load is called a partial constraint blast load. In a partial constraint blast load, the initial shock wave of the explosion is reinforced upon colliding with the structure’s fragile or rigid surface, and the explosion products are ejected into the atmosphere after a finite time, forming a vibration shock wave. Unlike free-face constraint blast loads, the constraint conditions of a partial constraint blast load result in the accumulation of internal temperatures, the formation of high-temperature conditions, and the generation of gas products, forming quasi-static gas pressure (static electricity). Compared to the pressure from the vibration shock wave, this pressure has a long duration of action.

Full Constraint Blast Load: When an explosion occurs inside a fully sealed or nearly fully sealed enclosure, this type of blast load is called a full constraint blast load. A full constraint blast load includes the vibration shock wave pressure from the free-face constraint blast load and the gas pressure with a very long duration of action from the partial constraint blast load. In contrast, the leakage pressure (i.e., the shock wave pressure propagated from the free-face or fragile surface) is usually much smaller than the first two.

In constraint blast loads, three forms of pressure need to be considered: leakage pressure, vibration pressure, and gas pressure.

When a constrained explosion occurs in a structure with a fragile surface or an opening facing the atmospheric environment, the pressure will experience a certain degree of leakage based on the degree of structural openness. This type of structure allows the explosion shock wave to overflow into the external atmospheric environment after the internal explosion, significantly reducing the internal pressure’s effect duration. In this explosive scenario, the external pressure on the structure is called leakage pressure. The shock wave pressure reflected and enhanced within the constrained structure is called vibration pressure. In constraint blast loads, there is also a pressure generated due to gas accumulation and the pressure generated by the increase in internal temperature in the constrained structure, called gas pressure. For most constraint structures with free faces, the influence of gas pressure can be neglected.

Different types of structures need to consider different pressure effects. As shown in the table below, under the action of constraint blast loads, free-face constraint blast loads need to consider internal vibration pressure and leakage pressure, and partial constraint blast loads need to consider internal vibration pressure, internal gas pressure, and leakage pressure. In contrast, full constraint blast loads must consider internal vibration and gas pressure.

Table 2-1: Blast Load Classification

2.1.2 Dual Blast Source Explosion Load

The blast waves interact when two or more similar types of explosive material (combustible materials) explode simultaneously or consecutively. This interaction is influenced by factors such as the initiation times of the explosions, the phase of blast wave propagation, and the direction of blast wave propagation. In the TM5-1300 manual, it was previously stipulated that if the time interval between two explosions is too long, the blast wave of the later explosion will rapidly propagate and couple with the blast wave of the earlier explosion. The distance of coupling depends on (1) the magnitude of the explosive load for each individual explosive, (2) the time interval between the explosive loads of each individual explosive, (3) the distance between the commercial explosives and the position of each explosive, and (4) obstacles or other structures affecting the propagation of the blast wave between such commercial explosives.

In this article, multiple explosive loads are called multiple blast source explosion loads. The internal interactions between the individual blast waves affect the pressure-time curve of a multiple-blast source explosion load. The individual explosive loads in the multiple blast source explosion load are coupled, resulting in a final pressure-time curve. The pattern of the pressure-time curve is similar to the pressure-time curve of an individual explosive when it explodes. At locations close to the blast source, the generated pressure-time curve exhibits a multi-peak shape caused by the internal interactions between multiple blast waves. The closer the position is to the blast source, the more the pressure-time curve resembles a multi-peak shape. In many cases, the multi-peak pressure occurs within a higher pressure range, especially when the distances between the blast sources are close.

Existing research on the detailed analysis of dual blast source explosion loads is scarce. This article mainly focuses on studying dual-blast source explosion loads, aiming to derive the propagation characteristics of dual-blast source explosion loads and identify the similarities and differences between dual-blast source explosion loads and single-blast source explosion loads. Before conducting the study on dual blast source explosion loads, research on single blast source explosion loads is conducted as a comparative reference.

2.2 Numerical Simulation of Free Air Explosion with Dual Blast Sources

One of the main research objectives of this paper is the explosion load of dual blast sources inside a fireworks factory building. To determine the propagation characteristics of the explosion shock wave when the internal dual blast sources explode, analysis needs to be conducted separately for free air explosion and the confined explosion of the dual blast sources. This chapter focuses on the free-air explosion of dual-blast sources.

The propagation characteristics of the explosion shock wave belong to the field of fluid mechanics. With the rapid development of computer hardware and numerical analysis software, the research methods for explosion problems have become more diverse, and an increasing number of explosion problems can be solved using finite element analysis software. LS-DYNA [35] is an explicit dynamic analysis software developed by LSTC, which can simulate and analyze complex real-world problems.

LS-DYNA software includes Lagrangian algorithms, Euler algorithms, ALE algorithms, and various functions such as fluid-structure interaction analysis, making it suitable for nonlinear dynamic analysis of collisions, explosions, and more. In this paper, a numerical simulation approach is employed to analyze the propagation of the explosion shock wave and its interaction with the surface of the structure using LS-DYNA software.

The peak pressure of the explosion shock wave will increase to its peak at a certain moment after detonation, then gradually decrease to atmospheric pressure and decrease to negative phase pressure. Its propagation speed is fast, with a large peak pressure, posing a significant risk (safety fuses) to surrounding structures. In recent years, most research has focused on the dynamic response analysis of civil engineering structures under explosive loads while neglecting the study of the explosive shock wave load. The dynamic response of civil engineering structures under explosive loads is influenced by factors such as structural type and material and the department of transportation explosive load itself.

As early as 1948, Stoner and Bleakney[1] conducted a study on the influence of the shape of very insensitive explosives on the pressure-time curve, comparing the peak pressures of spherical, cylindrical, and cubic explosives (vehicle transporting explosives), and proposed an empirical formula applicable to the range of scaled distances between 18＜Z＜110. Wisotski and Snyer[45] compared nine different-sized cylindrical explosive tests, ranging from 0.72kg to 3.70kg of TNT equivalent, using free air explosions.

They examined the influence of cylindrical explosives (very insensitive explosives) of different height-to-diameter ratios on the peak pressure of the explosion shock wave. Phillips[2] compared data from end-fired cylindrical explosives (vehicle transporting explosives) and center-fired cylindrical explosives and concluded that the end-fired cylindrical explosives produced higher peak pressures of the explosion shock wave than the center-fired cylindrical explosives.

The above-mentioned studies clearly focused on researching explosive form and shape, with only one initiation point and one explosive. There is limited research on the explosive load of dual and multiple explosives.

In this section, the propagation of the explosion shock wave from a dual explosion source with dual explosive arrangements is numerically simulated using the nonlinear finite element explicit dynamic analysis software LS-DYNA. The aim is to compare and analyze the propagation patterns of the explosion shock wave from dual explosion sources and identify similarities and differences compared to the propagation of the shock wave from a single explosion source through many numerical simulation results. The numerical simulations in this section also use the method of parameter variation analysis.

The compared parameter variables include the distance between the two explosion sources and the constraints of the explosion occurrence. The distances between the two explosion sources are 20cm, 40cm, 60cm, 80cm, 100cm, and 120cm. Due to the scarcity of research on dual explosion source explosive loads both domestically and internationally, systematic research is required for dual explosion source explosive loads. Starting from the free air explosion of dual explosion sources, the propagation characteristics in free air are analyzed and compared with the free air explosion of a single explosion source, comparing different distance intervals.

2.2.1 Finite Element Model

Shi Yanchao [11] previously utilized AUTODYN software to study the sensitivity of various parameters of explosion loads to the mesh size. AUTODYN established a one-dimensional axisymmetric model to simulate the three-dimensional diffusion of explosion shock waves. It was concluded that a mesh size of 50mm could provide reasonably accurate results for the numerical simulation of explosion problems in three dimensions.

This study shares similarities with that research, thus initially adopting a mesh size of 50mm for numerical analysis. The finite element model of free air explosion was established using the explicit dynamic analysis software LS-DYNA, as shown in Figure 2-4, to simulate the propagation of shock waves in a dual-source free air explosion. Based on evaluating the blast resistance performance of (propellant actuated power devices) the engineering foundation fireworks production facility in this study, the air model was designed as a 1/2 model structure with a mesh size of 50mm, forming a cubic structure with dimensions of 3m 3m 3m. The boundary keyword NON_REFLECTING was used to achieve explosive conditions in an infinitely free-air environment.

The primary explosive material was located at the geometric center of the finite element model structure. To accurately reflect the actual explosion situation, the primary explosive was set as a spherical TNT primary explosive with a radius of 10cm and an equivalent charge of 7.0kg was obtained through calculations. The keyword INITIAL_VOLUME_FRACTION_GEOMETRY was added, and symmetric boundaries were applied to the symmetrical surface. The model employed a single-point multiple-material ALE algorithm [36].

Figure 2-4: Finite Element Model of Single Explosive Source Explosion

The air material is modeled using the ideal gas state equation in numerical simulations. The sufficient density of air, ρ, is 1.29 kg/m3. Additionally, the keyword “NON_REFLECTING” is used to set the non-symmetric surfaces of the structure (i.e., the boundary surfaces of the structure) to a non-reflecting state, thereby achieving the free explosion state of the single explosive source in infinite air.

Throughout the numerical simulation, the material parameters of air are set using the MAT_NULL material model from the LSDYNA material library. The keyword “LINEAR.POLYNOMIAL” is used along with the material’s further description using a linear polynomial equation. The linear polynomial state equation for air material is given by [38]:

P = C0 + C1μ + C2μ^2 + C3μ^3 + C4μ^4 + C5μ^5 + C6μ^6 + CE^2 + CE^3 + CE^4 + CE^5 + CE^6 — (2.1)

In linear polynomials: P represents the detonation pressure of air; C0, C1, C2, C3, C4, C5, and C6 are coefficients of the state equation; μ = 1/V-1 = ρ/ρ0-1; E is the initial specific internal energy of the gas; V is the relative volume of the gas. The values of the air parameters calculated are shown in the following table [39-40].

In numerical simulations, the material of the explosive is represented by an equivalent TNT explosive, which belongs to the high-energy explosive combustion material model. The material parameters of the explosive are set using the MAT_HIGH_EXPLOSIVE_BURN material model from the LS-DYNA material library. The explosive is spherical, and the initiation point is located at the center of the explosive sphere. The initiation time is set at t=0. The sufficient density of TNT explosives is 1640 kg/m³.

Additionally, the JWL keyword is used to describe the material of the explosive further. The JWL state equation represents the relationship between the detonation pressure, initial specific internal energy per unit volume, and relative volume during the detonation process. The JWL state equation is as follows [37-38].

In the JWL state equation, P represents the detonation pressure of air; A, B, R1, and R2 are the coefficients of the JWL state equation; V represents the relative volume of the gas, and E represents the initial specific internal energy of the gas. The final calculated parameters of the explosive are represented in the table below [41]:

2.2.2 Analysis of Grid Size Effect

When using explicit dynamic analysis software to analyze explosion problems, it is generally necessary to analyze the convergence of the mesh size of the established finite element model. In the initial numerical simulation of explosion loads in this study, a grid size of 50mm was directly used for calculations. The results are generally accurate for simulating explosion problems with large-scale distances when using the same grid size. However, for simulating explosion problems with small-scale distances, the numerical results obtained may have significant errors. Therefore, it is necessary to validate the results obtained from numerical simulations to determine the accuracy of the finite element model established in the numerical simulation. At the same time, as the grid size becomes smaller, the results obtained will become more precise. However, reducing the grid size increases the number of finite element model elements. It decreases the minimum time step for computer calculations, which requires higher hardware capabilities and increases computation time. Therefore, to balance these two factors, a systematic analysis of the convergence of the grid size effect needs to be conducted for the studied explosion problem to obtain a reasonable grid size.

The calculation of airborne shock wave pressure for explosion in an ideal infinite gas has been determined by H.L.Brode [42-43]. H.L.Brode introduced international atmospheric standard values, dimensionless time, and dimensionless viscosity parameters to establish Lagrange’s motion equations, forming a system of difference equations. The calculation formula for shock wave front pressure is obtained by solving them using a computer. J.Henrych compared the calculation formulas derived by H.L.Brode, Naušmic, and Sadowiski [44], established formulas based on the theory of model similarity, and obtained a comprehensive calculation formula for explosion shock wave front pressure through experimental research [31]:

The symbol ∠R in the equation represents the scaled distance with units of kg/m3. R represents the actual distance from the center of the explosive to the monitoring point in units of meters. W is the equivalent TNT weight measured in kilograms. Pφ is the pressure on the shock front of the blast wave, and P0 represents atmospheric pressure. When using the above formula for calculations, converting the units according to the indicated instructions is necessary.

In TM5-1300, different positive and negative peak pressures, impulses, pressure duration, and other parameters corresponding to the proportional distance (Z = R/W1/3) from the monitoring point to the center of the explosive are provided. The proportional distance ranges from a minimum value of 0.136 ft/lb1/3 to a maximum value of 100 ft/lb1/3. The minimum proportional distance of 0.136 ft/lb1/3 represents a hemispherical TNT blast load (i.e., ground blast load). Some parameters are represented by dashed lines (not a straight line) in the graph, which indicate upper limits based on a large amount of discrete experimental and theoretical data. The blast wave parameters in the graph are functions of the proportional distance Z, with a maximum proportional distance considered up to 100 ft/lb1/3. When the proportional distance exceeds 100 ft/lb1/3, the structural damage is minimal for most protective structures and even lighter structures. Furthermore, the blast wave parameters are highly sensitive to the surrounding gas environment for proportional distances greater than 100 ft/lb1/3, and the peak pressure may deviate significantly from the values obtained through ideal gas calculations.

When comparing the peak pressure values on the blast wave front obtained from TM5-1300 with those calculated using the formula proposed by J. Henrych, it is found that the results from both approaches are very close when the proportional distance is greater than 1. However, the deviation between the two values increases gradually as the proportional distance decreases. This difference should be considered when conducting convergence analysis of the mesh size in finite element modeling.

Compare the pressure-time curves of the explosive load at different proportional distances using the finite element model established in 2.2.1. Grid sizes of 50mm, 40mm, 30mm, 20mm, and 10mm were used, and monitoring points were set at different distances from the center of the explosive. The equivalent TNT weight is 10kg, and the proportional distance is 1.5m/kg1/3. Compare the pressure-time curves of the monitoring point at this distance with the values obtained from the J.Henrych formula. The peak pressure values obtained from different grid sizes are analyzed and compared in Table 2-4.

Table 2-4: Comparison of peak pressure values at the target point with a proportional distance of 1.5m/kg1/3 under different grid sizes

As shown in the figure, it can be seen that (1) under different grid sizes, the finite element model exhibits a consistent trend of peak pressure variation with respect to the scaled distance. (2) When the scaled distance is large (i.e., in the case of a far-field explosion), the grid size effect on the peak pressure is relatively small. Considering the accuracy of numerical simulation results and the computational efficiency, this study adopts a grid size of 20mm for the finite element model.

As shown in Figure 2-5, the air model continues the grid size analysis results from section 2.2.2 and uses a grid size of 20mm. To improve computational efficiency based on structural symmetry, a 1/2 model structure is employed, designed as a cubic structure with dimensions of 3m3m3m. The explosive conditions of the double explosive in infinite free air are implemented using the keyword “NON_REFLECTING” for the boundary of the air model. The explosive positions are shown in Figure 2-5. Considering the actual processing process in a fireworks production facility, the two locations for storing gunpowder (like photographic flash powders) are above and below the facility’s center. Therefore, the explosives are arranged vertically at the center of the structure. To reflect the real explosion scenario more accurately, both explosives are set as spherical TNT explosives. At the same time, to eliminate the error caused by irrelevant variables when comparing with the explosion load of a single source, the total equivalent TNT weight of the two explosives should be equal to the total equivalent TNT weight of a single explosive in a single-source explosion. The calculated TNT equivalent weight of a single explosive is 7.0kg.

Figure 2-5: Finite Element Model of Dual Explosive Source Explosion

Therefore, the explosive radius of the freely propagating air explosion of the dual explosives is selected as 8cm. Such explosives are added using the keyword INITIAL_VOLUME_FRACTION_GEOMETRY, with symmetric boundaries used for the symmetrical plane. The model adopts the single-point, multi-material ALE algorithm.

The finite element model used in this section consists of two parts: spherical TNT explosive and air structure. The air structure is simulated using the Euler solid element SOLID164, and the single-point, multi-material ALE algorithm is used. The spherical TNT explosive is defined using the keyword INITIAL_VOLUME_FRACTION_GEOMETRY. In the numerical simulation, the explosive material consisting is set using the keywords MAT_HIGH_EXPLOSIVE_BURN and JWL equation, following the constitutive model of high-energy explosive ignition and combustion of explosive materials described in section 2.2.1. The air material is set using the keyword MAT_NULL and LINEAR_POLYNOMIAL equation, following the constitutive model of air material described in section 2.2.1.

2.2.3 Typical Simulation Results and Comparative Analysis

Through the numerical simulation in section 2.3.1, a large amount of data on the propagation of shock waves from dual explosive sources was obtained. This section will present the results obtained from the model that best represents the engineering reality of the explosion of dual explosives inside a fireworks production facility. Figure 2-6 shows the complete propagation process of the shock wave from dual explosive sources when the distance between the two sources is 100cm.

Figure 2-6: Pressure distribution variation cloud map of the free air explosion caused by dual detonation sources

At t=398.4us, the explosive shock wave propagates outward in a spherical shape. At t=599us, the first wavefronts of the explosive shock waves from the two detonation sources collide and couple together, propagating outward along the centers of the two sources. At t=898.6us, the formation of the coupled wave can be clearly observed at the collision surface of the explosive shock waves from the two sources. The pressure distribution cloud map of the coupled wave shows that the peak pressures at both ends of the coupled wave are greater than in the middle. At t=1195.2us, it is observed that the explosive shock waves from the two sources continue to propagate outward in a spherical shape, and the coupled wave formed at the collision surface also propagates outward in a spherical shape.

From Figure 2-6, it can be seen that the instantaneously formed shock waves propagate outward after the detonation of the two spherical explosives, following the same propagation pattern as the explosive load from a single detonation source, forming spherical shock waves propagating outward. During the propagation process, after the initial waves of the two explosive shock waves collide with each other, their propagation paths form two curved lines continuing to propagate outward.

Based on the pressure variation cloud map of the free air explosion with dual detonation sources shown in the above figure, the collision between the explosive shock waves generated by the two propellant explosives can be considered as the reflection of a single shock wave colliding with a rigid surface. Normal reflection occurs when the incident angle of the explosive shock wave is small. As the incident angle of the explosive shock wave gradually increases, the phenomenon of normal reflection transitions into Mach reflection.

This phenomenon can be referred to as the Mach reflection phenomenon. Similar to Mach reflection, the Mach reflection phenomenon occurs when the incident angle of the shock wave reaches a certain value, resulting in the coupling of the reflected shock wave generated by the collision of the explosive shock waves with their respective incident shock waves, forming a Mach wave. This pattern is consistent with the reflection phenomenon of explosive shock waves occurring on the ground, as described in TM5-1300 for air blast loads.

To monitor the propagation characteristics of shock waves at the collision surface of two explosive blasts and the propagation characteristics of shock waves at non-collision surfaces, monitoring points A0-A15 and B0-B11 are set as shown in Figure 2-7. A0-A15 are located on the perpendicular bisector of the line connecting the two explosive sources, with a spacing of 20cm between the measurement points. B0-B11 are located on the straight line where the explosive sources are located, with a spacing of 20cm between the measurement points.

Figure 2-7: Finite Element Model Measurement Point Arrangement

Figure 2-8 shows the time-pressure curve of the peak explosion pressure when the distance between the two explosive sources is 100cm, and the blast wave passes through measurement points A0-A15 in free air.

Figure 2-8: Pressure Peak Variation Chart for Measurement Points A0-A15

From Figure 2-8, it can be observed that during the propagation process of the shock waves generated by the dual explosive sources, there is an overall decreasing trend in the peak pressure of the shock wave front from measurement point A0 to measurement point A15, which represents the geometric center of the two explosives moving outward. Among them, the peak pressure of the shock wave at measurement point A2 is higher than that at points A1 and A0.

This phenomenon occurs because the incident angles of the shock waves at points A0 and A1 are smaller, resulting in a phenomenon known as positive and negative reflection when the two shock waves collide. It is only when reaching measurement point A2 that the incident angle of the shock wave is sufficiently large, and after the collision of the two shock waves, Mach reflection occurs, involving the coupling of incident waves, reflected waves, and Mach waves. Therefore, an enhanced effect in peak pressure occurs at measurement point A2.

The peak pressure at A2 is 5.49 MPa, while at A1, it is 5.07 MPa, indicating the enhanced effect of Mach reflection on shock wave pressure. Thus, in this section, the region at measurement points A0 and A1, where Mach reflection has not yet occurred, is defined as the reflection transition zone, with a range of 0-50 cm. In comparison, the region above 50 cm is designated as the Mach reflection zone.

Figure 2-9: Peak Pressure Variation at Different Positions with Source Spacing of 20cm-120cm

Figure 2-10: Variation of Peak Pressure along the Vertical Direction with Different Source Spacings

Figures 2-9 and 2-10 present the peak pressure curves on the collision surface of the double-source explosion shock waves under different source spacings. Figure 2-9 shows that the peak positive pressures of measurement points A0-A15 are obtained at source spacings of 20cm, 40cm, 60cm, 80cm, 100cm, and 120cm. On the collision surface of the two explosion shock waves, when the source spacing is 0cm, 20cm, or 40cm, the peak pressure-time curves of the explosion shock waves show an overall decreasing trend without a clear transition region of reflection. However, a distinct reflection transition region appears when the source spacing reaches 60cm.

Figure 2-10 presents the time curves of peak pressure of double-source explosion shock waves with source spacings of 60cm, 80cm, 100cm, 120cm, 140cm, 160cm, 180cm, and 200cm. Figure 2-10 shows that when the source spacing is 60cm, Mach reflection occurs at a distance of 41cm from the source, with 0-41cm being the reflection transition region and 41cm and above being the Mach reflection region.

When the source spacing is 100cm, Mach reflection occurs at a distance of 42cm from the source, with 0-42cm being the reflection transition region and 42cm and above being the Mach reflection region. When the source spacing is 140cm, 0-40cm is the reflection transition region, and 40cm and above is the Mach reflection region. When the source spacing is 180cm, 0-56cm is the reflection transition region, and 56cm above is the Mach reflection region.

When the source spacing is 200cm, 0-56cm is the reflection transition region, and 56cm and above is the Mach reflection region. With the increase of source spacing, the position of the reflection transition region remains unchanged, and the Mach reflection phenomenon begins to occur after the proportional distance reaches X. The overall trend of the pressure-time curves also remains consistent. However, the Mach reflection phenomenon gradually weakens as the source spacing increases.

When the source spacing is 60cm, the peak positive pressure at point A2 is 8.1MPa, and at point A1 is 7.2MPa, resulting in an enhancement effect of Mach reflection by 12.5%. When the source spacing is 100cm, the peak positive pressure at point A2 is 5.4MPa, and at point A1 is 5.0MPa, resulting in an enhancement effect of Mach reflection by 8%. When the source spacing is 180cm, the peak positive pressure at point A3 is 3MPa, and at point A2 is 2.9MPa, resulting in an enhancement effect of Mach reflection by 3.4%.

When the distance between the explosive sources reaches 200cm, Mach reflection occurs at the measurement point 60cm away. The peak pressure at measurement point A3, located at a proportional distance Y, is 2.6MPa, while the peak pressure at measurement point A2 is 2.42MPa. The enhancing effect of Mach reflection on the explosive shock wave is already very small. As the distance between the explosive sources increases, the enhancing effect produced by Mach reflection gradually weakens.

It can be concluded that the Mach reflection phenomenon in the double explosive source explosion load is similar to the Mach reflection phenomenon in the air explosion load described in TM5-1300. Mach reflection occurs at the collision surface after the detonation of the two blasting explosive. The coupling of the incident wave reflected wave and Mach reflection wave enhances the explosive shock wave.

The trajectory of the coupling point between the three waves moves symmetrically from the central axis towards both sides, consistent with air explosions. The collision surface of the double explosive source explosion load can be divided into a transition region and a Mach reflection region. As the distance between the explosive sources increases, the position of the Mach reflection region remains basically unchanged, located at a proportional distance Z, and Mach reflection starts to occur between 0 and Z in the transition region. As the distance between the explosive sources increases, the intensity of Mach reflection weakens, and the enhancing effect on the explosive shock wave also diminishes.

Figure 2-11: Variation of peak pressure along the horizontal direction with different explosive source spacing

Figure 2-11 shows the trend graph of the peak pressure of the explosion shock wave along the horizontal direction with proportional distances for different explosive source spacings of 20cm, 80cm, and 140cm. The minimum proportional distance occurs at point X when the explosive source spacing is 20cm. The peak positive pressure at measurement point B0 is 9.23MPa, at B1 is 1.8MPa, and at B2 is 1.5MPa. When the explosive source spacing is 80cm, the peak positive pressure at measurement points B0-B11 remains relatively unchanged.

By comparing with the single explosive source, it is found that the peak positive pressures at measurement points B0-B11 remain essentially constant for the same equivalent TNT charge. Figure 2-11 shows that the peak positive pressures at measurement points B0-B11 generally decrease as the proportional distance increases. The peak pressure of the explosion shock wave remains consistent for different two explosive charges and source spacing. It exhibits a similar trend to the peak positive pressure of the explosion load from a single explosive source concerning the proportional distance. In the case of a free-air explosion load with dual explosive sources, the Mach reflection effect does not affect the peak pressure in the non-collision zone. It can be calculated based on the peak pressure of the explosion load from a single explosive source.

2.3 Summary of This Chapter

This chapter introduced the research achievements of single explosive source explosion loads both domestically and internationally. It discussed the causes of explosion shock waves and classified them according to different environments (unconfined and confined explosion loads). Unconfined explosion loads include free-air explosion loads, air explosion loads, and ground explosion loads. Confined explosion loads are classified into structures with free surfaces, partially constrained structures, and fully enclosed structures. Various types of explosion pressures were listed for different confined explosions, and the research gap regarding dual explosive source explosion loads was addressed.

First, the propagation behavior of dual explosive source explosion loads in free-air explosions was analyzed through numerical simulations. The study revealed the occurrence of Mach reflection within the collision zone of dual explosive source explosions in free-air. This phenomenon resembles the Mach reflection observed during single explosive source air explosions. After dual explosive source explosions, collision and interaction between the incident and reflected waves occur within the collision zone, forming Mach reflection. The Mach reflection effect significantly enhances the explosion shock wave.

Secondly, a parameter analysis was conducted on nine different spacing distances between explosive sources in dual explosive source explosions. The sensitivity of dual explosive source explosion loads to the spacing distance was studied. The research showed that within the studied range of spacing distances, the region affected by Mach reflection remained mostly unchanged as the spacing distance increased.

The position of the transition region between the incident and reflected waves also remained relatively stable. The transition region was found to be within the range of 0-50cm, while the region beyond 50cm was considered the area affected by Mach reflection. With increasing spacing distance, the strength of the Mach reflection effect gradually decreased. The enhanced effect of Mach reflection between the two explosive sources had little impact on the peak positive pressure of the shock wave in the non-collision zone.

The peak positive pressure of the shock wave in dual explosive source explosion loads was unaffected by the Mach reflection between the two explosive sources. Therefore, the Mach reflection effect in dual explosive source explosion loads occurs within the collision zone of the two shock waves and only affects the collision zone, with no impact on the non-collision zone of the shock waves.

 Chapter 3: Determination of Internal Dual Blast Source Explosion Load

3.1 Introduction

So far, there is still a lack of practical application of research results on domestically and internationally dual blast source-constrained explosions. When determining the explosion load of internal dual blast sources, existing specifications are used as a basis, either treating the dual blast source as a single blast source for analysis or adding up the independent explosion loads of the two blast sources within the dual blast source. Both methods are based on assumptions in the specifications in such a manner, assuming that the explosion load acting on the structure’s interior is uniformly distributed along the wall surface.

However, as explained in Chapter 2 regarding the propagation of blast shock waves, the Mach reflection generated during dual blast source explosions enhances the blast shock waves, resulting in a highly uneven distribution of the explosion load along the structural walls. Furthermore, when dual blast sources explode within an enclosed structure, the explosion load is unevenly enhanced in certain areas (blast holes) due to different constraint conditions. The two load determination methods based on existing specifications (in such a manner) are not accurate enough.

In this chapter, numerical simulations are conducted to study the explosion of internal dual blast sources in a fireworks factory to determine the explosion load. The constrained explosion load distribution pattern along each wall surface is investigated. Based on the distribution pattern of the explosion load caused by internal dual blast sources, an enhancement coefficient λ is introduced to account for the uneven distribution phenomenon along the wall surfaces.

A more practical method for determining the explosion load of internal dual blast sources, applicable to engineering practice, is proposed based on the two aforementioned load determination methods. The structural wall surfaces are divided into nine regions according to the different types of reflections occurring in each region, with corresponding enhancement coefficients λ. Compared to the previous two methods, this method provides a more accurate calculation of the explosion load. Compared to fully employing numerical simulations to analyze the dynamic damage of the structure, this method is faster, more accurate, and more practical for engineering applications.

This chapter does not consider the presence of fragile surfaces or small-sized vent openings in the constrained explosion of dual blast sources. The study focuses on the explosion load of dual blast sources under the actual constraint conditions of a fireworks factory.

Figure 3-1: Classification of types of constrained explosion structures with dual explosion sources

According to the relevant specifications [33], in all types of cubic building structures (including ground), the constrained structures are classified based on the number of wall surfaces in the structure, as shown in Figure 3-1 [33]. In Figure 3-1 (a), the cubic structure has only one wall surface, referred to as the back wall according to the three-view of the structure. In Figure 3-1 (b), the cubic structure has two wall surfaces, with the addition of a side wall to the back wall shown in Figure 3-1 (a). In Figure 3-1 (c), the cubic structure has three wall surfaces: one back wall and two side walls.

In Figure 3-1 (d), the cubic structure has four wall surfaces, including one back wall, two side walls, and an additional roof structure. The letter “B” in the figure represents the position of the back wall, and the letter “N” represents the number of reflective wall surfaces in the structure. The structure of the fireworks factory studied in this chapter belongs to the category of a three-wall structure with a roof, and the dual explosion sources are located at the center of the constrained structure.

3.2 Numerical Simulation of Internal Dual Explosion Sources in Fireworks Factory

3.2.1 Finite Element Model

A finite element model is established based on the structural prototype of a fireworks production facility in Zhongzhou, Hunan. The numerical analysis of the fireworks factory structure is conducted using the nonlinear finite element explicit dynamic analysis software LS-DYNA. The model is built upon the free-air explosion finite element model with dual explosion sources described in Section 2.2.1, with the addition of rigid surface constraints for the ground, back wall, and roof. The distance between the two explosion sources is 150cm.

Figure 2-5 shows the finite element model, with a mesh size of 20mm, as used in Section 2.2.1. A half-model structure is employed, representing a cubic structure with dimensions of 3m3m3m. The keyword “NON_REFLECTING” is used on the surfaces of the asymmetric and non-rigid faces to implement the condition of non-reflecting boundaries. The explosive positions are shown in the figure, located at the center of the structure, with the explosion sources O1 and O2 arranged vertically.

Both explosives detonating are modeled as spherical TNT explosives detonating with a radius of 8cm. The equivalent TNT charge weight is 7.0kg. Symmetry boundaries are applied. This chapter only analyzes the load exerted on the back wall due to the explosion of the dual sources and does not consider the dynamic damage to the wall. Therefore, the keyword “MAT_RIGID” is used to treat the wall as a rigid structure. In the numerical simulation, the explosive material consisting is set using the keyword “MAT_HIGH_EXPLSIVE_BURN” and the JWL equation of state, following the constitutive model of high-energy explosive ignition and combustion material described in Section 2.2.1. The air material is set using the keyword “MAT_NULL” and the “LINEAR_POLYNOMIAL” equation of state, continuing the air material constitutive model described in Section 2.2.1.

3.2.2 Layout of Measurement Points in Fireworks Factory Building Structure

To facilitate the analysis of the impact of two Mach reflections on the explosion load on each wall of the fireworks factory building, measurement points are arranged as shown in Figure 3-2. Measurement points C0-C15 are arranged outward from the center of the rear wall horizontally, with a spacing of 20 cm between points. Measurement points D0-D15 are arranged upward from the centerline of the bottom of the rear wall vertically, with a spacing of 20 cm between points.

Measurement points F0-F15 are arranged on the vertical centerline of the side walls from the bottom to the top, with an interval of 20 cm between points. Measurement points E0-E15 are arranged outward from the rear wall on the horizontal centerline of the side walls, with an interval of 20 cm between points. Since the roof wall belongs to the non-collision zone of the two blast shock waves, it will not be analyzed systematically in this chapter.

Figure 3-2: Layout of Measurement Points on Rear and Side Walls

To study the differences between the dual explosive source blast load and the single explosive source blast load, based on the finite element model in section 3.2.1, the dual explosive sources were modified using the keyword INITIAL_VOLUME_FRACTION_GEOMETRY, changing it to a single spherical explosive. The explosive radius is 10cm, and the equivalent TNT mass is 7.0kg. The explosive is located at the geometric center of the constrained structure. The constitutive model for the explosive material follows the high-energy explosive ignition and combustion model in section 3.2.1.

3.2.2 Typical Simulation Results and Comparative Analysis

Figure 3-3 shows the typical pressure variation cloud map in the internal air units of the dual explosive sources.

Figure 3-3: Pressure Distribution Variation Cloud Map inside the Double Explosive Source Fireworks Factory

At 0 microseconds, both explosive sources detonate simultaneously, forming an explosive shockwave that propagates outward in a spherical shape due to chemical explosive reaction. At t=198.548 microseconds, the explosive shockwaves collide at three points: the shockwave from source 02 collides with the ground, the shockwave from source 01 collides with the roof, and interaction occurs between the two shockwaves in the intermediate region between the two sources.

During this process, three Mach reflection regions are formed: a Mach reflection occurs between source 02 and the ground, another Mach reflection occurs between source 01 and source 02 in the collision region, and a Mach reflection is formed between source 01 and the roof. The three Mach reflection shockwaves form circular rings and continue propagating outward near the plane of Mach reflection. At t=895.213 microseconds, the shockwave reaches the back wall, and three peak pressure regions are observed on the back wall, all located within the Mach reflection areas (blast holes). At t=2196.54 microseconds, the two Mach shockwaves couple together and propagate outward along the back wall. At 3297.67 microseconds, the explosive shockwaves from the double sources reach the side wall and initiate emission, while the reflected shockwaves start propagating in the opposite direction.

Figure 3-4: Pressure-time curves at different measurement points on the rear wall of the internal dual explosive source fireworks factory

Figure 3-4(a) presents the pressure-time curves of the explosion load at measurement points C0, C7, and C15 inside the fireworks factory structure. The pressure-time curves of these three points exhibit similar trends, conforming to the typical pressure-time curve of explosive loads at a specific point on the structure with a single explosive source. Based on the positions of the measurement points, it can be inferred that the curve at point C0 represents the pressure-time curve in the region of the direct and reflected Mach waves acting on the rear wall.

Figure 3-4(b) shows the pressure-time curves of the explosion load at measurement points C0, C7, and C15 inside the fireworks factory structure with a single explosive source. The pressure-time curves of these three points demonstrate consistent trends, aligning with the typical pressure-time curve of explosive loads at a specific point on the structure caused by a single explosive source. In the pressure-time curves, the peak protrusion at the peak-to-peak position corresponds to the jetting of chemical explosive products.

Figure 3-4(c) illustrates the pressure-time curves of the explosion load at measurement points D0, D3, D6, D9, D12, and C15 inside the fireworks factory structure with a dual explosive source. The pressure-time curves of points D0, D6, and D9 exhibit consistent trends, while points D3, D12, and D15 display characteristics of double peaks. It is due to the reflection of multiple waves, including the reflected wave and Mach reflection wave, at points D3, D12, and D15, resulting in multiple peak values.

Figure 3-4(d) displays the pressure-time curves of the explosion load at measurement points D0, D3, D6, D9, D12, and C15 inside the fireworks factory structure with a single explosive source. The pressure-time curves of these six points demonstrate consistent trends, conforming to the typical pressure-time curve of explosive loads on the structure caused by a single explosive source.

Figure 3-5: Distribution of Peak Pressure in the Vertical Direction of the Rear Wall

To visually compare the peak pressure distribution on the rear wall surface constrained by single and double explosion sources, the peak pressure values of the pressure-time curves at measurement points D0-D15 were extracted and compared with the single explosion source, as shown in Figure 3-5. On the rear wall surface, the peak positive pressure values from measurement point D0 at a distance of 0cm from the ground to measurement point D15 at a distance of 300cm from the ground exhibit a distribution with higher values on both sides and lower values in the middle when subjected to a single explosion source blast load. For the blast load constrained by double explosion sources, three peak pressure values appear from the bottom to the top of the rear wall, located at the bottom and top of the wall, and the collision area of the two explosion shock waves.

At measurement point D0, located at the bottom of the wall, the peak pressure for the blast load constrained by a single explosion source is 11MPa, while for the blast load constrained by double explosion sources, it is 7MPa. The peak pressure of the double explosion source blast wave is reduced by 36.4% compared to the single explosion source. At measurement point, D1, located 20cm above the bottom of the wall, the peak pressure for the blast load constrained by a single explosion source is 6MPa, while for the blast load constrained by double explosion sources, it is 5.8MPa. The peak pressure of the double explosion source blast wave is reduced by 3.3% compared to the single explosion source. At measurement points D2-D12, located between 40cm and 240cm above the bottom of the wall, the peak pressure for the blast load constrained by a single explosion source remains almost unchanged at approximately 3.9MPa.

In contrast, the peak pressure for the blast load constrained by double explosion sources in the region 60cm-80cm above the bottom of the wall is approximately 2.3MPa, 34% lower than the peak pressure of the single explosion source. As the distance from the bottom of the wall increases, the peak pressure of the blast load constrained by double explosion sources surpasses that of the single explosion source in the region 100 cm-200 cm away from the wall corner. The maximum peak pressure in this region is 89.7% higher than the peak pressure of the single explosion source. The higher peak pressure of the double explosion source compared to the single explosion source in this region is the Mach reflection effect generated between the two explosion sources.

Figure 3-6: Distribution of peak pressures in the horizontal direction on the back wall

To study the pressure distribution on the surface of the back wall caused by the explosion load from dual explosive sources, along with studying the vertical distribution of peak pressures along the back wall, it is also necessary to investigate the distribution of peak pressures along the horizontal direction of the back wall. Using the same research method, the first peak pressure values in the pressure-time curves of measuring points C0-C15 in the constrained explosion load from dual explosive sources were extracted and compared with those from a single explosive source, as shown in Figure 3-6.

For the positive peak pressures on the back wall surface, ranging from measuring point C0 at 0 cm distance from the back wall centerline to measuring point C15 at 300 cm distance from the back wall centerline, the explosion load from the single explosive source exhibits an overall trend of first decreasing and then increasing in the horizontal direction from the back wall centerline. Similarly, the explosion load from the dual explosive sources also shows a similar trend of first decreasing and then increasing in the horizontal direction from the back wall centerline. The difference in peak pressures between the two gradually decreases from 3.5 MPa at the back wall centerline to negative values.

Since the horizontal plane where measuring points C0-C15 is located in the plane where Mach reflection occurs, it can be concluded from the above findings that on the Mach reflection plane, the peak pressure caused by the explosion load from the dual explosive sources is greater than that from the single explosive source. It gradually decreases outward from the central region of the Mach reflection.

At measuring point C0, the peak pressure constrained by one wall of the back wall due to the dual explosive sources is 7.5 MPa, while for the single explosive source, it is 4 MPa. The peak pressure from the dual explosive sources is 87.5% higher than that from the single explosive source. At measuring point C1, the peak pressure constrained by one wall of the back wall due to the dual explosive sources is 7.1 MPa, while for the single explosive source, it is 3.9 MPa. The peak pressure from the dual explosive sources is 82% higher than that from the single explosive source.

At measuring point C2, the peak pressure constrained by one wall of the back wall due to the dual explosive sources is 6.6 MPa, while for the single explosive source, it is 3.7 MPa. The peak pressure from the dual explosive sources is 78% higher than that from the single explosive source. As the horizontal distance increases, the effect of Mach reflection gradually decreases. At measuring point C5, the peak pressure constrained by one wall of the back wall due to the dual explosive sources is 5.2 MPa, while for the single explosive source, it is 3.4 MPa.

The peak pressure from the dual explosive sources is 38% higher than that from the single explosive source. When the horizontal distance reaches 210 cm, the peak pressure constrained by the dual explosive sources is 2.0 MPa, the same as the peak pressure constrained by one wall of the back wall due to the single explosive source. For horizontal distances greater than 220 cm, the peak pressure from the dual explosive sources becomes smaller than the peak pressure from the single explosive source. After the horizontal distance exceeds 220 cm, multiple reflections occur due to the measurement points being close to the corner region between the back wall and the side wall, resulting in an upward trend in the explosion load.

From this, we can draw the following conclusions: With the horizontal distance gradually increasing within the Mach reflection plane, the intensity of the Mach reflection effect caused by the constrained explosion from dual explosive sources decreases. Within the 0-260 cm range, the peak pressure of the explosion load from the dual explosive sources is greater than that from the single explosive source. However, within the range exceeding 260 cm, the peak pressure of the explosion load from the dual explosive sources is smaller than that from the single explosive source, and the trend of the peak pressure along the horizontal direction shows a general pattern of first decreasing and then increasing.

Figure 3-7: Distribution of Peak Pressure in the Horizontal Direction of the Side Wall

The first wave peak pressure values extracted from the pressure-time curves of measurement points E0-E15 in the explosion load constrained by the double blast source are compared with those of the single blast source, as shown in Figure 3-7. The peak positive pressure of the explosion load on the side wall surface from measurement point E0 to measurement point E15 shows an overall decreasing trend for both the single and double blast sources.

Moreover, since the side wall is far from the center of the double blast source and the intensity of the Mach reflection is already weak, the difference between the explosion load pressures of the single and double blast sources is small. The peak pressures of the explosion load from the double blast source are almost all smaller than those from the single blast source. Therefore, it can be concluded that the peak pressure on the side wall surface constrained by the double blast source is smaller than that constrained by the single blast source.

Figure 3-8: Distribution of peak pressures in the vertical direction of the sidewall

Using the same research method, the peak pressure values of the first wave in the pressure-time history curves of measuring points F0-F15 are extracted for both the single-source and double-source constrained explosive loads, as shown in Figure 3-8. The distribution of explosive loads on the sidewall surface from a single source exhibits a trend of higher pressure on the two sides and lower pressure in the middle in the vertical direction. In contrast, the distribution of explosive loads from a double source shows two peak values in the vertical direction.

The peak positive pressure on the sidewall surface decreases gradually from 4.8 MPa at measuring point F0 as it moves away from the rear wall. At measuring point F5, the peak positive pressure is 1.5 MPa, and its decreasing trend becomes slower and symmetrically distributed along the center axis of the sidewall. Due to the Mach reflection effect, the peak pressures on the sidewall surface under the double-source constrained explosion are enhanced near measuring point F3, which is 60 cm away from the corner of the sidewall. At measuring point F0, the peak positive pressure is 1.7 MPa.

In contrast, at measuring point F3, due to the Mach reflection effect, the peak positive pressure increases to 2.5 MPa, forming a Mach reflection region. However, because the sidewall is relatively far from the center of the double source, the effect of Mach reflection has weakened. Therefore, the maximum region of peak pressures on the sidewall surface under the double-source explosive load is almost equal to that under the equivalent single-source constrained explosion. Due to the long distance between the sidewall and the center of the double source, the interaction between the two explosive shock waves minimally impacts the explosive load on the sidewall. Therefore, the peak pressures on the sidewall surface under the double-source constrained explosive load are lower than those under an equivalent single-source constrained explosion.

3.3 Traditional Methods for Determining Internal Dual Blast Source Explosion Loads

According to relevant experience, two methods exist for determining the explosion loads of internal dual blast sources.

The traditional method of assessing blast resistance performance (of propellant actuated power devices) utilizes computationally efficient software such as LS-DYNA, an explicit dynamic analysis program. It employs a fluid-structure coupling algorithm to simulate the explosive process and calculate the explosion loads and dynamic response of the structure. This method accurately calculates the explosion loads and structural dynamic response based on the simulation of the actual explosion scenario. However, due to the high time cost of the computational process and the demanding computational efficiency of the computer, its feasibility is relatively low when applied to engineering calculations.

Foreign blast resistance regulations provide detailed calculation methods for explosion loads of single blast sources in free air and various constrained structures. Treating the two blast sources of the internal dual blast as equivalent to a single blast source allows the explosion loads under various scenarios to be quickly calculated according to the relevant foreign regulations. However, the calculation process for explosion loads in the regulations is based on certain assumptions.

It assumes that the explosion loads are uniformly distributed along all wall surfaces, and the calculated explosion load represents the average load on the entire wall surface. Based on the research on the distribution pattern of internal dual blast source loads mentioned earlier, it can be observed that the explosion loads along the walls of the structure are highly uneven. If calculated according to the methods in the regulations, there is a certain safety risk via safety fuses. The areas where the structure experiences concentrated peak pressure may undergo shear failure, which the existing calculation methods in the regulations cannot consider.

3.4 Newly Proposed Method in This Chapter

The distribution pattern of explosion loads of internal dual blast sources in different constrained structures mentioned in section 3.3 shows that the explosion loads along the wall surfaces are highly uneven. This uneven phenomenon manifests in two aspects: (1) the Mach reflection enhancement effect occurring in the Mach reflection region of the dual blast sources significantly increases the explosion load compared to other areas (blast holes), and (2) within the constrained structure, the multiple reflections of the blast shock wave in the corner regions of the walls greatly enhance the explosion load compared to non-corner areas.

In this section, taking the typical three-wall constrained structure of a fireworks factory building (with a roof) as an example, the uneven distribution of internal double-source explosion loads is considered. The fireworks factory building’s back wall, side walls, and roof are divided into zones. The wall surfaces are divided into 9 zones based on the Mach reflection zone, wall corner zone, and non-corner zone, as shown in Figure 3-9. Using the enhancement coefficients for each zone, the explosion loads determined based on existing specifications are adjusted, obtaining more accurate reflected pressure (Pr) and impulse (ir) for the 9 zones of the wall surfaces of the fireworks factory building.

Figure 3-9: Diagram of Partitioning the Walls of the Fireworks Factory

LS-DYNA explicit dynamic analysis software was used to numerically simulate the explosion of the internal dual explosive source in the fireworks factory. The finite element model from version 2.3.1 was utilized, with a typical spacing of 150cm between the explosive sources and equal amounts of the two explosives. The total equivalent TNT weight was 7.0kg. The boundary conditions for the side walls, rear wall, and roof were fixed constraints, while the constitutive models and boundary conditions for other materials remained unchanged.

A multipoint integration method was employed to determine the loads on the rear wall, side walls, and roof surfaces of the fireworks factory structure in the 9 designated areas. Within each of the 9 areas on the walls, 25 measurement points were selected with a spacing of 20cm between each point. The reflected pressure Pr and reflected impulse ir at these 25 measurement points in each area were integrated to obtain the average explosion load rp and ri for that particular area. The load diagram results for the rear wall surface are shown in Figure 3-10.

Figure 3-10: Explosion Load of the Rear Wall Surface of the Fireworks Factory Building in Zone 9

Based on the explosion load obtained from the internal double explosion source of the fireworks factory building in Figure 3-10, the Mach enhancement factor λ is proposed.

In the equation, P represents the reflected pressure on the wall caused by the internal double explosion source simulated using LS-DYNA software. P+ is the pressure obtained by adding the average reflected pressure of the two internal explosion sources, calculated according to UFC3-340-02. ir represents the reflected impulse on the wall caused by the internal double explosion source simulated using LS-DYNA software. ri+ is the impulse obtained by adding the average reflected impulse of the two internal explosion sources, calculated according to UFC 3-340-02. λ is the amplification coefficient.

The formula is based on the typical overpressure time-history curve of explosion load. Since the typical overpressure time-history curve of explosion load is triangular, the uniformly distributed reflected pressure and reflected impulse on the wall calculated in the specifications can be superimposed for the case of internal double explosion sources. The superimposed explosion load of the internal double explosion sources in the fireworks factory building is a uniformly distributed load on the entire wall, without considering the Mach reflection effect between the two explosion sources and the multiple reflection effects in areas such as the internal explosion wall corner. In this section, the Mach amplification coefficient is introduced to consider the influence of these two effects on the uneven distribution of the explosion load. The amplification coefficients for the nine regions of the walls in the fireworks factory building are calculated as shown in Figure 3-11.

(a) Enhanced coefficient for the rear wall in Zone 9 (when γ is less than 1, λ=1)

(b) Enhanced coefficient for the roof in Zone 9 (when γ is less than 1, λ=1)

(c) Enhanced coefficient for the side wall in Zone 9

Figure 3-11: Enhanced coefficients for the nine zones of the fireworks factory building.

Considering the variation in distance between the two explosion sources, changes in the distance between internal dual explosion sources will affect the enhanced coefficients of the collision areas (IV, V, VI) while having little impact on the enhanced coefficients of non-collision areas. By introducing a chart depicting the variation of enhanced coefficients with the distance between explosion sources, the enhanced coefficients for zones IV, V, and VI can be determined.

Figure 3-12: Relationship Diagram between Explosion Load Enhancement Coefficient and Spacing Distance on Each Wall

In the equation, λpr represents the enhancement coefficient of reflected pressure, λir represents the enhancement coefficient of reflected impulse, x represents the spacing distance, and x ranges from 0cm to 240cm.

Based on the obtained enhancement coefficient diagram for each wall in the 9 regions, as well as the functional relationship between the enhancement coefficient λ and the spacing distance, the following steps are proposed for determining the internal dual-source explosion load:

1.   Calculate the uniformly distributed wall load for the independent explosions of sources O1 and O2, according to the UFC3-340-02 specification.

2.   Add the uniformly distributed explosion loads from the two sources and multiply them by the corresponding enhancement coefficient λ for each wall area.

3.   Obtain the explosion load (reflected pressure Pr and duration of action t0) for each wall area in the 9 regions.

Chapter 3.5 Summary

Based on the comparison between single and double explosive sources in Chapter 2, this chapter presents a method for determining the explosion load of the internal double explosive source fireworks factory. The concept of an enhancement coefficient is introduced, which is used to determine the uneven distribution of the explosion load in the internal double explosive source fireworks factory. According to the specifications, the explosion load generated by the independent explosion of the internal double explosive source is calculated and superimposed, multiplied by the enhancement coefficient, to determine the actual explosion load of the internal double explosive source.

Firstly, through numerical simulation, the explosion load of the internal double explosive source in the fireworks factory is studied and compared with the single explosive source to analyze the distribution pattern of the constrained explosion load of the double explosive source. The characteristics of the explosion load in the internal double explosive source fireworks factory are investigated.

Due to multiple reflections in the constrained structure’s internal partition and the Mach reflection between the two explosive sources, the reflected pressure of the double explosive source’s explosion shock wave in the Mach reflection region is greater than the explosion load of a single explosive source with the same TNT equivalent. In the structure’s multiple reflection region, the explosion load’s pressure-time curve exhibits a multi-peak phenomenon, significantly enhancing the reflected impulse in that area. On the side walls of the structure, the intensity of the constrained explosion load of the double explosive source is still less than that of a single explosive source with the same TNT equivalent, even in the area of strongest Mach reflection, due to the increased distance.

After preliminarily determining the propagation law of the constrained explosion load of the double explosive source, a new method for determining the explosion load of the internal double explosive source fireworks factory is proposed. This method considers the extremely uneven distribution of the explosion load in the internal double explosive source and is more accurate than the existing specification-based load determination method. The method divides the walls of the factory structure into different zones and calculates the corresponding enhancement coefficient for each zone. When determining the explosion load of the internal double explosive source in practical engineering, the explosion load after the independent explosion of the two sources is calculated according to the existing specifications and then multiplied by the corresponding enhancement coefficient of the respective zone to obtain the explosion load of the internal double explosive source fireworks factory.

Chapter 4: Evaluation Method for Blast Resistance of Internal Dual Explosion Source Fireworks Factory Buildings

4.1 Introduction

Under the action of explosive loads, the reflection process of shock waves within an enclosed space is relatively complex [33,46]. In the previous three chapters, research and analysis were conducted on the explosion loads of internal dual explosion sources, and the corresponding patterns of the explosion loads were obtained. Based on the previous discussions, this chapter will numerically simulate the structure of a fireworks production facility with three walls and a roof and evaluate its blast resistance.

According to traditional methods, the analysis of blast resistance for internal dual explosion source fireworks factory buildings uses LS-DYNA, an explicit dynamics analysis software, to establish a finite element model. Fluid and structural elements are coupled to simulate the real explosion process and analyze the blast resistance of the facility structure. This paper proposes a new evaluation method suitable for rapid assessment of the blast resistance of enclosed structures. This method is based on the blast resistance specification UFC 3-340-02 established by the U.S. Department of Defense.

It calculates the reflected pressure and duration on the internal surfaces of the structure under the action of internal explosion loads. The calculation method for internal explosion loads is based on semi-empirical data and many dynamic response experiments on reinforced concrete slab structures. The finite element model is simplified to enable rapid assessment of the structural blast resistance.

Under the internal dual explosion source explosion loads, the dynamic damage of the fireworks factory structure is different from that under single explosion source explosion loads due to the Mach reflection effect. In this chapter, numerical simulations are conducted using the LS-DYNA explicit dynamics software to establish a finite element model. The explosion loads are applied to the corresponding positions in different areas of the fireworks factory structure based on this paper’s proposed new load determination method.

4.2 Calculation of Explosion Loads Using the New Method

Based on semi-empirical data, the main types of structures affecting the load are shown in Figure 4-1.

Figure 4-1: Shape and Parameters of an Enclosed Structure

Based on Figure 4-1, taking the three-wall (with a roof) structure of a fireworks factory as an example, the main structure of the fireworks factory is a reinforced concrete structure with dimensions of 6000mm3000mm3000mm. The structure undergoes single-side explosion relief. Two explosion sources are arranged in parallel at the center of the structure, with a spacing of 150cm. First, calculate the uniformly distributed explosion load for each independent explosion source.

Referring to the annotations in Figure 4-1, calculate all the key parameters required to determine the explosion load on the back wall when the explosion source O1 independently explodes, as shown in Tables 1 and 2.

Table 4-1: Fireworks Factory Parameters

Proportional distance calculation formula:

Za = Ra / W^⅓ — (4-1)

Table 4-2: Key parameter values for fireworks factory buildings

According to the key parameters, refer to Fig.2-52 to Fig.2-149 in UFC. Using internal interpolation, obtain the peak overpressure and impulse of the explosive load acting on the inner surfaces of the structural walls and roof. Note the following: unit conversion, UFC uses US units (feet, pounds, psi).

The graphs in UFC, Fig.2-52 to Fig.2-149, are all in a logarithmic coordinate system. Pay attention to the coordinate system conversion. In the logarithmic coordinate system, the coordinates are lg1, lg2, lg3, lg4,…, lg9, 1+lg1(10), 1+lg2(20), 1+lg3(30), and so on. When performing internal interpolation via propellant actuated power devices (electrical or electromechanical device), the coordinate distance between two points is a logarithmic distance. For example, the distance between points 1 and 2 is lg2-lg1.

Taking the structure of a fireworks factory as an example, according to the key parameters obtained from Table 4-2, refer to UFC Fig.2-100 with N=4, l/L=0.5, h/H=0.75, interpolate to obtain L/H=2 and ZA=2.48, and calculate the peak pressure Pr.

Similarly, referring to UFC Fig. 2-149 (N=4, l/L=0.5, h/H=0.75), interpolate the value of ir/w when L/H=2.1/3:

The calculation formula for the duration of an explosion load:

The above method determines the peak average pressure (Pr) and duration (T0) on the side wall and roof, respectively.

Use the same method to calculate the uniformly distributed explosive load on the wall after independent explosions of the explosive sources O1 and O2. According to the Mach enhancement coefficient for the corresponding 9 regions of the rear wall surface in Figure 3-11 (a), multiply the calculated reflected pressure (Pr) and impulse (ir) from the independent explosions of sources O1 and O2 by the Mach enhancement coefficient. Determine the explosive load on the 9 regions of the rear wall of the internal dual-source fireworks factory.

Based on the above method, determine the explosive load on the rear wall, side wall, and roof wall surface of the 9 regions, respectively.

(a) Explosion Load Distribution in 9 Areas of the Roof Wall

(b) Explosion Load Distribution in 9 Areas of the Rear Wall

(c) Explosion Load Distribution in 9 Areas of the Side Wall

Figure 4-2: Explosion Load Distribution in 9 Areas of the Structure’s Walls in a Fireworks Factory

4.3 Finite Element Model

Using LS-DYNA, a software for explicit dynamic analysis, a finite element model of the main structure of the explosion-proof workshop is constructed. The structure is axisymmetric, and to improve computational efficiency, a 1/2 model is created along the length direction of the structure, as shown in Figure 4-3. The Z direction represents the length, the X direction represents the width, and the Y direction represents the height.

Solid 164 single integration point hexahedral elements are used for the concrete, while Beam 161 beam elements are used for the steel reinforcement. The mesh size for both the solid and beam elements is set to 50mm. After optimizing and analyzing the convergence of the mesh size, the results indicate that further refining the mesh can only provide limited improvement in computational accuracy while significantly increasing the computational cost. Therefore, a mesh size of 50mm is maintained for both the hexahedral and beam elements.

Figure 4-3: Fireworks Factory 1/2 Structural Finite Element Model

4.3.1 Definition of Material Properties

The bond-slip between steel bars and concrete is not considered in this study. The concrete adopts the MAT_CONCRETE_DAMAGE_REL3 (MAT72) model [35]. The Release III model proposed by K&C is an improvement over the PSEUDO TENSOR (TYPE 16) model, which includes initial yield failure surfaces, ultimate failure surfaces, and residual failure surfaces. It can simultaneously consider damage effects and strain rate effects [47], making it suitable for this study.

 MAT_CONCRETE_DAMAGE_REL3 improves the user input method based on MAT_CONCRETE_DAMAGE and simplifies the process. It only requires inputting data such as Poisson’s ratio, uniaxial compressive strength, and strain rate dynamic amplification curve [35,47,48]. Therefore, this study ultimately selects the MAT_CONCRETE_DAMAGE_REL3 (MAT72) model and inputs the standard value of uniaxial compressive strength for C30 concrete as 24.3 MPa.

The material properties of the steel bars are defined using the keyword MAT_PLASTIC_KINEMATIC. This model also considers the strain rate effects of the steel material. The steel bars used in the structure are HRB400 steel bars, with a sufficient density of 0.0078 g/mm3, Young’s modulus of 20 GPa, yield strength of 400 MPa for HRB400 steel bars, and a Poisson’s ratio of 0.3.

4.3.2 Strain Rate Effects

Due to the significant strain rate variations experienced by reinforced concrete structures under explosion loads, the high-strain process can affect the dynamic properties of concrete and steel bars [49,50]. Therefore, considering the strain rate effects is crucial. This study adopts the DIF dynamic amplification factor to represent the variation of material strength. The dynamic amplification factor for concrete strength is determined using the model proposed by K&C, and the dynamic amplification factor for tensile strength is determined by the following equation:

In the equation, FTD represents the dynamic tensile strength of concrete at strain rate εd and fits the static tensile strength of concrete at strain rate εts (εts = 10^(-6) s^(-1)). The term logβ is defined as 6δ-2, where δ = 1/(1+8fc’/fco’), fco’ = 10 MPa, and fc’ is the static uniaxial compressive strength of concrete (static electricity).

The dynamic amplification factor for the compressive strength of concrete is given by:

FTD/fc’ = 1 + (fts/fc’ – 1) exp(-logβ εd)

In the equation, FCD represents the dynamic compressive strength of concrete at a strain rate of εd. FCS represents the static compressive strength of concrete at a strain rate of εcs (where εts=30 × 10-6 s-1). logγ=6.15α-0.49, where α=(5+3fcu/4)-1, and fcu is the static compressive strength of concrete in cubic form (static electricity).

The dynamic increase factor for the strength of reinforcement is adopted as follows:

In the equation, ε represents the strain rate of the steel reinforcement, measured in (s-1), and fy is the yield strength of the steel reinforcement, measured in MPa. The applicable range of the equation is: 10^(-4) s^(-1) ≤ ε̇ ≤ 255 s^(-1); 270 MPa ≤ ε̇ ≤ 710 MPa.

4.4 Model Verification

A finite element model is established based on the internal blast load determined by UFC, and the model’s reliability needs to be verified. A series of long-distance blast tests on steel-reinforced concrete slab structures conducted by Razaqpur et al. [18, 51-52] were selected for simulation. The test specimens consist of two identical 1000×1000×70mm^3 double-layer reinforced concrete slab structures with a steel reinforcement diameter of 6mm and a steel reinforcement grid spacing of 152mm.

The yield strength of the steel reinforcement is 480MPa, and the ultimate strength is 610MPa. The compressive strength of the concrete after 28 days of curing is 40MPa. Specimen CS2 and specimen CS4 are subjected to blast loads from 33.4kg and 22.4kg ANFO, respectively, with the explosive source located at a vertical distance of 3m from the center of the steel-reinforced concrete slab structure. The steel-reinforced concrete slab is placed horizontally, and the boundary conditions on all sides are fixed constraints.

The finite element model proposed in this study is used for modeling. The steel-reinforced concrete slab’s nonlinear dynamic response and failure under blast loads are numerically simulated using the same material models and parameters. A concrete, plastic strain of 0.0018 is used as the criterion for erosion. Solid and beam elements are both meshed with a grid size of 50mm.

Table 4-3 compares the peak reflected pressure, pressure arrival time, and maximum deflection at the center of specimen CS2 and specimen CS4 between the experimental conditions and the finite element model established in this study. Since the blast loads in the numerical simulation are defined using the keyword LOAD_BLAST, the peak reflected pressure and arrival time are calculated using empirical formulas within LOAD_BLAST and are consistent with the experimental results; no comparison is made for the peak reflected pressure.

Table 4-3: Comparison of Finite Element Model Results and Experimental Data

Table 4-3 shows that the maximum deflection at the center of specimen CS2 obtained through numerical simulation is 11.4mm. Comparing it with the experimental results, there is an error of 2.63%. The center maximum deflection of specimen CS4 obtained through numerical simulation is 6.90mm, with an error of 7.44% compared to the experimental results.

Figure 4-4: Numerical Simulation Damage and Experimental Damage of Specimen CS2

Figure 4-3 compares failure modes between numerical simulation and experimental testing of a reinforced concrete slab structure. Due to the relatively small size of the finite element model mesh, the crack development trend is not obvious. According to the concrete erosion criteria set, the following observations were made: 1. A square damage region appears at the center of the bottom surface, extending towards the four corners of the reinforced concrete slab along a 45° direction. 2. Cracks develop along the positions and directions of the reinforcing bars. These results demonstrate that the finite element model proposed in this paper can accurately predict structural damage under explosive loads, confirming the correctness of the numerical model used in this study.

4.5 Simulation Results and Analysis

The computational scenario in this study involves internal dual explosive sources with an equivalent TNT weight of 7.0kg. Since this study focuses on blast resistance performance evaluation (propellant actuated power devices), concrete’s compressive strength and reinforcing bars’ strength are assumed to be standard values without considering explosion venting measures on the structure. The explosion load is calculated using the method proposed in section 3.4 for the explosion load of an internal dual explosive source fireworks factory. The explosion load is assumed to follow an inverted triangular load curve [34, 53], and it is assumed to be uniformly applied to each reinforced concrete wall surface of the structure using the keyword LOAD_SEGMENT-SET.

4.5.1 Stress and Strain Results Analysis

The LS-DYNA calculation results show that the concrete did not exhibit compressive yielding, and slight tensile cracks appeared at the ends of the roof and the mid-spans of each wall. Some of the reinforcing bars in the mid-span of the side walls entered the yielding stage, while the other reinforcing bars did not yield. Figure 4-5 shows the stress contour plot of the structural reinforcing bars at t=0.08s, and Figure 4-6 shows the plastic strain contour plot of the structural reinforcing bars at t=0.08s. Based on the results shown in the figures, some of the reinforcing bars in the mid-span and support locations of the main structural side walls and the mid-span of the roof are under relatively high stress.

The remaining steel bars did not reach the yield stage due to the hardening effect caused by the high strain rate.

Figure 4-5: Stress cloud map of structural steel bars (t=0.08s)

Figure 4-6: Plastic strain cloud map of structural steel bars (t=0.08s)

4.5.2 Analysis of Structural Deformation Results

Figure 4-7 shows the deformation contour map of the firework factory structure at time t=0.058s. Under the explosive action of the internal dual detonation source, the factory structure’s maximum displacement occurs at the rear wall’s mid-span position. Mild bending deformations appear on all wall surfaces, and the maximum deformations are located at the mid-span positions of each wall surface. Due to the Mach reflection effect caused by the dual detonation source explosion, the side wall surfaces experience two locations of maximum displacement. The displacement time-history curves are extracted for the selected nodes at the maximum displacement positions of the rear wall, side walls, and roof mid-span, as shown in Figures 4-8, 4-9, and 4-10. The node numbers are as follows: rear wall – Node 36591, side walls – Node 99458, Node 104237, and roof – Node 36594.

Figure 4-7: Deformation Contour Map of the Structure (t=0.058s)

Figure 4-8: Time History Curve of Nodal Displacement at the Maximum Displacement Location of the Roof Structure

The maximum displacement of the structure’s roof is located at its midpoint. The vertical displacement time history curve is obtained by selecting node 36594 at the midpoint of the roof span, as shown in Figure 4-7. At the location of the maximum displacement in the roof structure’s midpoint, the maximum displacement is 52.3mm, and the residual displacement is 20.1mm.

(a) Time History Curve of Nodal Displacement at Node 99458 of the Side Wall

(b) Displacement-Time History Curve of Node 104237 on the Side Wall

Figure 4-9: Displacement-time history curve of the node with the maximum displacement at the mid-span of the structure’s side wall

There are two locations with maximum displacements in the side wall. Similar to the single explosion source, the maximum displacement occurs at the mid-span region near the edge of the side wall. Node 99458, located at the mid-span near the edge of the side wall, was selected, and its maximum displacement was 58.8mm, with a residual displacement of 17.6mm. Unlike the single explosion source, the structural deformation in the Mach reflection region of the internal double explosion source fireworks factory will be amplified. The horizontal displacement-time curve of node 104237 in the Mach reflection region of the side wall was obtained and is shown in Figure 4-8b. At the location of maximum displacement in the Mach reflection region of the side wall, the maximum displacement was 64.9mm, with a residual displacement of 25.1mm.

Figure 4-10: Displacement-Time History Curve of Node with Maximum Displacement at the Mid-span of the Rear Wall

The maximum displacement of the rear wall also occurs at its mid-span position. Node 36594, located at the mid-span of the rear wall, was selected, and its horizontal displacement-time curve was obtained, as shown in Figure 4-10. At the location of maximum displacement in the mid-span of the rear wall, the maximum displacement was 76.7mm, with a residual displacement of 27.6mm.

4.5.3 Method Validation

LS-DYNA software was used with a fluid-structure interaction algorithm to validate this evaluation method’s reliability. The air units were enveloped outside the finite element model of the fireworks factory, with an air grid size of 20mm, to simulate the structural damage caused by the actual internal double-source explosion scenario. A comparison was made with the evaluation method proposed in this chapter.

The figure below shows the damage comparison between the two methods. The structural damage locations and forms are basically consistent between the two methods. The fluid-structure interaction simulation captures the real explosion process, although the arrival time of the explosion shock wave differs from the method proposed in this chapter. However, this method is primarily used for evaluating the structural blast resistance performance, focusing on the maximum deformation and deformation location of the structure. Therefore, it can be concluded that the method proposed in this chapter has a certain level of reliability.

(a) Methods of this chapter

(b) Fluid-structure coupling method

Figure 4-11: Structural damage comparison

4.6 Blast Resistance Evaluation

The American Society of Civil Engineers (ASCE) 59-11-2011 Blast Protection of Buildings [54] specifies the protection level for building structures. It sets the maximum allowable response limits for building structures corresponding to different defense levels. Under the action of blast loads, the components of a building structure can be divided into two categories: bending components and compression components.

Under the effect of long-distance explosions, the blast resistance evaluation of the structure is based on two indicators: the rotational angle of the support and the elongation of the components. ASCE 59-11 specifies the maximum allowable response limits for various forms of components to assess the structure’s capacity to withstand maximum dynamic loads, thereby establishing the corresponding protection level.

1.   According to the standards in ASCE 59-11, for a double-layer reinforced concrete slab structure without shear reinforcement, the blast resistance is evaluated based solely on the rotational angle of the support, as shown in Table 4-4:

Table 4-4: Maximum Allowable Response Limits for Structural Capacity

A support rotational angle close to 0° indicates mild damage, a rotational angle less than 2° indicates moderate damage, a rotational angle between 2° and 5° indicates severe damage and a rotational angle between 5° and 10° indicates critical damage.

According to the displacement-time curve provided in Figure 4-10 for the mid-span position of the rear wall of the fireworks factory building, the maximum displacement at the roof’s mid-span position is 76.7mm, and the residual displacement is 27.6mm. Based on the evaluation criteria for blast resistance of a double-layer reinforced concrete slab structure specified in ASCE 59-11-2011 Blast Protection of Buildings, the support rotational angle for the roof wall of this structure can be calculated.

Θ = arctan (76.7/3000) = 1.46 degrees

The rotation angle of the support is less than 2°, indicating mild damage to the main structure of the fireworks factory.

Chapter 4.7 Summary

In this chapter, numerical analysis was conducted on the reinforced concrete structure of the fireworks production factory. According to the dimensions of the structure and the number of explosives, the explosion load data acting on each wall of the structure was calculated using the method proposed in this paper for determining the internal double explosion source explosion load. Then, LS-DYNA finite element analysis was applied.

Chapter 5: Conclusion and Outlook

5.1 Main Conclusions

This paper used LS-DYNA to study the free-air and constrained structure explosion of dual explosive sources. Through pressure distribution maps, pressure-time curves, and the variation trend of peak pressures on various wall surfaces, the propagation laws of constrained explosion loads from dual explosive sources were investigated, and a method for determining the internal explosion load of a dual explosive source fireworks factory was proposed. Numerical simulations were conducted on typical fireworks and firecracker production plant structures to study their dynamic response. The main research findings are as follows:

(1) In the explosion load of dual explosive sources, shock waves are formed instantaneously after the two sources detonate and propagate outward in a spherical shape. After a certain period, the two shock waves reach the collision surface and collide, resulting in Mach reflection. The form of Mach reflection is the same as that in air explosions, and Mach waves uniformly propagate outward in the Mach reflection region. The intensity of Mach waves generated by the Mach reflection effect increases, enhancing the explosion load of dual explosive sources.

(2) With increasing distance, the peak pressure of the explosion shock wave shows an overall decreasing trend. In the collision area of the dual explosive source explosion load, there are regions of Mach reflection and a transitional region where Mach reflection has not occurred yet. As the distance between the sources increases, the position of the Mach reflection region remains basically unchanged, and the intensity of the Mach reflection effect gradually weakens in the explosion load of dual explosive sources. The non-collision area (i.e., the asymmetric region) is not affected by the Mach reflection in the collision area.

(3) In the constrained explosion load of dual explosive sources, the peak pressure on the rear wall surface shows a decreasing trend from the wall corner to the top along the vertical direction on the midline of the rear wall. The maximum pressure peak value is distributed at the wall corner and the center of the rear wall surface. The maximum pressure peak value at the wall corner is due to the multiple reflections and enhancement of shock waves at the wall corner, while the maximum pressure peak value at the center of the rear wall is due to the enhanced effect of Mach reflection.

The distribution of pressure peak values along the vertical direction of the rear wall shows the characteristics of peaks appearing at the wall corner and the wall center positions. Along the horizontal direction on the rear wall surface, the pressure peak values show a decreasing trend from the center to the edge of the rear wall. As the horizontal distance increases, the influence of the Mach reflection effect gradually weakens, and the distribution of pressure peak values along the horizontal direction of the rear wall shows the characteristics of smaller values on both sides and a larger value in the center.

(4) The peak pressure on the side wall surface decreases overall along the horizontal distance as the explosive load increases. Along the vertical direction from the wall corner to the top, the explosive load initially increases and then decreases, forming a peak region due to the influence of Mach reflection. Due to the relatively large distance from the side wall to the center of the dual explosive sources, the effect of Mach reflection is weakened. Therefore, at the location of the maximum Mach effect on the side wall surface, the explosive load from the dual explosive sources is smaller than that from a single explosive source, and the peak pressure of the dual explosive sources on the side wall surface is smaller than that of a single explosive source.

(5) In this study, the walls of the fireworks factory structure were divided into zones, and the concept of the enhancement coefficient λ was introduced. The enhancement coefficient is the ratio of the peak pressure at a certain proportional distance from the dual explosive sources to the peak pressure when the explosive sources explode individually. Based on this, a new method is proposed for determining the explosive load of internal dual explosive sources. Determining the explosive load from internal dual explosive sources involves adding the explosive loads from the two sources separately and multiplying them by the enhancement coefficient λ to obtain the explosive loads on each wall surface and in each zone of the fireworks factory structure.

(6) Based on the proposed method for determining the explosive load from internal dual explosive sources, an improved evaluation method for the blast resistance of fireworks factory structures with internal dual explosive sources is developed. The dynamic response of the fireworks factory with internal dual explosive sources is simulated. The load determination method proposed in this study is used to directly apply the explosive load to the structural surface and quickly assess the blast resistance of the structure. In the internal dual explosive source explosion with a TNT equivalent of 7.0 kg, the walls of the structure experience varying degrees of bending deformation, with the maximum deformation locations on the side wall and rear wall both located in the Mach reflection region.

5.2 Outlook

Although this study has conducted a detailed investigation into the propagation characteristics of shock waves from dual explosive sources in free-air explosions and explosions with different types of constraints, as well as the dynamic response of dual explosive sources in typical constrained structural explosions, achieving significant progress, there are still many remaining issues due to the diverse types of explosive loads from dual explosive sources and the complexity of the explosion process. These issues provide references for future researchers as follows:

(1) The classification of near-field and far-field explosions in the explosive loads from dual explosive sources studied in this paper is not strictly defined, and the proportional distance of the typical structural walls falls within a certain range. This study mainly focuses on explosions in free air and constrained explosions with free surfaces without considering partially constrained explosions and fully enclosed constrained explosions with dual explosive sources. Dual explosive source explosions are more common in real-life environments, and this study aims to investigate the propagation characteristics of their blast waves. The constrained structures studied in this paper are all rectangular reinforced concrete structures, without considering other types of structures, and the research on explosive loads from dual explosive sources and multiple explosive sources is becoming increasingly important.

(2) The simplified model for explosive loads from dual explosive sources proposed in this paper can be applied to the blast-resistant design and analysis of practical engineering. However, the duration of the shock wave effect, quasi-static gas pressure, and other factors related to the impact of the explosive load on the wall surface has not been within the scope of this study. Further research is needed when studying the explosive loads in fully constrained structures with dual explosive sources.

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