#### Abstract

A reliable estimate of free-field ground displacement induced by nuclear-air-blast is required for design of underground strategic structures. A generalized pseudostatic formulation is proposed to estimate nuclear-air-blast-induced ground displacement that takes into account nonlinear stress-strain behaviour of geomaterials, stress-dependent wave propagation velocity, and stress wave attenuation. This proposed formulation is utilized to develop a closed-form solution for linearly decaying blast load applied on a layered ground medium with bilinear hysteretic behaviour. Parametric studies of closed-form solution indicated that selection of appropriate constrained modulus consistent with the overpressure is necessary for an accurate estimation of peak ground displacement. Stress wave attenuation affects the computed displacement under low overpressure, and stress-dependent wave velocity affects mainly the occurrence time of peak displacement and not its magnitude. Further, peak displacements are estimated using the proposed model as well as the UFC manual and compared against the field data of atmospheric nuclear test carried out at Nevada test site. It is found that the proposed model is in good agreement with field data, whereas the UFC manual significantly underestimates the peak ground displacements under higher overpressures.

#### 1. Introduction

Underground siting of strategic structures [1] is an option to enhance the safety against nuclear-air-blast [2]. Nuclear explosions generate rapidly moving air-overpressures capable of producing significant ground displacements [3]. Most severe loading occurs within a close vicinity around the ground zero (GZ) known as superseismic zone. This zone is subjected to significantly high overpressures. The velocity of moving air-overpressure fronts is also more than the P-wave velocity of the ground [3–6]. Hence, a reliable estimate of nuclear-air-blast-induced ground displacement is required for design in superseismic zone [5, 7, 8]. Several studies using numerical approaches that account for realistic stress-strain behaviour and boundary conditions are reported for calculating the free-field response [9–16]. Such studies require specialized expertise in numerical tools and are unattractive to practicing engineers. Before the advent of computational tools, Wilson and Sibley [17] conceptualized a one-dimensional pseudostatic approach to estimate air-blast-induced ground displacement. Whitman [4] and Baron et al. [18] have also shown that air-blast-induced ground motion is predominantly one-dimensional and vertical in superseismic zone. In the present work, a generalized one-dimensional pseudostatic formulation is developed to estimate nuclear-air-blast-induced vertical ground displacement that accounts for (i) nonlinear stress-strain behaviour [e.g., 19, 20], (ii) stress-dependent wave propagation velocity, and (iii) stress wave attenuation. Using the proposed formulation, a closed-form solution is developed for linearly decaying blast load with negligible rise-time applied on a layered ground medium with bilinear hysteretic stress-strain behaviour. The proposed model is validated against the field data of an atmospheric nuclear test conducted at Nevada test site [21], and various parametric studies are carried out. In addition, performance of the closed-form solution is also compared with the model given in UFC [22].

#### 2. Generalized Formulation

The developed formulation is explained in detail below.

##### 2.1. Determination of Overstress Distribution (due to Air-Blast) in the Ground

Air-overpressure generated due to nuclear-air-blast consists of an initial rising portion followed by a decaying portion (Figure 1). Overpressure time-history can be visualized as consisting of multiple overpressure fronts. These overpressure fronts are transferred to the ground as P-waves. On arrival of an air-shock front at the point of interest, an initial pulse travels with seismic P-wave velocity and reaches a depth at time . Subsequently, other fronts arrive and penetrate into the ground. If the location of peak overpressure front at time is given as , then, the ground above the depth experiences compressive stresses lower than those caused by the peak overpressure front at that depth. Therefore, the depths above are referred to as “unloading zone.” However, the peak overpressure front is yet to reach at locations deeper than , and therefore, the zone lying beneath the depth is denoted as “loading zone” (Figure 1). It is noted that “unloading zone” corresponds to the “decay portion” of overpressure, whereas “loading zone” corresponds to the “rising portion.” Therefore, at times earlier than the rise-time, there does not exist any unloading zone as overpressure fronts from the decay portion do not arrive before the end of rise-time.

To determine the stress distribution in the ground at time , contribution from all stress fronts arrived before time is considered. If the *k*th overpressure front arrives at the point of interest on ground surface at time such that , then the *k*th overpressure front reaches at depth at time . Stress fronts propagating through ground are affected by attenuation (caused by hysteresis losses, viscosity of ground materials, and dispersion of energy in 3-dimensional space) and interference with reflected wave fronts (generated due to impedance variation with depth). In the present study, stress wave attenuation is taken into account through a geometrical attenuation parameter, whereas interference of incident and reflected stress waves is ignored. If the attenuation coefficient at depth is given as , then the stress generated at depth can be written as , where is the magnitude of the *k*th overpressure front. Depth of the penetration depends upon wave propagation velocity of the stress front, which in turn depends upon the tangential modulus of the geomaterial at the stress level at the depth of interest. If wave velocity of the *k*th stress front is denoted as (a function of depth “*z*”), then (1) can be solved to obtain as a function of , , and (2),where is the function obtained by solving (1). It is noted that for a given , is effectively a function of only because is also a function of . As soon as the air-blast slaps the ground surface, ground deforms elastically and an elastic wave is generated which propagates with seismic P-wave velocity. Upon subsequent arrival of higher overpressure fronts, inelastic waves propagate at velocities smaller than P-wave velocity. If the ratio of P-wave velocity to wave velocity corresponding to the *k*th overpressure front at depth “*z*” is denoted as , then at initial stress levels is equal to one. With increasing stress level, decreases [17, 23]; hence, it is logical to assume that increases with increasing stress level and the maximum value is attained at a peak stress level corresponding to the peak overpressure front. Therefore, a simplest choice is to assume a linearly increasing in direct proportion to overpressure (3). Overpressure fronts from the decay portion propagate through the media which is already stressed in a nonlinear range due to passage of the peak overpressure front. Therefore, it is assumed that wave velocity of overpressure fronts in the decay portion is the same as the peak overpressure front, and hence is the same as :where subscript “*z*” represents the depth coordinate. Using the average loading rate as , can be approximated as (3). As wave propagation velocity through ground media is proportional to the square root of the tangential modulus [24], is given by the following equation:where is the tangential modulus at initial stress level at depth “*z*” and is the tangential modulus at peak stress level at depth “*z*.”

It should be noted that the ground is in equilibrium under geostatic stresses and at rest before nuclear explosion and only the overstresses caused by nuclear-air-blast causes the ground displacement.

##### 2.2. Determination of Strain Distribution in Ground

Strain distribution with depth can be determined using stress distribution and an appropriate stress-strain relationship [e.g., 25–27] as given in the following equation:where functions and denote the loading and unloading branches of the stress-strain curve, respectively (Figure 2).

##### 2.3. Integration of Strains

To obtain the vertical ground displacement at time instant , the strain distribution is integrated (6) from the ground surface to the penetrated wavelength in the ground up to time ,

Substituting in (6) leads to the following equation:

Thus, the displacement time-history can be estimated using (7).

#### 3. Closed-Form Solution

Using (7), closed-form solutions can be obtained for several simplified cases [28]. In this article, a closed-form approximation is developed for the following simplifications:(a)Linearly decaying overpressure time-history with zero rise-time (Figure 3, (8)),where and are peak overpressure and equivalent positive phase duration, respectively. The reason behind choosing the special case with zero rise-time is the popularity of approximating the actual decay of the incidental pressure by an equivalent triangular pressure pulse among practicing engineers [8, 22]. It is to mention that design charts and empirical relations between the weapon yield, peak overpressure, and the equivalent positive phase durations are also available for equivalent triangular pulses with zero rise-time [5, 8, 29]. Thus, for practicing engineers, a linear decay model with zero rise-time is more useful.(b)Bilinear hysteretic stress-strain model [8] with the loading secant modulus (*M*_{L}) and strain recovery ratio (*r*) as parameters (Figure 2).(c)Attenuation coefficient is given by the following equation [8, 30]:where *W* is the yield of the explosion in kiloton and *V*_{L} is the wave propagation velocity of the peak overpressure front.(d)Using (4), is determined as 1 for the bilinear stress-strain model. However, due to nonlinear stress-strain behaviour, is usually greater than 1. Wilson and Sibley [17] and Batdorf [23] recommended a range of 1.5 to 2. Hereafter, is denoted as and assumed to be constant with depth.(e)A constant but representative P-wave velocity of ground media is assumed.

Using assumptions (a)–(e), integral for loading fronts in (7) (with ) between general time instants and is written as , where can be shown to be given by the following equation:

It is to clarify that the overpressure time-history defined by the piecewise function such that for and for converges to (8) in the limiting case when . Therefore, the expression in (10) is first obtained for a case of finite and then it is evaluated for the case when to determine the solution for overpressure time-history of (8). Similarly, the integral for unloading fronts in (7) (with ) between general time instants and can be written as , where can be shown to be given by the following equation:

By setting the appropriate values of and in (10) and (11), the closed-form integrals between desired time instants can easily be evaluated. Thus, the closed-form solution is given by the following equation:

The closed-form solution is further extended to accommodate multiple ground layers. This requires the determination of time instants and during which the wave fronts pass through a particular layer. For illustration, a layered ground medium with an interface at depth with the modulus of top layer as and modulus of bottom layer as is considered as shown in Figure 3. This leads to the three cases (Table 1) and corresponding displacement solutions (shown in Table 1).

#### 4. Validation

The proposed closed-form solution is validated against the field data from an atmospheric nuclear test conducted at Frenchman Flat (Nevada). Perret [21] presented the measured overpressure time-histories (Figure 4) along with corresponding peak ground displacements at different distances from GZ (Table 2) for a 37 kt nuclear explosion at a height of 214 m. Model input parameters for above mentioned nuclear test are determined as follows.

**(a)**

**(b)**

**(c)**

**(d)**

The recorded overpressure time-histories are converted to equivalent linearly decaying overpressure time-histories (Figure 4). Equivalent overpressure time-histories are given by (8) such that the peak overpressure is equal to the recorded peak overpressure, and equivalent positive phase duration is given by the following equation:where is the positive phase impulse (area under the recorded overpressure time-history). It is worth mentioning, though the rise-time is taken to be zero in the equivalent overpressure time-history in (8), that the effect of non-zero rising time has been accounted for by considering the total positive phase impulse (in (13)) which includes the area under the rising and decaying portions of actual overpressure time-history. Therefore, it is expected that setting the rise-time as zero in (8) would have a negligible impact on the magnitude of peak displacement. Furthermore, the ratios of length of rise-time to total positive phase duration for the four cases (P1, P2, P3, and P4) are 0.10, 0.21, 0.23, and 0.27, respectively. This indicates that rise-time is reasonably small near ground zero; however, the relative length of rise-time increases with increasing distance from ground zero. Thus, it is expected that the errors (if any) due to setting rise-time to zero would be more pronounced only at distances far away from ground zero. Even in such cases, a closed-form solution can be developed using (7) by setting a finite rise-time. However, it is worth noting that at far away distances (from ground zero), other important factors related to outrunning ground motion would start governing [31].

The initial pulse moving at seismic P-wave velocity penetrates to a depth of for equivalent case, whereas for actual case, it penetrates to a depth of . It is noted that the depth of the stressed zone in the equivalent case is smaller compared to the actual case as equivalent duration is smaller than the actual positive phase duration (Figure 4). Thus, all depth-dependent parameters are scaled by a factor , such that the total depth of the stressed zone becomes with the depth of the top layer being , and with a characteristic attenuation length of . Based on the variation of P-wave velocity with depth at Frenchman Flat (Figure 5), the average P-wave velocity is estimated as 658.69 m/s. Whitman [4] presented the representative constrained modulus for Frenchman Flat interpreted from different experimental techniques (Table 3). Whitman [4] also emphasized that all aspects of stress-strain behaviour of geomaterials under blast loading are not captured by a single test and advocated to choose the modulus judiciously. The stress attenuation coefficient is calculated using (9) with scaled (Table 4). Using the closed-form solution (shown in Table 1), parametric variations are studied (Table 5) and peak displacements (Table 6) are estimated.

**(a)**

**(b)**

**(a)**

**(b)**

**(c)**

**(d)**

The parametric studies (Table 5) highlight that the proposed model is most sensitive to the constrained modulus compared to other parameters and choice of the modulus is also subjective for engineers. However, based on the parametric studies, the last column “Recommended Value” of Table 5 provides some guidelines to select the constrained modulus. With increasing availability of similar case studies, these guidelines can further be refined.

#### 5. Comparison with UFC Model

The UFC manual [22] provides an expression for air-blast-induced peak vertical displacement based on one-dimensional elastic wave propagation: where is the bulk density of geomaterials. To estimate peak displacements using (14), positive phase impulse is taken from Table 2. Representative P-wave velocity and density are taken as 658.69 m/s and 1331 kg/m^{3}, respectively, taking into account the variation with depth (Figure 5). Computed peak displacements using the proposed model and UFC model are shown in Figure 9 along with the measured field values. It can be clearly seen that the predicted values of the proposed model are in good agreement with the measured values and the UFC model significantly underestimates the peak displacements under high overpressures (with increasing war head capacity).

#### 6. Conclusions

A closed-form expression is developed to estimate nuclear-air-blast-induced free-field ground displacement that takes into account peak overpressure, positive phase impulse, depth of layer interface, representative constrained modulus of each layer, strain recovery, stress attenuation, P-wave velocity, and velocity ratio. The solution is validated against a nuclear test conducted at Frenchman Flat (Nevada) and the following conclusions are arrived at:(i)Peak ground displacement estimates are quite sensitive to the constrained modulus, and a judicious selection of appropriate constrained modulus based on the magnitude of applied overpressure is recommended.(ii)Based on the presented case study, two guidelines are recommended to select the appropriate modulus: (a) shallow ground layers are likely to have modulus values determined by unconfined or triaxial compression test and deep ground layers are likely to have modulus values computed from seismic velocity test, and (b) constrained modulus increases with decreasing ratio of applied stress to overburden.(iii)Effect of attenuation should be accounted for low overpressures and may be neglected at higher overpressures (or high war head capacities).(iv)The velocity ratio affects mainly the time of occurrence of peak ground displacement and not the magnitude under higher overpressures. Therefore, the velocity ratio becomes important when design calculations utilize the complete displacement time-history such as in case of shock spectra.(v)A complete strain recovery is a better representation of actual conditions under low overpressure zones, and under higher overpressures, a partial strain recovery is recommended.(vi)The proposed model closely estimates the experimental values at all overpressures, whereas the UFC model significantly underestimates the peak ground displacements at high overpressures.

#### Notations

: | Attenuation coefficient at depth |

: | Strain at time at depth “z” |

: | Peak strain level in geomaterial at depth “z” |

: | Residual strain level in geomaterial at depth “z” |

: | Representative velocity ratio (between P-wave velocity and peak stress velocity) of ground media |

: | Ratio of P-wave velocity to wave velocity corresponding to the kth overpressure front at depth “z” |

: | Functional form for the loading branch of the stress-strain curve |

: | Ratio of P-wave velocity to wave velocity corresponding to the peak overpressure front at depth “z” |

: | Functional form for the unloading branch of the stress-strain curve |

: | Depth of the top ground layer |

: | Positive phase impulse of overpressure time-history |

: | Characteristic attenuation length |

: | Loading secant modulus |

: | Modulus of bottom layer |

: | Modulus of top layer |

: | Peak overpressure |

: | Pressure magnitude of the kth overpressure front |

: | Strain recovery ratio |

: | Bulk-density of geomaterials |

: | Scale factor |

: | Peak stress level in geomaterial at depth “z” |

: | Equivalent positive phase duration of linearly decaying overpressure time-history |

: | Time when the kth overpressure front arrives at the point of interest on ground surface |

: | Time of arrival of the last loading front to reach at depth H (by the time ) |

: | Positive phase duration of overpressure time-history |

: | Rise-time to peak overpressure in overpressure time-history |

: | Vertical ground displacement at time |

: | Representative P-wave velocity of ground media |

: | Representative wave propagation velocity of the peak overpressure front in ground |

: | Wave velocity of the kth stress front at depth “z” |

: | Yield of the explosion |

: | Depth penetrated by the kth overpressure front at time |

: | Stress at time at depth “z” |

: | Depth penetrated by initial pulse at time |

: | Depth penetrated by the peak overpressure front at time |

: | Tangential modulus at initial stress level at depth “z” |

: | Tangential modulus at peak stress level at depth “z.” |

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.