Performance of N-Z systems in the mitigation of underground blast induced vibration of structures

Abstract

Control of structural vibration due to wind and earthquake forces has been extensively researched. However, studies on vibration control of structures against underground blasts are limited. It is thereby important to study the usefulness of established vibration control technologies in the blast resistant design of structures. In the present work, an attempt is made to study the effectiveness of the New-Zealand (N-Z) type base isolator (BI) in mitigating structural vibration effects due to blast induced ground motion (BIGM). The BIGM is modeled by an exponentially decaying function representative of a typical rock blast, and the isolator by a bilinear model. The influence of the various BI parameters on its performance is examined in the time domain by considering a single-degree-of-freedom structural model and a realistic five-story building. Numerical studies reveal that the N-Z base isolation system is highly effective in reducing both the underground blast induced structural acceleration as well as the displacement. It is noted that the peak displacement and permanent deformation of the BI may be crucial design criteria. These may, however, be restricted by selecting a proper combination of the BI parameters without significantly compromising the response reductions obtained by it.

Keywords

Base isolation N-Z system bilinear model underground blast induced ground motion vibration control

1. Introduction

Underground explosions due to general engineering construction and mine blasting generate ground vibrations that can cause damage to nearby structures. Furthermore, due to the scarcity of land in many parts of the world, nowadays various facilities, like ammunition storage, air raid shelters, oil and water storage caverns, etc., need to be built underground to free the surface land for other uses. Accidental blasting of underground ammunition storage is another source of significant damage to adjacent structures. Depending upon the intensity of explosion, distance of the structure from the blasting source, soil characteristics, and the type of structures, the damage may range from severe to minor, such as that of nonstructural components, together with other effects on user comfort. Thus, the evaluation of the performance of structures under underground blast induced vibration is gaining importance. Constraints on space and cost as well as safety issues have limited the number of experimental investigations on structural response and damage due to blasting, though there are several simulation studies on this, such as by Dowding (1996), Hao and Wu (2001), Carvalho and Battista (2003), Wu and Hao (2005), and Hao and Wu (2005). To protect structures from the effects of underground explosions, various codes and regulations, like DIN 4150 (1984), DOD 6055 (1992), and the NATO (1993), recommend different criteria that were predominantly established from empirical correlations between observed damage and recorded peak velocities during field blast tests. These criteria have many lacunae as they do not consider the inherent structural condition, site condition, and ground motion features. Hence, the protection of structures subjected to BIGM needs careful attention.

Over the last few decades, there have been extensive efforts by researchers in the development of vibration control technologies, such as those of base isolation, viscous fluid dampers, and tuned mass and liquid dampers, for structural protection against earthquake and wind loading. However, the same is not the case for underground blast load. It may be noted that, even though earthquakes and underground blast both cause ground vibrations, blast pulses are extremely dissimilar from seismic motions as they are characterized by large amplitude, high frequency waves and are of extremely short duration. Moreover, as the epicentral distance in case of BIGM is much less than that of seismic ground motion, the attenuation and spatial variation of the underground blast action are more significant than those of the latter over the same propagation distance (Wu and Hao, 2005). While the structural response to earthquake ground motion is dominated mainly by the lower frequency modes with little contribution from the higher modes, in case of blast ground motion, during the major shock period the higher frequency modes are predominantly excited. Thereby, the higher frequency modes govern the structural response in the forced-vibration phase while the structural response in the free vibration phase is mainly due to the lower frequency modes (Lu et al. 2001, 2002). Consequently, a structure subjected to BIGM will experience large accelerations in the forced-vibration phase and, if it survives this phase, will undergo large displacements in the free vibration phase (Dhakal and Pan, 2003). Hence, specific studies are needed on the applicability of established vibration control methods to structures subjected to underground blast. There are some limited studies in the open literature on vibration control techniques for blast loading. Miyamoto and Taylor (1999) evaluated the effectiveness of fluid viscous dampers in the reduction of responses of steel buildings subjected to air blast. Wu et al. (2004) proposed the mitigation of ground shock effects on structures by the provision of a sand layer in the foundation of the structure. The sand base was found to be successful in reducing the structural response and damage. This particular study was a motivating factor for the present paper on exploring the effectiveness of base isolation in blast vibration protection.

One of the most established and widely implemented base isolator (BI) devices is the lead-plug or N-Z bearing which was first introduced in New Zealand and is basically a laminated rubber bearing with a central lead core. Large numbers of experimental as well as analytical studies by various researchers (Fan et al., 1991; Jangid, 2010; Kulkarni and Jangid, 2002; Matsagar and Jangid, 2004; Naeim and Kelly, 1999; Ramallo et al., 2002; Rao and Jangid, 2001, and others) have underlined the effectiveness of the N-Z system in controlling the structural vibration due to seismic excitation. Interestingly, there are some studies on base isolated structures subjected to near-fault earthquakes which are characterized by large, high-energy, pulse-like signals, similar to blast motions. Though earlier studies by Heaton et al. (1995), Hall et al. (1995), and others, were discouraging due to excessive bearing displacements, Jangid and Kelly (2001), Alhan and Gavin (2003), and Jangid (2007) reported that lead–rubber bearings with appropriate properties are quite effective for seismic isolation of structures for near-fault motions. While Jangid and Kelly (2001) showed that there exists an optimum damping of the isolation system for which the superstructure acceleration attains a minimum value, Alhan and Gavin (2003) observed that for a known near field earthquake record, an appropriate combination of isolation stiffness and isolation damping may be adopted to limit the base drift without significantly increasing floor acceleration and inter-story drift. Further, the optimum yield strength of the N-Z system under different system parameters was derived by Jangid (2007). Supplemental viscous dampers to mitigate the large isolator displacements under near-fault ground motion were studied by Providakis (2009). Noting the potential of base isolation in mitigating structural vibration due a to near-fault earthquake, it was felt that it would be meaningful to study this technology for the control of structural vibrations due to underground explosion.

In this paper, an attempt is made to study the performance of the N-Z type BI in mitigating structural vibration effects due to BIGM. In doing so, the superstructure is first modeled as a linear, viscously damped SDOF system. The force-deformation behavior of the N-Z BI is represented by Bouc–Wen’s bilinear model (Bouc, 1967; Wen, 1976). Parametric studies are carried out on the base isolated system and the influence of various important BI parameters such as the isolation period, yield strength, rigidity ratio, and damping ratio are observed. Variations in the intensity of the blast input and the flexibility of the superstructure are also taken up. The purpose of the parametric study is to find the existence of optimum parameters, if any, by considering the minimization of both the peak absolute acceleration and peak base displacement as the optimality criteria. The study is extended to a five-story building.

2. Modeling of the structure-base isolator system

For a multi-story shear type building on an N-Z type BI, as shown in Figure 1, the equation of motion of the system under BIGM can be written as

[M] {\overset{··}{x}} + [C] {\overset{\cdot}{x}} + [K] {x} = - [M] {1} ({\overset{··}{x}}_{g} + {\overset{··}{x}}_{b})

(1)

where

[M]

[C]

, and

[K]

are the mass, damping, and stiffness matrices of the superstructure of order NxN, where N is the number of floors above base,

{x} = {x_{1}, x_{2}, \dots, x_{N}} T

is the displacement vector of the superstructure, and x_j is the jth floor displacement measured with respect to the base mass.

{1}

is the influence coefficient vector, elements of which are all unity.

{\overset{\cdot}{x}}

and

{\overset{··}{x}}

are the velocity and acceleration vectors of the superstructure, respectively.

{\overset{··}{x}}_{g}

is the blast induced horizontal ground acceleration and

{\overset{··}{x}}_{b}

is the acceleration of the BI measured with respect to the ground. When the base isolation layer is added at the foundation level, the structure with N-DOF is augmented to a (N + 1)-DOF system. The equation of motion for the base mass can be written as

m_{b} {\overset{··}{x}}_{b} + F_{b} - c_{1} {\overset{\cdot}{x}}_{1} - k_{1} x_{1} = - m_{b} {\overset{··}{x}}_{g}

(2)

where

m_{b}

is the mass of the BI,

F_{b}

is the restoring force exerted by the N-Z system, and c₁ and k₁ are the first floor damping and stiffness of the superstructure, respectively. The ratio of mass of the BI to the total mass of the super-structure is given by μ and is expressed as

μ = \frac{m_{b}}{m_{s}} = \frac{m_{b}}{\sum_{j = 1}^{N} m_{j}}

(3)

where m_j is the mass of the jth floor.

Figure 1.

Idealized building model on N-Z bearing.

2.1. Mathematical model of the N-Z system

In the N-Z type base isolation system, the lead core provides an additional damping to reduce the bearing displacement, and energy dissipation occurs due to yielding of the lead core. Here, the force deformation behavior is nonlinear and is generally expressed by the bilinear model shown in Figure 2. This behavior can be approximated by Bouc–Wen’s equation (Bouc, 1967; Wen, 1976). The horizontal force ( $F_{b}$ ) induced in the N-Z system in the post-yield phase can be expressed as the sum of three forces acting in parallel as follows

F_{b} = c_{b} \overset{\cdot}{x}_{b} + α k_{b} x_{b} + (1 - α) F_{y} Z

(4)

where

c_{b}

represents the viscous damping and

k_{b}

is the initial stiffness of the N-Z bearing,

x_{b}

and

\overset{\cdot}{x}_{b}

, respectively, represent the displacement and velocity of the bearing measured relative to ground, α is the rigidity ratio, i.e., the ratio of the post- to pre-yield stiffness, F_y is the yield force of the lead plug. Z is the nondimensional hysteretic component and is expressed by the following nondimensional first order differential equation

q \overset{\cdot}{Z} = A {\overset{\cdot}{x}}_{b} - γ | {\overset{\cdot}{x}}_{_{b}} | Z | Z | η - 1 - β {\overset{\cdot}{x}}_{_{b}} | Z | η

(5)

where q is the yield displacement of the bearing and is given by

F_{y} / k_{b}

. β, γ, and A are the nondimensional parameters of the hysteresis loop. η is an integer which controls the transition from elastic to plastic response. α is the rigidity ratio, i.e., the ratio of the post-yield to pre-yield stiffness. The parameters α, β, γ, η, and A control the shape of the loop and are selected such that the predicted response from the model closely matches with the experimental results. Here the extent of nonlinearity depends on the rigidity ratio ‘α’ and the initial yield displacement ‘q’ which is dependent on the normalized yield strength of the N-Z system. It may be noted from equation (4) that the value of α is within the range

0 \leq α \leq 1

and a lower value of α is associated with greater elasto-plastic behavior, i.e., a greater degradation of stiffness beyond yield. For the same initial stiffness (

k_{b}

) yield displacement is proportional to the normalized yield strength. The parameters adopted for the present study are β = γ = 0.5, A = 1, and η = 1 as considered by Jangid (2010). α is selected in such a way that the design value of the post-yield stiffness provides the specific value of the isolation period,

T_{b}

, of the N-Z bearing and can be expressed as

T_{b} = 2 π \sqrt{\frac{M_{0}}{α k_{b}}}

(6)

where

M_{0} = (m_{b} + m_{s})

= total mass of the building. The damping ratio of the bearing (

ξ_{b}

) is expressed as

ξ_{b} = \frac{c_{b}}{2 M_{0} ω_{b}}

(7)

where

ω_{b} = \frac{2 π}{T_{b}}

is the base isolation frequency. Thus, the N-Z bearing is characterized by the isolation period

(T_{b})

, damping ratio

(ξ_{b})

, and normalized yield strength

(F_{0})

of the bearing where F₀ is expressed as

F_{0} = \frac{F_{y}}{W}

(8)

in which

W = M_{0} g

is the total weight of the isolated building and g is the acceleration due to gravity. Now equation (2) can be restructured by substituting the expression of

F_{b}

(as given by equation (4)) as follows

\begin{matrix} {\overset{··}{x}}_{b} = \frac{c_{1}}{m_{b}} {\overset{\cdot}{x}}_{1} + \frac{k_{1}}{m_{b}} x_{1} - \frac{c_{b}}{m_{b}} {\overset{\cdot}{x}}_{b} - α \frac{k_{b}}{m_{b}} x_{b} \\ - (1 - α) \frac{F_{0} (m_{b} + m_{s}) g}{m_{b}} Z - {\overset{··}{x}}_{g} \end{matrix}

(9)

By using equation (9), equation (1) leads to the following

\begin{matrix} {\overset{··}{x}} = [M] - 1 [C] {\overset{\cdot}{x}} - [M] - 1 [K] {x} - {1} (\frac{c_{1}}{m_{b}} {\overset{\cdot}{x}}_{1} + \frac{k_{1}}{m_{b}} x_{1} \\ - \frac{c_{b}}{m_{b}} {\overset{\cdot}{x}}_{b} - α \frac{k_{b}}{m_{b}} x_{b} - (1 - α) \frac{F_{0} (m_{b} + m_{s}) g}{m_{b}} Z) \end{matrix}

(10)

The fourth-order Runge–Kutta method is applied for the solution of the differential equations of motion of the base-isolated system for a given blast input. A small time step,

Δ t

, is required for convergence. In the numerical study carried out in Section 4,

Δ t = 0.002

is considered.

Figure 2.

Force–deformation behavior of N-Z bearing.

3. Modeling of blast induced ground motion

In the present study, the BIGM generated typically from an underground explosion is modeled by an exponentially decaying function by following Carvalho and Battista (2003). The blast loading on the structure, $F (t) = [M] {1} {\overset{··}{x}}_{g} (t)$ , is obtained from the blast induced ground acceleration, ${\overset{··}{x}}_{g} (t)$ , as expressed by

{\overset{··}{x}}_{g} (t) = - (1 / t_{d}) \bar{\overset{\cdot}{}} x_{g} \exp (- t / t_{d})

(11)

In the above equation,

\bar{\overset{\cdot}{}} x_{g}

is the peak particle velocity (PPV) and

t_{d} = R / C_{p}

is the arrival time. R and

C_{p} = \sqrt{E / ρ}

denote the distance to the charge center and propagation velocity of a wave in soil or rock, respectively. E is the Young’s Modulus and ρ is the average mass density of the rock or soil mass. The value of PPV is computed from the empirical attenuation relation given by Wu and Hao (2005). The relation, considering a granite site, is reproduced below in equations (12) and (13).

PPV = 2.981 f_{1} (R / Q (1 / 3)) - 1 . . 3375

(12)

with

f_{1} = 0.121 (Q / V) 0.2872

(13)

Here, f₁ is the decoupling factor for

PPV

, Q is the TNT charge weight in kilograms, V is the volume of the charge chamber in cubic meters, and R is measured in meters.

For the blast input model, typical values of the volume of charge chamber (V), charge center distance (R), and propagation velocity of wave (C_p) are taken as 1000 m³, 50 m, and 5280 m/s (for granites rock), respectively. Three different blast input acceleration time histories are considered for the present numerical study (named as cases A, B, and C) corresponding to three values of TNT charge weight (Q), namely 10 ton, 50 ton, and 100 ton. The PPV of the three types of input motions are calculated as 22.66 cm/s, 73.74 cm/s, and 122.56 cm/s, respectively. The resulting ground accelerations are shown in Figure 3.

Figure 3.

Input acceleration time histories.

4. Numerical study

In this section, the performance of the base isolation system in mitigating the underground blast induced ground vibrations of the superstructure modeled both as a SDOF system and a MDOF system is illustrated though some numerical examples. At first, the variations in the design parameters of the BI as well as the superstructure and blast input properties are considered to analyze their influences on the peak response reductions of the SDOF model of the superstructure. In the subsequent section, a five-story building idealized as a MDOF shear building is considered as a case study.

4.1. SDOF system on N-Z type BI

A SDOF system on N-Z type BI subjected to BIGM, as shown in Figure 3, is considered. The model of the isolated structural system under consideration is characterized by the parameters, namely the fundamental time period of the superstructure (T_s), the damping ratio of the superstructure ( $ξ_{s}$ ), the fundamental period of the base isolated system ( $T_{b}$ ), the damping ratio of the BI ( $ξ_{b}$ ),the mass ratio (μ), which is the ratio of mass of the BI to the total mass of the structure, the normalized yield strength of the bearing (F₀), and the rigidity ratio (α). Unless mentioned otherwise, the system parameters are taken as: T_s = 0.5 s, $ξ_{s}$ = 2%, $T_{b}$ = 2.5 s, $ξ_{b}$ = 5%, μ = 0.2, F₀ = 0.05, and α = 0.05. Figures 4 and 5 represent the displacement and acceleration time histories of the fixed base and isolated system with the aforesaid parameters subjected to type C blast input (see Figure 3), which has the highest intensity of the blast input acceleration. These two plots indicate that the peak acceleration and the peak displacement are very effectively reduced by the base isolation system. Similar observations are obtained for the response time histories of the other two cases of BIGM (not shown here). The superstructure comes to rest for types of input A, B, and C in about 16 s, 10 s, and 8 s respectively. As the blast intensity increases, the percent response reductions increase and the superstructure takes less time to come to rest. The quantitative estimate of the response reductions for all the three cases of blast input are presented in Table 1. As mentioned earlier, the peak acceleration response is very significant for the structure subjected to blast. It is clear from Table 1 that the BI is able to reduce the peak acceleration as well as the peak displacement of the structure subjected to BIGM. It may also be noted that the percent reduction of peak displacement and that of absolute acceleration are almost the same. This is apparently surprising as the isolator is nonlinear. However, Kelly et al. (2010) have found that the bilinear system (which has been the model for N-Z type BI here) may be described by an ‘effective’ period and an ‘effective’ damping factor, which, although approximate, work well in the determination of the displacements and base shear of an isolated structure. Thus, if it is possible to represent a bilinear isolator by an ‘equivalent’ linear isolator, then it may be reasonable to expect that the transmissibility of displacement and acceleration, which are identical in case of the classic linear system subjected to harmonic input, will show similar features in the case under discussion. Ramallo et al. (2002) have also reported similar patterns of reduction in structural drifts and structural acceleration in their work using lead–rubber bearings modeled using the Bouc–Wen model.

Figure 4.

Displacement time history for T_b = 2.5 s (case C input).

Figure 5.

Acceleration time history for T_b = 2.5 s (case C input).

Table 1.

Peak responses reduction and peak base displacement of 2-DOF base isolated system for three cases of blast input.

Type of blast input	Reduction in peak displacement of floor mass w.r.t. base (%)	Reduction in absolute peak acceleration of floor mass (%)	Peak base displacement (cm)
Case A	64.46	64.43	4.10
Case B	77.43	77.44	20.95
Case C	78.63	78.64	38.50

As discussed earlier, base isolated buildings are vulnerable to large pulse-like ground motions (similar to the impulsive BIGM) generated at a near-fault location due to large isolator displacements (Heaton et al., 1995; Jangid and Kelly, 2001). It is thus important to examine the peak bearing displacement in achieving high response reductions of the superstructure, as observed in Figures 4 and 5. Base displacement time histories for all three cases of blast input are shown in Figure 6 and the values of the peak base displacements are also given in Table 1. It is observed from Table 1 as well as from Figure 6 that for smaller intensities of blast input (case A), the peak base displacements are very small compared to those due to higher intensities of blast input. The peak base displacement of N-Z type BI under seismic vibration can be noted here to compare with that due to underground blast. For example, for the three earthquake records of Loma Prieta, Northridge, and Kobe with peak ground accelerations (PGAs) of 0.57 g, 0.6 g, and 0.86 g, respectively, peak base displacements of a N-Z type base isolated system (T_s = 0.5 s, $T_{b}$ = 2.5 s, q = 2.5 cm) have been reported as 69.34 cm, 57.06 cm, and 35.48 cm, respectively (Matsagar and Jangid, 2004). Hence, unless the blast loading intensity is very high (i.e., higher than in case C), peak base displacements are within the moderate to low ranges as compared to those from earthquake loading. Base displacement time histories in Figure 6 also show that there is a permanent deformation of the BI, the magnitude of which is not necessarily proportional to the corresponding blast intensity. This indicates the necessity of an optimization study for a suitable combination of BI parameters that would minimize the peak acceleration while restricting the peak base drift as well as the permanent deformation of the BI for a particular blast input. This will involve the solution of a constrained optimization problem which needs further study. It may also be noted that the present study is based on a deterministic description of the parameters, characterizing the structural and the blast load model. The robustness with respect to efficiency will be affected due to the unavoidable presence of uncertainty in the intensity and duration of the blast and in the structural system parameters. Therefore, the design of the BI system for robust performance should consider the effect of uncertain parameters. To address the problem in a more formal way, it is possible to formulate the problem as a bi-objective optimization where the mean value and associated variance of the response due to uncertain parameters will be the two objectives, i.e., the so-called robust optimization approach. In this way, one can achieve the desired optimum BI system from a set of Pareto solutions by ensuring the desired level of robustness in the design. This is beyond the scope of the present paper and needs advanced investigation.

Figure 6.

Base displacement time history for T_b = 2.5 s (cases A, B, and C input).

Now, the influence of various parameters on the performance of the base isolation system is studied. The parametric studies are chiefly presented for the case C blast input.

To study the effect of superstructure flexibility, the reductions in peak absolute acceleration due to base isolation for varying T_s are plotted in Figure 7 for three sets of isolation periods ( $T_{b}$ ), namely 1.5 s, 2.0 s, and 2.5 s. Other system parameters are taken as before. It is observed in Figure 7 that, with the increase in the flexibility of the superstructure, the peak absolute acceleration reduction decreases for all three cases of isolation period. This can be explained by the fact that for a fixed $T_{b}$ , the peak acceleration of the fixed base structure decreases with the increase in its flexibility and as a result a smaller amount of peak acceleration reduction is achieved for higher values of T_s. For example, for a particular value of isolation period, such as $T_{b}$ = 1.5 s, the peak absolute acceleration reduction is gradually decreased from 92% to 42% as T_s changes from 0.1 s to 1 s. Again, for a particular value of T_s, a larger amount of reduction can be achieved with a longer isolation period as a more flexible system is associated with a lower acceleration response. Here, for a fixed T_s (say 0.5 s) the acceleration response reduction is increased from 65% to 78% with the increase of $T_{b}$ from 1.5 s to 2.5 s. Figure 8 shows the variation of the peak base displacement with T_s for the three values of $T_{b}$ considered in Figure 7. The peak bearing displacement is observed to vary only slightly with T_s, as the peak drift of the BI drops a little with increasing T_s. However, the increase in the peak base displacement for higher value of $T_{b}$ , though expected, is quite significant. Similar observations are noted for the other two types of blast input.

Figure 7.

Variation of peak acceleration reductions with T_s for different T_b (case C input).

Figure 8.

Variation of peak base displacement with T_s for different T_b (case C input).

In order to understand the influence of the shape of the bilinear hysteresis loop of the BI, the normalized yield strength (F₀) and the rigidity ratio (α) are next considered in the parametric study. First, the variation of peak acceleration reduction with F₀ is presented in Figure 9 for three values of $T_{b}$ (1.5 s, 2 s, and 2.5 s). The results for both case A and case C blast inputs are presented. The other parameters remain the same as earlier. It is observed in Figure 9 that with the increase of F₀ the peak acceleration reduction decreases. For example, for case C input and $T_{b}$ = 1.5 s, the peak acceleration reduction achieved by the BI is brought down from 66% to 43% with an increase in the value of F₀ from 0.01 to 0.4. For case A, unlike case C, the effect of F₀ on the peak acceleration response is significant only for lower values of F₀ and negligible for higher values of F₀. This is because for case A, which corresponds to a lower blast input intensity base isolation, and therefore an acceleration response reduction is not achieved until the yield strength F₀ is low. However, in case C, as the blast input intensity is much more, base isolation takes place even for relatively higher values of F₀. This leads to a substantial acceleration response reduction at these higher values of F₀. For case B input, the reduction trend is similar to that of case C.

Figure 9.

Variation of peak acceleration reduction with F₀ for different T_b (cases A and C input).

The influence of F₀ on the peak base displacement is shown in Figure 10 for case C blast input. Here, the increase in F₀ proves beneficial as it reduces the bearing displacement. Similar trends are followed for the other two cases of blast input. The nature of these two figures can be explained by the fact that for higher values of F₀ (i.e., for higher yield strength F_y), the isolation system remains more in the elastic state (see Figure 2) which results in a lower flexibility in the structural system and smaller amount of energy dissipation. Consequently, the absolute acceleration of the superstructure increases while the bearing displacement decreases with increase of the isolator characteristic strength. Thus, the lower value of F₀ is more advantageous from a peak acceleration reduction point of view. But the resulting higher base drift at such lower F₀ values needs to be checked against the peak base displacement limit of that particular BI. Seismic codes such as UBC (1997) and IBC (2000) recommend the permissible limit of peak bearing displacement. Though no such code has been developed so far for the design of a base isolation system for blast vibration control, it can readily be recognized that the peak base displacement has to be controlled. Thus, a constrained optimization problem needs to be solved for the final selection of the value of F₀.

Figure 10.

Variation of peak base displacement with F₀ for different T_b (case C input).

As discussed earlier in Figure 6, the permanent set of the isolator is another parameter of concern for designing the base isolated system under BIGM. The base displacement time histories for the three cases of blast input are thereby plotted for three values of F₀ (= 0.05, 0.15, and 0.3) in Figure 11a–c. For this, only one value of $T_{b}$ (= 2.5 s) is taken and other system parameters are as in Figure 10. When the blast intensity is low (i.e., for case A input, as in Figure 11a), then lower values of F₀ leads to greater amounts of permanent set. But as the blast intensity increases (see Figure 11b and c for cases B and C, respectively), the vibrational response of the BI is such that the permanent set of the BI having lower values of F₀ is decreased, and for case C input the permanent set of the BI with the smallest F₀ (= 0.05) is the lowest, but it is not possible to predict the trend in which the permanent set will vary with F₀. Hence, in addition to the reduction of peak acceleration and peak base drift, the permanent set of the BI should also be considered while solving for F₀ in the constrained optimization problem for a particular case of blast input.

Figure 11.

Base displacement time histories for three values of F₀ for (a) case A blast input (T_b = 2.5 s) (b) case B blast input (T_b = 2.5 s), and (c) case C blast input (T_b = 2.5 s).

The rigidity ratio (α) is another factor that guides the shape of the hysteresis loop of the BI. The effects of α on the peak absolute acceleration reduction as well as on the base displacement are depicted in Figures 12 and 13, respectively, for the three sets of $T_{b}$ considered earlier. In Figure 12, results for both cases A and C type blast inputs are presented (as in Figure 9) while in Figure 13 results for case C type blast input are shown. It can be noted from Figure 12 that the peak absolute acceleration reductions for higher intensities of blast input remain almost unchanged for different values of α corresponding to a fixed isolation period. But for a lower intensity of blast input, as in case A, these reductions increase significantly with α. The trend for a case B type input (not shown) is the same as that of case C. From Figure 13 it is observed that the peak bearing displacement increases only slightly with α and the same is true for the other two cases of blast input as well. It thus appears from Figures 12 and 13 that, for higher intensities of blast input, the peak responses are almost independent of the rigidity ratio, though for lower intensities of input, a higher α is more effective in reducing the peak acceleration of the superstructure. However, as will be seen later in Figures 17–19 for higher values of F₀, the effect of rigidity ratio is more pronounced both for the peak acceleration reduction as well as for the peak base drift, irrespective of the blast input intensity.

Figure 12.

Variation of peak acceleration reduction with α for different T_b (case A and case C input).

Figure 13.

Variation of peak base displacement with α for different T_b (case A and case C input).

Figure 17.

Peak displacement reduction for different F₀ and α for five-story base isolated building (case C input).

Figure 19.

Peak base displacement for different F₀ and α for five-story base isolated building (case C input).

Finally, the effects of the damping ratio of the BI, $ξ_{b}$ , on the reduction of peak acceleration as well as on the resulting maximum base displacement are illustrated in Figures 14 and 15, respectively. The responses are plotted for three sets of $T_{b}$ as earlier. From Figure 14 it is noted that an optimum value of $ξ_{b}$ exists so far as minimization of peak absolute acceleration response is concerned. However, the peak bearing displacement decreases monotonically with increases of $ξ_{b}$ . Thus, the increase of $ξ_{b}$ beyond the optimum value indicated in Figure 14 may decrease the bearing displacement, but will transmit higher acceleration into the superstructure. It is also observed in Figure 14 that the optimum value of $ξ_{b}$ is dependent on the value of $T_{b}$ . The results for the other two cases of blast input are similar to those given above.

Figure 14.

Variation of peak acceleration reduction with ξ_b for different T_b (case C input).

Figure 15.

Variation of peak base displacement with ξ_b for different T_b (case C input).

4.2. MDOF system on N-Z type BI

A five-story base isolated building, modeled as a shear building, subjected to BIGM is studied further. The mass, stiffness, and damping coefficient of each floor as detailed in Table 2 are adopted from Kelly et al. (1987) and these structural parameters are kept constant in this study. The fundamental time period (T_s) of the first mode of the building is 0.3 s. The mass participation factors are obtained as −157.87, −62.94, −31.87, −21.07, and −12.36 for the five modes of vibration of the fixed-base building. For the BI, the base mass, post-yield stiffness (

α k_{b}

), and damping coefficients are also given in Table 2. The isolation period (

T_{b}

) is computed as 2.5 s. F₀ and α are each taken as equal to 0.05 and 0.167, respectively, as considered by Ramallo et al. (2002). The peak response reductions of each floor are presented in Table 3 for the three cases of blast input described in Figure 3. It is observed that a reduction of 82% to 96% in the peak top floor displacement can be achieved by a base isolation system. The peak absolute acceleration reduction of the top floor ranges from 90% to 94%. The peak base displacements for the three cases of blast input are also presented here. It is noted from Table 3 that the response reductions achieved by the BI for the different blast input intensities are close, with a marginal increase for the higher blast input intensity. The reason for this is the low value of F₀ (= 0.05) considered for the results in Table 3. This value is very near the optimal value of F₀ for all the three cases of blast input, more so for case C that corresponds to the highest input intensity. However, the concern lies with the base displacement that will be associated with low values of F₀ and so with the base displacement as the constraint, the actual response reduction achieved may well vary with the blast input intensity. Here, as expected, the peak base displacements increase largely with the increase in the blast input intensity. A comparative estimate of these peak bearing displacement values with those in case of seismic excitation may be made from the work of Ramallo et al. (2002), who have reported that for the same structural model parameters (as given in Table 2) and for F₀ = 0.05 and α = 0.167, recorded earthquakes such as El Centro (PGA = 0.3495 g), Hachinohe (PGA = 0.2294 g), Kobe (PGA = 0.8337 g), and Northridge (PGA = 0.8428 g) have resulted in peak base drifts of 8.24 cm, 9.32 cm, 26.51 cm, and 55.64 cm respectively. Thus the peak base displacement values obtained in the case of underground blast input in the present study are not excessively high and quite comparable with those due to seismic excitation.

Table 2.

Structural model parameters.

Floor	kg	Stiffness coefficients	kN/m	Damping coefficients	kN s/m
m_b	6800	$α k_{b}$	232	c_b	3.74
m ₁	5897	k ₁	33732	c ₁	67
m ₂	5897	k ₂	29093	c ₂	58
m ₃	5897	k ₃	28621	c ₃	57
m ₄	5897	k ₄	24954	c ₄	50
m ₅	5897	k ₅	19059	c ₅	38

Table 3.

Peak responses reductions of each floors and peak base displacement of five-story base isolated building for different blast input.

Case A input		Case B input		Case C input
Floor	Reduction in peak displacement (%)	Reduction in absolute peak acceleration (%)	Reduction in peak displacement (%)	Reduction of absolute peak acceleration (%)	Reduction in peak displacement (%)	Reduction of absolute peak acceleration (%)
5th	82.53	90.00	86.76	93.00	87.58	93.57
4th	84.67	91.58	88.26	93.59	88.95	94.04
3rd	87.67	92.70	90.44	94.18	90.97	94.48
2nd	91.27	92.44	93.15	94.15	93.51	94.35
1st	95.62	91.54	96.54	93.79	96.70	94.08
Peak base displacement (cm)	5.18		23.00		41.34

Another important aspect of study, especially in case of MDOF systems, is the inter-story drift. It is well known that very low inter-story drift is one of the main advantages in the base isolation design of buildings against earthquake loading. To investigate the inter-story drifts for the underground blast loading, the displacement–time histories of each floor of the five-story base isolated building described above subjected to case C type blast input are presented in Figure 16. It is observed that the inter-story drifts are very small. Hence, the superstructure mounted on the BI behaves practically like a rigid body when subjected to underground BIGM.

Figure 16.

Displacement time history of each floor of a five-story base isolated building (case C input).

The influence of the normalized yield strength (F₀) and the rigidity ratio (α) of the BI on the response of the same five-story base isolated building to the case C input are studied together in Figures 17--19. The peak displacement and peak absolute acceleration reductions of the top floor with respect to F₀ are plotted in Figures 17 and 18, respectively. The curves are obtained for three values of α, equal to 0.05, 0.15, and 0.3. The corresponding peak base drifts are shown in Figure 19. It is clear from these figures that for the case C input, at lower values of F₀ the sensitivity of the response reductions as well as that of the peak base displacement to α is low (this is also true for the example system in Figures 12 and 13). But for higher values of F₀, the peak response reductions and the peak base displacement are considerably affected by different values of α and then higher amounts of peak response reductions as well as peak base displacements can be obtained for higher values of α (i.e., = 0.3). Thus, as observed in the previous section, a lower value of F₀ is more effective from the peak response reduction point of view, but leads to a higher bearing displacement. If a higher value of F₀ is chosen, then a higher α will again lead to a greater response reduction with a compromise on the base drift. Depending upon the constraint on the peak displacement and permanent set of the BI, coupled with the response reduction requirement, a judicious combination of F₀ and α should be selected.

Figure 18.

Peak acceleration reduction for different F₀ and α for five-story base isolated building (case C input).

5. Conclusions

The study on the performance of the N-Z type base isolation system for structures subject to underground BIGM reveals that it can provide substantial reduction in the peak absolute acceleration, which is extremely significant for structures subjected to blast, as well as mitigate the peak displacement of the superstructure. The peak bearing displacements, which are often critical in base isolation design, are also computed and it is seen that, unless the blast loading intensity is very high, these values are within the moderate to low range as compared to those typically obtained in case of earthquake excitation.

A sensitivity study on the peak absolute acceleration reduction and peak base displacement to the various important design parameters, e.g., the flexibility of the superstructure, the yield strength, rigidity ratio, and damping ratio is performed for the base isolation system in which the superstructure is modeled both as SDOF and MDOF systems. Parametric studies are also carried out for different intensities of blast input by varying the TNT charge weight. In the parametric study for the SDOF system on N-Z type BI, it is noticed that greater peak absolute acceleration reductions can be achieved for stiffer superstructures for the same value of isolation period. These reductions are also high for higher values of the isolation period ( $T_{b}$ ) for a particular superstructure. For the base isolated system, the peak base displacement does not vary significantly with the different fundamental time periods of the superstructure (T_s), keeping other variables fixed. It is also observed that there is an optimum value of the damping ratio ( $ξ_{b}$ ) for which the reduction in peak absolute acceleration of the superstructure is maximized. The bearing displacement, however, decreases with the increase in the damping. Thus, increasing the damping beyond a certain value may decrease the base drift, but may cause higher acceleration transmission into the superstructure. The normalized yield strength and the rigidity ratio are the other two important guiding factors for designing the BI. Though the reduction in peak structural acceleration is greater at lower values of the normalized yield strength of the BI (F₀), it is noted that the peak base displacement also becomes high then. Again, depending upon the blast input characteristics, the nature of the permanent set of the BI is different for different values of F₀. For a smaller blast intensity, the lower value of F₀ leads to a greater amount of permanent set. But, as the blast intensity becomes very high (here case C), a lower value of F₀ results in a lower amount of permanent set in the system. The effect of the rigidity ratio is more pronounced both for the peak absolute acceleration reduction as well as for the peak base drift for higher values of F₀. But when F₀ is small, for a higher intensity of blast input, the peak responses are almost independent of α, though for a lower intensity of input higher α is more effective in reducing the peak acceleration of the superstructure.

For the multi-story base isolated building it is seen that a base isolated building almost behaves as a rigid superstructure as inter-story drifts are very small. If all other structural and blast input parameters are fixed, then the post-yield to pre-yield stiffness ratio and the normalized yield strength of the BI play an important role in base isolation design. Here also, by choosing an appropriate combination of the BI parameters, a proper balance between the peak absolute acceleration reduction and the peak base drift as well the permanent deformation of the BI can be achieved.

Thus, depending upon the constraints on the peak displacement and permanent set of the BI, coupled with the response reduction requirement, a judicious combination of F₀, $ξ_{b}$ , and α should be selected for designing the BI.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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