Blast Resistant Building Design, (Part 2 of 2): Calculating Building Structural Response
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
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.
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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 | psi | |||
| 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)/(L2*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:
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 6×6x1/2 | |||
| 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.
