Model validationMaterial modelReferring to previous studies18,30,31,32,33, the elastic–plastic material model *MAT_PIECEWISE_LINEAR_ PLATICITY was used to simulate the explosive mechanical behaviour of the polyurea. The stress–strain curves of polyurea at different strain rates (Fig. 10) were obtained using an MTS material testing machine and split Hopkinson tensile bar34. The stress–strain curves of polyurea exhibited significant nonlinearity and strain rate sensitivity. To characterize the strain rate effect of polyurea, the dynamic increase factor (DIF) of the stress was calculated and is plotted in Fig. 11. The DIF is the ratio of the stress at a specific strain rate to the stress at the reference strain rate (0.001 s−1) under the same strain (0.02).Fig. 10Stress–strain curves of polyurea.Fig. 11Dynamic increase factor of polyurea.The Cowper–Symonds model can be used to describe the strain rate effect of polyurea under extreme loads35:$$ {\text{DIF}} = 1 + \left( {\frac{{\dot{\varepsilon }}}{C}} \right)^{\frac{1}{p}} , $$
(1)
where \(\dot{\varepsilon }\) is the strain rate. Calculated by the least squares method, the constants C and p were 650.18 s−1 and 0.98, respectively.The stress–strain curve of polyurea at a strain rate of 0.001 s−1 was input into *MAT_PIECEWISE_ LINEAR_ PLATICITY model, and the Cowper-Symonds model was used to define the strain rate effect of polyurea. The main parameters of polyurea, including the parameters of the Cowper-Symonds model, are presented in Table 1. Fiber-reinforced concrete was described by the *MAT_BRITTLE_DAMAGE model, and the parameters of the fibre-reinforced cement are shown in Table 236. The galvanized steel plate and frame square steel were characterized by the *MAT_PLASTIC_KINEMATIC model. The material parameters of the galvanized steel plate and the frame square steel are shown in Table 337,38.Table 1 Polyurea material parameters.Table 2 Fiber-reinforced cement material parameters.Table 3 Steel material parameters.Numerical simulation modelNumerical simulations of the gas experiment were performed using the finite element software ANSYS/LS-DYNA. The finite element model is shown in Fig. 12, and the load curve method is used to directly apply explosive loads on the blast-resistant plate. During the experiment, there was a significant difference in the explosion overpressures between the upper and lower parts of the blast-resistant plate. In the simulation model, the explosion load was applied to the blast-resistant plate in the upper and lower zones. The pressure curve measured by sensor P1 was adopted for the upper load of the blast-resistant plate, and the pressure curve measured by sensor P2 was adopted for the lower load of the blast-resistant plate. Due to the failure of sensor P2 in the coated plate test, no valid data was collected. However, as mentioned above, the repeatability of the gas explosion test was acceptable. Therefore, the data of sensor P2 in the simulation was the same as that of the uncoated plate test.Fig. 12Finite element model of blast-resistant plate and steel frame.The galvanized steel plate and the square steel frame were meshed by shell elements, while the other structures were meshed using solid elements. The overall grid size of the model was 2.5 cm, and the total number of grid elements was 30,301. The square steel frame was anchored to the fixed frame through steel rods. Therefore, the square steel frame adopted fixed boundary conditions. The fixed bolts of the polyurea-coated blast-resistant plate were not damaged, and thus, bond contact was adopted between the polyurea-coated plate and the square steel frame.Comparison between numerical and experimental resultsThe structural response process of the polyurea-coated blast-resistant plate is shown in Fig. 13. The blast-resistant plate exhibited significant deformation under the explosive load, followed by a certain degree of deformation rebound. The experimental and numerical simulation comparison of the plastic deformation of the polyurea-coated blast-resistant plate is shown in Fig. 14. Due to the supporting effect of the central square steel frame, the blast resistance plate formed two concave deformations between the central square steel component and the boundary square steel.Fig. 13Simulation of structural response process of polyurea-coated plate.Fig. 14Comparison of plastic deformation of polyurea-coated blast-resistant plate.The maximum elastic–plastic deformation of the blast-resistant plate was 5.8 cm at 6.1 ms (Fig. 15). After a period of vibration, the residual deformation of the blast-resistant plate was 3.6 cm. The plastic deformation of the resistance plate measured in the experiment was 3.4 cm, and the relative error between the simulation and experimental values was 5.88%. The experimental results were in good agreement with the numerical simulation results.Fig. 15Maximum displacement time-history curve.Full-scale shelter simulationBlast loadingThe control room of a petrochemical enterprise was adjacent to the liquefied hydrocarbon loading and unloading platform. There was a risk of accidental leakage of liquefied hydrocarbons causing the control room to be exposed to the explosive wave. According to the architectural drawings, the building was a brick–concrete structure with a low resistance to progressive collapse. To reduce the risk of casualties in explosion accidents, a new blast-resistant steel structure shelter was proposed.Based on our previous simulation of liquefied hydrocarbon leakage and explosion39, it was determined that the peak incident overpressure (\(P_{so}\)) of the shock wave at the new shelter was 14.56 kPa, and the positive duration (\(t_{d}\)) was 174.56 ms. The design appearance size of the shelter was 6.2 × 2.8 × 2.7 m. The blasting load of each face of the shelter (Fig. 16) could be calculated by substituting the shock wave and building parameters into the following Formulas (2), (3), (4), (5), (6), (7), (8), (9), (10) 4.Fig. 16Blast loading curves of shelter.Front wallThe peak reflected overpressure (\(P_{r}\)) and the stagnation pressure (\(P_{s}\)) were calculated as follows:$$ P_{r} = \left( {2 + 0.0073P_{so} } \right)P_{so} , $$
(2)
$$ P_{s} = P_{so} + C_{d} q_{o} , $$
(3)
where Cd is the drag coefficient, and \(q_{o}\) is the peak dynamic wind pressure. For closed rectangular buildings, the Cd of the front wall was set as 1.0, and the values for the side wall, roof, and back wall were − 0.4. The effective duration (te) was calculated as follows:$$ {\text{t}}_{{\text{c}}} { = }3S/U, $$
(4)
$$ t_{e} = {{2I_{w} } \mathord{\left/ {\vphantom {{2I_{w} } {P_{r} }}} \right. \kern-0pt} {P_{r} }} = \left( {t_{d} – t_{c} } \right){{P_{s} } \mathord{\left/ {\vphantom {{P_{s} } {P_{r} }}} \right. \kern-0pt} {P_{r} }} + t_{c} , $$
(5)
where S is the clearing distance (m), U is the shock front velocity (m/s), and \({\text{t}}_{{\text{c}}}\) is the reflected overpressure clearing time (s).Side wall and roofThe equivalent peak overpressure (\(P_{a}\)) and rise time (\(t_{r}\)) were calculated as follows:$$ P_{a} = C_{e} P_{so} + C_{d} q_{o} , $$
(6)
$$ t_{r} = L_{1} /U, $$
(7)
where \(C_{e}\) is the equivalent peak overpressure, and L1 is the length of the structural member in the direction of the shock wave (m).Rear wallThe equivalent peak overpressure (\(P_{{\text{b}}}\)) and time of arrival (\(t_{a}\)) were calculated as follows:$$ P_{b} = C_{e} P_{so} + C_{d} q_{o} , $$
(8)
$$ t_{a} = {{\text{L}} \mathord{\left/ {\vphantom {{\text{L}} U}} \right. \kern-0pt} U}, $$
(9)
where \({\text{L}}\) is the building width (m).The rise time (\(t_{r}\)) is calculated as follows:$$ t_{r} = {S \mathord{\left/ {\vphantom {S U}} \right. \kern-0pt} U} $$
(10)
Numerical simulation modelThe shelter consisted of a steel frame and polyurea-coated blast-resistant plates. The main beam and column of the frame were 100 × 100 × 4 mm square steel, and the secondary beam and column were 100 × 50 × 4 mm square steel. The steel grade of the frame beam column was Q345. To facilitate the installation of 1.2 m-long blast-resistant plates, the maximum spacing between the beams and columns was 0.6 m (Fig. 17a). Consistent with the gas explosion testing, 3 mm polyurea was sprayed on the rear face of the blast-resistant plate. The full-scale shelter model is shown in Fig. 17. Galvanized steel plates and steel frames were modelled by shell elements, while other components were modelled using solid elements. For the structural simulation of a mobile refuge chamber under an explosion load37, the mesh size of the shelter model was set to 2.5 cm, and the total number of grid elements was 515,120. The shelter was fixed to the precast concrete foundation through anchoring, and thus, the bottom of the shelter was set as a fixed boundary.Fig. 17Full-scale shelter model. (a) Steel frame. (b) Steel frame and panel.Structural dynamic response analysisDisplacementThe deformation process of the shelter under an explosion load is shown in Fig. 18. The overall deformation of the shelter was the central depression of the wall, and the shelter wall remained intact without cracking. The displacement time-history curves of the centre of each wall are shown in Fig. 19, and the maximum deformation values of the walls are summarized in Table 4. The shelter steel frame showed inward bending deformation, and there were no fractures in the square steel beams and columns (Fig. 20). The displacement curves of the maximum deformation points on each side of the frame are shown in Fig. 21, and the maximum deformation values are listed in Table 4.Fig. 18Deformation process of shelter.Fig. 19Displacement time-history curves of the centre of each wall.Table 4 Simulation results.Fig. 20Deformation process of steel frame.Fig. 21Displacement curves of the maximum deformation points on each side of the frame.During the deformation process of the shelter, the blast-resistant plate did not detach from the steel frame. Therefore, as presented in Table 4, the maximum displacement of the wall was basically the same as the maximum displacement of the frame (except for the right side). The number of horizontal square steel components on the right side was less than that on the left side, and the maximum displacement of the right-side square steel components was basically the same as that of the left-side square steel components. However, the displacement of the right-side wall increased significantly (Table 4), which indicated that the horizontal square steel component mainly affected the deformation of the wall.StressThe stress variations of the shelter frame are shown in Fig. 22, where the upper limit of the scale is the dynamic yield strength of steel (452 MPa)4. According to Fig. 22, only a few front columns had stresses higher than the yield strength of the steel, and the overall frame was in the elastic stage. The stress curves of the front column with the largest deformation are shown in Fig. 23. Three stress points at the bottom (A), middle (B), and top (C) of the column were selected for analysis. The maximum stress at the bottom was 572 MPa, the maximum stress in the middle was 443 MPa, and the maximum stress at the top of was 296 MPa. Since the bottom of the shelter was fixed, the column stress decreased from the bottom to the top, and the maximum stress was located at the bottom of the column.Fig. 22Process of framework stress variation.Fig. 23Stress curves of the front column with the largest deformation.Parametric studyThere are two typical blast shock waves in the explosions of the blast-resistant structures in the petrochemical industry. One was the peak overpressure of 21 kPa with a duration of 100 ms, and the other was the peak overpressure of 69 kPa with a duration of 20 ms4. Based on the peak overpressure and its duration, the wall load of the shelter was calculated by Formulas (2), (3), (4), (5), (6), (7), (8), (9), (10).Under the action of the explosion wave (69 kPa, 20 ms), the cloud map at the moment when the shelter displacement and stress reached the maximum values is shown in Fig. 24. The blast face of the shelter was subjected to the greatest reflected overpressure, and the maximum displacement and stress occurred on the front side. The maximum displacement time-history curve of the shelter is shown in Fig. 25, and the maximum displacement reached 24.1 cm at 19 ms. The maximum stress time-history curve of the shelter is shown in Fig. 26, and the maximum stress reached 783 MPa at 12 ms.Fig. 24Shelter structure response (69 kPa, 20 ms). (a) Maximum displacement (19 ms). (b) Maximum stress (12 ms).Fig. 25Maximum displacement curves of shelters under different loads.Fig. 26Maximum stress curves of shelters under different loads.Under the action of the explosion wave (21 kPa, 100 ms), the cloud map at the moment when the shelter displacement and stress reached the maximum value is shown in Fig. 27. Different from the high explosion load (69 kPa, 20 ms), the maximum displacement of the shelter appeared on the right wall, but the wall was not damaged. The blast-resistant wall load would be transferred to the steel frame, which was the main bearing member of the shelter and directly determined the safety of the shelter. In order to compare the responses of shelter structures under different loads, the maximum displacement and stress of the shelter were uniformly characterized by the maximum displacement and stress of the steel frame. The maximum displacement time-history curve of the shelter is shown in Fig. 25, and the maximum displacement reached 3.72 cm at 15 ms. The maximum stress time-history curve of the shelter is shown in Fig. 26, and the stress reached its maximum value of 634 MPa at 12 ms.Fig. 27Shelter structural response (21 kPa, 100 ms). (a) Maximum displacement (15 ms). (b) Maximum stress (12 ms).The simulation results showed that the shelter was not damaged under the action of two typical explosion waves. The deformation and stress of the shelter under a typical high explosive overpressure (69 kPa, 20 ms) were higher than those under a typical low explosive overpressure (21 kPa, 100 ms). Therefore, the dynamic response analysis of the shelter under a high explosion load (69 kPa) and different durations (5, 10, and 20 ms) was carried out. To compare with the results of subsequent FAE explosion tests, the simulated load also included the maximum load of the FAE explosions. The maximum displacement and maximum stress time-history curves of the shelter under different loads are shown in Figs. 25 and 26, respectively, and the maximum displacement and stress are summarized in Table 5.Table 5 Simulation results.According to Table 5, the structural response of the shelter was closely related to the incident overpressure and duration. The maximum displacement and stress of the shelter under a high incident overpressure significantly decreased with decreasing duration. When the duration decreased from 30 to 5 ms, the maximum displacement decreased from 28.8 to 4.81 cm, and the maximum stress was reduced from 814 to 679 MPa. However, the maximum displacement and stress of the shelter under a high explosion overpressure for a low duration (69 kPa, 5 ms) were still higher than those under a low explosion overpressure for a high duration (21 kPa, 100 ms).
Experimental and numerical study on explosion resistance of polyurea-coated shelter in petrochemical industry
