Blast-Resistant Design Considerations for Precast Prestressed Concrete Structures

PCI Journal November–December 2007 53

Editor’s quick points

n As part of this issue's theme, the authors present prevalent simplified methods for the blast-resistant design of precast/prestressed concrete components.

n This paper is valuable for those who would like a better understanding of blast loads, material behavior under dynamic loads, and design of precast/prestressed concrete structures to resist blast loads.

n It is part of the work in progress of the PCI Blast Resis-tance and Structural Integrity Committee.

Blast-resistant design considerations for precast, prestressed concrete structuresSanaa Alaoui and Charles Oswald

Blast-resistant design is becoming more common in the precast con-crete industry as more blast-resistant buildings are constructed with precast/prestressed concrete components. This is occurring primar-ily because many large government and U.S. Department of Defense buildings now require some level of blast-resistant design. Blast design has been performed for many years for the chemical and petrochemi-cal industry and explosive storage and manufacturing facilities, which have inherent accidental explosion hazards.

Based on both theoretical analysis and testing, blast design guidelines and methods have been developed for many common types of building components, including steel members, concrete masonry unit walls, and reinforced concrete members.1,2 Much of this blast design guidance is applicable to precast/prestressed concrete components, though it is not widely understood by designers within the precast concrete indus-try. Some of the design guidance is restricted to official government use only or is based on proprietary research, but most of this informa-tion resides in the public domain.

This paper presents prevalent blast-resistant design information that can be used for precast/prestressed concrete elements and structures. It is part of work in progress of the newly formed PCI Blast Resistance and Structural Integrity Committee.

Objective

The information in this paper focuses on understanding blast loads, material behavior under dynamic loads, and design of precast/prestressed concrete structures to resist blast loads. An example is included in which a precast concrete panel is designed to resist a blast load. Only blast loads from high explosives (HE) are addressed. Other types of explosions differ primarily in the calculation of the blast loads, but the basic dynamic structural response and blast-resistant design ap-proach is similar for all types of blast loads.

54 PCI Journal November–December 2007

Related industry guidelines, specifications, and codes

Documents are available that generally discuss blast design consider-ations for precast/prestressed concrete components, such as the De-signer’s Notebook, Blast Consideration by PCI.3 However, no specific standards or guidelines for blast design exist in American Concrete Institute or PCI documents. An abundant amount of information for calculating blast loads and structural component response is provided in the Army Technical Manual TM 5-1300, Structures to Resist the Effects of Accidental Explosions, including information on the design of precast/prestressed concrete components.2 The American Society of Civil Engineers’ (ASCE) Task Committee on Blast-Resistant Design has also published Design of Blast Resistant Buildings in Petrochemi-cal Facilities.1

The available documents provide a wide range of information. Army Technical Manual TM 5-1300 focuses on the blast design of compo-nents subject to HE blast loads, whereas the ASCE document focuses on blast design of components subject to industrial explosions, for ex-ample, vapor-cloud explosions. From its website, the U.S. Army Corps of Engineers distributes single-degree-of-freedom blast effects design spreadsheets for the design and analysis of structural components sub-ject to blast loads and a detailed methodology manual.

Overview of building blast design

The primary objective of blast-resistant design is to provide an accept-able level of safety to building occupants in the event of an explosion. Significant damage is usually acceptable as long as the components re-main attached to the building and the building remains stable after the explosion. Most fatalities and injuries sustained by building occupants during explosions are due to falling or flying building debris from failed structural and nonstructural components or are due to build-ing collapse. People outside of the building may also be vulnerable because components may rebound and fail during their initial response phase due to inadequate connections or lack of reinforcement for stress reversals.

For a given explosion scenario, precast/prestressed concrete compo-

nents and their connections to the building should be designed to resist the resulting blast load. Fram-ing members directly supporting precast concrete components need to be designed to resist the dynamic reaction forces from the components. Al-ternatively, they can be designed to resist the blast load acting over their tributary area.

Unlike seismic and wind loads, blast loads have a short duration, typically milliseconds. Therefore, the large mass associated with overall building response often provides enough inertia so that the building lateral framing system does not need to be strengthened to resist blast loads. The fact that a blast wave applies positive pressure on all sides of the building also limits the effect on the lat-eral framing system, though the pressure is much higher on the sides facing the explosive source. The lateral-load-resisting system on smaller one- or two-story buildings must be checked, considering the combined effects of overall frame sway and di-rect blast load effects on frame members. Conven-tionally designed foundation systems for large and small buildings almost always have adequate mass and strength to resist the short-duration reaction loads from the building response to the blast load.

Understanding blast loads

Many variables influence the resulting overpres-sures from an HE blast event. The type of explo-sive, charge shape, height above ground surface, and level of confinement are only a few. The most common case for an external—outside of the building—HE explosion is a surface burst, which assumes that the explosive charge is close to the ground relative to its standoff distance from the structural component of interest. Part of the shock wave reflects off the ground surface in this case and coalesces with the rest of the shock wave, thus reinforcing the wave. Any surrounding surfaces that reflect the blast wave toward the component of interest can also contribute to the blast load.

In an internal HE explosion, many reflections of the shock wave occur off the walls of the explo-sion room that all add to the total blast load. An additional, longer-duration phase of blast load-ing known as the gas, or quasistatic, blast load is caused by the confinement of the products and heat from the explosion within a confined volume. Blast loads from industrial explosions, such as vapor-cloud explosions, include more complicating fac-tors and are generally the most difficult to predict.

The simple case of an external surface burst of HE is applicable for most vehicle-bomb scenarios where the explosion occurs in the open near a building. Two parameters are typically used to de-fine blast loads for design purposes: peak pressure Figure 1. Idealized blast pressure history typically used for design.


member relative to the blast wave path. The free-field blast pressure history in Fig. 2 occurs in open air where no obstacles impede propaga-tion of the blast wave. Blast loads on the sides of a building not facing the explosive source, that is, the roof and the leeward wall, are known as side-on or incident blast loads and are usually assumed equal to the free-field blast load.

The blast pressure on building surfaces that reflect the blast wave is known as reflected pressure, which is much higher than the cor-responding free-field pressure. Figure 3 shows the reflected pressure and impulse resulting from 100 lb (45 kg) of TNT at 50 ft (15 m) standoff distance. The reflected pressure is 2.4 times higher than the incident pressure for the same explosion case given in Fig. 2, while the reflected impulse is 2.2 times higher than the free-field impulse. This reflection factor is never less than 2.0 for the typical design case

and impulse. Figure 1 shows the simplified shape of blast loads commonly used for blast design. The blast load rises immediately to its peak pressure and then decays linearly to ambient pressure over a duration td. The impulse, which is the shaded area under the pressure and time curve, is a measure of the total energy in the blast load.

A blast load cannot be adequately defined for blast design purposes with a single parameter, such as only the peak pressure, unless there is an understanding that the blast load duration is long compared with the response time of the building’s structural components. This case only occurs for some industrial explosions and large nuclear explo-sions. Also, the additional dynamic load from blast-generated fragments can be significant compared with the blast pressure load, but this is not usually considered when designing structural components against vehicle bomb explosions.

The resulting overpressures and impulse in the free field for a surface-burst explosion depend on two parameters: charge weight and standoff distance. Charge weight is typically given in equivalent weight of trinitrotoluene (TNT) explosive, while standoff distance is the length from the center of the explosion to the point of interest. Blast loads on a building surface also depend on the orientation of the surface relative to the shock wave’s direction of travel.

Figure 2 shows the calculated free-field overpres-sure history from 100 lb (45 kg) of TNT at a 50 ft (15 m) standoff distance. All blast loads are typically expressed in terms of gauge pressure, so that ambient pressure is a zero blast-load pressure. Figure 2 shows the actual shape of the blast load, including the negative phase (suction pressure), which has a much lower magnitude but a longer duration than the positive phase (inward pressure). The negative phase is often neglected for design (Fig. 1). This approach is generally conservative because component response calculated without consideration of the negative phase can be signifi-cantly greater than measured component response in blast tests.

A comparison of Fig. 1 and 2 shows how the blast-load decay from peak pressure is idealized as a straight line for design purposes where the positive phase impulse is preserved. A blast wave always engulfs a building and applies positive blast loads to all surfaces, including the leeward face, fol-lowed by a negative phase. Therefore, the shape of the blast load in Fig. 2 applies to all surfaces of a building.

The blast pressures imparted on any given struc-tural member depend on the orientation of the

Figure 2. Incident blast load for 100 lb TNT at 50 ft. Note: 1 ft = 0.3048 m; 1 lb = 0.45 kg; 1 psi = 6.9 kPa.

Figure 3. Reflected blast load for 100 lb TNT at 50 ft. Note: 1 ft = 0.3048 m; 1 lb = 0.45 kg; 1 psi = 6.9 kPa.

Blast loads can often be minimized to some extent by site planning.


charge weight is equal to the largest amount of explosive that is assumed to be undetected by the search. Unified Facilities Criteria (UFC) 4-010-014 has recommended minimum standoff distances from the nearest parking areas and roadways to conventionally designed buildings, that is, no blast design, with upgraded windows that are intended to protect the building occupants from mass casualties from assumed vehicle bomb explosions.

This overview is only intended to demonstrate some basic concepts of blast loads. Other refer-ences are available with more detailed information.2 Many important factors that also affect blast loads have not been discussed. In general, blast loads should always be calculated, or at least checked, by engineers with experience in this area. In some cases, a nominal design blast load for building cladding and window components is defined by the project specifications in terms of the positive phase peak pressure and impulse, for which no calcula-tions to determine blast loads are required.

Dynamic material properties

Blast load durations are generally milliseconds, causing component response times to be similarly short durations. Therefore, the effects of dynamic material properties should not be ignored. Under dynamic loads, materials exhibit increased yield strengths due to strain rate effects, which can con-siderably improve the ultimate load capacities of blast-loaded components.

In general, the higher the strain rate is, the greater the increase in strength should be. Quantifying the strain rate and computing the material properties based on that exact rate is a tedious process and is usually not warranted in a design in which many parameters are not known with a great degree of certainty.

To account for the strain rate effects, a dynamic increase factor (DIF) is used when computing the strength of members. Default values for DIF are presented in numerous blast design documents.1,2 Typical values are shown in Table 1 for reinforcing steel and concrete. Dynamic yield strengths, signi-fied with a d in the subscripts of the yield strength parameters in Table 1, are used for blast design.

Also, the minimum specified yield strength for rein-forcing steel fy is typically increased by a static in-crease factor (SIF) of 1.1 for blast design to account for the difference between actual reinforcing steel static yield strengths and the minimum specified values obtained from many tensile tests. Therefore, the dynamic yield strength fdy of Grade 60 (410 MPa) reinforcing steel is 77,000 psi (530 MPa).

where the building surface facing the explosion is perpendicular to the path of blast wave propagation from the explosive source, that is, fully reflected blast load.

Blast loads can often be minimized to some extent by site planning. As mentioned previously, the two factors governing blast loads are charge weight and standoff distance. While charge weight is difficult to control, identifying considerations during the early planning stages that increase the standoff distance for potential vehicle bombs can be beneficial because this is the best way to reduce blast loads and miti-gate the amount of required blast design.

Figure 4 illustrates these benefits. Increasing the standoff distance by a factor of 2, for the same charge weight, can reduce the overpressure by approximately a factor of 3. Many measures can be taken to ensure a desirable standoff distance, including landscape planning (for example, the use of planters, retaining walls, bollards, and the like), increasing the distance between driving surfaces and the face of the structure, per-forming vehicle inspections at a controlled perimeter, and limiting the access of large vehicles in areas near the building. The charge weight of a vehicle bomb can be assumed to be proportional to the vehicle size,3 or it can be based on the vehicle search criteria, where the design

Figure 4. Pressure and impulse versus range for 100 lb TNT surface-burst explosion. Note: 1 ft = 0.3048 m; 1 lb = 0.45 kg; 1 psi = 6.9kPa.

Stress type

Dynamic increase factorReinforcing bars* Concrete Masonry

fdy / fy fdu / fu f 'dy / f '

c f 'dm / f '

m

Flexure 1.17 1.05 1.19 1.19

Compression 1.10 1.00 1.12 1.12

Diagonal tension 1.00 1.00 1.00 1.00

Direct shear 1.10 1.00 1.10 1.00

Bond 1.17 1.05 1.00 1.00

*Applicable for Grade 40 and Grade 60 reinforcing steel only. Note: DIF for all prestressing steel = 1.0. DIF = dynamic increase factor. 1 ksi = 6.9 MPa.

Table 1. Dynamic increase factors for reinforcing bars, concrete, and masonry


steel and concrete without any strength reduction factor. It is intended to be a realistic estimate of the actual dynamic moment capacity at the maximum moment regions of the component.

Yield line theory is used to determine the ultimate resistance for indeterminate components. These components have multiple yield loads, where the reinforcing steel at different maximum moment regions yields as the resisted load increases, which causes multiple slopes in the resistance-deflection curves as illustrated in Fig. 6. The resistance-deflection relationship of an indeterminate component can be simplified, as shown in Fig. 6, using an equivalent stiffness kE and yield deflection xE that cause the simplified resistance-deflection curve, with one slope, to have the same strain energy as the actual resistance-deflection curve out to the deflection where the component becomes a mechanism xp. Procedures for determining the resistance-deflection relationships for blast-loaded components are described in detail in a number of references.1,2

The resistance-deflection relationship in Fig. 5 is arbitrarily stopped at

Component resistance-deflection function

Unlike conventional design, components designed for blast are allowed to undergo a controlled amount of plastic deformation. The component absorbs strain energy during elastic and plastic response that must equal the energy imparted by the blast load, or the component will fail. Typically, a well-designed ductile component will absorb most of the blast load energy with plastic strain energy, but the maximum component deflection will only be half, or less, of the deflection corresponding to failure.

The strain energy absorbed by a component during response to blast load can be measured as the area under its resistance-deflection curve at any given deflection where the strain energy increases with deflection. The resistance-deflection curve relates the resisted load to the midspan deflection of the blast-loaded component. It can be derived with conventional static calculation methods, including applicable DIF and SIF factors. The resisted load in the resistance-deflection curve has the same spatial distribution as the applied blast load, typically a uniformly distributed pressure load. In a ductile reinforced concrete component, the resisted load, or resistance, increases approximately linearly with deflection until the reinforcing steel yields in the maximum moment regions, and then the resistance remains relatively constant with increasing deflec-tion until failure.

The slope in Fig. 5 shows the resistance-deflection curve for precast concrete components with simple supports. The initial slope is an average elastic flexural stiffness. Typically this is calculated for reinforced concrete components using an average of the gross moment of inertia and the fully cracked moment of inertia. The stiffness goes to zero when yielding occurs at the maximum moment region. The resistance does not degrade after yielding in Fig. 5, implying ductile response. It is assumed that the stress in the reinforcing steel remains constant at fdy after yielding, ignoring the small amount of strain hardening that occurs so that the resist-ing moment and resistance remain constant with increasing midspan deflection out to a limit deflection.

In Fig. 5, a uniform pressure of 2.3 psi (16 kPa) is resisted by the component when the applied moment at midspan equals the ultimate dynamic moment capacity and therefore causes yielding. Thus, the ultimate resistance of the panel, equal to the ultimate load capacity, is 2.3 psi (16 kPa) at a midspan deflection of 0.45 in. (11 mm). The ultimate dynamic moment capacity is calculated using applicable DIF and SIF factors for reinforcing

Figure 5. Resistance-deflection relationship for simply supported precast concrete panel. Note: 1 in. = 25.4 mm; 1 psi = 6.9kPa.

Figure 6. Resistance-deflection relationships for determinate and indeterminate boundary conditions.


stress is not used in blast design. Instead, allowable design limits are set on component deflections. Al-lowable maximum dynamic deflections for compo-nents subject to blast loads are typically specified in terms of two parameters; the support rotation θ and the ductility ratio μ. These parameters are illustrat-ed in Fig. 7 and Eq. (1), respectively. The ductility ratio is the ratio of maximum deflection under the applied blast load to the yield deflection. The value xE in Fig. 6 is typically used as the yield deflection for indeterminate components. The support rotation measures the maximum rotation at the supports and relates the maximum deflection to the span of the member.

Maximum support rotation and ductility ratio values allowed for design are based primarily on blast tests in which the values of these parameters corresponding to component failure have been determined. These values are usually specified for government buildings based on the building’s intended level of protection. In privately owned buildings, the owners may set acceptable design criteria for given explosion scenarios. Often, the allowable support rotation and ductility ratio are limited to no more than one-half the values corre-sponding to failure for design, implying a factor of safety of at least 2.0.

µ =x

m

xe

(1)

where

xe = deflection causing yield of determinate com-ponent or yield of equivalent elastic slope of indeterminate component (Fig. 6)

xm = maximum component deflection

μ = ductility ratio

Based on testing and analysis, the support rotation correlates much better with failure for conven-tionally reinforced concrete, while the ductility ratio correlates better with failure for prestressed concrete. Allowable support rotations for blast design of reinforced concrete components are typi-cally in the range of 2 degrees to 4 degrees.1,2,5 This corresponds to an allowable maximum dynamic deflection between 2.5 in. and 5.0 in. (63 mm and 127 mm) for a 12 ft (3.7 m) span. Allowable ductil-ity ratios for blast design of prestressed concrete components are typically in the range of 1 to 3.2,5 For a 40 ft (12 m) prestressed concrete T-beam, this can correspond to an allowable maximum dynamic deflection between 1.3 in. and 4 in. (33 mm and 100 mm), though this varies depending on the yield deflection and, therefore, the prestressed steel area.

a given deflection. At a larger deflection, the rotation of the member at midspan will typically cause the strains at the compression face of the concrete to exceed the concrete failure strain of about 0.003, and the member will fail. This assumes that the component reinforcing ratio is less than the balanced steel ratio, as is typically required for both conventional and blast design. The maximum acceptable deflection for blast design is less than this failure deflection by a safety factor.

This discussion has assumed that the component response is con-trolled by ductile flexural response. It is important that the component does not fail in shear or due to connection failure before reaching its ultimate flexural resistance (for example, 2.3 psi [16 kPa] in Fig. 5) and that it deflects in a ductile manner out to a failure. A much lower amount of deflection past yield occurs prior to failure when shear or connection failure controls the ultimate resistance or load capacity of a component because these are typically nonductile response modes. However, this will not occur if a component’s ultimate flexural resis-tance is less than the ultimate resistances based on the component’s shear and connection strengths, that is, component response is con-trolled by ductile, flexural response. Therefore, a goal of blast-resistant design is to design components so that their responses are controlled by ductile flexural response.

Finally, flexural response is not ductile for all component types. Pre-stressed concrete components have much lower ductility than conven-tionally reinforced concrete because prestressing strands have a lower failure strain than conventional reinforcing steel. Prestressed concrete components fail at a lower maximum deflection and must possess a higher ultimate flexure resistance and ultimate dynamic moment capacity to absorb the same strain energy as conventionally reinforced concrete components. Their higher ultimate flexural resistance also implies that they must have higher shear strength and connection strength. However, the higher shear strength and connection strength requirements can still be manageable from a practical design viewpoint in some cases.

Deformation limits

As mentioned in the previous section, blast design involves allow-ing component stresses to exceed yield. Therefore, allowable design

Figure 7. Component support rotation.


does not usually affect maximum deflections of blast-resistant compo-nents when there is significant ductile, plastic response. Also, rigorous FEAs tend to be time consuming and require sophisticated software packages and a high level of knowledge from the user.

It is generally accepted that the blast design of most building com-ponents can be performed by modeling the dynamic response of the component with an equivalent single-degree-of-freedom (SDOF) system.1,2 This is referred to as the SDOF method for blast design. It should also be noted that commercially available software for apply-ing FEA to blast design has become much more accessible and user friendly in recent years and is being used for more complex blast design cases.

General design concepts

Rigorous analysis methods, such as dynamic nonlinear finite element analyses (FEAs), may be used to obtain the dynamic response of blast-loaded components. These analyses can consider more-complex effects, such as dynamic interaction between cladding and framing, multiple modes ex-cited during frame sway response, and highly non-uniform blast loading over the area of a component. However, blast loads are usually relatively uniform over a component, so it is generally conservative to ignore interaction effects. Higher mode response

Figure 8. Equivalent spring-mass system representing dynamic response of beam loaded by blast.

Figure 9. Deflected shape functions for simply supported beam.

Blast Load P(t)


Component design

The SDOF method idealizes the structural component into a mass-spring system with stiffness and mass related to those in the blast-loaded structural component (Fig. 8). The equivalent SDOF system is defined such that its deflection at each time step is equal to the maxi-mum deflection of the structural component, assuming the response of a structural component can be described in terms of an SDOF, that is, the point of maximum deflection, and an assumed shape function that defines motion at all other points on the component relative to the maximum deflection.

Figure 9 shows the shape functions that are usually assumed for a sim-ply supported beam before and after yielding in the maximum moment region. Similar shape functions are defined for other boundary condi-tions and for two-way spanning components.1,2,6,7 Components that are n degrees indeterminate have n + 2 sets of shape functions to represent response prior to yield, response after successive yielding at each of n maximum moment regions, and response after the component becomes a mechanism. If the analysis of an indeterminate system is simplified with an equivalent elastic resistance-deflection curve, as shown in Fig. 6, then only two shape functions are used in the dynamic analysis, as shown in Fig. 9.

Equation (2) shows the equation of motion for the equivalent spring-mass system. The terms in Eq. (2) are related to the mass, stiffness, damping, and load on the structural component by transformation factors, such as KL, KM, and others. These transformation factors are calculated such that the work energy, strain energy, and kinetic energy of the component expressed in terms of its SDOF and assumed shape functions equal the work, strain, and kinetic energies of the equivalent spring-mass system at each time step.6

KMMw"(t) + Kd Cw'(t) + KRR(w(t)) = KL F(t) (2)

where

C = viscous damping constant of component

F(t) = total blast load on component with respect to time

Kd = viscous damping factor

KL = load factor

KM = mass factor

KR = resistance factor = KL

M = mass of blast-loaded component

R(w(t)) = resistance of component based on resistance and deflection curve and deflection at each time step

t = time variable

w(t) = deflection of SDOF system with respect to time

w'(t) = velocity of SDOF system with respect to time

w"(t) = acceleration of SDOF system with respect to time

Equation (3) shows how the load factor is derived for a one-way spanning component. The left side of the equation is the work energy for the equivalent SDOF system, and the right side is the work energy of the structural component. This equation is solved for the load factor, as shown in Eq. (4), assuming that the deflection of the equivalent SDOF system is equal to the maximum deflection of the compo-nent at each time step, as stated previously.

KLF t( )w t( ) = P t( )w

0t( ) p x( )! x( )dx

0

L

" (3)

where

F t( ) = P t( ) p x( )dx

0

L

!

KL=

p x( )! x( )dx

0

L

"

p x( )dx

0

L

" (4)

where

F(t) = total blast load on component with respect to time

KL = load factor

L = length of blast-loaded component

p(x) = blast load shape function of component rela-tive to distance along its length with P(t) = 1.0 for a uniformly applied blast load on component

P(t) = blast load at given point along length of component at each time step

t = time variable

w(t) = deflection of SDOF system with respect to time

= w0(t)

w0(t) = maximum deflection of component at each time step

ø(x) = deflected shape function of component

Load distribution Response rangeLoad factor

KL

Mass factor* KM

Single point load at midspan

Elastic 1.0 0.49

Plastic 1.0 0.33

Two point loads at third-points

Elastic 0.87 0.52

Plastic 1.0 0.56

Uniformly distributed load

Elastic 0.64 0.5

Plastic 0.5 0.33

* For a component with uniformly distributed mass

Table 2. Load and mass factors for one-way components with simple supports


Additional design considerations

In almost all cases, the allowable design criteria are based on assumed ductile flexural response for the component. Therefore, the shear capacity and connection capacity of blast-loaded components must be designed to exceed the ultimate resistance of the component based on flexural capacity. The required connection capacity is typically much higher than that required for conventional loads. Also, the required shear capacity may necessitate shear reinforcement, though this is typically not necessary for precast/prestressed concrete slabs. Some government agencies state that shear strength of the component must exceed its dynamic flexural strength by 20%.

Typically, the component shear capacity is based on the static shear strength with no DIF. This is a conservative assumption. However, no strength reduction factor is typically used. The capacity of connections for blast-loaded components may include a small DIF, on the order of 1.05, because connections typically have high-strength steel and the DIF decreases with increasing steel strength.2 However, this DIF is often neglected. In this case, the connection capacity based on load and resistance factor design (LRFD) must be greater than the reaction loads from a static load equal to the ultimate resistance of the component (2.3 psi [16 kPa] in Fig. 5) with no load factor. The connection capacity can be based on 1.7 times the allowable connection capacity.

These requirements for the shear and connection capacity imply that the flexural resistance of blast-loaded components should not be over- designed to the extent that this causes corresponding increases in the required shear and connection capacities that are expensive or difficult to construct. The more the component relies on ductility, or deflection past yield, within the allowable design criteria to develop the required strain energy capacity to resist the blast load, the more economic and constructible the resulting design will be.

In cases where these required shear and connection capacities cannot be provided, the ultimate resistance of the component must be based on

The mass and resistance factors are derived from similar energy equivalency equations for the kinetic and strain energies, respectively, between the equivalent SDOF system and the blast-loaded com-ponent. However, the load and resistance factors are equal because the strain energy can be treated as internal work. The internal work from the spring in the SDOF system is set equal to the internal work from the resisting force of the component, which is assumed to have the same shape function as the ap-plied load, and the derived resistance factor is equal to the load factor.6

Equation (5) shows the result for the derivation of the mass factor. Table 2 shows load and mass fac-tors for one-way components with simple supports. The calculation of transformation factors for other boundary conditions and blast-load distribution cases, including two-way spanning components, is explained elsewhere.2,6,7,8 Morison7 provides the most thorough discussion of transformation factors.

KM=

m x( )!2x( )dx

0

L

"

m x( )dx

0

L

" (5)

where

KM = mass factor

m(x) = mass distribution along component span

It is common to conservatively neglect the damping term in Eq. (2) because the definition of this term is not well understood for a component responding to blast load. Also, the amounts of damping assumed for other types of dynamic structural response (earthquake loading, for example) typically do not have a significant effect on blast-loaded compo-nents because these components reach their maxi-mum response during their first response cycle.

Equation (6) shows the simplified form of the equa-tion of motion for the equivalent SDOF system that is typically used for blast design. This equation can be solved using any number of time-stepping nu-merical integration techniques, such as the New-mark-Beta method.1,2,6,8 The maximum deflection from the solution of Eq. (6) is used to calculate the ductility ratio or the support rotation for the compo-nent as described previously, which are compared with the allowable design values.

KLMMw"(t) + R(w(t)) = F(t) (6)

where

KLM = load-mass factor = KM /KL Figure 10. Wall panel designed to resist blast load. Note:

1” = 25.4 mm; 1’ = 0.3048 m; 1 psi = 6.9 kPa.


the lower resistance provided by the shear or connection capacity, and a significantly lower allowable design criteria must be used.

It is also common to provide symmetrical reinforcing in concrete sec-tions, such as doubly reinforced sections, subject to blast loads. Struc-tural components initially respond to the positive pressure from the blast load and then rebound out from the building in a similar manner to a spring rebounding from a short-duration applied load. The negative phase of the blast load can accentuate the rebound response phase.

During rebound, there is stress reversal at all maximum stress regions of the component. All compressive stress areas during inbound re-sponse become tensile stress regions during rebound and require suf-

ficient reinforcing steel to prevent rebound failure. Also, the connections must have the capacity to resist reaction forces from component rebound re-sponse. Often the maximum forces and stresses that develop during rebound are on the order of 70% to 80% of those during inbound response, though 100% is possible. It is typical for blast design to provide equal reinforcing steel and connection ca-pacity for inbound and rebound phase responses.

Wall panel with opening design example

As an example, Fig. 10 shows a wall panel with an opening. The wall panel will be designed to resist the positive-phase blast load from a 100 lb (45 kg) TNT surface-burst explosion at a straight-line standoff distance of 50 ft (15 m) from the center of the panel. It is assumed that the panel is facing toward the explosive source so that a reflected blast load is applied. The resulting pressure and time relationship of the blast load for design purposes is equal to the positive phase of the blast load in Fig. 3. The wall panel is assumed to span vertically between spandrel beams at each floor. The 4 ft (1.2 m) width of the panel that spans vertically is as-sumed to carry the blast load from the full 8-ft-wide (2.4 m) panel, where the opening is assumed to be covered with a blast-resistant window that transfers the full blast load into the panel on either side.

The calculated ultimate flexural resistance of the panel is equal to 2.17 psi (15.0 kPa) based on simple support conditions, a clear span of 8.25 ft

Figure 11. Resistance-deflection curve for example panel. Note: 1 in. = 25.4 mm; 1 psi = 6.9 kPa.

Figure 12. Calculated deflection history for reinforced concrete panel. Note: 1 in. = 25.4 mm; 1 psi = 6.9 kPa.


2.17 psi (15.0 kPa) in ductile response.

After the component reaches its maximum inward deflection, that is, positive deflection, at 33 msec in Fig. 12, the rebound phase begins and the resistance begins decreasing in Fig. 13 until maximum rebound resistance and deflection occur at 52 msec. Figure 12 shows successive cycles of positive and rebound response that are somewhat unrealistic because damping was not included in this analysis and other simplify-ing assumptions, such as a constant stiffness in spite of increased con-crete cracking. The resistance will eventually return to zero in an actual blast loading case. There will be some permanent deflection based on the amount of plastic deflection past the yield deflection that occurs during dynamic response.

The maximum shear force is calculated based on an equivalent static load equal to the ultimate resistance of 2.17 psi (15.0 kPa) acting over the 8 ft (2.4 m) width of the panel. The calculated maximum shear stress at a distance d from the supports is therefore 98 lb/in. (17 N/mm) acting along the width of the panel. This shear force would typically be used for design even if the maximum component resistance from the applied blast load in Fig. 13 were less than the ultimate resistance. This approach helps ensure that the shear strength of the component will always be stronger than the maximum load that the component can ap-ply before it yields in flexure, even if the component is actually loaded by a blast load that is larger than the design blast load.

The corresponding ultimate shear strength of the panel is 600 lb/in. (105 N/mm), assuming the whole 8 ft (2.4 m) width of the panel is available to resist shear. Therefore, the panel load capacity is not controlled by shear. If this were not the case, the panel would require shear reinforcement, higher concrete compression strength, or possibly a greater thickness. The maximum connection force of 107 lb/in. (19 N/mm) over the whole panel width is also calculated based on a load equal to the ultimate resistance. The panel can be attached to the

(2.5 m), no. 4 (13M) Grade 60 (410 MPa) vertical reinforcing bars at 12 in. (300 mm) on center at each face, dynamic increase factors for the reinforc-ing steel, and concrete yield strengths as shown in Table 1. An SIF of 1.1 is applied to the minimum specified yield strength of 60,000 psi (410 MPa) for the reinforcing steel. The minimum specified compression strength for the concrete is 5000 psi (35 MPa). The flexural stiffness for the panel is based on an average of the gross moment of inertia and the fully cracked moment of inertia. Figure 11 shows the resistance and deflection curve for the wall panel.

The maximum allowable dynamic response for this example will be 3 degrees of support rotation, equal to 2.6 in. (66 mm) of midspan deflection. Figure 12 shows the calculated midspan deflection versus time relationship calculated by solving the equa-tion of motion in Eq. (6) of the equivalent SDOF system for the wall panel at each time step. The maximum calculated deflection is 2.36 in. (60 mm), corresponding to 2.7 degrees of support rotation. Therefore, this design is acceptable compared with the maximum allowable dynamic response.

Figure 13 shows the calculated resistance and time relationship for the panel. As this figure shows, the resistance and corresponding resisting moment in the panel increase quickly during the first 5 msec until the maximum moment at midspan reaches the ultimate dynamic moment capacity. The panel continues to deflect, but the maximum moment and corresponding resistance remains constant at

Figure 13. Calculated resistance and time relationship for reinforced concrete panel. Note: 1 psi = 6.9 kPa.


Tedesco, J. W., W. G. McDougal, and C. A. 8. Ross. 1999. Structural Dynamics. Menlo Park, CA: Addison Wesley. J

Notation

C = viscous damping constant of component

d = distance from extreme compression fiber to centroid of tension reinforcement

f 'c = compressive strength of concrete

f 'dc = dynamic compressive strength of con-

crete

f 'dm = dynamic compressive strength of ma-

sonry

f 'm = compressive strength of masonry

fdu = dynamic ultimate strength of reinforcing steel

fdy = dynamic yield strength of reinforcing steel

fu = ultimate strength of reinforcing steel

fy = yield strength of reinforcing steel

F(t) = total blast load on component

kE = equivalent stiffness

K = load-mass factor = KM /KL

Kd = viscous damping factor

KL = load factor

KM = mass factor

KR = resistance factor

L = length of blast-loaded component

m(x) = mass distribution along component span

M = mass of blast-loaded component

p(x) = blast load shape function of component relative to distance along its length with with P(t) = 1.0 for a uniformly applied blast load

P(t) = blast load at given point along length of component at each time step

re = resistance causing yielding in initial maximum moment region

ru = resistance causing plastic hinges at all maximum moment regions (component becomes mechanism)

R(w(t)) = resistance of component based on resistance versus deflection curve and deflection at each time step

t = time variable

building framing with a variety of connections, including drilled anchor bolts. Only drilled anchor bolts recommended for dynamic load applica-tions should be used.

Unfortunately, there has been little testing to investigate the blast re-sponse of connections for precast/prestressed concrete components. Usu-ally, the same types of connections are used for conventional and blast loads, except that the connections must be stronger for blast loads. The connection must be able to resist the maximum connection force for both inbound and rebound response. In most cases, the maximum connection design force for rebound response is assumed equal to that from inbound response. An additional safety factor may be provided if the connections are designed so that a ductile response mode controls the overall connec-tion strength. This occurs, for example, when the overall strength of a drilled anchor bolt connection is controlled by yielding of the bolt, rather than pullout from the concrete. If a lower allowable support rotation were specified for this example, the panel would require more reinforc-ing steel. It would, therefore, have a larger ultimate resistance and higher corresponding maximum shear forces and connection forces. The higher shear forces would not be a problem in this case. Also, the panel could easily accommodate more reinforcing steel and the connections could be designed for a higher maximum connection force.

Conclusion

Existing references have blast design guidance that is applicable to pre-cast/prestressed concrete components, though this guidance is not widely understood within the precast concrete industry. The guidance is based on modeling components as equivalent SDOF systems, which is the most prevalently used simplified blast-design method. An understanding of blast loads, dynamic component response to the blast loads, and allow-able design criteria are all critical to blast-resistant design of precast/pre-stressed concrete components.

References

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

U.S. Department of the Army. 1990. Structures to Resist the Ef-2. fects of Accidental Explosions. TM 5-1300, Navy NAVFAC P-397, AFR 88-2. Washington, DC: Departments of the Army, Navy, and Airforce.

Precast/Prestressed Concrete Institute (PCI). 2004. Designer’s Note-3. book: Blast Considerations. Pts. 1, 3, and 4. Ascent (Spring 2004): pp. 12–18; (Summer 2004): pp. 18–24; (Winter 2005): pp. 38–43.

Unified Facilities Criteria (UFC). 4. DoD Minimum Antiterrorism Stan-dards for Buildings. UFC 4-010-01. www.wbdg.org/ccb/DOD/UFC/ufc_4_010_01.pdf.

U.S. Army Corps of Engineers. 2006. 5. Single Degree of Freedom Structural Response Limits for Antiterrorism Design. PDC-TR 06-08. U.S. Army Corps of Engineers.

Biggs, J. M. 1964. 6. Introduction to Structural Dynamics. New York, NY: McGraw-Hill Book Co.

Morison, C. 2006. Dynamic Response of Walls and Slabs by Single-7. Degree-of-Freedom Analysis: A Critical Review and Revision. International Journal of Impact Engineering, V. 32, No. 5 (August 2006): pp. 1214–1247.


td = blast load duration

w´(t) = velocity of single-degree-of-freedom system with respect to time

w˝(t) = acceleration of single-degree-of-freedom system with respect to time

w0(t) = maximum deflection of component at each time step

w(t) = deflection of single-degree-of-freedom system with respect to time

xe = yield deflection of determinate components

xE = equivalent yield deflection of inde-terminate components for simplified resistance-deflection relationship

xm = maximum deflection of component

xp = deflection of component when all plastic hinges are formed

x(t) = deflection as a function of time

θ = support rotation

μ = ductility ratio

ø(x) = deflected shape function of component

About the authors

Sanaa Alaoui is a lead engineer with ABS Consulting in San Antonio, Tex.

Charles Oswald, Ph.D., P.E, is senior principal with Protection Engineering Consultants LLC in Spring Branch, Tex.

Synopsis

Blast design guidelines and methodologies have been de-veloped for many common types of building components, including reinforced concrete members, based on both theoretical analysis and testing. Much of this blast design guidance is applicable to precast/prestressed concrete components, though it is not widely understood within the precast concrete industry. This paper presents prevalent simplified methods for blast-resistant design of precast/

prestressed concrete components, assuming such compo-nents can be modeled as equivalent single-degree-of-free-dom (SDOF) systems. This paper includes discussions of blast loads, material behavior under dynamic loads, and the design of precast/prestressed concrete components to resist blast load. A number of design considerations are included (such as design of connections) with a design example. The process of defining an equivalent SDOF system for components is also discussed, though other references are available that provide specific detailed information.

Keywords

Blast, dynamic increase factor, dynamic load, load factor, mass factor, single degree of freedom.

Review policy

This paper was reviewed in accordance with the Precast/Prestressed Concrete Institute’s peer-review process.

Reader comments

Please address any reader comments to PCI Journal editor-in-chief Emily Lorenz at [email protected] or Precast/Prestressed Concrete Institute, c/o PCI Journal, 209 W. Jackson Blvd., Suite 500, Chicago, IL 60606.

Blast-Resistant Design Considerations for Precast Prestressed Concrete Structures

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