Blast Resistant Building Design: Defining Blast Loads (pt. 1 of 2)
Explosions occur when an explosive material, either in a solid, liquid or gaseous state, is detonated. Detonation refers to chemical reaction that rapidly progresses, at supersonic speeds, through the explosive material. The material is converted to a high temperature and high pressure gas that quickly expands to form a high intensity blast wave. Structures in the path of the blast wave are entirely engulfed by the shock pressures. At any location away from the blast, the pressure disturbance has the following shape:
Almost instantaneously following the blast, the pressure within the blast radius rises to a peak overpressure, Pso, (side-on, incident, or free field overpressure). The side-on overpressure decays to ambient after the positive phase duration after which a negative phase duration occurs where pressure falls below ambient to a minimum value, -Pso. The negative pressure phase of a blast wave is usually significantly smaller and longer in duration than the positive phase and is consequently generally ignored in blast resistant design. Correspondingly, a typical design blast load is represented by a triangular loading with side-on overpressure, Pso, and a duration, td, characterized by the following graph:
The area under the pressure-time curve is the impulse, Io, of the blast wave and is defined for the positive phase as follows:
For a triangular blast wave, the impulse is calculated as 0.5 Pso td.
The side of a structure facing a blast is subject to an increase in pressure over the side-on overpressure due to a reflection of the blast wave. For side-on overpressures of 20 psi or less and with an angle of incidence of normal to the structure face, the reflected pressure can be estimated as:
Pr = (2 + 0.05 Pso)Pso (psi)
In order to determine the loading on a building from a blast, several characteristics must be calculated. The Dynamic Pressure (Blast Wind), qo, is caused by air movement as the blast wave propagates through the atmosphere and is dependent on the peak overpressure of the blast wave. For low overpressures at normal atmospheric conditions:
qo = 0.022 Pso2 (psi)
The Shock Front Velocity, U, represents the speed at which the blast wave travels. Again, at low overpressures at normal atmospheric conditions:
U =1130(1 + 0.058 Pso)0.5 (ft/sec)
The Blast Wave Length, Lw, is the radial distance from the leading edge of the blast wave, at the highest overpressure, to the point at which the pressure dies out to atmospheric pressure. The direction of the Blast Wave Length is outward from the source of the explosion. For low overpressures, Lw is calculated as:
Lw = U td (ft)
Blast Loading Example:
From the basic blast design parameters, the engineer must determine the distribution of the blast loads on the overall building and then disburse the loading to the individual structural members. In order to be able to accomplish this, a general understanding of the propagation of the blast wave over the building is required. The following example attempts to illustrate this. The procedure illustrated is for vapor cloud explosions and is consistent with the method shown in Reference 1. The procedure for high energy explosives is similar though the equations are somewhat different and is explained in detail in Reference 2.
NOTES: Notations in parenthesis are from Reference 1.
Calculations assume the following:
1. The angle of incidence of the blast (angle between radius of blast from
the source and front wall) is 0 degrees.
the source and front wall) is 0 degrees.
2. A triangular blast load is assumed
3. The blast load is uniformly distributed across the building front wall.
Building Data:
Building Width, B = 12.00 ft (in direction of blast)
Building Height, H = 11.00 ft
Building Length, L = 40.00 ft (perpendicular to direction of blast)
Building Height, H = 11.00 ft
Building Length, L = 40.00 ft (perpendicular to direction of blast)
Blast Loading:
Peak Side-on Overpressure, Pso = 8.00 psi
Blast Duration, td = 200.0 ms
Blast Duration, td = 200.0 ms
Impulse, Io = 0.5*Pso*td = 800 psi-ms (Eq. 3.2 – Triangular Wave)
Peak Reflective Pressure, Pr = (2 + 0.05*Pso)*Pso = 19.2 psi (Eq. 3.3)
Dynamic (Blast Wind) Pressure, qo = 0.022*Pso2 = 1.408 psi (Eq. 3.4)
Shock Front Velocity, U = 1130*(1 + 0.058*Pso)0.5 = 1,367 ft/s (Eq. 3.5)
Blast Wave Length, Lw = U*td/1000 = 273.5 ft (Eq. 3.6)
Peak Reflective Pressure, Pr = (2 + 0.05*Pso)*Pso = 19.2 psi (Eq. 3.3)
Dynamic (Blast Wind) Pressure, qo = 0.022*Pso2 = 1.408 psi (Eq. 3.4)
Shock Front Velocity, U = 1130*(1 + 0.058*Pso)0.5 = 1,367 ft/s (Eq. 3.5)
Blast Wave Length, Lw = U*td/1000 = 273.5 ft (Eq. 3.6)
Front Wall Loading:
The side of the building facing the blast initially sees the reflected overpressure, Pr, as discussed above. At the clearing time, tc, the reflected overpressure decays to the stagnation pressure, Ps, the pressure resulting from the deceleration of moving air particles. The resulting pressure/time curve is the merging of two triangles as illustrated below.
In order to use the dynamic response charts for a triangular loading configuration (to be discussed in Part 2), an equivalent loading triangle is required to be developed. The equivalent impulse, Iw, with the corresponding effective duration te needs be calculated. The equivalent loading curve is illustrated below.
Drag Coefficient, Front Wall, Cd = 1 (Para. 3.3.3)
Stagnation Pressure, Ps = Pso + Cd*qo = 9.41 psi (Eq. 3.7)
Clearing Distance, S = 11 ft (Para. 3.5.1)
Clearing Time, tc = 3*S/U = 0.0241
s, which must be less than td O.K. (Eq. 3.8)
Front Wall Impulse, Iw = 0.5*(Pr-Ps)*tc + 0.5*Ps*td/1000 = 1.059 psi-s (Eq. 3.9)
Iw = 1059.0 psi-ms
Effective Duration, te = 2*Iw/Pr = 0.1103
s = 110.31 ms (Eq. 3.10)
Stagnation Pressure, Ps = Pso + Cd*qo = 9.41 psi (Eq. 3.7)
Clearing Distance, S = 11 ft (Para. 3.5.1)
Clearing Time, tc = 3*S/U = 0.0241
s, which must be less than td O.K. (Eq. 3.8)
Front Wall Impulse, Iw = 0.5*(Pr-Ps)*tc + 0.5*Ps*td/1000 = 1.059 psi-s (Eq. 3.9)
Iw = 1059.0 psi-ms
Effective Duration, te = 2*Iw/Pr = 0.1103
s = 110.31 ms (Eq. 3.10)
Side Wall Loading:
As the blast wave moves across the length of the building in a direction perpendicular to the blast, the side-on overpressure will not be applied in a uniform manner. The peak side-on overpressure exists at the leading edge of the blast and, as it travels across the side walls, the blast-side end of the side wall sees a lower pressure. Correspondingly, a reduction factor, Ce, is used to account for this difference.
In the application of the side wall loading to a nonlinear dynamic finite element analysis, it is useful, and conservative, to divide the side wall loading into a unit width (strip loading). Application of the loading to a finite element model is thus made easier and will be discussed in detail in Part 2.
Drag Coefficient, Side & Rear Walls + Roof , Cd = -0.4 (Para. 3.3.3)
For a Unit Width (Strip Loading): L = 1 ft
Lw/L= 273.45
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Rise Time, t1 = L/U = 0.0007
s = 0.73 ms (Fig. 3.8)
Decay Time, t2 = L/U + td = 0.2007
s = 200.73 ms (Fig. 3.8)
Equivalent Peak Overpressure, Pa = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
For a Unit Width (Strip Loading): L = 1 ft
Lw/L= 273.45
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Rise Time, t1 = L/U = 0.0007
s = 0.73 ms (Fig. 3.8)
Decay Time, t2 = L/U + td = 0.2007
s = 200.73 ms (Fig. 3.8)
Equivalent Peak Overpressure, Pa = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
For Wall Width: L = 12.0 ft
Lw/L= 22.79
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Rise Time, t1 = L/U = 0.0088
s = 8.78 ms (Fig. 3.8)
Decay Time, t2 = L/U + td = 0.2088
s = 208.78 ms (Fig. 3.8)
Equivalent Peak Overpressure, Pa = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
Lw/L= 22.79
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Rise Time, t1 = L/U = 0.0088
s = 8.78 ms (Fig. 3.8)
Decay Time, t2 = L/U + td = 0.2088
s = 208.78 ms (Fig. 3.8)
Equivalent Peak Overpressure, Pa = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
Roof Loading:
The roof loading is calculated in the same manner as the side wall loading.
Drag Coefficient, Side & Rear Walls + Roof , Cd = -0.4 (Para. 3.3.3)
For a Unit Width (Strip Loading): L = 1 ft
Lw/L = 273.45
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Rise Time, t1 = L/U = 0.0007
s = 0.73 ms (Fig. 3.8)
Decay Time, t2 = L/U + td = 0.2007
s = 200.73 ms (Fig. 3.8)
Equivalent Peak Overpressure, Pa = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
For Roof Width: L = 12.0 ft
Lw/L = 22.79
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Rise Time, t1 = L/U = 0.0088
s = 8.78 ms (Fig. 3.8)
Decay Time, t2 = L/U + td = 0.2088
s = 208.78 ms (Fig. 3.8)
Equivalent Peak Overpressure, Pa = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
For a Unit Width (Strip Loading): L = 1 ft
Lw/L = 273.45
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Rise Time, t1 = L/U = 0.0007
s = 0.73 ms (Fig. 3.8)
Decay Time, t2 = L/U + td = 0.2007
s = 200.73 ms (Fig. 3.8)
Equivalent Peak Overpressure, Pa = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
For Roof Width: L = 12.0 ft
Lw/L = 22.79
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Rise Time, t1 = L/U = 0.0088
s = 8.78 ms (Fig. 3.8)
Decay Time, t2 = L/U + td = 0.2088
s = 208.78 ms (Fig. 3.8)
Equivalent Peak Overpressure, Pa = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
Rear Wall Loading:
In a similar manner to the side walls and roof, an equivalent peak overpressure, Pb is calculated for the rear wall. A delay in time, known as the arrival time, ta, exists from the time the blast wave hits the front wall until the wave reaches the rear wall. The overpressure takes some time to reach the peak (rise time, tr) and the blast decays over the duration, td.
Drag Coefficient, Side & Rear Walls + Roof , Cd = -0.4 (Para. 3.3.3)
Equivalent Loading, S = MIN(H,B/2) = 11 ft
Lw/S = 24.86
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Equivalent Peak Overpressure, Pb = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
Arrival Time, ta = L/U = 0.0088
s = 8.78 ms (Fig. 3.10.b)
Rise Time, tr = S/U = 0.0080
s = 8.05 ms (Fig. 3.10.b)
Duration, td = 200.00 ms (Fig. 3.10.b)
Total Positive Phase Duration,to= 208.05 ms
Total Time, t = ta+to = 216.82 ms (Fig. 3.10.b)
Equivalent Loading, S = MIN(H,B/2) = 11 ft
Lw/S = 24.86
From Fig. 3.9, Reduction Factor, Ce = 0.92 (Fig. 3.9)
Equivalent Peak Overpressure, Pb = Ce*Pso + Cd*qo = 6.80 psi (Eq. 3.11)
Arrival Time, ta = L/U = 0.0088
s = 8.78 ms (Fig. 3.10.b)
Rise Time, tr = S/U = 0.0080
s = 8.05 ms (Fig. 3.10.b)
Duration, td = 200.00 ms (Fig. 3.10.b)
Total Positive Phase Duration,to= 208.05 ms
Total Time, t = ta+to = 216.82 ms (Fig. 3.10.b)
