Introduction

An unexpected bomb explosion within or in the near vicinity of a commercial or military structures can cause severe damage on the building’s external and internal structural frames. Generally, these external structures are strengthened using reinforced concrete walls. However, the entrances cannot be completely sealed off under consideration of free pass. Thus, a variety of blast resistant doors that can withstand explosion impact were designed for various operational requirements. Such a blast resistant door could be constructed as stiffened steel plate, reinforced concrete structure or even a sandwich structure.

The design and analysis of steel plate with different stiffeners subjected to varied blast loads require a detailed understanding of blast phenomena and the dynamic response of various structural elements. In the design of blast resistance structure, it is well recognized that two important design considerations, including structural dynamic analysis and blast aerodynamics, are involved. Especially, the dynamic response to explosive loading should be taken into consideration more critically during design stage. On the other hand, the identification of the explosive loading characteristics such as peak pressure and loading duration of blast wave is a prerequisite to mechanical design and analysis. Although some studies on blast loading model are available in literatures, quantification of the blast parameters has been a highly complicated task with advanced techniques.

Literature

The literature on blast load resistant steel plates is quite limited. Further, most of the available literature is either classified in nature or circulated among a few focused groups. There are a few experimental and/or numerical studies that are carried out by the previous investigators are reported in [1–6]. The studies characterize different failure modes and the relationship between deformation and tearing of clamped blast loaded plates. Numerical results of nonlinear analysis on both stiffened and un-stiffened plates and nonlinear dynamic response and failure of clamped circular plates have been described [7, 8]. Experimental and numerical studies have been conducted on the response of quadrangular stiffened plates subjected to uniform blast loads and the case of localized blast loading have been analyzed also [9, 10]. Previous studies reported the effect of thickness on deformation of plates subjected to blast loading and numerical results for square clamped, stiffened steel plates subjected to blast loading [11, 12]. The failure modes were first defined for the case of impulsively loaded clamped beams [13].

In the past there have been some experimental studies that were conducted on plates subjected to blast loads. The experimental results for the prediction of tearing of clamped circular plates subjected to uniformly loaded air blast were reported [14]. The scaling of metal plates subjected to impact were investigated for application of geometrical scaling [15]. The researchers also studied the scaling aspects of quadrangular plates subjected to localized blast loads. A study on close-range small blast loading on a steel disc and the behavior of aluminum foam panels subjected to close-range blast loading were reported in previous studies [16, 17]. The validity of the results presented so far is limited to small plate geometries and loading conditions. Hence, there is a need for numerical studies/simulations for larger explosions. These numerical studies and simulations provide vital information on the consequences of unexpected large explosions. Further, these results provide the necessary precautions to be taken for conducting safe experiments on structures subjected to explosions. Both numerical and experimental investigations on the blast resistance of stiffened door structures were conducted earlier [18].

Accordingly, most of the blast door structures were designed as per the guidelines provided in the military technical manual UFC 3-340-02(2008) [19], NAVFAC P-397 [20] and the analysis work was performed following the official design criteria. Essentially, the structural stiffness of blast door was estimated under different blast loads and then the blast resistance was evaluated according to the material strength, making sure that the structure could meet the requirement when subjected to explosion.

Air Blast

The typical characteristic of a shock wave is to create an instantaneous pressure rise and an exponential fall. The parameters of interest in the damage process are the peak overpressure (i.e. the pressure above the atmospheric pressure), the positive duration and impulse with respect to the scaled distance. The negative phase of the blast wave is generally ignored in the analysis.

The characteristics of a blast for a variety of explosives can be obtained from empirical charts or formula in military technical manual UFC 3-340-02 or statistical analysis [3, 19, 21] during design procedure. As an example in theorem [3], the peak overpressure Ps and duration of positive pressure ts can be expressed as the function of the scaled distance Z and explosive charge weight W, respectively. That is,

$$ \frac{{{\text{P}}_{\text{s}} }}{{{\text{P}}_{\rm o} }} = \frac{{808[1 + \left(\frac{Z}{4.5}\right)^{2}] }}{{\sqrt {1 + \left(\frac{Z}{0.048}\right)^{2} } \cdot \sqrt {1 + \left(\frac{Z}{0.32}\right)^{2} } \cdot \sqrt {1 + \left(\frac{Z}{1.35}\right)^{2} } }} $$
(1)
$$ \frac{{{\text{t}}_{\text{s}} }}{{{\text{W}}^{\frac{1}{3}} }} = \frac{{980[1 + \left(\frac{Z}{0.54}\right)^{2}] }}{{\sqrt {1 + \left(\frac{Z}{0.002}\right)^{3} } \cdot \sqrt {1 + \left(\frac{Z}{0.74}\right)^{6} } \cdot \sqrt {1 + \left(\frac{Z}{6.9}\right)^{2} } }} $$
(2)

The equation P(t) is often simplified with a triangular pressure–time curve [22].

$$ P(t)\; = \;P_{ \hbox{max} } \left(1 - \frac{t}{{t_{d} }}\right) $$
(3)

Conventional high explosives tend to produce different magnitudes of peak pressure. As a result, the environment produced by these chemicals will be different from each other. In order to have a basis for comparison, various explosives are compared to equivalent TNT values. With a scaling parameter, it is possible to calculate the effect of an explosion as long as the equivalent weight of charge in TNT is known,

$$ Z = \frac{\text{R}}{{{\text{W}}^{\frac{1}{3}} }} $$
(4)

where R is the distance from the detonation and W is the equivalent weight of TNT. It is very difficult to simulate a situation that can fully and adequately represent the ground reality. So we employ a simpler model in which the blast waves propagate out wards as if they travel in the open air and not being affected by the surrounding objects. As a result, the simulated damage that occurs on the object may not be fully identical to the damage due to real-life explosion.

Problem Definition

The weight of the TNT (explosive) applied is varied from 100 to 500 kg with an increment of 100 kg at a stand-off distance 11 m. It is assumed that plate is fully fixed in all directions. The problem addressed in the present study has the following specifications. ASTM A515 GRADE 50 Material for Cover Plate–Young’s Modulus, E = 200 × 109 N/m2, Poisson ratio, ν = 0.3, Density, ρ = 7,830 kg/m3, Static yield stress, σyield = 265 × 106 N/m2, Ultimate Tensile strength, σut = 492 × 106 N/m2. Blast door with dimensions 5.142 × 2.56 m with thickness 20 mm is taken and three different stiffeners T, I and HAT are chosen for stiffening of the cover plate of the door structure.

Two separate materials for cover plate and stiffeners are chosen [18], in which the materials are chosen from a manual which is specially designed by Department of the Army, “Structures to resist the effects of accidental explosive,” UFC 3-340-02, Washington, D.C. [19]. The FEM analysis is carried-out using ABAQUS 6.6-3. ASTM A36 Steel Material for Stiffener-Young’s Modulus, E = 200 × 109 N/m2, Poisson ratio, ν = 0.26, Density, ρ = 7,800 kg/m3, Static yield, σyield = 248 × 106 N/m2, Ultimate Tensile strength, σut = 400 × 106 N/m2.

The plate is subjected to a blast load that varies as a function of time and the Fig. 1 shows only the positive peak overpressure and its decline with respect to time. This is because the positive phase is more important in studies of blast wave effects on structures due to its high amplitude of the overpressure and the concentrated impulse. The pressure is applied on the face of the plate, where as the stiffeners are attached on the wake side of the plate. Such a pressure load will place the outer layers of the stiffeners in tension (Tables 1, 2).

Fig. 1
figure 1

Positive duration of load

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Table 1 Different stiffener shapes
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Table 2 Characteristics of blast wave with different W-R combinations
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Results and Discussion

The duration of the positive phase of the blast is 6 ms. The 3D computational domain of the cover plate with stiffeners is modeled. The deflection time-history of the cover plate subjected to blast loading is obtained in the dynamic explicit analysis using ABAQUS. The corresponding results are furnished in the following Tables 3, 4, 5, 6 and 7.

Table 3 Comparison of the maximum stress and deflection for door structure with different stiffener sizes and shapes
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Table 4 Comparison of the maximum stress and deflection for door structure with different stiffener sizes and shapes
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Table 5 Comparison of the maximum stress and deflection for door structure with different stiffener sizes and shapes
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Table 6 Comparison of the maximum stress and deflection for door structure with different stiffener sizes and shapes
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Table 7 Comparison of the maximum stress and deflection for door structure with different stiffener sizes and shapes
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From Tables 3, 4, 5, 6 and 7 summarizes the maximum principal stress for door structures with different stiffener shapes and stiffener sizes. For 100 kg blast load the maximum tensile stresses generated at critical regions are in the range of 29.08–152.58 MPa for cover plate and 328.03–655.66 MPa for stiffener respectively. The stresses are in the range of 144.3–258.11 MPa for cover plate and 562.89–1167.09 MPa for stiffener respectively for 200 kg of Blast load. The stresses vary in the range of 198.1–360.15 MPa for cover plate and 774.64–1548.87 MPa for stiffener respectively for 300 kg of Blast load. Similarly, stresses are in the range of 247.97 MPa to 415.5 MPa for cover plate and 969.87–2,010.87 MPa for stiffener respectively for 400 kg of Blast load. Stresses are highest in the range of 296.14–493.15 MPa for cover plate and 1155.44–2395.71 MPa for stiffener respectively for 500 kg of Blast load.

The effect of stiffener shape and size on the rigidity and stress of door structure can further be investigated from Figs. 2, 3, 4, 5 and 6. It is obvious that increasing the stiffener size can remarkably increase the rigidity of door structure and reduce the stress generated in stiffeners. HAT shape stiffener has larger area moment of inertia resulting in higher flexure stiffness to resist blast loads.

Fig. 2
figure 2

100 kg of Dynamic blast load

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Fig. 3
figure 3

200 kg of Dynamic blast load

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Fig. 4
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300 kg of Dynamic blast load

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Fig. 5
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400 kg of Dynamic blast load

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Fig. 6
figure 6

500 kg of Dynamic blast load

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Results of Stress Variation in Cover Plate with H-shape Stiffener of 8.5 mm Thick Subjected to Blast Load

In addition, the predicted peak stresses for stiffeners are apparently higher than the yield strength and ultimate tensile strength of the materials. Also the stress induced on the central region of the cover plate is lower than its strength indicating the initiation of local yielding at these locations, rather than the whole structure.

Mid-Point Displacement at Center Node

The following graphs show the maximum displacements plots at center node for T stiffener, I stiffener and HAT stiffener of 5.5, 6.5, 7.5 and 8.5 mm thickness.

Observations for Mid-Point Displacement at Center Node

Maximum mid-point displacement at the center can be investigated from Figs. 12, 13, 14, 15 and 16. It is noted that the maximum displacements are in the range of 8.71–23.6 mm respectively for 100 kg of Blast load, displacements are in the range of 14.96–33.86 mm respectively for 200 kg of Blast load, displacements are in the range of 20.58–55.24 mm respectively for 300 kg of Blast load, displacements are in the range of 25.77–58.36 mm respectively for 400 kg of Blast load, displacements are in the range of 30.7–69.52 mm respectively for 500 kg of Blast load. The increase in thickness of the stiffener results in the decrease in mid-point displacement. HAT shape stiffener has a larger area moment of inertia yielding higher flexure stiffness to resist blast loads.

Observations for Strain Energy Absorbed

Strain Energy absorbed can be investigated from Figs. 12 and 13. It is noted that the maximum strain energy is in the range of 4.53–5.724 kN-m respectively for 100 kg of Blast load, strain energy is in the range of 7.769–10.21 kN-m respectively for 200 kg of Blast load, strain energy is in the range of 14.71–19.33 kN-m respectively for 300 kg of Blast load, strain energy is in the range of 23.06–30.3 kN-m respectively for 400 kg of Blast load, strain energy is in the range of 32.74–43.01 kN-m respectively for 500 kg of Blast load (Figs. 7, 8, 9, 10, 11]. The increase in thickness of the stiffener results in the decrease in strain energy absorbed. HAT shape stiffener has a larger area moment of inertia yielding higher flexure stiffness to resist blast loads (Figs. 12, 13, 14, 15, 16, 17, 18).

Fig. 7
figure 7

Blast load of 500 kg of TNT

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Fig. 8
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Blast load of 400 kg of TNT

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Fig. 9
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Blast load of 300 kg of TNT

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Fig. 10
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Blast load of 200 kg of TNT

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Fig. 11
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Blast load of 100 kg of TNT

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Fig. 12
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100 kg of Dynamic blast load

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Fig. 13
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200 kg of Dynamic blast load

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Fig. 14
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300 kg of Dynamic blast load

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Fig. 15
figure 15

400 kg of Dynamic blast load

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Fig. 16
figure 16

500 kg of Dynamic blast load

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Fig. 17
figure 17

Energy absorbed by HAT shape stiffener

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Fig. 18
figure 18

Strain energy absorbed

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Validation of Results

To establish the validity of numerical results of the present investigation, predicted values of displacements and maximum principal stresses were compared with published numerical results of earlier reports [18]. The published numerical results that are taken as reference for comparison, in this study, were chosen such that in turn they have been validated with prior experimental data. The following Table 8 presents the comparisons of cover plate displacements, maximum principle stresses in the cover plate and stiffener for different thicknesses of the stiffener. The deviation of the present numerical values from the reference values are presented as percentage deviation from the reference values. The deviation is in the range of 2–8 % in the case of displacements. However, the deviation in case of maximum principal stresses is in the range of 5–20 %. In general, the measurement of deflections will be done either by micro-meter dial gauges or displacement transducers (LVDT). These measurements can be very accurate. In the case of displacements, the numerically predicted values are very close to the experiments results. But in the case of stress measurements, strain is measured with strain gauges, and stress is estimated. Therefore, the measurement of stress is not as much accurate as displacement measurement. The predicted values of maximum principle stresses in this investigation were comparable in the range of 5–20 % with reference data [18]. This comparison establishes the predictive capabilities of the present modeling and analysis. Table 8 shows the comparison and validation of numerical results.

Table 8 Comparison and validation of the results of present investigation with published numerical and experimental results [18]
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Conclusions

In the present investigation, an attempt is made to predict the failure of a fully fixed stiffened cover plate of a blast door, subjected to air blast loading. The numerical analysis is performed using ABAQUS. In the entire analysis, a cover plate (20 mm thick) with T, I and HAT shaped stiffeners is considered with the thickness of the plate and height of the stiffener(140 mm) being constant. The thickness of the stiffeners is varied from 5.5 mm to 8.5 mm with an increment of 1 mm. The blast load is varied from 100–500 kg with an increment of 100 kg.

The blast load is applied based on the Friedlander equation. For the analysis, the pressure is assumed to be uniform within each finite element. However, the variation of the pressure across the length and breadth of plate is considered and is estimated based on spherical shock wave spreading pattern. With respect to time, pressure is assumed to be exponentially decaying. From the validation of numerical results of the present analysis, it is concluded that the methodology employed in this work can adequately predict the mid-point displacements and maximum principal stresses developed in the door structures designed to withstand the blast loads due to unexpected explosion.

Form the analysis of the results, it is observed that the stiffener would absorb the major component of the blast loads, hence reduces the damage to cover plate. It is concluded from the results of dynamic explosive loading, that the blast doors seem to be adequately qualified with a valid protection level. However to avoid substantial failure at critical regions, the door structure should be strengthened with suitable stiffener size and stiffener shapes.

The blast response of stiffened doors with three different stiffeners subjected to constant blast load was examined systematically and successfully. The results of blast load analysis of stiffened blast door structure for stress, mid-point displacements for T, I and HAT stiffeners were compared against each other. The stress levels in the plate are found to be well within the safe limits. Whereas, the maximum stress predicted in the stiffener are very high and are in the increasing order from HAT, I and T shape. It is found that the door structures with HAT stiffener performs better in this application.