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A Designer's Approach for Estimation of Nuclear-Air-Blast-Induced Ground Motion.

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-over-pressure 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 AirBlast) 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 [Z.sub.0i] at time [t.sub.i]. Subsequently, other fronts arrive and penetrate into the ground. If the location of peak overpressure front at time [t.sub.i] is given as [Z.sub.pi], then, the ground above the depth [Z.sub.pi] experiences compressive stresses lower than those caused by the peak overpressure front at that depth. Therefore, the depths above [Z.sub.pi] are referred to as "unloading zone." However, the peak overpressure front is yet to reach at locations deeper than [Z.sub.pi], and therefore, the zone lying beneath the depth [Z.sub.pi] 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 [t.sub.i], contribution from all stress fronts arrived before time [t.sub.i] is considered. If the kth overpressure front arrives at the point of interest on ground surface at time [t.sub.k] such that [t.sub.k] [less than or equal to] [t.sub.i], then the kth overpressure front reaches at depth [Z.sub.ki] at time [t.sub.i]. 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 [Z.sub.ki] is given as [alpha] ([Z.sub.ki]), then the stress generated at depth [Z.sub.ki] can be written as a([Z.sub.ki])x P([t.sub.k]), where P([t.sub.k]) is the magnitude of the kth overpressure front. Depth of the penetration [Z.sub.ki] 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 kth stress front is denoted as [V.sub.kz] (a function of depth "z"), then (1) can be solved to obtain [Z.sub.ki] as a function of [t.sub.i], [t.sub.k], and [V.sub.kz] (2),

[mathematical expression not reproducible], (1)

[Z.sub.ki] = g([t.sub.i] - [t.sub.k], [V.sub.kz]) = [phi] ([t.sub.k]), (2)

where g is the function obtained by solving (1). It is noted that for a given [t.sub.i], [Z.sub.ki] is effectively a function ([phi]) of [t.sub.k] only because [V.sub.kz] is also a function of [t.sub.k]. 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 kth overpressure front at depth "z" is denoted as [f.sub.kz] = ([V.sub.pz]/[V.sub.kz]), then at initial stress levels [f.sub.kz] is equal to one. With increasing stress level, [V.sub.kz] decreases [17, 23]; hence, it is logical to assume that [f.sub.kz] 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 [f.sub.kz] 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 [f.sub.kz] is the same as [f.sub.rz]:

[mathematical expression not reproducible], (3)

where subscript "z" represents the depth coordinate. Using the average loading rate as [P.sub.o]/[t.sub.r], P([t.sub.k])/[P.sub.o] can be approximated as [t.sub.k]/[t.sub.r] (3). As wave propagation velocity through ground media is proportional to the square root of the tangential modulus [24], [f.sub.rz] is given by the following equation:

[mathematical expression not reproducible], (4)

where [mathematical expression not reproducible] is the tangential modulus at initial stress level at depth "z" and [mathematical expression not reproducible] 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 ([[epsilon].sub.iz]) can be determined using stress distribution ([[sigma].sub.iz]) and an appropriate stress-strain relationship [e.g., 25-27] as given in the following equation:

[mathematical expression not reproducible], (5)

Caption: Figure 1: Schematic representation of dilatational stress distribution in ground at time [t.sub.i]

where functions [f.sub.L] and [f.sub.U] 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 [u.sub.i] at time instant [t.sub.i], the strain distribution is integrated (6) from the ground surface to the penetrated wavelength in the ground up to time [t.sub.i]([Z.sub.0i]),

[mathematical expression not reproducible]. (6)

Substituting Z = [phi]([t.sub.k]) in (6) leads to the following equation:

[mathematical expression not reproducible]. (7)

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)),

P([t.sub.k]) = [P.sub.o](1 - [t.sub.k]/[t.sub.eq]), (8)

where [P.sub.o] and [t.sub.eq] 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.sub.L]) and strain recovery ratio (r) as parameters (Figure 2).

(c) Attenuation coefficient ([alpha]) is given by the following equation [8, 30]:

[alpha] = [(1 + z/[L.sub.w]).sup.-1], (9)

where [L.sub.w] - 0.971 x [W.sup.1/3] x [P.sup.-1/2.sub.o] x [V.sub.L],

where W is the yield of the explosion in kiloton and [V.sub.L] is the wave propagation velocity of the peak overpressure front.

(d) Using (4), [f.sub.rz] is determined as 1 for the bilinear stress-strain model. However, due to nonlinear stress strain behaviour, [f.sub.rz] is usually greater than 1. Wilson and Sibley [17] and Batdorf [23] recommended a range of 1.5 to 2. Hereafter, [f.sub.rz] is denoted as f and assumed to be constant with depth.

(e) A constant but representative P-wave velocity [V.sub.p] of ground media is assumed.

Using assumptions (a)-(e), integral for loading fronts in (7) (with [t.sub.r] [right arrow] 0) between general time instants [t.sub.x] and [t.sub.x+1] is written as [mathematical expression not reproducible], where L([t.sub.x] [right arrow] [t.sub.x+1]) can be shown to be given by the following equation:

[mathematical expression not reproducible]. (10)

It is to clarify that the overpressure time-history defined by the piecewise function P ([t.sub.k]) such that P ([t.sub.k]) = [P.sub.o][t.sub.k]/[t.sub.r] for [t.sub.k] [less than or equal to] [t.sub.r] and P([t.sub.k])= [P.sub.o] (1 - ([t.sub.k] - [t.sub.r]))/ ([t.sub.eq] - [t.sub.r]) for [t.sub.r] [less than or equal to] [t.sub.k] [less than or equal to] [t.sub.eq] converges to (8) in the limiting case when [t.sub.r] [right arrow] 0. Therefore, the expression L ([t.sub.x] [right arrow] [t.sub.x+1]) in (10) is first obtained for a case of finite [t.sub.r] and then it is evaluated for the case when [t.sub.r] [right arrow] 0 to determine the solution for overpressure time-history of (8). Similarly, the integral for unloading fronts in (7) (with [t.sub.r] [right arrow] 0) between general time instants [t.sub.x] and [t.sub.x+1] can be written as [mathematical expression not reproducible] where U ([t.sub.x] [right arrow] [t.sub.x+1]) can be shown to be given by the following equation:

[mathematical expression not reproducible]. (11)

By setting the appropriate values of [t.sub.x] and [t.sub.x+1] 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:

[mathematical expression not reproducible]. (12)

The closed-form solution is further extended to accommodate multiple ground layers. This requires the determination of time instants [t.sub.x] and [t.sub.x+1] during which the wave fronts pass through a particular layer. For illustration, a layered ground medium with an interface at depth H with the modulus of top layer as [M.sub.1] and modulus of bottom layer as [M.sub.2] 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.

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 ([P.sub.o]) is equal to the recorded peak overpressure, and equivalent positive phase duration [t.sub.eq] is given by the following equation:

[t.sub.eq] = 2[I.sub.p]/[P.sub.o], (13)

where [I.sub.P] 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 ([V.sub.p]) penetrates to a depth of [V.sub.p][t.sub.eq] for equivalent case, whereas for actual case, it penetrates to a depth of [V.sub.p][t.sub.p]. It is noted that the depth of the stressed zone in the equivalent case is smaller compared to the actual case as equivalent duration ([t.sub.eq]) is smaller than the actual positive phase duration ([t.sub.p]) (Figure 4). Thus, all depth-dependent parameters are scaled by a factor SF = [t.sub.p]/[t.sub.eq], such that the total depth of the stressed zone becomes SF x [V.sub.p][t.sub.eq] = [V.sub.p][t.sub.p] with the depth of the top layer being H/SF, and with a characteristic attenuation length of [L.sub.w]/SF. Based on the variation of P-wave velocity with depth at Frenchman Flat (Figure 5), the average P-wave velocity ([V.sub.p]) 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 [L.sub.w] (Table 4). Using the closed-form solution (shown in Table 1), parametric variations are studied (Table 5) and peak displacements (Table 6) are estimated.

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:

[u.sub.i|max] = [I.sub.p]/[rho][V.sub.p], (14)

where [rho] is the bulk density of geomaterials. To estimate peak displacements using (14), positive phase impulse ([I.sub.p]) is taken from Table 2. Representative P-wave velocity ([V.sub.p]) and density ([rho]) are taken as 658.69 m/s and 1331 kg/[m.sup.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

[alpha]([Z.sub.ki]):     Attenuation coefficient at
                         depth [Zk.sub.i]
[[epsilon].sub.iz]:      Strain at time ti at depth "z"
[[epsilon].sub.pz]:      Peak strain level in geomaterial
                          at depth "z"
[[epsilon].sub.rz]:      Residual strain level in
                         geomaterial at depth "z"
f:                       Representative velocity ratio (between
                         P-wave velocity and peak
                         stress velocity) of ground media

[f.sub.kz]:              Ratio of P-wave velocity to
                         wave velocity
                         corresponding to the kth
                         overpressure front
                         at depth "z"
[f.sub.L]:               Functional form for the loading
                         branch of the stress-strain curve
[f.sub.rz]:              Ratio of P-wave velocity
                          to wave velocity
                         corresponding to the peak
                         overpressure front at depth "z"
[f.sub.U]:               Functional form for the
                         unloading branch of
                         the stress-strain curve
H:                       Depth of the top ground layer
[I.sub.p]                Positive phase impulse of
                         overpressure time-history
[L.sub.w]                Characteristic attenuation length
[M.sub.L]:               Loading secant modulus
[M.sub.2]:               Modulus of bottom layer
[M.sub.1]:               Modulus of top layer
[P.sub.o]:               Peak overpressure
P([t.sub.k]):            Pressure magnitude of the
                         kth overpressure front
r:                       Strain recovery ratio
[rho]:                   Bulk-density of geomaterials
SF:                      Scale factor
[[sigma].sub.pz]         Peak stress level in
                         geomaterial at depth "z"
[t.sub.eq]               Equivalent positive phase
                         duration of linearly
                         decaying overpressure time-history
[t.sub.k]                Time when the kth overpressure
                         front arrives
                         at the point of interest
                         on ground surface
[t.sub.kH]:              Time of arrival of the last
                         loading front to
                         reach at depth H (by the time tf)
[t.sub.p]                Positive phase duration of overpressure
                         time-history
[t.sub.r]:               Rise-time to peak overpressure in
                         overpressure time-history
[u.sub.i]:               Vertical ground displacement
                         at time [t.sub.i]
[V.sub.p]:               Representative P-wave velocity of ground
                         media
[V.sub.L]:               Representative wave propagation
                         velocity of the peak overpressure
                          front in ground
[V.sub.kz]:              Wave velocity of the kth stress
                         front at depth "z"
W:                       Yield of the explosion
[Z.sub.ki]               Depth penetrated by the kth overpressure
                         front at time [t.sub.i]
[[delta].sub.iz]         Stress at time ti at depth "z"
[Z.sub.0i]:              Depth penetrated by initial pulse
                         at time [t.sub.i]
[Z.sub.pi]:              Depth penetrated by the peak
                         overpressure front at time [t.sub.i]
([partial derivative]    Tangential modulus at initial stress
 [[sigma].sub.z]/        level at
 [partial derivative]    depth "z"
 [[epsilon].sub.z])|
 .sub.[sigma]=0]:

[mathematical not       Tangential modulus at peak
reproducible]           stress level at depth "z."


https://doi.org/10.1155/2018/3029837

Conflicts of Interest

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

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Shashank Pathak [ID] and G. V. Ramana

Department of Civil Engineering, Indian Institute of Technology Delhi, New Delhi, India

Correspondence should be addressed to Shashank Pathak; shashankpathaks@gmail.com

Received 19 August 2017; Accepted 5 December 2017; Published 8 March 2018

Academic Editor: Chiara Bedon

Caption: Figure 1: Schematic representation of dilatational stress distribution in ground at time [t.sub.i].

Caption: Figure 2: Schematic diagram for stress-strain relation at a depth "z."

Caption: Figure 3: Scaling of equivalent ground media.

Caption: Figure 4: Measured and equivalent (linear) overpressure time-histories for shot Priscilla at stations (a) P1, (b) P2, (c) P3, and (d) P4 (measured overpressure time-histories [21]).

Caption: Figure 5: P-wave velocity and geomaterial density profile at Frenchman Flat, NTS [21].

Caption: Figure 6: Variation of error in estimated peak displacement using constrained modulus determined from different experimental techniques: (a) T3 and T4 and (b) T1, T2, T5, and T6. T1: deduced from observed ground motion; T2: calculated from seismic velocity; T3: triaxial tests: initial loading; T4: dynamic 1-D compression test: initial loading; T5: triaxial tests: reloading; T6: resonant column test.

Caption: Figure 7: Variation of peak ground displacement with attenuation characteristic length [L.sub.w].

Caption: Figure 8: Effect of (a) attenuation; (b) strain recovery ratio; (c) velocity ratio on estimated ground displacement; (d) velocity ratio on time of occurrence of peak displacement.

Caption: Figure 9: Comparison between measured and estimated peak ground displacements.
Table 1: Displacement solutions for the double-layered media.

Case              Explanation              Displacement
                                           equation

1            When all loading and
        unloading fronts are in layer-1
             ([t.sub.i] [less than
           or equal to] H/[V.sub.p[).
           When some loading fronts
            have crossed layer-1 and
           remaining loading fronts        [mathematical
           along with all unloading        expression not
      fronts are in layer-1 (H/[V.sub.p]   reproducible]
         < [t.sub.i] < fH/[V.sub.p]).
                   Here, the
          first task is to determine
          those loading fronts which
            have crossed layer-1.          [mathematical
       This can be obtained by equating    expression not
2           loading front velocity         reproducible]
       multiplied with travel time with
           the depth of layer. The
         loading front velocity of the
              kth front can be
             written using (3) as
        [V.sub.p]/{ (f - 1)([t.sub.k]/
       [t.sub.r]) + 1}, and travel time
          is ([t.sub.i] - [t.sub.k]).
            Thus, the last loading
       front to reach at depth H by the
       time [t.sub.i] would be given by
       [t.sub.kH] = [t.sub.r] ([V.sub.p]
           [t.sub.i] - H)/{H(f - 1)
            + [V.sub.p] [t.sub.r]}.
           When all loading fronts
            are in layer-2 and some
            unloading fronts have
             crossed layer-1 (fH/
            [V.sub.p] < [t.sub.i]).
3        Similar to case 2, the last
          unloading front to reach at
             depth H by the time
          [t.sub.i] would be given by
            [t.sub.kH] - [t.sub.i]         [mathematical
             - fH/[V.sub.p].               expression not
                                           reproducible]

Table 2: Details of recording stations.

Station          Range          Peak        Positive phase
designation       (m)        overpressure   duration (sec)
                                (kPa)

P1                198            1863            0.196
P2                259            1290            0.179
P3                320            828             0.307
P4                411            408             0.442

Section         Positive      Equivalent      Scale factor
designation   phase impulse     duration          (SF)
               (kPa-sec)        (sec)

P1               88.32          0.0948           2.07
P2               62.18          0.0964           1.86
P3               50.54          0.1221           2.51
P4               48.34          0.2371           1.86

Table 3: Constrained modulus of playa silt at Frenchman Flat [4].

Method                                Notation   Constrained modulus
                                                      (MPa)

                                                  For         For
                                                depths       depths
                                                 less         more
                                                than 18 m   than 18 m

Deduced from observed ground motion     T1        96.6        552
Calculated from seismic velocity        T2        242        1035
Triaxial tests: initial loading         T3        55.2       103.5
Dynamic 1-D compression test:
 initial loading                        T4         69         138
Triaxial tests: reloading               T5        276        448.5
Resonant column test                    T6        276         759

Table 4: Scaling of attenuation characteristic length ([L.sub.w]).

Station         W    [P.sub.o]   [V.sub.p]   [L.sub.w]     Scaled
designation   (kt)    (kPa)        (m/s)       (m)       [I.sub.w]
                                                           (m)

P1                    1863        658.69      48.58       23.47
P2           37       1290        658.69      58.38       31.44
P3                     828        658.69      72.88       28.98
P4                     408        658.69     103.84       55.70

Table 5: Parametric studies carried out on the proposed
closed-form solution.

Parameter      Parametric study

[M.sub.1]      Parameters [M.sub.1] and [M.sub.2] based on
and            six different methods are adopted from
[M.sub.2]      Table 3.

               Other parameters kept constant at
               r = 0.6; f = 2; [L.sub.w] adopted from Table 4.

               Variation of absolute percentage errors
               in estimated peak ground displacements
               is plotted against peak overpressures in
               Figures 6(a) and 6(b).

[L.sub.w]      [L.sub.w] is varied from 10m to 250m.
               Other parameters kept fixed at [M.sub.1] and
               [M.sub.2] adopted from Table 6; f = 2; r = 0.6.
               Estimated peak displacements are
               plotted against [L.sub.w] as shown in
               Figure 7. Errors in estimated peak
               displacements for the attenuating medium and
               non-attenuating medium are also plotted
               against peak overpressure in Figure 8(a).

r              Two cases are considered: (i) full strain
               recovery r = 1 and (ii) partial strain
               recovery r = 0.6.

               Other parameters fixed at [M.sub.1] and
               [M.sub.2] adopted from Table 6; f = 2;
               [L.sub.w] adopted
               from Table 4.

               Errors in estimated peak displacement
               for the two cases are plotted as shown in
               Figure 8(b).

f              f is varied from 1.0 to 2.2.

               Other parameters fixed at [M.sub.1] and
               [M.sub.2] adopted from Table 6; r = 0.6;
               [L.sub.w] adopted from Table 4.
               Estimated peak displacements are
               plotted against f (Figure 8(c)).
               Error in occurrence time of peak
               displacement are also plotted against f
               (Figure 8(d)).

               Observation

[M.sub.1]      Peak ground displacements are very
and            sensitive to the constrained modulus
[M.sub.2]               value.

               Average coefficient of variation in peak
               displacement estimates = 83%.

               Estimates are close to measured values if
               higher modulus values are used for
               smaller overpressures.

[L.sub.w]      As [L.sub.w] increases (or [alpha] decreases), peak
               displacement increases.

               Beyond [L.sub.w] of 250m, the peak ground
               displacement does not increase, and
               [L.sub.w] [greater than or equal to] 250 is
               considered as the
               non-attenuating medium.

r              Assumption of full strain recovery gives
               less errors as compared to partial strain
               recovery under low overpressures.

f              Under high overpressure (P1 and P2),
               error in occurrence time of peak
               displacement reduces significantly with
               increasing f.

               Error increases marginally under lower
               overpressures as the velocity ratio is
               close to 1 under lower stress (in (3) when
               [t.sub.k] [right arrow] 0).

               Recommended value

[M.sub.1]      Observed results are consistent with the
and            observations of Wilson and Sibley [17].
[M.sub.2]
               (1) 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 or resonant column test.+e
               justification to this variation in selection
               of modulus values can be attributed to
               the small strains associated with deeper
               layers and higher strains at shallow
               depths.

               (2) Constrained modulus increases with
               decreasing ratio of applied stress to
               overburden. For higher overpressures,
               the overstress ratio would be higher and
               therefore modulus value will be lower
               compared to lower overpressures. An
               optimal choice of constrained modulus
               values is adopted as shown in Table 6.
               The computer displacements are found
               to be in good agreement with measured
               displacements (Table 6).

[L.sub.w]      Attenuation has to be taken into account
               under low overpressures, and it can be
               ignored under high overpressures.

r              Under low overpressure, ground is not
               stressed beyond its elastic limit, and
               hence full strain recovery is a better
               representation of actual conditions in
               low overpressure zones.

               A lower strain recovery causes higher
               permanent deformations and increases
               peak ground displacement compared to
               elastic case (i.e., unit strain recovery
               ratio). Under higher overpressures, the
               ground is stressed beyond its elastic limit
               and the assumption of partial strain
               recovery is recommended.

f              f has insignificant effect on magnitude of
               estimated peak displacements. However,
               as the velocity ratio increases, the rise-
               time of overstress pulse with depth also
               increases and affects the occurrence time
               of the peak displacement.

Table 6: Suggested combination of modulus values and corresponding
estimated peak displacements.

Station       [M.sub.1] (MPa)   [M.sub.2] (MPa)   Estimated
designation    (depths less      (depths more     peak
                 than 18 m)        than 18 m)     displacement
                                                   (mm)

                   55.2              552
P1           triaxial tests:   deduced from        235.59
             initial loading     observed
                                ground motion

                    69               759
P2             dynamic 1-D                          147.68
               compression     resonant column
              test: initial          test
                 loading

                    69               759
P3             dynamic 1-D     resonant column      74.79
               Compression          test
              test: initial
                 loading

                   96.6              1035
P4            deduced from     calculated from      38.49
                 Observed      seismic velocity
              ground motion

               Experimental           %
               records (mm)         error

P1                250.95             6.12

P2                157.56             6.27

P3                 68.75              8.79

P4                 33.95             13.37
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Title Annotation:Research Article
Author:Pathak, Shashank; Ramana, G. V.
Publication:Advances in Civil Engineering
Date:Jan 1, 2018
Words:5913
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