Investigating the influence of plate geometry and detonation variations on structural responses under explosion loading: A nonlinear finite-element analysis with sensitivity analysis

2.1 Shell buckling

In many engineering constructions, plate and shell structures are used as fundamental components. Throughout their service life, these structures are frequently exposed to complicated loading circumstances that might alter their dynamic properties and result in buckling failure [14]. Furthermore, the existence of fractures significantly affects their quality and degrades their performance. Shell buckling can be caused by a number of things, including blast loading. It is believed that structures are prone to compressive stress, which might result in abrupt failure without any prior indication of deformation [15]. Because of the tremendous pressure that a blast wave creates, shell structures are susceptible to collapsing during an explosion, which can degrade or even destroy their structural integrity. The unequal distribution of pressure over the geometry can lead to increased stress at certain areas in a blasted shell structure, which can soon cause the damaged structure, for example, a plate with a stiffener, to buckle. The formula for shell buckling is provided in Eq. (1):

The critical shear buckling load is determined by multiplying the critical buckling stress by the web area. The elastic modulus, Poisson’s ratio, web depth, and web thickness are represented by the letters E , v , h , and t , respectively. To calculate the shear buckling stress, the shear buckling coefficient ( k v ), which varies depending on whether the web is stiffened or not, is also dependent on whether the entire cross-section or only the web is taken into account. When a plate with a long stiffener is subjected to a compressive stress, σ, it will eventually reach the critical stress and cool down. The plate will proceed to the post-bending stage. A feature of plates is that the applied load tends to increase in tandem with the deflection, resulting in stable post-bending. We define this in Eq. (2):

where the buckling deflection’s amplitude is indicated by the letter w for a plate thickness deflection of the same magnitude. The design takes into account the effects of imperfections and plasticity along with the post-buckling strength of the plates in the elastic range. Note that the post-buckling strength of plates is limited to thin plates. To specify the non-dimensional parameter of slenderness, the following equation was used:

with f y representing the yield stress of the material. A plate’s geometrical slenderness b t is directly proportional to its non-dimensional slenderness parameter, λ p ̅ . For a narrow plate, σ cr ≪ f y (high λ p values), whereas for a stocky plate, σ cr ≫ f y (low λ p values).

2.2 Finite-element analysis (FEA) using the conventional weapons effect method

Through the division of a structure into several small elements, the finite-element method (FEM) may solve a difficult issue [16,17,18]. As a result, the structural response to various applied-load circumstances may be thoroughly analyzed. When a system is in equilibrium, the total energy in it is equal to the sum of the external potential energy and the strain potential energy. This is the fundamental idea of an FEA’s indirect approach to the lowest potential energy. Eqs. (4) and (5) demonstrate the fundamental concept of the FEM.

The total amount of potential energy is expressed as follows:

The total potential energy of the discretized individual element is expressed as follows:

The above equations explain that a system will reach equilibrium when the total potential energy is at a minimum. [F] is the external force vector applied to the structure, while [K] explains the structure’s matrix, and the variable [U] explains the displacement vector. With these, a structure’s behavior can be predicted under a load such as pressure or an external force.

An FEA can be used to document a problem in various cases, such as explosions. When an FEA is combined with the CONWEP method, which is used to calculate the effects of explosions based on the distance and mass of the explosives, the CONWEP method serves to provide initial data on the explosion pressure on the structure and is involved in the FEA model regarding the dynamic response of the structure to the explosion. An explosion is a mechanism consisting of the generation of a shock wave that, upon impacting the unexploded material, activates it through a shock pressure force. Materials that explode through explosions are called explosives or high explosives, such as TNT [19]. Both the relationship between the reflected pressure and the incoming shock wave are crucial for understanding and predicting how a blast will affect a structure when employing the CONWEP method [20]. A surface made up of solid or shell elements, or a collection of predetermined segments, is affected by the load. According to Eq. (7), the pressure p acting on a segment takes into account the pressure wave’s angle of incidence θ , where p i is the incident pressure and p r is the reflected pressure

A complex analysis is needed to determine the equivalent weight of TNT associated with deflagration. Models predicting mid-field deflagration effects are impractical for the characterization of far-field deflagrations and vice versa. Estimations of near-field deflagration effects via TNT equivalence methods generally lack accuracy [21,22,23]. The modified Friedlander Eq. (8) accurately describes the blast pressure–time history at a fixed point:

where the absolute pressure recorded at the point of interest (time t ) following the detonation is represented by P ( t ) , the undisturbed atmospheric pressure is identified by P 0 , the peak overpressure is identified by P s , the blast wave’s arrival time at the point of interest is indicated by t a , the positive phase duration is indicated by t d , and the decay is represented by the b coefficient. Figure 1 illustrates the conventional curve that Eq. (4) describes.

Figure 1

Blast wave pressure–time history (illustration based on the study of Povey [24]).

The pressure–time history is divided into two separate phases, as shown in Figure 1. The positive phase begins at time instant P ( t ) , with a peak overpressure, and rapidly decays to a value below the undisturbed air pressure, which is generally referred to as the negative phase, after the requisite time span for the blast wave to reach the place of interest. The blast force crushes the affected structure during the positive phase, and a reversed blast wind arises during the subsequent negative phase to further impact the target [24].

In recent decades, a number of empirical techniques for characterizing the primary characteristics of the blast wave pressure–time history curve have been developed. A scaling law was introduced to assess the effects of large-scale explosions because tests are typically carried out at small scales. The scaled distance, which connects the explosion’s center to a specific point using the energy released by the explosive, is one of the crucial metrics used to assess the effects of an explosion. It serves as a gauge for the explosion’s impact on a structure at a given distance. Eq. (9) expresses the scaled distance:

where R is the place of interest’s distance from the explosion site and W TNT is the weight of TNT that was previously introduced. Eq. (7) can be used to analyze the structural response to explosions at various possible scales.

2.3 von Mises stress

von Mises stress is a concept that refers to the internal force generated through external loads applied to an object or structure. Contextually, von Mises stress comes from the distribution force or pressure received by external forces applied to a structure or object. In a structure, a component can receive different forces. Forces with various combinations will cause different stresses at different points, depending on the material and the resulting stress. The normal and shear stress components are contained in a symmetric matrix called the stress matrix, often known as the stress tensor, in three dimensions when it comes to a stress analysis.

In addition, von Mises stress emphasizes giving an important role to the cooling of the structure. The von Mises stress principle is defined in Eqs. (10) and (11).

The principal stresses or the stress components represented by the three stress tensors can be used to compute the von Mises stress. For a given stress state, the same von Mises stress, σ _v , can be obtained using either of the formulas in Eqs. (12) and (13).

These formulas represent the stress value that results from combining a component’s shear and normal stresses at a given position. We can determine the amount of a certain stress, which is crucial for providing strength and preventing material failure, by computing the equivalent stress. von Mises stress frequently employs stress in structural analysis to predict when an object may fail or fail permanently. Furthermore, there is an indirect relationship between the von Mises stress and the occurrence of deflection and dissipation energy [25].

When a structure is shifted or distorted as a result of external loads such as vibration, momentum, or temperature, energy dissipation frequently occurs [26]. Furthermore, in the analysis of a structure, energy dissipation is frequently used for mitigation [27]. Friction between surfaces, material deformation, and energy transmission to the surrounding medium are all factors that contribute to the process of energy dissipation. Therefore, the structural responses to applied stress are evaluated using energy dissipation as a key indication. Eqs. (14) and (15) can be used to define the dissipation energy.

F(δ) describes the function of deformation or displacement when force is applied to a structure, ED describes the total energy dissipation, and m is the total mass of the deformed structure. In general, several main mechanisms cause energy dissipation to occur in a system. First, plastic deformation is a process where mechanical energy is used to change the structure of a material so that deformation occurs. Second, friction is a phenomenon when mechanical energy is lost, and two surfaces move against each other. Third, heating is the result of mechanical energy being converted into thermal energy due to friction or deformation.

3.1 Testing profile

The testing profile comprehensively discusses various aspects of the numerical simulation, ranging from the geometry to boundary conditions. In this study, the geometry used was a sandwich panel measuring 31 mm × 31 mm × 5 mm. The plate stiffener used a tie constraint as a preventive measure to prevent movement between the plate and the honeycomb core [28]. The mesh element used was reduced integration (C3D8R) on the front and back of the plate. Then, S4R elements were used in the sandwich geometry. The load used on the plate was with ENCASTRE, so that it could not move in any direction. The symmetry condition about the x-axis (XSYMM) was applied to the plane in the direction of the y-axis. In addition, the symmetry condition of the y-axis (YSYMM) was applied to the plane parallel to the x-axis, as shown in Figure 2.

Figure 2

Sandwich panel boundary condition.

3.2 Results

The benchmarking results from the CONWEP helped provide essential results in evaluating models for their ability to simulate the effects of TNT explosions with masses of 1, 2, and 3 kg on the structural response that occurs in sandwich panels. This was achieved by involving empirical data from trials carried out in previous studies and comparing them with the current study. The data from the numerical simulation results between the previous and present studies are presented in Table 1.

The results of the numerical simulations carried out in the previous study and present study used the same variations, namely varying the mass of TNT used for the sandwich panels from 1 to 3 kg. The benchmarking results that were carried out with the results of the previous study showed a displacement of 69.19 mm with a TNT mass of 1 kg. At the same mass, the difference in displacement between the previous and present studies was 1.03 mm with an error of 1.4%. The results of the numerical simulations carried out by both studies showed that the mass or load given to the sandwich panel can influence the structural response of the sandwich panel. Contour illustrations of the numerical simulation results of the two studies are shown in Figures 3 and 4.

Figure 3

Deflection in sandwich panels with a TNT mass of (a) 1 kg, bar 3; (b) 2 kg; and (c) 3 kg in the previous study.

Figure 4

Deflection in sandwich panels with a TNT mass of (a) 1 kg, bar 3; (b) 2 kg; and (c) 3 kg in the present study.

3.3 Mesh convergence study

Mesh convergence is one of the crucial components in an FEA when the convergence value can reach the actual solution. It involves combining the elements in the geometry and analyzing the impact of the process on the accuracy of the results or the achievement of stability (convergence). At this stage, further changes no longer significantly impact the results, so determining the optimal element size is very important in order to obtain accurate results [29]. In this study, a mesh element size of 6 was used because the mesh reached a convergence or stability value, so at this value, the mesh convergence showed the achievement when the simulation results were no longer affected by the level of mesh fineness used. This is shown in the mesh convergence graph in Figure 5.

Figure 5

Mesh convergence graph.

4.1 Geometrical model

This study used five types of circular plate geometries: no stiffener, one stiffener, one cross-stiffener, two stiffeners, and two cross-stiffeners. The plate was modeled by representing it as a component in the form of a circle with a radius of 73 mm, whereas the open area had a radius of 53 mm. The S4R shell element was used for this representation. An additional radius of 20 mm served to delineate the clamping area. The test plate was designated as a ‘shell revolve’ component, with a specific section thickness of 2 mm or 3 mm using a 1.3 mm mesh, as shown in Figure 6.

Figure 6

A 2D model of the circular plate geometry.

The clamp was modeled as a 3D “analytical rigid body” with a revolved cross-section shell. It was designed as a holder for the test plate to simulate the boundary conditions used in this experiment. The clamp was given a clamping surface, which limited the clamp area that interacted with the test plate. Figure 7 illustrates the 3D models of various geometric variations.

Figure 7

The 3D models of geometry variations.

4.2 Applied material

The Johnson–Cook material model is often used in numerical simulation analyses to describe the response of a material to the geometry of complex mechanical loads. The Johnson–Cook model uses three main parameters for materials: the strain hardening rate, thermal parameters, and strength. These parameters can help accurately describe a material’s response to varying conditions and loads. This study applied the Johnson–Cook model to two types of materials: Domex 700 steel and Domex 1100 steel. Domex 700 steel, with a thermal conductivity (m) of 500 and a melting temperature (T _melt) of 1,370 K, has been effectively used in previous research, such as in the study by Yuen et al. [30]. Similarly, 1100 (D1100) steel, with a thermal conductivity (m) of 1.03 and a melting temperature (T _melt) of 1,795 K, has shown promise in research, such as in the work by Pratomo et al. [31]. The Johnson–Cook relation, as expressed in Eqs. (16) and (17), was instrumental in capturing the material behavior in these loading scenarios, demonstrating its practical application in real-world situations.

Here,

Before considering the standards for Domex 700 steel and 1100 steel, it is necessary to define the chemical composition of the steel, as shown in Tables 2 and 3. These two types of steel have different composition types and percentages. The mechanical property parameters are also different for Domex 700 (a yield strength of 700 N/m 1100 steel and a tensile strength of 750–950 N/m 1100 steel) and circular plate steel Domex 1100 (a yield strength of 1,100 N/m 1100 steel and a tensile strength of 1,250–1,550 N/m 1100 steel). The steel structure standards for ships generally use the ASTM A 131 M standard, which covers structural steel forms, plates, bars, and rivets. Materials with specifications in ASTM A 131 M are divided into two categories: ordinary strength – Classes A, B, D, DS, CS, and E, with a specified minimum melting point of 235 MPa – and higher strength – Classes AH, DH, and EH, with the minimum melting point determined as 315 MPa or 350 MPa.

Under standard test parameters, different references were also used; for Domex 700, EN-10149-2 [32], the European standard for hot-rolled flat products made of high-yield-strength thermo-mechanically rolled steels for cold forming was used, and the impact strength test used the Charpy V-test. Notching was carried out in accordance with EN 10045-1 [33]. Meanwhile, for Domex 1100 circular steel, the tensile test was performed according to EN 10002-1 [34] and the impact strength test also used EN 10045-1 [33].

The corresponding constants used to describe a complex material response are comprehensively detailed in Table 4. They ensure a comprehensive understanding of a material’s dynamic response to local blast-loading conditions. This approach accurately represents the material behavior of Domex 700 steel and Domex 1100 steel in a given context, as demonstrated in this study.

4.3 Finite-element settings

Boundary conditions, the mesh, and the scenario design are three aspects that are quite important in the numerical simulation process. Boundary conditions are used to regulate system behavior within predetermined boundaries. In contrast, the mesh divides the geometric domain into elements for the numerical simulation. Meanwhile, the scenario list is a series of scenarios or conditions planned using code to make it easier to identify simulation results from different variations. By paying attention to these three aspects, the simulations can be made more accurate and relevant for analyzing the responses to various predetermined situations and conditions.

4.3.1 Mesh configuration

In numerical simulations, the meshing process is the first step and is crucial in providing accurate results. Dividing the meshing elements into more minor elements allows for a simpler representation of complex geometries. Meshing has a vital role in determining the accuracy and reliability of the simulation results, so the choice of mesh type, mesh size, and mesh production are important factors that can influence the quality of the resulting simulation model. In addition, the meshing of the geometry has an important role in the simulation efficiency, because the efficiency depends on the quality of the mesh [35,36]. The mesh in this study used S4R mesh shell elements with a size of 1.3 mm. The number of elements used in the test plate affected the pressure distribution on the test plate as a whole. The mesh applied used a global size of 1.3 mm and had 11,727 total elements. It was chosen because this mesh size achieved convergence or stability in the numerical simulation results. A mesh illustration is shown in Figure 8.

Figure 8

Mesh illustration for geometry variations.

4.3.2 Boundary conditions

Boundary conditions are an important component in preparing numerical simulations; they are parameters that determine the behavior of the system at the boundaries in a numerical simulation. This is quite crucial, because it allows the user to simulate situations according to the actual physical conditions of the object being tested. Boundary conditions include external forces applied to a bounded structure or boundary, as they can influence the results [37]. When appropriate boundary conditions are set, the simulation results can provide more accurate results regarding the structure’s response to the environment and the load conditions that are determined. The boundary conditions must be carefully determined, and the actual physical conditions must be considered along with the analysis objectives [38].

The investigation of the structural response of Domex 700 steel and Domex 1100 (D1100) steel circular plates used the ABAQUS approach with the CONWEP approach. The clamping mechanism was assigned a defined clamping surface, representing the region where the mechanism engaged with the test plate. Next, a reference point with certain zero-velocity boundary conditions (X, Y, Z) was established on the clamp, ensuring its stability in space during plate loading, as shown in Figure 9.

Figure 9

Illustration of boundary conditions on a circular plate.

The numerical model was subjected to blast loading using the CONWEP interaction, defined as the interaction of the incident wave with the blast area on the test plate. The explosion source was determined at a normal distance of 150 mm from the explosion area. The CONWEP interaction characteristic was defined as an air blast with a mass equivalent to TNT for each specific charge mass ranging from 10 to 40 g.

4.4 MADM method

MADM is an analytical approach used to make complex decisions by considering several relevant alternative criteria or attributes [41,42,43,44,45,46,47,48]. By using the MADM method, decision makers can evaluate and compare various alternatives based on different criteria (such as crashworthiness, fire safety, etc.), making it possible to obtain results that are in accordance with the objectives [49,50,51,52,53,54,55,56].

The simple addition weighted (SAW) method is one type of the MADM method that is used to weight the values obtained for each criterion. The SAW method is quite commonly used. It determines the performance weight for each alternative owned by each attribute. This method requires a data normalization process and compares the data with all the result rankings. Data normalization is shown in Eq. (19), and the value of each alternative is calculated using Eq. (20).

The calculation model using MADM is explained by the equation. The matrix’s alternative values and criteria are described by x ij , and their relationships are established by Max i x ij , which specifies the maximum value of each alternative and criterion. Next, w r describes the assigned weight value and r ij defines the matrix normalization so that v i , the defined alternative’s final value, may be obtained.

5.1 Results based on material variation

The simulation carried out on the geometry used several variations, one of which was a variation in the material between Domex 700 (D700) steel and 1100 (D1100) steel, as shown in Table 4. The data from the CONWEP simulation results with a TNT mass of 30 g showed that the geometry with the Domex 700 (D700) steel material produced a more significant deflection compared to the Domex 1100 (D1100) steel material, with a difference of 0.92 mm at 0.5 ms. The data are presented in Figure 10.

Figure 10

Deflection graph of Domex 700 steel and 1100 steel.

The graph in Figure 11 shows that Domex 700 (D700) steel experienced a more significant deflection compared to 1100 (D1100) steel. The difference in the deflection values for the two different materials was analyzed through the stress value produced by the geometry. The geometry with the Domex 700 (D700) steel material received a greater von Mises stress than 1100 (D1100) steel, with a difference in the value of 388 Pa. Therefore, the force distribution produced by Domex 700 (D700) steel was more significant, so the energy absorbed by the material was relatively high. This resulted in the deflection being greater than that of 1100 steel. The difference in the deflection values between the two materials was influenced by the ability of each material to accept the force obtained from the explosion on the geometry. This shows the significant influence of the explosion on the geometric response of different materials, where the energy absorbed increases with increasing time until it reaches the maximum stress and energy dissipation values, as illustrated in Figures 11 and 12.

Figure 11

von Mises stress contour, one cross-stiffener: (a) Domex 700 steel and (b) 1100 steel.

Figure 12

Energy dissipation contour, one cross-stiffener: (a) Domex 700 steel and (b) 1100 steel.

5.2 Results based on thickness variation

Thickness variations in structures can significantly impact the structural response to explosions. Varying the thickness of a structure can cause a different stress distribution when the structure is exposed to an explosion. An analysis of varying thicknesses can help determine the structural behavior in response to blast loads. In the numerical simulation, thicknesses ranging from 2 to 6 mm were used with the same blast load mass of 30 g. Data on the effects of the explosions on geometries with varying thicknesses are presented in Figure 13.

Figure 13

Deflection graph with thickness variations.

The data resulting from deflections that occurred at varying thicknesses showed that the geometry’s thickness significantly influenced the explosion. The largest deflection was obtained at a thickness of 2 mm, with a deflection of 13.47 mm, and the slightest deflection occurred at a thickness of 6 mm, with a deflection of 5.69 mm. As the thickness increased, the resulting deflection became smaller compared to the thickness, which tended to be small. Thus, there was a correlation between the thickness of the material and the level of deformation that occurred due to the explosion. Therefore, the ability of a given thickness to withstand the forces received from an explosion influences the structural response of the geometry.

Deformations in geometries with varying thicknesses can occur because there is stress received by the geometry, resulting in dissipated energy, which is absorbed by the geometry and produces deflection. The impact of this dissipated energy is then reflected in the form of deformation that occurs in the geometry. The factors that differentiate each thickness are the von Mises stress value obtained and the energy produced by the von Mises stress on the geometry, which are very important parameters. This causes each thickness to have different characteristics when responding to explosions and different abilities to receive force and absorb energy, resulting in different deflections from each thickness. This is shown by different structural responses with the same von Mises stress and energy dissipation values at each thickness, as illustrated in Figure 14.

Figure 14

(a) von Mises stress and (b) energy dissipation contour for a circular plate with thickness variations.

5.3 Results based on varying the mass of TNT

The effect of varying the TNT mass on the geometry can produce significantly different structural response results. Different TNT masses provide differences in the explosive energy produced by the explosion on the geometry and can influence the stress received by the material and the force absorbed during the explosion. This CONWEP simulation used variations in the explosion load given to the geometry with two cross-stiffeners using the 1100 (D1100) steel material. The explosion load ranged from 10 to 40 g of TNT to determine the structural response exhibited by the geometry. The explosion results for variations in the TNT mass are shown in Figure 15.

Figure 15

Deflection graph with varying masses of TNT.

Based on the deflection values obtained for the geometries with varying explosion masses, it was revealed that the explosion mass has a crucial role in the impact on the geometry. It was observed that an explosion mass of 10 g produced a deflection of 3.51 mm, which was the smallest deflection compared to an explosion mass of 40 g, which produced a deflection of 11.34 mm and was the most significant deflection obtained for the geometry due to the explosion. This shows that the mass of TNT has a crucial role in providing significant deflection changes to the geometry’s ability to receive force and then absorb the force to produce a deflection.

The structural response of the geometry to the received von Mises stress and the energy dissipation were 1,421 Pa and 629.8 J. The analysis results showed that a TNT mass of 10 g obtained after the explosion indicated that the energy dissipation capability and the von Mises stress peaked at these values. However, a TNT mass of 15–40 g showed a different structural response because these values were obtained when the explosion was not complete relative to the geometry. The illustration of the von Mises stress and the energy dissipation contours shown in Figures 15 and 16 can explain that a higher explosion mass caused a higher von Mises stress value, thus affecting the geometry’s ability to absorb energy and influencing the deflection that occurred.

Figure 16

(a) von Mises stress and (b) energy dissipation contour for a circular plate with a varying mass of TNT.

5.4 Results based on variations in geometry

Variations in the geometry significantly impacted the blast’s structural response. In the numerical simulations with variations in the geometry, a stiffener on a circular plate was used to analyze the influence received. These used aspects of deflection, von Mises stress, and energy dissipation. The number, position, and spacing of the stiffener can substantially affect the structural response to blast loading. Figure 17 shows the deflection results obtained for each geometric variation.

Figure 17

Deflection graph based on geometry variations.

Numerical simulations using the CONWEP method were carried out on the geometry using Domex 1100 (D1100) steel material with an explosion load of 30 g. The results of the deflection data showed that the largest deflection was obtained with an unstiffened circular plate, at 9.37 mm, and the smallest deflection was obtained for a circular plate with one cross-stiffener, with a deflection value of 9 mm. Through the deflection value obtained due to the explosion, the von Mises stress value and the energy absorbed by the geometry played a role in the deflection of the geometry. It can be explained that the structural response to geometric variations significantly influenced the explosion, as illustrated in Figure 18.

Figure 18

(a) von Mises stress and (b) energy dissipation contour for a circular plate with geometry variations.

The results of the numerical simulation data analysis showed that variations in the geometry with stiffeners on circular plates resulted in quite impressive differences in the structural response in terms of the von Mises stress and the energy dissipation at values of 1,120 Pa and 289 J. This shows that the number and distance of stiffeners in the geometry affected the performance. The geometry defines the ability of the structure to absorb energy when receiving stress from outside. The areas that receive the highest stress, on average, are at the end of the stiffeners and the edge of the geometry due to being connected to the clamp and experiencing energy absorption, resulting in plastic deformation.

5.5 MADM analysis

MADM is an approach that is used to make decisions and evaluate several alternatives based on relevant criteria. The impact of the circular plate on blast loading was analyzed to validate the results of the numerical simulations using the CONWEP method. There is an initial stage in carrying out MADM, namely determining the score from the parameters used and proceeding to the next stage by entering the test result data according to the parameters that have been determined. The parameters used in analyzing MADM were the deflection, von Mises stress, and energy dissipation by taking the value of each criterion when it reached 0.5 ms. The value data for each criterion from the numerical simulation results are shown in Table 7.

In the next stage, the inputted data were subjected to a data normalization process for further analysis to avoid data anomalies. For each criterion, which consisted of C1 (deflection), C2 (von Mises stress), and C3 (energy dissipation), the smallest data value was taken. This was chosen because the smaller the deflection value, the smaller the value of the von Mises stress and the energy dissipation, as the force received and the energy absorbed did not cause the deflection to become more significant. Therefore, the structural response that occurred was considered better. The data normalization results for each numerical simulation data result are presented in Table 8.

After the data were normalized, the next stage was to carry out assessment calculations on MADM, assuming that the numerical simulation results with the highest score are considered the best results. Then, after each data result had been given a score, ranking was carried out to determine the numerical simulation results with the most significant score values. The calculation results and data ranking of the numerical simulation results are shown in Table 9.

The ranked results were divided into fifths, and each fifth consisted of 14 numerical simulations, from the smallest to the largest. The larger value of the total weight gained in each fifth indicated that the results obtained were worse. Based on the above data, the best-ranked results were in the first fifth, with annotations from LXXI to LVIII and the lowest score of 6.176 × 10⁻⁰⁹. The worst-ranked result was in the last fifth, with annotations from XV to I and a top score of 3.659 × 10⁻⁰¹.

5.6 Sensitivity analysis

The sensitivity analysis in this research used the regression method to obtain the coefficient values, standard error, R-squared value, P-value, and significant F-value, which helped to analyze the impact of the variations in the material, the mass of TNT, the thickness, and the geometry on the explosions. In applying the sensitivity analysis, the carbon content, mass of TNT, stiffener volume, and thickness became the input variables for the variations in the numerical simulation, which resulted in the deflection, von Mises stress, and energy dissipation. A high R-squared value indicates that a variable significantly influences the geometric structural response. The coefficient value describes how much a change in the independent variable contributes to the average change in the dependent variable. The greater the coefficient value, the more significant the influence of variation on the numerical results.

The standard error value shows how far the average value is from the regression line and how accurate the regression line is at predicting the response variable. The significant F-value in the regression analysis shows the extent to which the regression model fits the observed data compared to models that do not use predictor variables. The lower the significant F-value, the greater the impact of the variations on the final results produced by the regression model. The results of the sensitivity analysis of the criteria are shown in Table 10.

Based on the data above, the deflection criterion showed that the mass of TNT had a significant effect, as it obtained the highest R-squared value and the smallest significant F-value. In addition, the larger coefficient, standard error, and P-values indicated that the mass of TNT had little influence on the final results. The sensitivity analysis results on the von Mises stress criterion showed that the carbon content had a significant influence due to having the highest R-squared value and the smallest significant F-value. In addition, the smaller coefficient, standard error, and P-values for the von Mises stress criterion indicated that the carbon content significantly influenced the final result. The energy dissipation criterion showed that the carbon content had a significant effect because it had the largest R-squared value and the smallest significant F-value. In addition, a larger coefficient value indicated that the carbon content had little influence on the results. Meanwhile, the P-value and standard error, which became smaller, indicated that the different carbon content had an influence and an impact on the final results. Thus, changes in the mass of TNT significantly impacted the deflection. Meanwhile, the carbon content had an impact on the stress and energy dissipation in the geometry.