Blast Resistant Building Design: Defining Blast Loads (pt. 1 of 2) An Explosion Primer:
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, P so, (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.
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)
Blast Loading:
Peak Side-on Overpressure, Pso = 8.00 psi 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)
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)
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 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)
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)
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)
Part 2 of this article,”Calculating Building Structural Response” will be published in the coming weeks.
Pat Lashley, PE, MBA –Vice President of Engineering MBI
References: 1. American Society of Civil Engineers (1997), “Design of Blast Resistant Buildings in Petrochemical Facilities”, ASCE Task Committee on Blast Resistant Design. 2. Unified Facilities Criteria (2008), “UFC 3-340-02 Structures to Resist the Effects of Accidental Explosions“, U.S. Army Corps of Engineers, Naval Facilities Engineering Command & Air Force Civil Engineer Support Agency (Superseded Army TM5-1300, Navy NAVFAC P-397, and Air Force AFR 88-22, dated November 1990).
Blast Resistant Building Design, (Part 2 of 2): Calculating Building Structural Response Basic Information:
After determining the distribution of the blast loads on the overall building (see Blast Resistant Building Design – Part 1, Defining the Blast Loads), the engineer must distribute the loading to the individual structural member. The response of building to a blast load may be analyzed using dynamic structural analyses ranging from the basic single degree of freedom analysis (SDOF) method to nonlinear transient dynamic finite element analysis (FEA). In this article, the SDOF method is defined and an example calculation is illustrated.
All structures, regardless of how simple the construction, posses more than one degree of freedom. However, many structures can be adequately represented as a series of SDOF systems for analysis purposes. The accuracy obtainable from a SDOF approximation depends on how well the deformed shape of the structure and its resistance can be represented with respect to time. Sufficiently accurate results can usually be obtained for primary load carrying components of structures such as beams, girders, columns, wall panels, diaphragms and shear walls. However, it is very difficult to capture the overall system response if a building is broken into discrete components with simplified boundary conditions using the SDOF approach, with the result that the SDOF method may be overly conservative.
Nonlinear finite element analysis methods may be used to evaluate the dynamic response of a single building module or a multi-module assembly to blast loads. This global approach can remove some of the conservatism associated with breaking the building up into its many components when using the SDOF approach. Geometric and material non-linearity effects are normally utilized in such analyses. These analyses are typically carried out using a finite element program capable of modeling nonlinear material and geometric behavior in the time domain. The following shows a finite element model for a six-module complex:
SDOF Analysis:
All structures consist of more than one degree of freedom. The basic analytical model used in most blast design application is the single degree of freedom (SDOF) system. In many cases, structural components subject to blast load can be modeled as an equivalent SDOF mass-spring system with a nonlinear spring. This is illustrated below:
The accuracy obtainable from a SDOF approximation depends on how well the deformed shape of the structure and its resistance can be represented with respect to time. Sufficiently accurate results can usually be obtained for primary load carrying components of structures such as beams, girders, columns, and wall panels. However, it is very difficult to capture the overall system response if a building is broken into discrete components with simplified boundary conditions using the SDOF approach, with the result that the SDOF method may be overly conservative.
The properties of the equivalent SDOF system are also based on load and mass transformation factors, which are calculated to cause conservation of energy between the equivalent SDOF system and the component assuming a deformed component shape and that the deflection of the equivalent SDOF system equals the maximum deflection of the component at each time. The mass and dynamic loads of the equivalent system are based on the component mass and blast load, respectively, and the spring stiffness and yield load are based on the component flexural stiffness and lateral load capacity.
The “effective” mass, damping, resistance, and force terms in Equation 1 cause the equivalent SDOF system to represent a given blast-loaded component responding in a given assumed mode shape such that the SDOF system has the same work, strain, and kinetic energies at each response time as the structural component.
M a + C v + K y = F(t)
where:
M = effective mass of equivalent SDOF system a = acceleration of the mass C = effective viscous damping constant of equivalent SDOF system v = velocity of the mass K= effective resistance of equivalent SDOF system y = displacement of the mass F(t) = effective load history
When damping is ignored, where damping is usually conservatively ignored in the blast resistant design, elastic system then becomes,
M a + K y = F(t)
In the blast analyses, the resistance (R) is usually specified as a nonlinear function to simulate elastic-plastic behavior of the structure.
M a + R = F(t)
For convenience, the Equation is simplified through the use of a single load-mass transformation factor, KLM, as follows:
KLMM a + K y = F(t)
Where, KLM = KM/KL
The transformation factors for common one- and two-way structural members are readily available from several sources (Biggs 1964, UFC 3-340-02 2008). Blast loadings, F(t), act on a structure for relatively short durations of time and are therefore considered as transient dynamic loads. Solutions for Equation are available in the UFC 3-340-02 (2008) and Biggs (1964).
The response of actual structural components to blast load can be determined by calculating response of “equivalent” SDOF systems. The equivalent SDOF system is an elastic-plastic spring-mass system with properties (M, K, Ru) equal to the corresponding properties of the component modified by transformation factors. The deflection of the spring-mass system will be equal to the deflection of a characteristic point on the actual system, i.e. the maximum deflection. To perform equivalent SDOF, the assumption of a deformed shape for the actual system is required.
The majority of dynamic analyses performed in blast resistant design of petrochemical facilities are made using SDOF approximation. The dynamic responses of all structures were calculated in accordance with the procedures in the ASCE and Department of Army’s Technical Manual. The following figure (from UFC 3-340-02) shows the maximum deflection of elasto-plastic, one-degree of freedom system for triangular load and this figure is typical graphical solution of SDOF.
Additionally, P-I diagrams can be developed using SDOF analysis. The concept of P-I diagram method is to mathematically relate a specific damage level to a range of blast pressure and corresponding impulses for a particular structural component. When the P-I diagram is available for a structural component, for a given blast load, the damage level can be obtained directly from the P-I diagram.
The following table from ASCE 1997 shows the response criteria used to define damage levels.
SDOF Example:
This example shows the SDOF analysis for 40’(L)X12’(W)X11’(H) single module blast resistant enclosure. The building is designed to resist a free field overpressure of 8 psi with 200ms duration for “medium damage”. The SDOF analysis combines both dynamic analysis and structural evaluation into a single procedure which can be used to rapidly assess potential damage for a given blast load.
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 or roof plate) is 0 degrees. 2. A triangular blast load is assumed 3. The blast load is uniformly distributed across the building front wall and roof plate. It is conservatively assumed in the analysis that the blast load can be from any direction around the building. As a result, all walls can be subjected to reflected pressure during a blast event. For analysis purpose, the free field overpressure is converted into local pressure loads for the building front wall, side wall, rear wall, and roof (see Blast Resistant Building Design – Part 1, Defining the Blast Loads).
Roof Joists:
Member
W6×15
Area, A
4.43
in^2
Plastic Modulus, Z
10.8
in^3
Moment of Inertia, I
29.1
in^4
Weight/ft, Wt
15
lbs/ft
Support Weight, Ws
7.67
psf
Total Weight, Wtotal
13.295
psf
Elasticity, E
29000000
psi
Yield Strength, Fy
50000
Dynamic Increase Factor, DIF
1.19
Strength Increase Factor, SIF
1.1
Spacing, w
32
in
Length, L
132
in
Gravitational Constant, g
386*10^-6
in/ms^2
Dynamic Strength, Fdy=DIF*SIF*Fy = 239.2 psi-ms2/in
Elastic Stiffness, Ke =(384*E*I)/(5*L4*w)= 6.67 psi/in
Dynamic Strength, Fdy=DIF*SIF*Fy = 65,450 psi
Ultimate Bending Resistance, Ru = 8(Mpc+Mps)/(L 2*w) = 20.3 psi
Equivalent Mass, Me = KLM*M = 184.17 psi-ms2/in
Natural Period, tn = 2*pi*SQRT(Me/K) = 33.00 ms
Equivalent Elastic Deflection, Xe = Ru/Ke= 3.04 in
td/tn = 6.06
Ru/P = 2.54
Xm/Xe=
0.9
from the figure below:
psi
Ductility Factor, m =0.9 which must be less than Allowable, ma = 10, Design O.K.
Maximum Deflection, Xm = m*Xe = 2.7 in
Rotation Factor, q = atan(Xm/(0.5*L)) = 2.4, which must be less than Allowable, qa =
6, Design O.K.
Intermediate Column:
Member
HSS 6×6×1/2
Area, A
9.74
in^2
Plastic Modulus, Z
19.8
in^3
Moment of Inertia, I
48.3
in^4
Weight/ft, Wt
35.11
lbs/ft
Supported Weight, Ws
7.67
psf
Total Weight, Wtotal
22.7
psf
Elasticity, E
29,000,000
psi
Yield Strength, Fy
46,000
psi
Dynamic Increase Factor, DIF
1.1
Strength Increase Factor, SIF
1.21
Spacing, w
28
in
Length, L
125
in
Gravitational Constant, g
386*10^-6
in/ms^2
Mass, Wtotal/g = 408.7 psi-ms2/in
Elastic Stiffness, Ke =(384*E*I)/(5*L4*w)= 15.74 psi/in
Dynamic Strength, Fdy=DIF*SIF*Fy = 61,226 psi
Ultimate Bending Resistance, Ru = 8*(Mpc+Mps)/(L2*w) = 44.3 psi
Equivalent Mass, Me = KLM*M = 314.70 psi-ms2/in
Natural Period, tn = 2*pi*SQRT(Me/K) = 28.08 ms
Equivalent Elastic Deflection, Xe = Ru/Ke= 2.82 in
Ductility Factor, m = 0.4 which must be less than Allowable, ma = 2, Design O.K.
Maximum Deflection, Xm = m*Xe = 1.1 in
Rotation Factor, q = atan(Xm/(0.5*L)) = 1.0 which must be less than Allowable, qa = 1.5, Design O.K.
Each structural member of the building must be analyzed in a similar fashion for the applied blast load and compared against the respective damage levels.
Pat Lashley, PE, MBA –Vice President of Engineering MBI
Minkwan Kim, PhD — Design Engineer MBI
References:
1. American Society of Civil Engineers (1997), “Design of Blast Resistant Buildings in Petrochemical Facilities”, ASCE Task Committee on Blast Resistant Design.
2. Unified Facilities Criteria (2008), “UFC 3-340-02 Structures to Resist the Effects of Accidental Explosions“, U.S. Army Corps of Engineers, Naval Facilities Engineering Command & Air Force Civil Engineer Support Agency (Superseded Army TM5-1300, Navy NAVFAC P-397, and Air Force AFR 88-22, dated November 1990).
3.John M. Biggs (1964), “Introduction to Structural Dynamics”, McGraw-Hill Companies.
