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The study of blast effects on structures has been an area of formal technical investigation for over 60 years. There are numerous texts, guides and manuals on the subject, with continuing research and technical reporting occurring at a brisk pace. However, there is limited guidance available in the literature on the direct application of established blast effects principals to structural design. Numerous efforts are under way to develop comprehensive guides and standards to fill this void. This article presents a general overview of key design concepts for reinforced concrete structures.

## Blast Resistance and Progressive Collapse

Progressive collapse-resistant design mitigates disproportionately large failures following the loss of one or more structural elements. Progressive collapse-resistant design is system-focused, and is often divided into two approaches, direct and indirect. The direct method designs the structural system to respond to a specific threat either by providing an alternate load path in the event of failure of one or more members, or by specific local-resistance improvements of key elements. This method is similar to blast-resistant design. The indirect method provides general systemic improvements to toughness, continuity and redundancy; tension ties are an example of an indirect detailing technique.

Blast-resistant design is element-focused. It enhances toughness, ductility, strength and dynamic characteristics of individual structural elements for resistance to air-blast induced loading. This article is devoted to blast-resistant design, though there is overlap with progressive collapse-resistant design.

## What’s Special About Blast Loading?

This article specifically addresses the affects of shock loading from airblast. This type of load is applied to the perimeter structural elements of a building due to a high explosive blast event external to the building. The pressure wave applied to the building is characterized by short duration and high intensity (Figure 1).

The blast wave duration, td , is typically in the range of 0.1 - .001 seconds. This is often much shorter than, or at most on the order of, the natural period, Tn , of typical structural elements. For situations where td < 0.4Tn (some sources advise td < 0.1Tn), the blast wave effectively imparts an initial velocity to a structural element and the element then continues to respond at its natural frequency. The magnitude of that initial velocity, for a single-degree-of-freedom (SDOF) model, is v = ƒ0td/2m , where ƒ0 and td are shown in Figure 1 and m is the mass. Thus, in this response regime, the mass of the structural element is the only system parameter that controls the magnitude of the initial motion of the system - the more massive the structural element, the less it will be excited by the impulse from the blast wave. In this regard, the greater mass of concrete structures can be used to great advantage.

This load response to a blast is significantly different from the load response to a seismic event, for which the natural frequency of the structure, rather than the mass, is the primary factor in the response.

## Response Limits and Member Analysis

The extreme nature of blast loading necessitates the acceptance that members will have some degree of inelastic response in most cases. This allows for reasonable economy in the structural design and provides an efficient mechanism for energy dissipation. This also requires the designer to understand how much inelastic response is appropriate. Greater inelastic response will provide greater dissipation of the blast energy and allow for the sizing of smaller structural elements, but it will also be accompanied by greater damage and, at some point, increased potential for failure of the element.

The U.S. Army Corps of Engineers Protective Design Center (PDC) has developed response criteria for many typical structural elements in terms of maximum allowable support rotation, qmax , or ductility ratio, mmax , as shown in Tables 1 and 2. These limits were developed in conjunction with experts in the field of blast effects and are based on existing criteria and test data. The limits can be correlated to qualitative damage expectations ranging from no damage with elements responding elastically to severe damage with elements responding far into the inelastic regime. Table 3, provides a sampling of damage expectations for specific structural components, and Table 4 provides guidance on overall structural damage that the Department of Defense (DoD) equates with varying levels of protection.

These limits are calibrated to an equivalent single degree of system (SDOF) model of the structural member with lumped mass and stiffness, and should only be compared to responses determined in that manner. The SDOF method assumes the response of the member can be appropriately modeled as a single mode, neglecting contributions from all other modes. The calibration process used for the PDC limits incorporates mapping the idealized SDOF to actual structural response.

The undamped SDOF equation of motion is written:

me x(t) + R(x,t) = f (t) where f (t) is the blast load, x(t) is the acceleration response, me is the equivalent or activated mass of the structural element, and R(x,t) is the internal resistance as a function of time and displacement. Assuming elasto-plastic material behavior, the resistance is divided into three phases:

1) Elastic response until yield: R(x,t) = ke x(t), where ke is the equivalent stiffness and x(t) is the displacement response.

2) Plastic deformation after yield when deformation continues without increase in resistance: R(x,t) = Rm , where Rm is the maximum resistance.

3) Elastic rebound after reaching the maximum displacement: R(x,t) = Rm - ke[xm - x(t)], where xm is the maximum displacement.

While closed form solutions exist for some simple load profiles, it is often necessary to solve the SDOF equations of motion numerically. Such methods and a more complete treatment of equivalent SDOF systems can be found in texts on structural dynamics.

## Design

The design procedure includes:

1) Blast load definition

2) Response limit selection

3) Trial member sizing and reinforcing

4) Nonlinear dynamic SDOF analysis of the member

5) Comparing the calculated SDOF response with the response limit and adjusting the trial member as necessary

As noted above, some amount of inelastic response is generally anticipated when designing members for blast response. Economy of design is achieved by selecting smaller members and allowing greater inelasticity. Where greater protection is warranted, larger members are selected, potentially even such that the response to the design blast threat remains elastic. While member sizes can be scaled to match the desired level of protection, proper detailing of joints, connections and reinforcing should always be provided so that the members can achieve large, inelastic deformations even if the intent is for elastic response (thus providing greater margins against an actual blast that is larger than the design blast). Without proper detailing, it is uncertain whether a structure intended for blast resistance will achieve the design intent. The January, 2007 STRUCTURE® article Concrete Detailing for Blast provides effective recommendations for concrete detailing. In addition to that article, general design and detailing considerations include:

#### Beams

1) Balanced design often leads to a strong column - weak beam approach, with the intent that beam failure is preferable to column failure.

2) Provide sufficient shear transfer to floor slabs so that directly applied blast loads can be resisted by the diaphragms rather than weak-axis beam bending.

3) Transfer girders should be avoided in regions identified as having a high blast threat.

#### Columns

Design critical columns to be able to span two stories, in the event that lateral bracing is lost, particularly when using a weak beam approach.

#### Detailing and Connections

1) Use special seismic moment frame details.

2) Avoid splices at plastic hinge locations.

3) Provide continuous reinforcing through joints.

4) Used hooked bars where continuous reinforcing is not possible (particularly at corners).

## Example

Consider an exterior panel wall measuring 12 feet tall by 30 feet long, attached to the primary structural framing system at its top and bottom. The wall is to be designed to resist the effects of a high explosive blast resulting in a 12 pounds per square inch (psi) peak reflected pressure and a positive phase pulse duration, td = 50 milliseconds.

Since the wall is attached at its top and bottom, the vertical reinforcement will provide the primary load-path and blast resistance; as such this example will be limited to design of the vertical reinforcement. As an initial trial, an 8-inch thick wall with #4 reinforcing bars spaced every 6 inches at each face will be considered. For each trial section, the bending and shear (yield) strength of a unit strip are computed, applying strength increase factors (SIF) to account for the actual (rather than code minimum) strength of materials and dynamic increase factors (DIF) to account for the increased strength of materials exhibited under fast load application rates. SIF and DIF values for reinforced concrete design are suggested in Design of Blast Resistant Buildings in Petrochemical Facilities (ASCE 1997) and TM5-1300, Structures to Resist the Effects of Accidental Explosions (USACE 1990). The lesser of the computed bending or shear strengths is used as the maximum resistance, Rm, in the elasto-plastic resistance function. Rm = 10 kips for the 8-inch thick unit strip trial section.

The equivalent SDOF is then computed. The effective stiffness in this case would be computed based on the center deflection of a simply supported beam. Since both elastic and plastic response is anticipated, the moment of inertia used for the stiffness calculation is taken as the average of the gross and cracked moments of inertia. Load (stiffness) and mass transformation factors may be applied to compute the effective mass of the trial section. The effective mass can be thought of as the portion of the total mass of the section that participates in the SDOF response. A more complete treatment of mass participation and load-mass factors used to compute the effective mass can be found in Introduction to Structural Dynamics (Biggs 1964). The 8-inch thick unit strip trial section has an equivalent stiffness, ke = 27.7 kip/in, and an equivalent mass, me = 2.24 pounds-seconds2/inch, giving a natural period of vibration of the equivalent SDOF of

Since the pulse duration and natural period are similar (i.e. td / Tn = 0.05 sec/0.057 sec ≈ 1) in this case, the assessment of the response requires solution of the SDOF equation of motion. Numerical solution of the SDOF equation of motion gives a peak displacement response of xm = 3.1 inches with a permanent deformation after rebound of xp = 2.7 inches and a ductility ratio of m = xm / (xm - xp) = 7.75. The peak displacement corresponds to rotations at the top and bottom of the wall section of q = tan-1 (xm / 0.5hwall) = 2.5 degrees, which exceeds the response limit for flexural members of qmax = 2.0 degrees. Hence, the analysis must be conducted again with a new trial section.

Using the same reinforcing steel spacing, but increasing the wall thickness to 10 inches, increases the maximum resistance to 13.4 kips, the equivalent stiffness to 53.5 kip/inch, and the effective mass to 2.8 pounds-seconds2/inch. This results in a natural period of 0.045 seconds for the new trial section. Numerical solution of the equivalent SDOF with these parameters gives a peak displacement response of 1.4 inches with a permanent deformation of 1.1 inches, or a ductility demand just over 4.5 times the elastic limit. Rotations at the top and bottom of the wall are reduced to 1.1 degrees, which is now within the response limit. Figure 2 shows the applied force and internal resistance time histories for each of the trial sections. Figure 3 shows the SDOF response for each trial in three dimensions, with two-dimensional projections of the resistance-displacement curves and the displacement time history.

## Summary

Reinforced concrete can provide substantial protection from even extreme blast loading. The relatively large mass of concrete elements provides an inherent resistance to impulsive loads. Structural design considerations include sizing members to provide an expected degree of deformation and associated damage and optimizing the structure to resist and transfer blast loads in a reliable manner. Proper detailing is the final critical component of the design process to ensure that the structural elements have sufficient toughness to achieve the desired inelastic deformations.▪

The authors wish to thank Professor Andrew Whitaker and Mr. Jon Schmidt for their contributions to the tables in this article.

## References

[1] American Society of Civil Engineers (1997) Design of Blast Resistant Buildings in Petrochemical Facilities, Reston, VA.

[2] Biggs, John M. (1964) Introduction to Structural Dynamics, McGraw-Hill, New York, NY.

[3] Clough, Ray W. and Penzien, J. (1993) Dynamics of Structure, 2nd edition, McGraw-Hill, New York, NY.

[4] Mays, G. C. and Smith, P. D. (1995) Blast Effects on Buildings: Design of Buildings to Optimize Resistance to Blast Loading, Thomas Telford, New York, NY.

[5] U.S. Army Corps of Engineers (1990) TM 5-1300, Structures to Resist the Effects of Accidental Explosions, U.S. Army Corps of Engineers, Washington, D.C., (also Navy NAVFAC P-397 or Air Force AFR 88-22).

[6] Schmidt, Jon A. (2003), Structural Design for External Terrorist Bomb Attacks, STRUCTURE®magazine, March issue.