Dynamic Response and Damage Analysis of a Large Steel Tank Impacted by an Explosive Fragment

The domino effect refers to the phenomenon in which the thermal radiation, shock waves, and fragments generated by the explosion of storage tanks in chemical industrial parks cause perforations or dents in adjacent tanks and trigger secondary accidents, leading to more severe casualties and property losses [1–3]. Numerous accidents have shown that fragment projection is a critical trigger of the domino effect [4,5]. For example, the 1984 Mexico City National Petroleum Company explosion, the 2017 Shandong Linyi petrochemical explosion, etc., are typical accidents caused by the domino effect triggered by explosive fragments. Therefore, investigating the dynamic response of tanks impacted by explosive fragments is crucial for reducing the likelihood of the domino effect and mitigating accidental losses.

Currently, numerous scholars have researched this topic. With respect to the theoretical analysis, Gubinelli and Cozzani [6] analyzed 143 tank explosion accidents and summarized that the highest proportion of accident tanks are horizontal, amounting to 70.6%. Holden [7] discovered that the fragments produced by the explosion of horizontal cylindrical pressure vessels are mainly categorized into long end-cap, end-cap, and plate. Hauptmanns [8] analyzed the number of explosive fragments generated by the explosion of horizontal tanks and found that their number follows a lognormal distribution. Mébarki et al. [9] investigated the parameters of the number, mass, velocity, and shape of explosive fragments with the maximum entropy principle, and their probability distributions are summarized.

With respect to the experimental research, Chen et al. [10] conducted experiments on pointed bullets impacting small tanks; the theoretical formula for calculating the residual velocity of pointed fragments penetrating tank walls is deduced to be applicable to axial impact angles ranging from 0 deg to 45 deg. Sun et al. [11] evaluated the protective effect of tank protective layers under varying thicknesses, surface densities, and fragment impact velocities. They considered that the energy absorption properties of the protective layer are related to the critical thickness of the material.

With respect to numerical analysis, Hu et al. [12] established a finite element model of tanks impacted by explosive fragments and analyzed the damage process of the tanks. Li et al. [13] investigated the coupling effects of fragment impact and adjacent pool-fire on the thermal buckling of a fixed-roof tank. The effects of impact velocities, impact angles, and fragment shapes on the failure mode and fire resistance of a fixed-roof tank were discussed. Liu et al. [14] investigated the penetration of cubic, spherical, and conical fragments into tanks with the coupled Smoothed Particle Hydrodynamics-Finite Element method and summarized the four typical phases of the penetration process: extrusion, hole breaking, penetration, and inertia flight. Feng et al. [15] simplified the inner liquid as internal pressure and studied the failure process of tanks impacted by explosive fragments. The results indicate that the core factor affecting the damage is the impact velocity perpendicular to the impact surface. Zhu et al. [16] established a finite element model of tanks impacted by long cylindrical fragments. The three failure modes of the tank are summarized: local failure, overall failure, and rupture failure.

Analysis reveals that most existing investigations have limitations, such as idealizing fragment parameters and neglecting or simplifying the inner liquid, which is somewhat different from the real situation. Additionally, relatively few studies have investigated the process of explosive fragment impact on tanks. Therefore, this paper selected a reasonable fragment shape based on the statistical law of accidents, determined the fragment mass and velocity according to the empirical formula, then established a finite model of large steel tanks impacted by explosive fragments in line with the real scenario, and the fluid–structure interaction method was considered. The dynamic process of large steel tanks impacted by fragments was simulated, the typical stages of the impact process were summarized, the characteristics of each stage were elucidated, and the damage laws of tanks were analyzed under various filling coefficients, volumes, fragment velocities, and impact angles. The conclusions of this study can serve as a reference for optimizing the distribution of tanks on tank farms and designing impact protection measures for tanks.

The dome roof tank is a representative type of steel tank. Thus, according to the Chinese standard “GB 50128-2014 Construction Code for Vertical Cylindrical Steel Welded Storage Tanks” [17], large steel dome roof tanks with volumes of 2000 m³, 3000 m³, 5000 m³, and 10,000 m³ were regarded as potential target tanks, and the inner liquid of the tanks is petroleum. The geometric parameters of the large steel dome roof tanks are shown in Table 1.

A dome roof tank is primarily constructed from the dome roof, shell plate, bottom plate, and accessory structure. The thickness of the shell plates varies with the liquid pressure. In order to focus on the structural dynamic response of the tank itself, accessory structures were ignored when modeling. The structure of the large steel dome roof tank is shown in Fig. 1, while the thicknesses and heights of the shell plates are shown in Table 2.

The generation of explosive fragments predominantly hinges on the characteristics of the pressure vessel. Gubinelli and Cozzani [6] revealed that horizontal cylindrical pressure vessels account for the greatest percentage of explosion accidents, at 70.6%, with accident tank volumes typically falling within 50 m³ to 300 m³. Therefore, according to the Chinese standard “NB/T 47001-2009 steel liquefied petroleum gas horizontal storage tank type and basic parameters” [18], the volume of a 150 m³ horizontal cylindrical tank was chosen as the source of explosive fragments. The horizontal tank material is 16Mn, and the inner liquid is liquefied petroleum gas. The geometrical parameters are shown in Table 3.

The determining factor for the shapes of explosive fragments is the crack expansion pattern of the pressure vessel. According to research by Holden [7], taking a chemical industrial park tank distribution scenario as an example, when a horizontal cylindrical pressure vessel explodes, its crack expansion pattern is shown in Fig. 2. As a result of the explosion, the horizontal tank will tear along the cracks into long end-cap, end-cap, and plate fragments; these fragments will project into the surrounding tank farms, which long end-cap, end-cap, and plate fragments accounted for 46%, 23.5%, and 30.5%, respectively. The geometry of the explosive fragments is given in Fig. 3, where R₁ and R₂ are the head radius of the long end-cap and end-cap fragments, respectively; L₁ is the extension length of the long end-cap fragment; and L₂ and L₃ are the length and width of the plate fragment, respectively.

where $Ms$ is the mass of the explosive fragment, kg; $Vs$ is the volume of the explosive fragment, m³; $ρs$ is the density of the explosive fragment, kg·m⁻³; $Ri$ is the radius of the head, m; $Vec$ is the volume of the end-cap explosive fragment, m³; $ε1$ is the thickness of the head, mm; $ε2$ is the thickness of the cylinder, mm; and L₁ is the extension length of the long end-cap fragment, m.

where $λ$ is a scaling factor of the formula; $v0$ is the velocity of the explosive fragment, m/s; and $Em$ is the energy of the tank explosion, MJ.

where $Vt$ is the tank volume, m³; $P0$ is the environmental pressure, MPa; $P1$ is the tank explosion pressure, 0.9–1.1 times the design pressure, MPa; $ξ$ is the specific heat capacity, J/(kg·°C); and $ke$ is the energy coefficient.

According to the investigation by Hu et al. [12], plate fragments cause less damage to tanks than end-cap fragments at the same kinetic energy. For the long end-cap fragments, on the one hand, although their impact behavior is similar to end-cap fragments, their mass is much greater than the end-cap fragments, and the deformation of the target tank is so severe that it is difficult to conclude the damage law. On the other hand, its extension length is difficult to determine. Thus, only the end-cap fragment was considered in this paper. Taking R₂ = 1.6 m and L₁ = 0.0 m, the design pressure of the horizontal tank is 1.77 MPa, with an environmental pressure of $P0$ = 0.1 MPa. The specific heat capacity is 1.44 J/(kg·° C). The parameters of the end-cap fragment from the 150 m³ horizontal tank are calculated, as shown in Table 4.

In fact, because of aerodynamic drag, the impact velocity of fragments will be affected by the distance between the accident tank and the target tank. The fragment velocity attenuation is small when the accident tank is close to the target tank. On the contrary, as the distance between the tanks increases, velocity attenuation becomes more pronounced. The magnitude of the fragment drag is mainly related to the aerodynamic coefficients (lift coefficient and drag coefficient), in which the drag coefficient mainly depends on the shape of the fragments and surface roughness. Liu and Deng [21] showed that the drag coefficient of the end-cap fragments is uniformly distributed in the range of [0.8, 1.1]; the lift coefficient mainly depends on the shape of the fragments and the windward angle, and Nguyen et al. [19] considered that the lift coefficient of the end-cap fragments is uniformly distributed in the range of [0.351, 0.468].

Although the aerodynamic coefficient has some effect on the projection distance and velocity of the fragment, the research of Pula et al. [22] pointed out that this effect is limited. Therefore, in this paper, the initial velocity of the fragment was taken as the impact velocity, and the velocity loss in the projection process was ignored so that the fragment impacted the tank with a velocity slightly higher than the actual velocity, representing the most unfavorable case. Meanwhile, the distance between the fragment and the tank was set close enough to save calculation time.

The finite element model includes four parts: tank, explosive fragment, petroleum, and air, as shown in Fig. 4. The tank and explosive fragment were established with Shell163 elements. Air and petroleum were established with Euler elements; the Arbitrary Lagrangian-Eulerian multimatter coupling algorithm was used. To balance calculation accuracy and efficiency, the air domain dimensions were extended outward by 2.5 m depending on the radius of the tanks and 1.5 m above the tanks. The fluid–structure interaction relationship between the four parts was defined by model keywords, and specific settings are shown in Ref. [23]. The interaction between the explosive fragment and the tank was defined as erosive contact [24], maintaining a certain gap between them to prevent initial penetration. Finally, in order to reflect the dynamic response characteristics of the tank structure comprehensively and to ensure the universality and representativeness of the research conclusions, the impact location was kept in the middle of the tank wall, fixed constraints were applied to the bottom surface of the tank [12], and no-reflection boundary conditions were set on the outer surface of the air domain.

Selecting a reasonable mesh size is especially important to produce accurate results. In order to ensure the fluid–structure interaction effect, this paper kept the same mesh size for each part of the model and compared four mesh sizes of 0.2 m, 0.3 m, 0.4 m, and 0.5 m. Figure 5 gives the displacement curves of the impact center varying with different mesh sizes for the partially filled tank impacted by the end-cap fragment at 65 m/s. It can be seen that the displacement values gradually stabilize when the mesh size is less than 0.3 m, whereas the calculation time for a mesh size of 0.2 m is approximately twice as long as for 0.3 m. Therefore, the mesh size of the finite element model was determined to be 0.3 m.

where $σy$ is the flow yield stress, MPa; $C$⁠, $P$⁠, and $β$ are the material constants; $σ0$ is the initial yield stress, MPa; $εPeff$ is the effective plastic strain; $ε˙$ is the effective plastic strain rate; and $EP$ is the plastic hardening modulus.

where $C0−C6$ are the material constants.

The simulation of the experiments conducted by Chen et al. [10] on the penetration of small tanks by pointed bullets was carried out to verify the accuracy of the finite element simulation. The exact model dimensions and boundary conditions of the experiments were used. The pointed bullet and small tank material were Q235 and modeled using the Cowper–Symonds model. The finite element model is shown in Fig. 6.

As shown in Table 8, the residual velocities of the pointed bullet are compared for the seven typical working conditions in the experiment, and the deformation patterns of the tank are compared in Fig. 7. The deformation patterns of the two small tanks are similar. The residual velocities of the pointed bullet are closer, with a maximum error of 2.8%. These results show that the modeling approach is effective and that the Cowper–Symonds model is able to simulate the mechanical behavior of steel under impact well and can be applied to analyze the dynamic response of large steel tanks impacted by fragments.

The proportion of liquid filling in the tank height was defined as the filling coefficient φ. Tanks with volumes of 5000 m³, corresponding to φ = 0 (empty tank) and φ = 0.5 (half-filled tank), impacted by the end-cap fragment with a velocity of 65 m/s were considered. These scenarios represented the impact of fragments on tanks without and with inner liquid content, respectively.

The displacement distribution of the tank impacted by the end-cap fragment in the two scenarios is shown in Fig. 8. Without the inner liquid, the deformation of the tank is mainly concentrated in the triangular area defined by points A, B, and D. The maximum deformation occurs at the impact center, point C. The tank is overall dented, with symmetrical fold deformations in the dented region, where AC and BC are depression folds; EC, FC, and the lines AD and BD, which demarcate the deformation regions, are convex folds.

To further analyze the deformation, Fig. 9 shows the radial and circumferential strain–time curves at point-1 and point-2 located at the depression and convex folds of the empty tank, respectively. It can be seen that the deformation at point-1 is predominantly radial. In contrast, point-2 is predominantly circumferential. This indicates that the deformation is caused by the circumferential and radial stresses resulting in the tank wall alternating between tensile and compressive states during the impact process. The depression folds in the deformed region of the tank are caused by high tensile forces, while the convex folds are caused by high compressive forces, which agree with the results of Li et al. [13].

With the inner liquid, as shown in Figs. 8 and 10, the deformation of the tank is concentrated at the center of the impact, and the degree of the dent is less than that of the empty tank; no significant folds are observed in the deformation region. The reason is that the wall of the tank is concave inward after impact, resulting in the inner liquid generating a reaction force on the tank wall, then limiting the development of deformation, which is called the “added mass” effect by Jones and Birch [30]. Furthermore, noticeable sloshing and splashing of the inner liquid can be observed near the impact location. However, the sloshing is not violent due to the constraints of the tank and the inertia of the inner liquid.

For a more detailed analysis of the effect of the inner liquid on the dynamic response of the tank, the acceleration–time curves of the tank wall node are presented in Fig. 11. Both curves show significant negative acceleration peaks at the beginning of the impact. Without the inner liquid, the acceleration curves exhibit large fluctuations with more peaks and valleys, indicating that the acceleration change is violent and gradually stabilizes after 0.1 s. In contrast, with the inner liquid, the fluctuations in the acceleration curves are smaller and stabilize faster than those in the empty tank. Clearly, the inner liquid significantly improves the impact resistance of the tank and effectively limits the development of tank wall deformation.

To specifically analyze the impact process of the end-cap fragment on the tank, the velocity variation curves and the impact force–time curves of the tank wall node and the fragment node, with and without inner liquid, are shown in Figs. 12 and 13, respectively. There are significant nonlinear characteristics of both the velocity and impact force curves, which are oscillating. The velocity curve crosses several times, and there are three prominent peaks in the impact force curve. These results indicate multiple collisions between the fragment and the tank wall during impact. Moreover, due to the mass and inertial resistance of the inner liquid, the peak impact force is much greater with the inner liquid than without it.

In addition, the impact process goes through approximately three stages, as shown by the velocity variation curve. Stage 1 is the initial collision (without inner liquid: 0.00–0.04 s, with inner liquid: 0.00–0.02 s). The fragment collides with the tank wall for the first time, and the impact force curve increases, reaching the first peak at 0.012 s. At this time, the kinetic energy of the fragment is heavily dissipated, resulting in rapid velocity decay. Without the inner liquid, the tank wall node velocity increases quickly and exceeds the fragment velocity for a short time. With the inner liquid, the fragment velocity decay increases due to the inertial resistance provided by the inner liquid, and the tank wall node gains a smaller velocity. Subsequently, the velocities of the fragment and the tank wall decrease, and the impact force curve enters a decreasing phase.

Stage 2 is the crushing and collision (without inner liquid: 0.04–0.22 s, with inner liquid: 0.02–0.18 s). The trend of the velocity decay of the fragment and the tank wall slows compared to that in stage 1, with multiple crush collisions of the fragment against the tank wall. The impact force curves reach second peaks at 0.072 s and 0.068 s, respectively. Without the inner liquid, the bottom plate and other nonimpact area shell plates constrain the tank wall node, resulting in a lower velocity than that of the fragment. However, with the inner liquid, the fragment velocity decays more rapidly, causing the wall node velocity to exceed that of the fragment. When the velocities of the fragment and the tank wall decrease to 0 m/s, the rebound effect of the tank wall, since the fragment does not penetrate the tank, causes the two to obtain the reverse velocity. The impact force curves reach the third peak at 0.208 s and 0.164 s, respectively.

Stage 3 is the separation flight (without inner liquid: 0.22–0.40 s, with inner liquid: 0.18–0.40 s). During this stage, the fragment detaches from the tank wall and moves in the opposite direction at a constant velocity, while the impact force gradually decreases to 0 kN. When the tank wall stops rebounding, under the damping effect, the tank wall velocity gradually returns to 0 m/s, marking the completion of the impact process. It is evident that the inertial resistance provided by the inner liquid reduces the impact duration, leading to shorter tank contact with the fragment and decreased energy absorption of the tank.

The 5000 m³ tank was used as the target tank, and the velocity of the end-cap fragment was set to 65 m/s. The four filling coefficients, φ = 0 (empty tank), φ = 0.3 (partially filled tank), φ = 0.5 (half-filled tank), and φ = 0.8 (close to full tank), were selected.

The damage to the tank caused by the fragment was characterized by the magnitude of the displacement at the impact center in the paper. The displacement distribution of the tank with different filling coefficients impacted by the end-cap fragment is shown in Fig. 14. The deformation value and area of the tank gradually decrease with increasing filling coefficient. Additionally, no deformation of the tank wall occurred in the area covered by the liquid level, which is attributed to the lateral pressure of the liquid limiting the development of deformation.

The displacement curves of the tank impact center with different filling coefficients are given in Fig. 15. There is a significant negative correlation between the impact center displacement and the filling coefficient. Additionally, a notable sharp decrease in the impact center displacement, reaching an amplitude of 1.90 m, is observed as the filling coefficient increases from 0 to 0.3. However, the displacement curve exhibits an inflection point and subsequently shows a gradual decreasing trend, with further increases in the filling coefficient. Thus, the vacant tanks in tank farms can be filled with a small amount of liquid to keep them at a low liquid level (0–30% filled) to increase their resistance significantly.

The deformation energy–time curves of the end-cap fragment are shown in Fig. 16. From Fig. 13, the peak impact force is greatest at the initial collision, which largely determines the deformation volume of the fragment. Figure 16 shows that when the filling coefficient is increased from 0 to 0.3, the change in the fragment deformation energy is not obvious, while the deformation energy increases sharply when the filling coefficient is increased to 0.5, and the fragment deformation energy is maximized when the filling coefficient is 0.8. The reason is that at the moment of impact when the impact center is above the inner liquid surface, only the tank wall directly undergoes the impact load, while when the impact center is below the inner liquid surface, the inner liquid and the tank wall undergo the impact load simultaneously. Therefore, it can be concluded that when the impact center is below the inner liquid surface, the inner liquid can provide more direct inertial resistance. The higher the liquid level in the tank is, the greater the safety is when fragments impact it and the lower the impact damage is.

The filling coefficient of the tank is uncertain when an accident occurs, so according to the relative position of the impact center and the inner liquid surface, this paper selected three typical conditions: no inner liquid, the impact center located above the inner liquid surface, and the impact center located below the inner liquid surface. The filling coefficient was set to φ = 0 (empty tank), φ = 0.3 (low liquid level), and φ = 0.8 (high liquid level), ensuring that the impact location was situated in the middle of the tank, corresponding to working conditions I, II, and III in Fig. 17, respectively. Subsequent analyses were conducted based on this setup.

Tanks with volumes of 2000 m³, 3000 m³, 5000 m³, and 10,000 m³ impacted by the end-cap type fragment with a velocity of 65 m/s were considered to determine the influence of tank volume on the impact response.

The displacement distribution of the tank with different volumes impacted by the end-cap fragment is shown in Fig. 18. The curves of the tank impact center displacement varying with the volume are shown in Fig. 19. The deformation value and area of the tank gradually decrease with increasing volume under working conditions I and III. The curve in Fig. 19 shows a decreasing trend. The most significant reduction in displacement is achieved when the tank volume is increased from 3000 m³ to 5000 m³, with reductions of 14.9% and 27.3%, respectively. Moreover, when the tank volume increases from 5000 m³ to 10,000 m³ under working condition II, the impact center displacement increases instead, and the curve in Fig. 19 shows a downward and then upward increasing trend. The reason is that as the volume increases, the structural resistance increases while also increasing the distance at which fragment impact affects the dome roof, bottom plate, and inner liquid surface. This effect is more significant when the tank volume increases to 10,000 m³, meaning that the time at which the inner liquid limits the 10,000 m³ tank wall deformation will lag behind that of the 5000 m³ tank, resulting in increased displacement.

The velocity decay of the end-cap fragment impacting the 5000 m³ and 10,000 m³ tanks under working condition II is shown in Fig. 20. The velocity of the fragment impacting the 10,000 m³ tank decays more rapidly before 0.028 s. However, after 0.028 s, the inner liquid in the 5000 m³ tank begins to exert a limiting effect, causing the velocity of the fragment impacting the 5000 m³ tank to decrease to 0 m/s earlier. This indicates that the 5000 m³ tank had a shorter contact time with the fragment and smaller displacement values, consistent with the above results. Therefore, it is concluded that the larger the tank volume is, the lower the damage under working conditions I and III (i.e., empty and high liquid level). However, the damage increases in working condition II (i.e., low liquid level) after the volume exceeds 5000 m³. In practical engineering, it is recommended to use tanks with a volume exceeding 5000 m³ when the stored liquid reaches a higher liquid level (50–80% filled) in the selected tank; conversely, at low liquid level (0–30% filled), tanks with volumes less than 5000 m³ are recommended.

The fragment velocity is an essential factor affecting the dynamic response of tanks. In this subsection, the 5000 m³ tank was used as the target tank, and the end-cap fragment velocities were set to 65 m/s, 75 m/s, 85 m/s, and 95 m/s.

The displacement distribution of the tank impacted by the end-cap fragment with different velocities is shown in Fig. 21. The deformation value and area of the tank increase with the fragment velocity under all working conditions. This is attributed to the higher kinetic energy of fragments at higher velocities, resulting in increased tank damage.

To specifically analyze the sensitivity of the tank to the fragment velocity, curves depicting how the impact center displacement and tank deformation energy varying with the fragment velocity are presented in Fig. 22. It is observed that displacement and deformation energy exhibit linear relationships with the fragment velocity across all working conditions. As the fragment velocity increases, so do the displacement and deformation energy of the tank. Furthermore, it is noteworthy that the slope of the curve in Fig. 22 is the largest for working condition I and the smallest for working condition III. This indicates that the higher the liquid level is, the lower the sensitivity of the tank to the fragment velocity. In other words, the more liquid is stored, the greater the impact resistance of the tank. Given the uncertainty of explosive fragment velocities, it is recommended that empty or low-liquid level tanks (0–30% filled) be avoided to reduce the probability of damage to tanks.

The impact angle influences the area of fragments in contact with tanks. Considering that fragments may impact tanks from any direction, three typical impact angles were chosen for this subsection. The end-cap fragment was set at a velocity of 65 m/s to impact the 5000 m³ tank at angles of γ = 0 deg (frontal impact), γ = 45 deg (diagonal impact), and γ = 90 deg (vertical impact). The relative positions of the fragment and the tank are depicted in Fig. 23.

The displacement distribution of the tank impacted by the end-cap fragment with different impact angles is shown in Fig. 24. As the impact angle increases from 0 deg to 90 deg, the deformation values of the tanks decrease and then increase under working conditions I and II, but decreases gradually under working condition III. In addition, the tank deformation value reaches a maximum when γ = 90 deg under working condition I, whereas the deformation value reaches a maximum when γ = 0 deg under working conditions II and III. Figure 25 shows the impact process of the end-cap fragment on the tank under working condition I. Figure 25 shows that when γ = 45 deg, the fragment is significantly flipped and detached from the tank wall. When γ = 90 deg, the fragment undergoes severe curling deformation due to the structural characteristics of the thin wall. However, when γ = 0 deg, the fragment neither flipped nor curled.

To analyze the reason for this, the impact force–time curves with different impact angles are shown in Fig. 26. The peak impact force when γ = 0 deg is greatest for all working conditions. In working condition I, when γ = 45 deg, the peak impact force is the lowest and the fragment earliest detachment of the tank wall; the tank absorbs the least amount of energy; when γ = 90 deg, although the end-cap fragment curling deformation occurs, the contact area with the tank wall increases, and thus, the friction increases, resulting in fragment difficulty detaching from the wall; the tank absorbs the most energy, so the deformation value of the tank is the largest. In working condition II, due to the inertial resistance of the inner liquid to shorten the impact duration, the detachment times of the fragments from the tank wall when γ = 90 deg and γ = 0 deg are closer compared to the working condition I. The damage to the tank impacted by the curled fragment is smaller at this time, so the deformation of the tank is the largest when γ = 0 deg. In working condition III, the inner liquid provides more direct inertial resistance, and the end-cap fragment is severely curled and deformed within a short time when γ = 90 deg and detached from the tank wall, and its damage on the tank is the smallest at this time.

From the perspective of fragment deformation energy, because at γ = 45 deg (oblique impact), the fragment detaches from the tank wall, only the γ = 0 deg and γ = 90 deg conditions are discussed here. Figure 27 shows the deformation energy curves for the end-cap fragment with different impact angles. It is observed that the fragment deformation energy when impacting the tank at γ = 90 deg far exceeds that at γ = 0 deg under all working conditions, with a maximum of 2 orders of magnitude. This indicates that the end-cap fragment undergoes severe deformation when impacted at γ = 90 deg, which is consistent with the above results. It is also noted that the deformation energy of the end-cap fragment tends to increase with the inner liquid level in the tank, supporting the conclusion that the higher the liquid level in the tank is, the greater the safety when impacted by fragments.

Therefore, the conclusion can be reached that the peak impact force of the end-cap fragment is the greatest when γ = 0 deg (frontal impact); as the vertical impact angle increases, the fragment impact mode changes from initial frontal impact to flip detachment and finally to curling deformation; when there is an inner liquid at the impact center, due to the direct inertial resistance of the inner liquid, the end-cap fragment is severely curled and detached from the tank wall in a short time when impact at angles of γ = 90 deg, and the damage on the tank is smallest at this time. It is recommended that the axial direction of horizontal tank heads be perpendicular to the radial direction of large steel tanks. This configuration ensures that large steel tanks are more likely to be impacted by plate fragments or nonfrontal impacts of end-cap fragments during an explosion. Additionally, using barrier nets or similar methods is advised to further reduce the probability of damage to large steel tanks.

This paper investigated the dynamic response of large steel tanks impacted by explosive fragments using the finite element method. The typical impact processes and results were analyzed, and the damage laws of tanks under various filling coefficients, volumes, fragment velocities, and impact angles were also examined. The following conclusions are obtained:

1. The depression and convex folds in the deformation region of the tank are caused by high tensile and compressive forces, respectively. Multiple collisions occurred between the fragment and the tank wall during impact, and the impact process involved approximately three stages: initial collision, crushing and collision, and separation flight. The inertial resistance provided by the inner liquid shortens the impact duration and mitigates tank deformation.

2. With the same fragment impact, the impact center displacement shows a fast and then slow reduction trend as the liquid level height increases, so the higher the liquid level in the tank is, the greater the impact resistance of the tank. In addition, the sensitivity of tanks to fragment velocity decreases as the liquid level height increases, and tanks with high liquid levels offer greater safety in accidents due to the uncertainty of the explosive fragment velocity.

3. The larger the volume is, the greater the impact resistance when the tank is empty or at a high liquid level. When the tank is at a low liquid level, the volume increases to 10,000 m³; due to the increased distance that fragment impact affects the dome roof, bottom plate, and inner liquid surface, the impact resistance of the tank decreases instead.

4. The peak impact force of the end-cap fragment is the greatest when γ = 0 deg (frontal impact). Fragments are flipped and curled at γ = 45 deg (oblique impact) and γ = 90 deg (vertical impact), respectively. As the vertical impact angle increases from 0 deg to 90 deg, the fragment impact mode changes from initial frontal impact to flip detachment and finally to curling deformation. Due to the direct inertial resistance of the inner liquid, the end-cap fragment is severely curled and detached from the tank wall in a short time when γ = 90 deg (vertical impact), and the damage on the tank is smallest at this time.

• Natural Science Foundation of Heilongjiang Province Joint Guidance Program (No. LH2020E018; Funder ID: 10.13039/501100005046).

The authors attest that all data for this study are included in the paper.

$a$ =

first-order volume correction

$C$ =

material constant of Cowper–Symonds formula

$C0−C6$ =

material constant of air equation of state

$E$ =

Young's modulus, MPa

$Em$ =

tank explosion energy, MJ

$EP$ =

plastic hardening modulus

$Et$ =

tangent modulus, MPa

$E0$ =

initial relative volume, J/m³

$K$ =

material constant of petroleum equation of state

$ke$ =

energy coefficient

L₁ =

extension length of the long end-cap fragment, m

L₂ =

length of the plate fragment, m

L₃ =

width of the plate fragment, m

$Ms$ =

mass of the explosive fragments, kg

$P$ =

material constant of Cowper–Symonds formula

$P0$ =

environmental pressure, MPa

$P1$ =

tank explosion pressure, MPa

R₁ =

head radius of the long end-cap fragment, m

R₂ =

head radius of the end-cap fragment, m

$S1−S3$ =

material constant of petroleum equation of state

$vmax$ =

maximum velocity of the end-cap explosive fragment, m/s

$vmin$ =

minimum velocity of the end-cap explosive fragment, m/s

$v0$ =

velocity of the explosive fragment, m/s

$Vec$ =

volume of the end-cap explosive fragment, m³

$Vs$ =

volume of the explosive fragment, m³

$Vt$ =

volume of the tank, m³

$V0$ =

initial relative volume

$β$ =

material constant of Cowper–Symonds formula

γ =

impact angle, deg

$γ0$ =

Gruneisen constant

$ε˙$ =

effective plastic strain rate

$ε1$ =

thickness of the head, mm

$ε2$ =

thickness of the cylinder, mm

$εPeff$ =

effective plastic strain

$λ$ =

a scaling factor of formula proposed by Mébarki et al.

$μ$ =

relative density

$ξ$ =

specific heat capacity, J/(kg °C)

$ρ$ =

current density of the material, kg m⁻³

$ρs$ =

density of the explosive fragment, kg m⁻³

$ρ0$ =

initial density of the material, kg m⁻³

$σy$ =

flow yield stress, MPa

$σ0$ =

initial yield stress, MPa

φ =

filling coefficient