Explosion is defined as a very rapid release of a large amount of energy within a limited space [39] . An explosion is typically categorized as a nuclear, physical or chemical event. Eruptions of a volcano, catastrophic failure of pressure vessels or violent mixing of liquids at different temperatures are considered physical explosions. The energy released from the formation of different atomic nuclei during the redistribution of protons and neutrons within different interacting atomic nuclei forms an explosion. Most practical explosives in a solid or liquid state are also known as condensed explosives [39] .

When an explosion initiates in air the following sequence of events happens [39] [40] :

· High temperature and pressure are released from the center of explosion.

· Gas expands violently and high pressure released pushes surrounding air out of the volume that it occupies.

· Most of the energy released by the explosive is transmitted to the air pushed out (now in a compressed state) and forms a blast wave in front of the gas from the explosive reaction.

· The blast wave moves outward from the source of explosion and air pressure at the blast wave decays with distance.

· The air pressure drops and falls below atmospheric pressure before it return to a state of equilibrium where there is no gas or air being pushed away from the source.

Blast loads can be classified into confined and unconfined explosions. Confined explosions occur when the blast initiates inside a structure and the shock load initiated from the explosion is amplified with reflections of the internal surfaces. When the explosive is detonated in an open source and the blast waves propagate away from the source towards the structure, this is considered an unconfined explosion. Unconfined explosions can be classified into three types: free air burst explosions, air burst explosions, and surface burst explosions. This classification is based on their locations relative to the surrounding surfaces [41] .

Calculating TNT equivalence, written as (W_TNTeq ), is the first step in the quantification of blast waves. When determining the blast wave parameters generated by an explosive other than TNT, the initial step is to convert the mass of the explosive to an equivalent mass of TNT. For calculating the equivalent mass of TNT (W_TNTeq , the mass of the explosive (W_e) is scaled by a conversion factor that is based on its specific energy (Q_e) and the specific energy of TNT(Q_TNT ) [40] :

W T N T e q = ( Q e / Q T N T ) W e (1.1)

A list of conventional explosives used in military and commercial applications can be seen in Table 1, along with values for mass specific energy and the TNT equivalent.

The parameters of the blast wave in a free air burst explosion are important for characterizing the loads that will be used to design a structure. Figure 1 illustrates the free air burst wave front. Numerical analysis by Brode [43] produced the relationships (i.e. Equations (1.2) and (1.3)) between peak side-on overpressure, P_so and scaled distance Z.

P s o = 6.7 Z 3 + 1 ( P s o > 10   bar ) (1.2)

P s o = 0.975 Z 3 + 1.445 Z 3 + 5.85 Z 3 − 0.019 ( 0.1   bar < P s o < 10   bar ) (1.3)

The scaled distance (Z) is given by:

Z = R c W 1 / 3 (1.4)

where, R_c is the distance from the center of the charge (spherical) and W is the TNT equivalent charge weight ( W T N T e q ) defined by Equation (1.1).

Newmark and Hansen [45] introduced Equation (1.5) to calculate the maximum blast pressure (P_so), in bars, for a high explosive charge detonates at the ground surface as:

P s o = 6784 W R 3 + 93 ( W R 3 ) 1 2 (1.5)

Another expression of the peak overpressure in KPa was introduced by Mills [46] , given in Equation (1.6), in which (W) is expressed as the equivalent charge weight in kilograms of TNT, and Z is the scaled distance:

P s o = 1772 Z 3 − 114 Z 2 + 108 Z (1.6)

A typical pressure-time history plot for a blast wave in air is shown in Figure 2. At the time of arrival (t_a), the rise from ambient pressure (P_o) to the incident overpressure (P_so) is assumed to be instantaneous. The pressure decays to a minimum under pressure ( P s − ) before returning ambient conditions. The duration of the positive phase (t_p) occurs in the time span where the pressure remains greater than ambient. The pressure remains decaying until it falls below ambient pressure, during the negative phase duration (t_n). The exponential pressure decay was described using the Friedlander equation [39] [40] , as:

P ( t ) = P s o ( 1 − t t 0 ) e − ( b t t 0 ) (1.7)

where b is the waveform parameter.

Another parameter important to the analysis of blast waves is specific impulse (i_s ), which is the area under the positive phase of the pressure-time curve and this is defined using:

i s = ∫ t a t p P s o ( t ) d t (1.8)

For simplified analysis, the decay may be approximated as linear from t_a to t_p the impulse may be written as:

i s = 1 2 t p P s o (1.9)

The duration of the blast wave profile when the pressure dips below than ambient pressure is called the negative phase. According to Brode [40] , the minimum under pressure, P s o − , of the negative phase can be determined utilizing:

P s o − = − 0.35 Z ( bar ) (1.10)

where the pressure is in bar and the scaled distance is greater than 1.6. The impulse of the negative phase is denoted i − and can be calculated using:

i − ≈ i s [ 1 − 1 2 Z ] (1.11)

The parameters of the negative phase are often unimportant in the design of structural components because the positive phase pressures and impulses for threats of interest are significantly higher and have a more significant effect on the structure response. Other important parameters of the blast wave are the shock front velocity (U) the wave length of the positive phase (L_w), reflected pressure (P_r) , and reflected impulse (i_r ).

Graphical methods are convenient for determining the blast wave parameters described above. Several references such as TM-5 [44] , provide plots of the different parameters for a free air burst (Y-axis) versus the scaled distance (X-axis) as plotted in Figure 3, where this figure can be used to determine the different blast wave parameters in the free air during the positive phase only using the scaled distance.

The magnitude of the reflected pressure (P_r ) and impulse (i_r) will be greater than the over peak pressure and impulse. This enhancement occurs because in addition to the potential energy stored as a pressure differential in the blast wave, the air particles that comprise the blast wave have velocity and thus, kinetic energy. In the case when the wave encounters an infinitely large rigid wall wherein the particles are brought to rest and are compressed at the surface resulting in an increase in pressure [47] . An idealized pressure-time history on a

structure is shown in Figure 4, denoting the incident overpressure (P_s) and the reflected blast wave overpressure ( P_r).

When the angle of incidence is zero the peak reflected pressure (P_r ) can be calculated by the Rankine-Hugoniot prediction [39] using:

P r = 2 P s o + ( γ + 1 ) q s (1.12)

where, γ is the specific heat ratio of a real gas and q_s is the dynamic pressure which can be determined using:

q s = 1 2 ρ s u s 2 (1.13)

where, ρ_s is the density of the air, and u_s is the particle velocity behind the wave front. The particle velocity ( u_s) can be computed utilizing:

u s = a o P s γ P o [ 1 + [ γ + 1 2 γ ] P s P o ] − 1 2 (1.14)

where, a_o is the speed of sound at ambient conditions. Substitutions of Equations (1.12) and (1.13) into (1.14) gives:

P r = 2 P s [ 7 P o + 4 P s 7 P o + P s ] (1.15)

When γ = 1.4 for air. The reflected pressure will be twice the incident side-on pressure for the case when P_s is small relative to P_o ; which arises when the explosion is small and at long range. The reflected pulse can be as much as eight times the incident side-on pressure for the case when the P_s is large compared to P_o which arises when the charge is large and close in. It should be noted that these equations are only applicable when Z is greater than 0.134 ft/lb^1/3 (0.053 m/kg^1/3) which represents the radius of a spherical TNT explosive and the surface of the explosive [39] .

The following methods are available for prediction of blast effects on building structures:

1) Empirical methods are related to experimental data. Most of these approaches are limited by the extent of the underlying experimental database. The accuracy of all empirical equations diminishes as the explosive event becomes increasingly near field [48] .

2) Semi-empirical methods are based on simplified models of physical phenomena. The attempt is to model the underlying important physical processes in a simplified way. These methods depend on more extensive data and case study than that of empirical methods. Thus, the predictive accuracy is generally better than that provided by the empirical methods [48] .

3) Numerical (or first-principle) methods are based on mathematical equations that describe the basic laws of physics governing a problem. These principles include conservation of mass, momentum, and energy. In addition, the physical behavior of material is described by constitutive relationships. These models are commonly termed computational fluid dynamics (CFD) models [48] .

There are other attempts to predict the blast effects, among them, Khadid [49] who studied the fully fixed stiffened plates under the effect of blast loads to determine the dynamic response of plates with different stiffener configurations. This study included the effect of mesh density, time duration and strain rate sensitivity. The author used the finite element method and the central difference method for the time integration of the nonlinear equations of motion to obtain numerical solutions.

Blance et al. [50] presented an empirical approach for the external blast load on structures. The authors used models based on TM5-1300 in pure Lagrange approach for blast evaluation leads to more precise and conservation load.

Pandey [51] studied the effects of an external explosion on the outer reinforced concrete shell of a typical nuclear containment structure. The analysis was performed using non-linear material models till the ultimate stages. An analytical procedure for nonlinear analysis by adopting the used model was implemented into a finite element code DYNAIB.

Ngo et al. [52] gave an overview on the analysis and design of structures subjected to blast loads for understanding the blast loads and dynamic response of various structural elements.

Characteristics of the blast wave generated in an explosion depend on both the explosive energy release and the nature of the medium through which the blast propagates. These characteristics are measured under controlled conditions in experiments (providing a reference set of explosion data) that can be used to obtain data on other explosions using scaling laws. The most common form of blast scaling is Hopkison according to ref. [39] or cube root scaling law. This law states that when two charges of the same explosive and geometry, but of different size are detonated in the same atmosphere, the shock waves produced are similar in nature at the same scaled distances. The scaled distance Z, is defined as:

Z = R W 3 , m / kg − 1 / 3 (1.16)

where R is the distance from the center of the explosion to a given location and W is the weight of the explosive. In addition to this, the explosive yield factor λ, is useful in blast scaling. This is defined as:

λ = W W r 3 (1.17)

where W_r is the weight of the reference explosion. Thus similar shock effects are seen at similar scaled distances:

Z = R W 3 = R r W r 3 (1.18)

where R_r is the distance of the reference explosion, and is related to R through:

R = λ R r (1.19)

Scaling can be applied to time parameters. The scaled time τ_sc for a time t is defined as:

τ s c = t W 3 (1.20)

When a blast wave strikes a surface, which is not parallel to its direction of propagation, a reflection of the blast wave takes place. The reflection can be either normal reflection or an oblique reflection. There are two types of oblique reflection, either regular or Mach reflection; the type of reflection depends on the incident angle and shock strength [53] [54] .

The simplest discretization of transient problem is achieved by using of the SDOF approach. The actual structure is replaced by an equivalent system of one concentrated mass and one weightless spring representing the resistance of the structure against deformation [48] . Figure 8 illustrates the idealized spring mass system. The structural mass (M) is under the effect of an external force (F(t) ) and the structural resistance(R_m ) is expressed in terms of the vertical displacement (y) and the spring constant (k).

The blast load can be represented as a triangular pulse having a peak force (F_m ) and positive phase duration (t_d ). The forcing function can be identified as:

F ( t ) = F m ( 1 − t t o ) (1.39)

The blast impulse is defined as the area under the force-time curve, and is given by:

i = 1 2 F m t d (1.40)

The equation of motion of the un-damped elastic SDOF system for a time ranging from zero to positive phase duration (t_d), is given by Biggs [62] as:

M y ¨ + K y = F m ( 1 − t t o ) (1.41)

where, y ¨ is the acceleration.

Solving this equation of motion, the general solution can be expressed as:

y ( t ) = F m K ( 1 − cos ω t ) + F m K t d ( sin ω t ω − t ) (1.42)

y ˙ ( t ) = F m K [ ω sin ω t + 1 t d ( cos ω t − 1 ) ] (1.43)

where, y(t) is the displacement function, while y ˙ ( t ) is the velocity function.

The natural circular frequency of vibration of a structure (ω) can be calculated as:

ω = 2 π T = K M (1.44)

where, T is the natural period of vibration of a structure.

The maximum response is defined as the maximum dynamic deflection (Y_max ) which occurs at time tm. This deflection can be evaluated by setting y ˙ ( t ) in Equation (1.43) equals to zero, when the structural velocity is zero. The dynamic load factor (DLF) is defined as the ratio of the maximum dynamic deflection (Y_max ) to the static deflection (Y_st ) resulted from the static application of the peak load (F_m ) [48] and it is defined by:

DLF = Y max Y s t = Y max F m / K = ψ ( ω t d ) = ψ ( t d T ) (1.45)

Structural elements are expected to experience large inelastic deformation under blast load or high velocity impact [48] . Exact analysis of dynamic response is then only possible using step-by-step numerical analysis requiring nonlinear dynamic finite-element software.

The maximum displacement is presented in chart form TM 5-1300 [44] , as a family of curves for selected values of R_u/F_m showing the required ductility μ as a function of t_d/T in which R_u is the ultimate or maximum resistance of the structure and T is the natural period. The maximum response of elasto-plastic SDOF system subjected to a triangular load is shown in Figure 9.

The mechanical properties of concrete subjected to dynamic loading are quite different from those due to static loading. While dynamic stiffness does not vary a lot from the static stiffness, the stresses that are preserved for a certain period of time under dynamic conditions may gain values that are remarkably higher than the static compressive strength as shown in Figure 11 [52] . In compression

strength magnification factors as high as 4 in compression and up to 6 in tension for strain rates in the range: 102 - 103/sec have been observed [64] .

For the increase in peak compressive stress ( f ′ c ), a dynamic increase factor (DIF) is introduced in the Euro International Committee for Concrete-International Federation for Pre-stressing (CEB-FIP) model [65] as shown in Figure 12 for strain-rate enhancement of concrete as follows:

DIF = ( ε ˙ ε ˙ s ) 1.026 α     for   ε ˙ ≤ 30   s − 1 (1.46)

DIF = γ ( ε ˙ ε ˙ s ) 1 / 3     for   ε ˙ > 30   s − 1 (1.47)

where, the strain rate is denoted by ε ˙ and the quasi-static strain rate ε ˙ s equal 30 × 10⁻¹ s⁻¹ . Also the coefficients (γ) and (α) can be computed from:

log γ = 6.156 α − 2 (1.48)

α = 1 / ( 5 + 9 f ′ c / f c o ) (1.49)

where f c o = 10   MPa = 1450   psi .

Metallic materials subjected to dynamic loading have an elastic and inelastic response that can easily be monitored and assessed due to their isotropic properties. Norris et al. [66] tested steel with two different static yield strengths of 330 and 278 MPa under tension at strain rates ranges of 10⁻⁵ to 0.1 s⁻¹. Strength increases of 9% - 21% and 10% - 23% were reported for the two steel types, respectively. Dowling and Harding [67] conducted tensile experiments using the tensile version of Split Hopkinton’s Pressure Bar (SHPB) on mild steel using strain rates varying between 10⁻³ s⁻¹ and 2000 s⁻¹. It was concluded from their experiments that materials of Body-Centered Cubic (BCC) structure, such as mild steel, showed the greatest strain rate sensitivity. It was also found that the lower yield strength of mild steel can almost be doubled; the ultimate tensile strength can be increased by about 50%; and the upper yield strength can be considerably higher. In contrast, the ultimate tensile strain decreases with increasing strain rate. Malvar [68] studied enhancing the strength of steel reinforcing bars under the effect of high strain rates. This was described in terms of the Dynamic Increase Factor (DIF), which can be evaluated for different steel grades and for yield stresses ( f y ) ranging from 290 to 710 MPa as represented by Equation (1.48) [63] .

DIF = ( ε ˙ 10 − 4 ) α (1.50)

where, for calculating yield stress α = α f y and

α f y = 0.074 − 0.04 ( f y / 414 ) (1.51)

For ultimate stress calculation α = α f u and

α f u = 0.019 − 0.009 ( f y / 414 ) (1.52)