#### Abstract

In this paper, a new method is developed to find the ductility ratio in blast walls, resulted by hydrocarbon explosions. In this method, only the explosion energy and distance from the centre of explosion are required to find the damage by using simple predictive models in terms of empirical-type formulas. The explosion model herein is a TNO multiphysic method. This provides the maximum overpressure and pulse duration in terms of the explosion length and distance from explosion centre. Thereafter, the obtained results are combined with the SDOF model of the blast wall to determine the ductility ratio and the damage. By using advanced optimisation techniques, two types of predictive models are found. In the first model, the formula is found in terms of 2 parameters of explosion length and distance from explosion centre. However, the 2^{nd} model has 3 parameters of explosion length, distance, and also the natural period of the blast wall. These predictive models are then used to find explosion damages and ductility ratio. The results are compared with FEM analysis and pressure-impulse (P-I) method. It is shown that both types of models fit well with the outputs of the simulation. Moreover, results of both models are close to FEM analysis. The comparison tables provided in this paper show that, in the asymptotic region of P-I diagrams, results are not accurate. Therefore, this new method is superior to classical pressure-impulse (P-I) diagrams in the literature. Advantage of the new method is the easy damage assessment by using simple empirical-type formulas. Therefore, the researchers can use the method in this paper, for damage assessment in other types of blast resistive structures.

#### 1. Introduction

Blast walls are sacrificial barriers to protect offshore structures when subjected to hydrocarbon explosions. Substantial research has been performed to develop a code of practice for design of such structures [1]. The theoretical foundations for designing blast-resistive structures and blast walls can be found in [2, 3].

An analytical method based on plate theory for blast wall design [4] is rarely used in the literature since the results of those studies cannot be used directly as a design code. However, linear and nonlinear finite elements have been used significantly (for example, [5, 6]). They are applicable in cases where batch simulations enable cost-benefit analysis [7].

Presently, the dominant approach is a single degree Of freedom (SDOF) method [1–3] and leads to some design curves known as Bigg’s chart. They appeared first in a well-known book [8] but originated from the initial attempt by Newmark [9]. This SDOF method enables the famous pressure-impulse (P-I) diagrams which was first introduced in [9] to be constructed [1–3].

These P-I diagrams strongly depend on pressure versus time expression (pulse shape) of the explosion [10], and together with SDOF modelling, they are used to find the blast response of complex of structures, such as cable-supported facades [11]. Both SDOF-type model [12] and continuous beam model [13] are used for developing P-I diagrams. Recently, it is shown that batch finite element simulations [14] cannot lead to P-I diagram unless preliminary information regarding SDOF parameters is available.

Regardless of importance of the P-I diagrams in the damage assessment, they are not straightforward and the designer needs substantial information about the calculation of the explosive loads and pulse shape to be able to use P-I diagrams in damage assessment. There is not any attempt (or new method) that directly connects intensity of explosion to the resulted damage and deflection in the blast walls. Recently, the author looked at this important issue where, in the vicinity of box girders [15], TNT explosions may occur. Since the possibility of hydrocarbon explosions are much higher than any terrorist activities, blast walls are used in many offshore structures. Therefore, any research regarding this topic is justifiable.

In this article, the explosive physics known as the multienergy method known as TNO [16], and further models fitted into it [17], is combined with the SDOF method for deformation of the blast walls. Thereafter, the deformation and ductility, for both rigid plastic models and elastic-plastic models, are determined in each distance and explosion length. Then outcomes of the batch simulations are exported to advanced optimisation programs to develop two types of predictive models expressed by using simple empirical-type formulas.

Using any of the models in this paper, the designer can find the deformation (or ductility) from the intensity of explosions (explosion length), distance of the blast wall from explosion centre, and natural period of the blast wall. As far as the author is aware, this new method is the easiest one for predicting the damage in the blast wall, thereby declaring the explosion resistance. The knowledge about explosion physics is embedded in the formulas. Therefore, it is an excellent tool for preliminary analysis of the blast wall.

In a case study, in the asymptotic region of the P-I diagram, it is shown that, while P-I provides inaccurate results, this method leads to accurate results, when it is compared with FEM simulation of the blast wall. Therefore, the approach herein can be extended to other types of structures in future to replace P-I diagrams (or FEM) for predicting the damage.

#### 2. Overpressure History in Explosions

When hydrocarbon mass (in kg) with heat energy (Joule/kg) causes an explosion with efficiency , the resulted explosive energy will be

In the TNO multienergy method [16], an explosion length is defined bywhere is the atmospheric pressure (in Pa); therefore, truly has units of the length (m). If is the distance from the explosion centre (m), dimensionless will be defined by

Then, the overpressure (in bar, i.e., dimensionless) and explosion pulse duration (dimensionless) can be found from TNO charts [16]. In those charts, the overpressure and duration can be found from the curves designated by the level of the explosion. The charts are developed from computer simulations performed in eighties and are strongly applicable to hydrocarbon explosions. Due to the importance of the TNO charts, researchers produced curve fitted formulas for the data in those charts. These formulas are given in [17] via the following equation:

It should be reminded that another valuable software is provided for blast waves (for example, [18, 19]) but not reformulated for designers yet (such as (3)). In the above expressions, is dimensionless overpressure pulse duration given in [16]:where is the overpressure duration in sec and is the sound velocity at atmospheric conditions in m/sec. Majority of explosions will fall into all of the three levels in (3). It is recommended that the overpressure and duration should be computed in each level and the average value should be taken into consideration [17].

The author herein produced the overpressure contours in terms of and which are two important parameters in any explosions. They are shown in Figure 1 and are used in the next part of the paper, for developing the new method.

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#### 3. SDOF Model for Blast Walls

The typical geometry of the cross section of a blast wall [1] is similar to (a) in Figure 2.

The finite element analysis shows [5] the deformation pattern resulted by an explosion by using shell elements which is similar to Figure 3.

The front view of a typical blast wall [1] is shown in (b) in Figure 2. The main parameter is the pitch that is shown in (a). The wall is connected to the structure by upper and lower supports shown in (c). When overpressure is applied to the wall with uniform distribution, the upper and lower supports with thicknesses and (in m) (see (c) in Figure 2) have equivalent lengths and shown in (Figure 15). They will yield since they have limited yield stress (Pa). The total length is (in m), and and are the yield moments (per length, i.e., in N) of the upper and lower supports and are given by the following equations [1–3]:

or the plastic bending moment (per unit length) of the main wall is given by (6). It depends on the details of the cross section in (a) in Figure 2, which are designated by two parameters, (plastic section modulus) and (material yield stress) of the cross section:

and in (6) are flattening and shear correction factors described in [1]. The equivalent length of the blast wall is less than the total length and can be found by

Derivation of (7) is shown in Appendix A, and instead of total length, will be used in all calculations, regarding the blast wall. For example, the stiffness per unit length will be given as shown in [1–3] as follows:

The corrected stiffness of wall is recommended in [1] to correct (8) resulted from beam theory which is

Equations (8), (9), and others that follow are true when the SDOF method is chosen as a route of the analysis, where the beam simplification and can be justified. This is also current practice for the preliminary design of blast walls [1–3]. However, for the detail of the buckling pattern similar to Figure 3, the beam model simplification is not appropriate. According to rigid plastic theory in structures, the maximum resistance of a beam cross section [2, 3] is given by

This is defined for finding maximum elastic deformation of the wall [1–3] by using the following formula:

However, if the maximum blast load given by the following equation exceeds , the wall deforms plastically:

In (12), is the projected blast area per pitch in Figure 4. For further clarification, this area with the pressure applied to it is shown in Figure 4.

The deformation is allowed up to the ductility limit. The ductility is very important in design of structures under extreme and blast loading [2, 3, 8] and is the ratio of maximum plastic deformation to the elastic limit given by

The backbone of the SDOF model relies on the natural period of free structural vibration , [2, 3, 8] of the blast wall which will be given by

In (14), is the blast wall mass (for one pitch) and is the correction factor for the distributed mass. In Appendix B, it shows that for rigid plastic theory based on plastic hinge assumption [2, 3], we find that . However, in the current practice [1], designers use higher values without any justification. Part of this article investigates how this apparent inconsistency can affect the ductility results.

The SDOF modelling is well known by Biggs’ chart since it appeared in a famous book [8]. However, the initial research is done by Newmark who is one of the pioneers in structural dynamic. He summarised Bigg’s chart a decade before it is seen in [8], in his famous paper [9] by using the following formula:

All the parameters in (15) are described in previous formulas. When an explosion with length occurs at distance , one can find the preliminary ductility curves. For a particular blast wall that is designed by a manufacturer, geometrical and material details are available. Therefore, the ductility contour can be constructed easily from (15), without using the pressure-impulse diagram of the blast wall.

#### 4. Numerical Example

For a steel blast wall with pitch meter, the cross-sectional dimensions are shown in Figure 5. It is one of the existing profiles of the blast wall that is described in [1].

The second moment of the cross section , the section modulus , mass per pitch , thicknesses of the upper and lower supports and , and Young’s modulus , and yield stress , the length , and the correction factors [1] and . In Figure 6, the ductility is shown, which is the result of substantial simulations of the SDOF model for this blast wall.

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Figure 6 is prepared for as recommended in [1] and is not the result of rigid plastic theory. Figure 6 is drawn in range and , and the contours seem linear and visible. However, for higher ranges, visibility and linearity cannot be observed.

#### 5. Model with Two Parameters

A nonlinear predictive model of Figure 6 with two parameters and (both explosion related) can be suggested in this form:

For example, the higher range estimation of ductility for can be replaced by the following approximate expression:

In Figure 7, the computed ductility ratio and the estimated ductility ratio in (17) are drawn together. It can be concluded that, in higher ductility ratios, where severe plastic deformation occurs, the estimated ductility is very close to the computed ductility. In (17), only explosion-related parameters are used. Three parameter models will be discussed as well.

#### 6. Rigid Plastic Modelling

Rigid plastic theory [2, 3] assumes plastic hinge at the midlength of the blast wall. In appendix B, it is shown that, in such situation, the equivalent mass and . The damage calculation will be straightforward because the calculations regarding overpressure and duration remain the same as the ones used for producing Figure 7. Obviously if we assume , the results will change which is shown in Figure 8. The region in which ductility ratio is below 1 remains elastic, and by producing such contour maps, the pressure-impulse diagram is not required. If we compare Figure 6 in which peak deformation with Figure 8 in which , we can conclude that considering (rigid plastic model) provides conservative estimation for ductility.

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#### 7. Model with Three Parameters

A nonlinear predictive model with three parameters , (explosion related), and in (14) which are blast wall related can be suggested as in the following form:

The parameters in (18) can be found by taking the logarithm for that expression that will change it into

The above expression enables the linear regression techniques to be implemented for finding the parameters . These parameters can be found by using nonlinear regression analysis. Moreover, the powerful Nelder–Mead algorithm [20] which is built in MATLAB is also used, to find the fractional powers *α*, *β*, and *γ* in (18). Finally, the numerical expression of (18) when (rigid plastic modelling) will be in the following form:

In Figure 9, the computed ductility ratio and the estimated ductility ratio in (20) are drawn together. It can be concluded that, in higher ductility ratios, where severe plastic deformation occurs, the estimated ductility is very close to the computed ductility. In (20), explosion-related parameters plus blast wall natural period are used. Three-parameter models use (rigid plastic modelling) because of its conservativeness in estimation of the maximum ductility.

The author has suggested many other forms for the regression analysis, using advanced optimisation techniques [20], and so far, he has not found better forms than (20) for the 3-parameter-type model and (17) for the 2-parameter-type model. It is quite possible that some other forms with closest fit may be found by further research.

#### 8. Comparison of the Results

Consider that an explosion with effective energy occurs at distance from the explosion centre. According to parameters (2a), (2b) and (3). The overpressure is the average value of the explosion levels 3 and 9 and 6, in (3).

The elastic deformation from (11) is , whereas the maximum deflection at the middle section, , in (12) can be found by knowing about the ductility ratio.

Since the velocity of sound in the room temperature condition is , from formula (4), we have duration of the explosion pulse , whereas the natural period of the blast wall herein which is given by using (14) is .

The pressure-impulse curve that introduced before is still used for damage assessment for many structures. They are a series of the asymptotic curves inscribed in the vertical and horizontal asymptotes. To find the points on the curves, either we use analytical methods [21, 22] or numerical methods [23] and sometimes FEM analysis [24]. In the *x*-*y* plane, the vertical axis displays , whereas horizontal axis displays , *I* is the impulse, and is the maximum explosion forces. With uniform overpressure, they are

In [2, 3], it can be shown that the equations of the vertical and horizontal asymptotes are in terms of the ductility ratio *μ* that is defined in (13), i.e.,

Typical curves for elastic-plastic structures are shown in Figure 10 in which the ductility ratio can be found via interpolation. The snapshot designated by the point shows the coordinates and that correspond to this particular explosion, and we can find the ductility as a result of this explosion.

However, the direct simulation in this paper shows that . It shows that the P-I method particularly in asymptotic ends are significantly inaccurate. The two approximated models in this paper that are expressed by (17) and (20) to replace the P-I method give much closer results. The comparison is shown in Table 1.

Further comparison can be done by using FEM technique via ABAQUS modelling [25] of the blast wall in this example. The meshing is shown by a snapshot in Figure 11. In this model 6500-shell-type S4R elements, each with nine internal integration point are used. Obviously substantial FEM outputs, including the local buckling details in bottom flanges are available. However, the one that can be compared with in (12) has been extracted. Since ductility ratio is not defined in ABAQUS, Table 2 is provided to compare the (maximum deflection) in each approach.

The last row of Table 2 is found from history of the displacement of the middle of the top flange of the blast wall. This history for , , and *A* is shown in Figure 12. It is obvious that velocity in mm/s and acceleration in m/s^{2} are big numbers since *T* in (14) is very low.

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Figure 12 is prepared by using history of nodes. However, the history of stress and strain in any location of the blast wall can be prepared by element output files. Similar to Figure 11, Figure 13 shows the Mises stress map that is scaled in Pa.

Obviously, the yield stress is , and the material is assumed elastic-perfectly plastic (E-P-P); all similar to the SDOF model. Since the blast wall is modelled with shell elements, Poisson’s ratio of the material is also required. The history of the Mises stress and also the maximum principal strain can be found from the element file. To do this, the shell element corresponding to middle of the top flange of the blast wall is chosen. The history file for stress and stain for that location is shown in Figure 14. It is obvious that stress does not exceed 400 MPa. However, for strain after quick jump at the beginning of the explosion, the fluctuations are not significant. From the model in this paper, we can check and verify the displacement as shown in Table 2. This suits the purpose of this paper in developing a simple and accurate model for checking high-fidelity FEM analysis.

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In Table 3, the material properties and also maximum velocity, acceleration, and stress and strain are shown. The maximum displacement is shown in Table 2 for the comparison purposes. The maximum stress in Table 3 exceeded slightly above 400 MPa because the E-P-P material model is ABAQUS, which is expressed via a very low plastic Young’s modulus (not zero).

The outcomes of this section are shown in Table 4. This table compares the advantages and disadvantage of each method that is discussed. It can be seen that there are many advantages of using the method in this paper, particularly when we compare with the pressure-impulse diagram. However, it should be used together with high-fidelity FEM analysis to achieve more details about the response of the blast wall to the explosion.

#### 9. Conclusions and Remarks

It this paper, a new method for damage assessments in blast walls are developed. It is much easier than the classical method of the pressure-impulse diagram and FEM analysis. As shown in [21–24], and also in this paper, the high-fidelity analytical or FEM models cannot predict explosion response without knowledge about explosion overpressure and pulse duration.

The inaccuracy of the P-I diagram in the asymptotic region is clearly shown in this paper via Tables 1 and 2. Regardless of that, the P-I diagram is an active field of research for blast-resistive structure as seen in recent publications [22–24]. Therefore, an alternative method is required to replace the P-I diagram in asymptotic region. This approach should be much easier than FEM analysis and can produce a result accurate enough to be compared with FEM. The author believes that he has found an alternative in this paper.

When overpressure-time history is not available, both of the SDOF and FEM cannot predict the damage. The advantage of this new method is the combination of SDOF method and overpressure-time history of explosion. Herein the TNO method (that provides overpressure history) with SDOF (that provides deflection) has been combined together. Thereafter approximate formulas have been produced that easily predicts the ductility ratio without using P-I diagrams or doing SDOF calculations or FEM analysis. Therefore, it will be very useful for preliminary design applications.

#### Appendix

#### A. Equivalent Lengths and Bending Moment Distribution

According to Figure 15, the total length of the blast wall consist of 3 parts:

The overpressure as a result of explosion produces a uniform load that results a parabolic type of bending moment as follows:

When we place the origin of the coordinate system at the middle of the wall, then we have

Moreover, the shear force at maximum bending moment is zero, i.e.,

The segment with length acts as the simply supported beam such that, in its two ends, the bending moment is zero such that

Then, (A.2) can simplified into

The upper and lower supports of the blast wall act as cantilevers such that maximum bending moments of the supports occur at the corners such that

The expressions (A.7) and (A.8) can be simplified into

Substituting (A.9) into (A.1), and after simplification, we have equation (7):

#### B. Rigid-Plastic Beam Model

In rigid plastic type of modelling, the plastic hinge occurs at the middle of the beam where the maximum lateral deflection will occur (Figure 16).

Obviously, the lateral deformation and velocity pattern will be linear and are given by

Then, considering the form in (B.1), the overall kinetic energy of the beam will be

The equivalent mass located at the plastic hinge position should possess the same kinetic energy in (B.2), i.e.,

Comparing (B.2) with (B.3) will result

#### Symbols

: | Cross-sectional area |

: | Constants of the parabolic function |

: | Effective overpressured area |

: | Velocity of sound |

, : | Explosive energy and modulus of elasticity |

, : | Steel yield stress and total applied force |

: | The maximum explosion force |

: | Second moment of cross section (in bending) |

: | Impulse of the explosion pulse |

: | Stiffness and reduced stiffness (in bending) |

, : | Flattening and shear correction factors |

: | Mass correction factor |

: | Total length and equivalent length |

: | Lengths of the upper and lower supports |

: | Mass and equivalent mass of the blast wall |

: | Hydrocarbon mass |

, : | Yield bending moment in the upper and lower supports |

: | Yield bending moment in the blast wall |

: | Maximum overpressure |

, : | Projected blast area per pitch and atmospheric pressure |

, : | Explosion length and dimensionless explosion length |

: | Maximum elastic resistance of beam cross section |

: | Distance from explosion centre |

: | Dimensionless pulse duration and pulse duration |

and : | Thicknesses of the upper and lower supports |

: | Natural period of the structure |

: | Plastic section modulus |

: | Maximum deflection of the midspan |

: | Maximum elastic and maximum plastic deformation |

: | Heat energy |

: | Constants in the predictive model |

: | Ductility ratio |

: | Efficiency of explosion. |

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The author declares no conflicts of interest.

#### Acknowledgments

The author appreciates Aberdeen University for the time provided to him for doing independent research as part of his duties of an academic post.