analytical assessment of blast resistance of precast

PCI Journal November–December 2007 67

Editor’s quick points

n As part of this issue’s theme on blast, this paper pres-ents a series of four explosive detonations that were conducted on precast concrete wall panels at Tyndall Air Force Research Laboratory (AFRL) in Panama City, Fla.

n The analysis method presented is validated with ex-perimental results from the Tyndall AFRL experiments.

n Research presented in this paper was partially funded by a PCI Daniel P. Jenny Fellowship.

Analytical assessment of blast resistance of precast, prestressed concrete componentsNicholas Cramsey and Clay Naito

Protection against blast has become a high priority for many build-ing owners. Blast retrofits and structural hardening, much like earthquake retrofits, can be costly. For this reason, it is important to understand that any structural element has an inherent capacity to absorb energy and resist some level of blast demand.

A general evaluation that allows a designer to realize the absorption capacity of a structural element may preclude the need for a blast-specific retrofit. To illustrate this concept, the blast resistances of non-loadbearing precast, prestressed concrete sandwich wall panels are examined. These components are used extensively in modern construction for cladding of building systems and often provide a significant level of protection from blast events.

This paper investigates the behavior of precast, prestressed con-crete sandwich wall panels subjected to blast loads. Four explosive experiments were performed on four sets of wall panels. An analyti-cal model was developed and validated with the measured blast demands and peak displacements. The analytical model is used to predict wall-panel damage for varying levels of peak pressures and impulses. An extension of this method is proposed for assessing the blast resistance of horizontal diaphragm elements, such as double tees or hollow-core. An example analysis for a double-tee floor system with a localized blast is provided.

Background on blast demands

Blast demands on structures are generated by the detonation of a high-explosive charge. Detonation occurs over a short period of time—nanoseconds—and results in the generation of elevated pres-sures and temperatures. The pressure loading that is generated is complex and depends on many factors, including the type and size of explosive, the location of the explosive relative to the structure, and the objects between the high explosion and the structure.

68 PCI Journal November–December 2007

The load effects generated by an explosion can be divided into four components: overpressure, primary fragment impact, secondary frag-ment impact, and reflected pressure.1 Primary fragments are charac-terized as high-velocity pieces of the explosive casing. Secondary fragments are objects that lie between the structure and the high explo-sion that are propelled as the blast wave expands from the detonation. Although both of these demands can result in significant loss of life, they are not commonly considered in the design or assessment of the structural system due to their limited size.

As the pressure loading expands outward from the detonation location, a rise in the ambient pressure occurs. The magnitude of this pressure increase, or overpressure, is dependent on both the amount of and radial (standoff) distance from the explosive. When the expanding pressure wave comes into contact with an object in its path, an instan-taneous rise in pressure above the overpressure occurs. This pressure increase is called reflected pressure. The reflected pressure is typically the controlling demand for analysis and design of structural compo-nents subjected to blast. Figure 1 shows reflected pressure demand.

The pressure arrives at time ta after the detonation. The pressure instantaneously rises to the maximum reflected pressure Pmax and decreases exponentially to a negative reflected pressure Pmin. The positive pressure occurs over a duration t0. The energy of the blast demand can be characterized by the area under the pressure-time curve. The area is divided into positive impulse Ip and negative impulse In.

Typically, pressure and time curves are approxi-mated for dynamic analysis. For simplicity, the negative impulse is often ignored and the shape is approximated as a triangular pulse. The pulse is assumed to have the same Ip and Pmax as the actual demand. A corresponding duration td can be determined (Fig. 1). Typical blast demands have td values on the order of 10 msec to 20 msec. Due to the short duration of load event, blast evaluation requires a structural dynamics approach.

Static and dynamic evaluation

Equation (1) shows that static design requires that the resistance of the structural component R be greater than the applied load demand F. For conventional demands such as wind or earthquake, an equivalent static demand is often used. For this case, the accelerations and velocities of the struc-ture are assumed to be zero.2 When subjected to an explosive pressure demand, the accelerations ÿ and velocities y· of the structure are not zero. For this highly dynamic event, the demand F(t) varies with time. Equation (2) presents the equation of equilib-rium that must be satisfied under a dynamic condi-tion. As a consequence, the mass M and damping C of the structure can provide significant contribution to the resistance.

R > F (1)

Mÿ + Cy· +R(y) = F(t) (2)

Approximate SDOF method

To analyze a structural system under blast- generated dynamic demands, the equilibrium equa-tion is often incrementally evaluated. Dynamic analysis methods, such as the approximate single-degree-of-freedom (SDOF) method, provide an efficient means for incremental evaluation. The approximate SDOF method is widely accepted for blast analysis and is used in practice. A brief over-view of the procedure is presented in the next sev-eral sections. A detailed discussion of this method is presented in Biggs3 and is outlined in Naito and Wheaton.4

Figure 1. Reflected pressure demand typically controls the analysis and design of components that are subject to blast.

Figure 2. Wall panel and double-tee components are uncoupled to allow for an idealized, ap-proximate, single-degree-of-freedom analysis when subjected to blast load. Note: 1” = 1 in. = 25.4 mm; 1’ = 1 ft = 0.3048 m; 1 lb = 0.453 kg; 1 psi = 6.895 kPa.


the given curvature distribution is calculated using structural-analysis principles. This procedure is repeated for each increment of the applied uniform load p(t) to develop an entire F-y relationship.

Structural damping

In most blast analysis and design, determination of the peak displace-ment is the primary goal. For these cases, damping is typically ignored. To model the post-peak response of a structural element, some form of damping must be included. A linear, viscous, classic damping model was assumed. An approximate damping ratio for the fundamental mode shape was found by applying the log-decrement method to the measured response. Equation 4 is the equivalent damped dynamic equation of equilibrium. Damping varies as damage accumulates in the structural component, but the damping is considered constant for this application. For dynamic evaluation, Eq. (3) or (4) can be solved incrementally using numerical integration.

KLMMÿ + Cy· + R(y) = F(t) (4)

Numerical integration

Numerical integration was performed using Newmark’s constant ac-celeration method for non-linear systems.6 A time step of 0.1 msec was used for all models. The midspan displacement as a function of time was found for the actual system subjected to blast loading. The ap-proximate SDOF method will be validated using experimental results in the next several sections.

Equivalent component

In an SDOF evaluation, the structure is examined at the component level. Each member of interest is un-coupled from the rest of the structure and analyzed individually. Appropriate boundary conditions are used, but, for simplicity, simple support conditions are typically assumed. The use of simple supports results in the lowest stiffness, which provides a conservative estimation of element deformation.

As an example, Fig. 2 shows how the first-floor wall panel and the second-floor double-tee panel can be uncoupled from the rest of the structure. Once uncoupled, an approximate SDOF analysis can be conducted. In the analysis for Fig. 2, the real wall panel and double-tee components that contain an infinite number of degrees of freedom (DOFs) are reduced to an equivalent component character-ized by an SDOF. Reduction to an SDOF system is based on the assumption that the fundamental mode shape, or a close approximation of it, governs system behavior. This greatly reduces the computa-tional effort required to perform the analysis.

In this paper, mode shapes were derived from the deflected shape that resulted from static applica-tion of a uniform pressure load. In Fig. 2, mode shapes Φ1 and Φ2 correspond to elastic and plastic response, respectively.

The SDOF equivalent component has an equiva-lent mass Me, applied force F(t), and component dynamic resistance R(y), which depend on the assumed mode shape, boundary conditions, and externally applied loading of the actual component. These aspects of the structural component are ac-counted for with a load-mass factor KLM. Table 1 presents the wall-panel load-mass factors that were formulated based on a uniform pressure load acting over the surface area. The double-tee load-mass factors for different spans are also included. Equa-tion 3 shows the equivalent undamped equation of dynamic equilibrium.

KLMMÿ + R(y) = F(t) (3)

Resistance of structural component

The resistance of the structural component is char-acterized by a load-displacement F-y relationship computed according to standard procedures.5 The moment-curvature M-Φ relationship of the cross section is determined using a fiber analysis. For a given applied pressure, the distribution of moment along the wall element is determined by static equilibrium. From the moment-distribution and moment-curvature relationship, the curvature along the wall can be determined. The displacement for Figure 3. Wall-panel elevations and cross sections used in this testing program. Note:

CFRP = carbon fiber-reinforced polymer. 1” = 1 in. = 25.4 mm; 1’ = 1 ft = 0.3048 m.

Cross section Φ1(x) Φ2(x)Wall panel 0.780 0.660

40-ft-span double tee 0.496 0.368




Note: 1 ft = 0.3048 m.

Table 1. Load mass factors based on uniform pressure load acting over surface area


exterior and interior wythes so that composite ac-tion could occur. The solid-zone panel contained solid zones of concrete connecting the exterior and interior wythes at eight locations on the face of the panel and at each end. The solid zones at each end extended the full width of panel. The CFRP panel contained only carbon-fiber reinforcement between the wythes. No other mechanical connectors were used for shear transfer between the wythes. The solid-zone panel and CFRP panel provided 80% and 100% composite action, respectively.

Material properties

The 28-day concrete compressive strength was 7.6 ksi (52 MPa), 8.9 ksi (61 MPa), and 8.6 ksi (59 MPa) for the control panel, solid-zone panel, and a CFRP panel, respectively. It was assumed that the no. 4 (13M) reinforcing bars met ASTM A706 specifications,7 and that the WWR met ASTM A185 specifications.8 The prestressing tendons were 3⁄8-in.-diameter (9.5 mm), seven-wire, low-relaxation strands with a strength of 270 ksi (1860 MPa). The mild steel reinforcement and WWR were modeled from recent mill certifications of similar material. Elastic moduli of the prestress-ing steel, mild reinforcing steel, and WWR were assumed to be 29,000 ksi (200,000 MPa).

Reaction structure

Figure 4 shows the reaction structure, which sup-ported two panels per detonation, one in position A and the other in B. Table 2 describes the place-ment of each wall panel for each blast detonation. The structure was enclosed on all sides with access doors located on the rear of the structure. Simple support conditions were assumed for the wall pan-els. The fixture was designed to have a 0.25 in. (6 mm) gap on either side of the panel. Some bind-ing was noted in the bottom 5 ft (1.5 m) of posi-tion A. The reaction structure was fitted with 14 external pressure gauges, denoted with P1 through P14, and six displacement gauges, denoted with D1 through D6. Displacement gauges were placed at the one-quarter, half, and three-quarters height of each panel.

Blast demands

Information on the amount of explosive and stand-off distance used in the experimental program is restricted. The same amount of explosive was used in each detonation. The maximum standoff distance occurred in the first experiment (test 1) and was subsequently decreased for the three remaining experiments (tests 2, 3, and 4). The combination of standoff distance and amount of explosive used in the final evaluation (test 4) exceeded the high

Experimental program

A series of four explosive detonations were conducted on precast con-crete wall panels at Tyndall Air Force Research Laboratory (AFRL) in Panama City, Fla.

Panel geometry

Three precast concrete wall-panel types were studied: two different pre-cast concrete sandwich panels, a solid-zone panel and a C-grid carbon fiber-reinforced polymer (CFRP) panel, and a solid control panel. Each precast concrete sandwich wall panel was tested twice, and each time it was paired with a control panel for testing. Figure 3 shows the wall-panel elevations and cross sections.

The control panels had a solid cross section and were conventionally reinforced with no. 4 (13M) reinforcing bar and welded-wire rein-forcement (WWR)—no prestressing was used. The cross section of the control panel was designed to provide the same mass as that of the sandwich panels.

The solid-zone panel and the CFRP panel were conventional, insulated, precast concrete sandwich wall panels. They both contained prestress-ing strand and WWR. Eight strands, each with an initial jacking force of 16.1 kip (71.6 kN), were pretensioned. A 25% prestress loss from jacking to effective stress was assumed. The sandwich panels were constructed in specific ways to provide shear transfer between the

Figure 4. The reaction structure supports two panels, A and B, per blast detonation. Note: D# = displacement gauges; P# = pressure gauges. 1” = 1 in. = 25.4 mm; 1’ = 1 ft = 0.3048 m; 1 lb = 0.453 kg; 1 psi = 6.895 kPa.

Test Position A Position B1 Solid-zone panel Control panel 1

2 Solid-zone panel Control panel 1

3 C-grid CFRP panel Control panel 2

4 C-grid CFRP panel Control panel 2

Note: CFRP = carbon fiber-reinforced polymer.

Table 2. Wall panel placement for each detonation

The combination of standoff distance and amount of explosive used in the final evaluation (test 4) exceeded the high level of protection requirements of government standards.


variation over the wall was small, with a coefficient of variation of less than 3% on the pressure and 10% on the positive impulse. Conse-quently, a uniform pressure load was assumed for all evaluations. The measured peak pressures ranged from 8 psi to 29 psi (55 kPa to 200 kPa), with a positive duration from 16.4 msec to 15.1 msec and a posi-tive impulse from 69.1 psi-msec to 144 psi-msec (476.4 kPa-msec to 992.8 kPa-msec) over the four experiments. Designing wall panels for a static load of 8 psi (55 kPa) would be overly conservative and cost prohibitive.

Figure 6 presents post-test cracking diagrams for test 2 and test 3. Cracking was more extensive for the control panels than for the sand-wich panels. The crack size was largest at the midheight and decreased toward the supports in accordance with one-way action behavior. A vertical crack, however, was observed in the CFRP panel after test 3. This is indicative of two-way action at the bottom of the panel and may be due to binding between the panel and reaction structure. In all cases the panels performed well, with cracking distributed over the height, no de-lamination between wythes, and a permanent displacement (de-formation) of less than 3 in. (76 mm) over the 30 ft (9 m) span.

level of protection requirements of government standards. Because no significant structural distress was apparent in the solid-zone panel or control panel after test 1, they were reevaluated in test 2 at a closer standoff distance. The CFRP panel and control panel evaluated in test 3 were also reevalu-ated in test 4.

Experimental results

Figure 5 presents results from the four blast deto-nations. The control panels exhibited consistently higher peak displacements and permanent displace-ment (deformation) than the sandwich panels. The first peak displacement was the largest and was fol-lowed by a rebound displacement of smaller magni-tude. The sandwich panels exhibited much greater damping than the control panels did, which may be attributed to the prestressing or to partial binding of the panels against the reaction structure. Due to the questionable source of damping in the sandwich panels, the damping of the control panel was used for the subsequent analysis of these panels.

In all cases, the peak displacements occurred during or after the negative region of the impulse. The uncracked natural period of the walls was rela-tively long compared with the pressure demands. It ranged from 0.10 sec to 0.30 sec, while the entire pressure demand lasted 30 msec to 50 msec. Shorter panel lengths of 10 ft to 20 ft (3 m to 6 m), typical of traditional floor heights, would signifi-cantly increase the stiffness and natural period of the walls. For these systems, the peak displacement would occur earlier, resulting in less sensitivity to the negative pressure phase of the blast. Due to the occurrence of the peak displacement after the posi-tive pressure phase, the damping present in the wall systems, and yielding of the walls in flexure, the first peak displacement was the critical displace-ment. For systems with a significant difference in positive and negative flexural strength, this may not be the case.

Displacement was not symmetric about the mid-height of the wall. Figure 3 shows the differences between the one-quarter- and three-quarters-point displacements: D17, D20 and D19, D22. Because the explosive charge was at ground level, the pressure wave reached the lower portion of the wall first. Measurements on the reflected pressure gauges indicated that the time of arrival varied from 1.5 msec to 2.0 msec over the wall. In addition, because the distance to the three-quarters point was greater, the pressure was lower at the top of the wall than at the bottom.

The peak pressure and positive impulse varied over the height and width of the wall, with the highest value at the bottom center pressure gauge P6. The

Test 1 Solid zone panel Test 1 Control panel

Test 2 Solid zone panel Test 2 Control panel

Test 3 CFRP panel Test 3 Control panel

Test 4 CFRP panel Test 4 Control panel

Figure 5. Displacement-time response histories are from displacement gauges, and pressure-time histories of central pressure are from gauge P7. Secondary plots are magnifications of the responses up to the first peak displacements. Note: CFRP = carbon fiber-reinforced polymer. 1 in. = 25.4 mm; 1 psi = 6.895 kPa.


marizes the DIFs for different materials and strain rates from the Protective Structures Automated Design System.1 DIFs were not available for the ultimate strength of the WWR or for the prestress-ing strands. To be conservative, a value of 1.0 was used for both.

Moment curvature approximation

The moment curvature M-Φ analyses were per-formed using a fiber-analysis technique. The con-trol panels were modeled as solid concrete sections containing three hundred 0.02-in.-thick (0.5 mm) fibers. The solid-zone panel and CFRP panel were modeled as two solid concrete sections with fibers 0.02 in. thick. Figure 3 shows where the layers of reinforcement were concentrated. The fibers were assigned material properties as presented in Fig. 7.

For the sandwich panels, fully composite and non-composite analyses were performed to provide a boundary for the response under each of these cases. It was assumed that the insulation wythe remained intact during the analyses and provided no structural resistance.

Figure 8 presents the estimated moment-curvature responses of the panels. The sandwich panels were designed to have a nominal capacity greater than the cracking strength. However, a decrease in strength occurred as the panel transformed from the gross to the cracked section. After this decrease, the tension reinforcement became effective, resulting in yielding of the tension reinforcement and frac-ture of the WWR. The sandwich panels’ ultimate strength was controlled by fracture of the prestress-ing strands in the tension wythes. The control panel failed due to crushing of the concrete.

Approximate moment-curvature responses were developed from the non-linear estimates to sim-plify the SDOF analyses. The approximate curves characterize the behavior of the wall panels to ultimate curvature as shown in Fig. 8. In estimating the ultimate response, a conservative equal-energy approach was used. That is, the area under the pre-dicted moment-curvature diagram curve was found and used to develop an elastic, perfectly plastic ultimate curve. The elastic stiffness and the ultimate curvature were kept consistent. It is important to note that by maintaining an equal-energy approach in determining the ultimate curve, energy- absorption capacity of the system was not exaggerated.

Load-displacement approximation

Figure 8 shows the theoretical load-displacement relationships that were determined using the ap-proximate moment-curvature responses. Although

Predictive modeling of wall panels

The measured response was examined using an equivalent SDOF model based on the as-built conditions of the panels. The resulting moment-curvature and load-displacement responses are presented in this section.

Dynamic increase factors

Figure 7 shows the assumed in-place, static, steel stress-strain models and the linear, piecewise, approximate, concrete stress-strain models. Under high strain rates, the yield strength and ultimate strength of the concrete and steel increase. To account for this increase, a dynamic increase factor (DIF) was used. The strain rate was computed from the measured displacement time history and the predicted moment cur-vature at the face of the wall and at the location of the reinforcement. Strain rates varying from 0.05/sec to 0.1/sec were estimated. A strain rate of 0.1/sec was used to compute the DIF values. Table 3 sum-

Material Yield UltimateNo. 4 reinforcing bar 1.15 1.05

Welded-wire reinforcement 1.05 1.00

Prestressing strand 1.00 1.00

Concrete, compression n.a. 1.20

Concrete, tension/shear n.a. 1.10

Source: Data from Protective Structures Automated Design System (PSADS) Note: n.a. = not applicable. No. 4 = 13M.

Table 3. Dynamic increase factors from protective structures automated design system

Figure 6. Post-blast cracking patterns are shown for test 2 and test 3. Cracking was more extensive for the control panels than for the sandwich panels. Note: CFRP = carbon fiber-reinforced polymer. 1 in. = 25.5 mm.

0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

0 0 .0 5 0 .1S tra in

Stre

ss k

si

W W R

P restress ing

# 4 B ar

-2000

0

2000

4000

6000

8000

10000

-0.001 0.004 0.009 0.014 0.019

Strain

Stre

ss p

si

Concrete

x DIF

Figure 7. This shows the in-place, assumed, static stress-strain model for steel and the linear, piecewise, approximate stress-strain model for concrete. Note: DIF = dynamic increase factor; WWR = welded-wire reinforcement; # = no. 1 psi = 6.895 kPa; 1 ksi = 6.895 MPa; no. 4 = 13M.


vals. During the initial positive displacement, the wall panel follows the elastic and strain-hardening stiffness K1 and K2 that was previ-ously determined from the load-displacement analysis in Fig. 8. After maximum displacement is reached, the wall panel rebounds with a new stiffness K3, softer than K1 due to the damage incurred, until a zero displacement is reached at midspan. At this displacement, it is assumed that the wall-panel stiffness K4 returns to the original stiffness K1 as the uncracked exterior face, initially under compression, experiences tension. Once the wall panel midspan rebounds to zero displacement, a final stiffness K5 is used. The K5 stiffness should be comparable to K3 if no additional damage is incurred during the rebound displacement.

The initial rebound stiffness K3 and the reloading stiffness K5 are determined from the measured experimental results of the control panel. Stiffness K3 was determined by minimizing the error between the measured displacement and the damped model at yreb. Two methods were used to determine stiffness K5. In the first method, a fast Fourier transform (FFT) was performed on the data as the wall-panel midspan rebounded to zero displacement.

The FFT provides the natural circular frequency ωn inherent to the be-havior of a wall panel rebounding to zero displacement, which allows the stiffness K5 to be calculated using Eq. (5). In the second method, one half of the wall panel’s natural period Tn was estimated by calcu-lating the time between a positive and subsequent negative peak dis-placement as the wall-panel midspan rebounded to zero displacement. The K5 stiffness can be calculated from the natural period estimation Tn. Stiffness calculated using the two methods correlated to within about 5% for test 1.

the wall panels were designed to have the same mass, it is apparent from the load-displacement response that the wall panel flexural capacities differ. The fully composite solid-zone panel and CFRP panel sustained about 33% and 66%, respectively, more load than did the control panels. The extensive transverse cracking of the control panel shown in Fig. 6 can be attributed to the lower flexural capacity.

To determine the ultimate deformation, a plastic hinge length—that is, the length over which maxi-mum damage occurred—was estimated. A hinge length of 18 in. (460 mm) located at midheight was assumed. This corresponds with the region of con-centrated transverse cracking observed in the panels as shown in Fig. 6. The panel shape at the ultimate deflection level is consistent with the shape func-tion Φ2 shown in Fig. 2. The estimated theoretical ultimate displacements yult were 19.2 in. (488 mm) and 16.5 in. (419 mm) for the solid-zone panel and CFRP panel, respectively.

The solid-zone panel had ultimate deflections cor-responding to support rotations of 6 degrees (0.1 rad) with about 2.7 in. (70 mm) of shortening. The CFRP panel had ultimate deflections corresponding with support rotations of 5.1 degrees (0.09 rad) with about 1.6 in. (40 mm) of shortening. The calculations did not account for load-deflection effects, which may have reduced the ultimate displacement. This should be accounted for in loadbearing wall systems.

The control panel’s ultimate deflection was limited by the size of the supports. Based on the wall-panel boundary conditions, it was assumed that the wall panels could shorten 4 in. (100 mm) before stability of the top support was compromised. Using the ultimate deflection mode shape Φ2 as a basis for calculation of midspan deflection, this amount of shortening was equivalent to a midspan deflection of 26.9 in. (683 mm) and a support rotation of 8.4 degrees (0.15 rad). It is important to note that this deflection limit was a characteristic of the setup. Due to the high flexibility of the control panels, additional deflection capacity was available with enhanced supports.

Once the maximum displacement ymax for the given demand was achieved, the wall panels rebounded (Fig. 5). Due to the damage that occurs in achieving maximum displacement, the rebound stiffness was not equal to the initial stiffness. To determine the appropriate stiffness to use for rebound and subse-quent cycles, the measured response was examined.

Rebound stiffness estimation

Figure 9 presents a schematic of a resistance and displacement history for a typical wall panel sub-jected to blast load as well as the stiffness inter-

0

250

500

750

1000

1250

0.000 0.010 0.020 0.030Curvature 1/in.

App

lied

Mom

ent k

ip-i

n

Predicted M-Curv.Ult. M-Curv Approx.

(a)

SZP(NC)CFRPP(NC)

CP

CFRPP(FC)

SZP(FC)

WWR Failure

0

2 0

4 0

6 0

8 0

0 1 0 20 30

D ispla cement inches

Pres

sure

Loa

d ps

f

yc r

CF R P P (F C )

S ZP (F C )

C P

C F RP P (N C)

S ZP (N C)

yu l t

(b)

Figure 8. Estimated wall-panel moment-curvature responses and theoretical load-displacement relationships are developed from the estimated moment-curve responses. Note: CFRPP = [C-grid] carbon fiber-reinforced polymer panel; CP = control panel; FC = fully composite; NC = noncom-posite; SZP = solid-zone panel; WWR = welded-wire reinforcement. 1 in. = 25.4 mm; 1 kip-in. = 0.13 N-m; 1 psf = 47.9 Pa.

Figure 9. This resistance-displacement history is for a typical wall panel subjected to blast load. It includes stiffness intervals. Note: K# = stiffness; R = resistance. 1 in. = 25.4 mm; 1 lb = 0.453 kg.


plicable to different systems or configurations.

Critical damping ratio estimation

Based on the measured responses, a significant amount of damping was present in the panels. To model the displacement-time history accurately, a damping ratio was determined from the measured re-sults. The log-decrement method was used to calculate the damping ratio whereby the peak displacements in free vibration yf,max 1 and yf,max 2 (Fig. 10) were mea-sured relative to the permanent displacement (defor-mation) yperm. The peak displacements, permanent displacements (deformations), and damping ratios are summarized in Table 5 for tests 1 and 2. The lower critical damping ratio x of 10% is used in all analyti-cal models to provide a conservative estimation of displacement.

Experimental validation for control panels

Using the approximate load-displacement response in Fig. 8, the stiffness values in Table 4, and the ap-plied reflected pressure-time history from gauge P7, the approximate SDOF analysis was performed. The analyses were conducted for two models, one that was undamped and one with a 10% damping ratio. In Fig. 10, the predicted control-panel displacement-time histories are compared with the measured re-sponse for tests 2 and 3. The undamped predictions were truncated after the first rebound displacement.

The predicted peak positive displacement ymax, peak rebound displacement yreb, and permanent displace-ment (deformation) yperm are compared with the measured displacements in Table 6. The damped

!n

2=

K5

KLM

M (5)

Table 4 summarizes the resulting stiffness values for the control panels. The resulting percentage decrease in stiffness from K1 is shown. Tests 2 and 4 were conducted on damaged panels. Thus, the initial stiffness for those two tests was assumed to equal the K5 stiffness from the preceding test. The same stiffness degradation values were used for the sandwich panels. These values are specific to the walls studied and may not be ap-

Variable Test 1 Test 2yf, max 1 1.02 in. 1.84 in.

yf, max 2 0.74 in. 1.49 in.

yperm 0.52 in. 1.11 in.

x 13% 10%

Note: 1 in. = 25.4 mm.

Table 5. Peak displacements, permanent displacements (deformations), and critical damping ratios

StiffnessTest 1 Test 2 Test 3 Test 4

Value, lb/in. Initial, % Value, lb/in. Initial, % Value, lb/in. Initial, % Value, lb/in. Initial, %K1 15,564 100 5060 33 15,528 100 3882 25

K2 356 2 356 2 416 3 416 3

K3 7159 46 4046 26 4813 31 3571 23

K4 15,564 100 15,564 100 15,528 100 n.a. n.a.

K5 5060 33 3950 25 3882 25 n.a. n.a.

Note: n.a. = not applicable. 1 lb/in. = 175.2 N/m.

Table 4. Control panel stiffness values

-1.5

0.0

1.5

3.0

4.5

0 500 1000 1500Time [msec]

Dis

plac

emen

t [in

ches

]

ActualModel - 10% dampingModel - Undamped

yf ,m a x 1 yf , m a x 2

yp e r m

(a)

L/2

y

-1.5

0.0

1.5

3.0

4.5

0 500 1000 1500Time [msec]

Dis

plac

emen

t [in

ches

]

ActualModel - 10% dampingModel - Undamped

yf ,m a x 1 yf ,m a x 2

yp e r m

(b)

L /2

y

Test 2 Control panel Test 3 Control panel

Figure 10. Predicted control-panel displacement-time histories are compared for tests 2 and 3. Note: 1 in. = 25.4 mm.

Test

Experimental 10% damping model Undamped model

ymax, in. yreb, in. yperm, in.ymax, in.

(% variation)yreb, in.

(% variation)yperm, in.

(% variation)ymax, in.

(% variation)yreb, in.

(% variation)1 2.02 -0.44 0.52 1.82 (-9.9) -0.45 (-2.3) 0.65 (25) 2.05 (1.5) -0.58 (-31.8)

2 3.92 -0.02 1.11 3.85 (-1.8) - 0.02 (0.0) 1.25 (12.6) 4.21 (7.4) -0.25 (-1150)

3 4.98 0.59 1.58 4.10 (-17.7) 0.63 (6.8) 2.20 (39.2) 5.19 (4.2) 0.65 (10.2)

4 8.36 -0.11 2.25 7.92 (-5.3) -0.12 (-9.1) n.a. 9.11 (9.0) -0.45 (309)

Note: n.a. = not applicable. 1 in. = 25.4 mm.

Table 6. Control panel comparison of predicted peak positive and rebound displacements to measured displacements


model consistently underestimated the peak displacement, while the undamped model consistently overestimated the peak displacement. To be conservative, it is recommend-ed that damping be ignored when maximum displace-ment is to be determined. The first rebound displacement was predicted more accurately with the damped model. The damped model also provided a conservative estimate of the permanent displacement (deformation).

Experimental validation for sandwich panels

The analytical model developed using the approxi-mate SDOF method was extended to the sandwich panels. Unless otherwise stated, the same tech-niques used for the control panels were used for the sandwich panels. The results of the approximate undamped SDOF analyses for tests 2 and 3 are compared with the experimental response in Fig. 11. The peak and rebound displacements for all four experiments are compared to the fully compos-ite and non-composite model estimates in Table 7.

For all experiments, the assumption of full com-posite action provided a reasonable correlation with the measured peak displacement. Modeling the sandwich panels as non-composite significantly overestimated the peak displacement. The rebound displacement was not accurately captured in either of the models. This may be attributed to a number of causes, including, but not limited to, binding of the panel in the reaction structure or variation in composite action over displacement history.

Reflected pressure approximation

As mentioned previously, reflected pressure and time curves are typically approximated for dy-

namic analysis as a triangular pulse. To determine the adequacy of this approximation, the wall panels were subjected to a triangular pulse equivalent to the positive phase of the measured reflected pressure and time curves from Fig. 5. The resulting maximum displacements are compared in Table 8 to the displacements from the analytical model developed using the measured reflected pressure-time history in Tables 6 and 7.

It is apparent that the triangular pulse approximation was overly con-servative for the wall panels in this study. To accurately estimate the peak displacement of these panels, the negative pressure phase must be included. This may not be the case for shorter panels. Recall from Fig. 5 that all peak displacements occurred during or after the negative re-gion of the impulse. Shorter panels will have increased stiffness, which will cause the peak displacements to occur earlier. This will decrease the sensitivity of the peak displacement to the negative phase.

Generalized blast capacity assessment

To determine the blast capacity of a structural component, the equiva-lent SDOF modeling technique is used to develop isodamage curves.9 A flexural isodamage curve (FIDC) provides combinations of peak pressure Pmax and a corresponding impulse Ip that cause the flexural capacity of the structural element to be achieved.

-4.0

-2.0

0.0

2.0

4.0

6.0

0 200 400 600 800Time [msec]

Displacement [inches]

ActualModel: Fully Comp.Model: Non-Comp.

L/2

y

(a) -4.0

-2.0

0.0

2.0

4.0

6.0

0 200 400 600 800Time [msec]

Displacement [inches]

ActualModel: Fully Comp.Model: Non-Comp.

L/2

y

(b)

Test 2 Test 3

Dis

plac

emen

t in.

Dis

plac

emen

t in.

Figure 11. Sandwich panel approximate, undamped, single-degree-of-freedom analyses are compared with experimental responses. Note: msec = millisecond. 1 in. = 25.4 mm.

TestActual Fully composite model Non-composite model

ymax, in. yreb, in. ymax, in. (% error) yreb, in. (% error) ymax, in. (% error) yreb, in. (% error)1 1.52 -0.91 1.86 (22.4) -0.09 (90.1) 2.19 (44.1) -2.89 (-218)

2 2.87 -0.95 3.22 (12.2) -0.75 (21.1) 5.05 (76.0) 1.12 (218)

3 4.46 -0.49 4.32 (-3.1) 1.89 (-486) 5.97 (33.9) -3.75 (-665)

4 6.50 -0.31 6.11 (-6.0) -0.75 (-142) 14.81 (128) 0.32 (203)

Note: 1 in. = 25.4 mm.

Table 7. Sandwich wall panel analytical models and measured peak and rebound displacements

TestSandwich panel Control panel

Actual P-t, in. (% error) Approximate P-t, in. (% error) Actual P-t, in. (% error) Approximate P-t, in. (% error)1 1.86 (22.4) 5.35 (252) 2.05 (1.5) 8.04 (298)

2 3.22 (12.2) 10.03 (249) 4.21 (7.4) 16.54 (322)

3 4.32 (3.1) >16.5 (>270) 5.19 (4.2) >26.9 (>440)

4 6.11 (6.0) >16.5 (>154) 9.11 (9.0) >26.9 (>222)

Note: P-t = pressure time. 1 in. = 25.4 mm.

Table 8. Analytical model displacements and measured maximum displacements


Flexural isodamage curves

To formulate a specific point on an FIDC using the triangular pulse ap-proximation, a td value from Fig. 12 was assumed. Peak pressure Pmax was incrementally increased within the approximate SDOF method until the ultimate displacement of the wall was achieved. For the walls examined, the following ultimate displacements were computed: 26.9 in. (683 mm), 19.2 in. (488 mm), and 16.5 in. (419 mm) for con-trol panels, solid-zone panel, and CFRP panel, respectively. The com-bination of maximum pressure and positive impulse that caused the ultimate displacement to be achieved represents one point on the FIDC. The td was then increased and the procedure was repeated. A schematic of increasing triangular pulses and the corresponding displacement history for one value of td are shown in Fig. 12. Figure 12 also presents the peak pressure and impulse causing failure on the FIDC and the FIDCs for the panels.

Given the amount of explosive and standoff distance, the pressure and impulse demand can be computed using well-established methods.1 The peak pressure and impulse for a given amount of explosive can be computed for multiple standoff distances and be incorporated into the FIDC as shown in Fig. 12. Pressure and impulse values to the left and below the FIDC will not cause failure, whereas values to the right and above the FIDC will cause failure. For a given amount of explosive, the panels will fail in a flexural mode at a standoff distance defined as Xcloser. As the distance is increased to X and Xfurther, the demands de-crease. This trend is captured in the explosive demand curve in Fig. 12. Classified information, such as actual standoff distances and explosive charges, are not available for publication.

It is important to note that the conservatism of the triangular pulse ap-

proximation is nested within the FIDC. To remove this conservatism, the negative pressure phase of the blast loading can be included in the formulation.

Flexural isodamage curves and negative reflected pressure

Unlike the standard method, a complete positive and negative pressure-time history is required. The relative size and shape of the positive and negative pressure phase are dependent on the standoff dis-tances and explosive used. Consequently, in place of a simplified triangular pressure-time history, a series of pressure-time histories can be developed and used for a specific explosive charge at numer-ous standoff distances.

The reflected pressure-time history corresponding to a small amount of explosives at a large standoff distance can be generated using one of the tools available. The standoff distance is decreased until the reflected pressure-time history causes the wall panel ultimate displacement to be achieved. The amount of explosive is then increased and the pro-cedure repeated. This procedure was conducted on the solid-zone panel. Because a realistic pressure-time history was used, both positive and negative impulses were applied. To characterize the response on an FIDC, only the positive terms Pmax and Ip are included. The resulting FIDC in Fig. 12 indicates that the panel has a greater capacity than expected compared with the triangular pulse approximation FIDC. The solid-zone panel, which initially was expected to fail at a standoff distance Xcloser, was now capable of resisting the same explosive at an even closer standoff distance. At close standoff dis-tances, the uniform pressure load assumption may begin to break down. To account for these pressure distributions, new KLM factors may be required.

Shear capacity

For the flexural capacity to be achieved, premature shear failure must be prevented. This evaluation is accomplished by comparing the dynamic support reactions with the dynamic shear capacity. The dynamic reaction is determined from equilibrium of the dynamic system. Using the previously discussed mode shapes, the shear demand can be found with respect to time. Biggs3 summarizes this procedure, and Eq. (6) and (7) present it for the elastic and plastic responses, respectively. The dynamic sup-port reaction V(t) is a function of the applied load F(t) and resistance R at that time. The dynamic re-actions were calculated using numerical integration and compared with the shear capacity. The shear capacity of the panels was found from the static shear strength of concrete

2 f

c

' with the DIF from Table 3 applied over the gross concrete area.

-0.6

0

0.6

1.2

1.8

2.4

3

3.6

0 40 80 120Time [msec]

Pmax increased

td

(a)

0 600Time [msec]

Displacement [in]

26.9 in.

0 .0 1

0 .1

1

1 0

1 0 0

1 0 0 0

0 .0 1 0 .1 1 1 0 1 0 0 1 0 0 0P o sitive Im p ulse , I p [p si-s]

CPSZP

CFRPPSZP w/ Neg. PhaseExplosive Demands

X closer

X

X further

P m ax a nd t d fro m 1 2 (a)

F A ILU R E

S A F E

(b )

Pres

sure

psi

Pres

sure

, Pm

ax p

siFigure 12. Flexural isodamage curves are given for the three types of panels tested in this study. Note: CFRPP = [C-grid] carbon fiber-reinforced polymer panel; CP = control panel; SZP = solid-zone panel. 1 in. = 25.4 mm; 1 psi = 6.895 kPa.

-25

0

25

50

75

100

125

40 140

CFRP Shear CapacityCP Shear Capacity

CFRP Support Reaction

CP Support Reaction

90

T ime msec

Supp

ort R

eact

ion

& S

hear

C

apac

ity k

ips

Figure 13. Shear capacity and reaction demands for test 3. Note: CFRP = carbon fiber-reinforced polymer; CP = control panel. 1 kip = 4.45 kN.


Double-tee resistance

Figure 14 shows a 15 ft (4.6 m) double-tee section. Sixteen 0.5 in. (13 mm) special, low-relaxation, grade 270 ksi (1860 MPa) prestress-ing strands were used with a constant eccentricity. Each strand had an initial jacking stress of 60% of ultimate stress, and 18% prestressing losses were assumed. The double tee was assumed to have the same steel material properties as the previously examined panels in Fig. 7. The concrete was assumed to have a compressive strength of 6 ksi (41 MPa). The double-tee members were designed to support their self-weight D and a live load L of 40 psf (1.9 kPa) in accordance with ASCE 7-05.2 They were also designed to remain uncracked under service loads.

Load displacement results

Moment-curvature analyses were performed and load-displacement results generated for each double-tee span length. A plastic hinge length of twice the depth of the double tee was assumed for ultimate displacement calculations. The ultimate strength of the double tee was controlled by fracture of the lowest layer of prestressing strand. It was assumed that the panels have adequate supports to prevent unseating. Prior to application of the blast load, a factored load of 1.2D + 0.5L, consistent with the Unified Facilities Criteria, was applied.10 The re-maining double-tee capacity was available for blast resistance.

Analytical model application

Figure 15 presents the FIDCs for the four double-tee span lengths. The curves were formulated using the triangular pulse approximation as shown in Fig. 12. For the low-standoff-distance case examined, the negative phase was negligible to that of the positive phase as shown in Fig. 2. Using only the positive phase for this case should provide an accurate estimate of the panel capacity. These curves can be used to

V(t) = 0.39R(t) + 0.11F(t) (6)

V(t) = 0.38R(t) + 0.12F(t) (7)

Figure 13 presents capacity and reaction demands for test 3. The support reactions for both panels were similar until about 55 msec. Because the resistance is low and demand is highest at the time of arrival, the applied pressure load dominated the reaction force. The panel capacity was greater than the shear demand at all times.

The techniques presented in the previous sections allow a designer to effectively analyze the blast ca-pacity of precast, prestressed concrete wall panels. The FIDCs that include the negative pressure phase can be used as an effective blast evaluation tool.

Blast capacity assess-ment of double tees

The analysis techniques presented in the previous sections can be modified to simulate a close-in explosion with a 4 ft (1.2 m) standoff distance on double-tee floor diaphragms. Double tees of vary-ing span lengths were examined, and FIDCs were developed for each case.

Localized demand on double tee

For a close-in explosion acting over the surface area of a double tee, the blast loading is non-uniform. Simplified loading was developed to ease dynamic evaluation.

A prototype structure was developed in BlastX similar to the structure in Fig. 2. Both direct pres-sures on the double tee and reflections off of over-head double-tee members were considered. Seven different amounts of explosive, ranging from a satchel to a vehicle-sized explosive, were detonated at midspan of the double tee, 4 ft (1.2 m) above the surface. For each detonation, the peak pres-sures along the double tee were found, resulting in a non-uniform distribution. This was approximated with a uniform distributed load at midspan. For the uniform load, the maximum pressure was kept constant and the width was chosen to produce the same total force as the non-uniform distribution. A uniform distributed load width of 7.5 ft (2.3 m) was found by averaging the results from each amount of explosive. Figure 2 shows a schematic of the origi-nal non-uniform and uniform approximation.

Span lengths of 40 ft (12 m), 45 ft (14 m), 50 ft (15 m), and 55 ft (17 m) were examined in this study. To reduce the actual system to an equivalent system, KLM factors were determined based on the approximate loading, double-tee length, and simply supported boundary conditions. Table 1 summa-rizes the KLM factors for the varying span lengths.

Figure 14. Double-tee cross sections were examined in this study. The double tee was assumed to have the same steel material properties as the panels in Fig. 7. Note: WWR = welded-wire reinforcement. 1” = 1 in. = 25.4 mm; 1’ = 1 ft = 0.3048 m.

1

10

100

1000

1 10 100 1000Positive Impulse, I p psi-sec

40 ft.45 ft.50 ft.55 ft.

SAFE

FAILUREE

Pres

sure

, Pm

ax p

siFigure 15. This flexural isodamage curve is for double-tee span lengths. Note: 1 ft = 0.3048 m; 1 psi = 6.895 kPa; 1 psi-s = 6.895 kPa-s.


Future work

The analysis method presented is validated with experimental results from the Tyndall AFRL precast concrete wall panel experiments. To fully validate the method for other applications, ad-ditional experiments of alternate wall panel types and floor diaphragm members are imperative. In addition, the experimental and evaluation method discussed assumes that the supports are capable of resisting the dynamic forces and resulting deforma-tions imposed. Experimental validation of connec-tion performance under blast demands should be conducted.

With further validation of double-tee, hollow-core, and other precast concrete panels, the methods pre-sented can be used to develop a database of FIDCs, which could facilitate effective blast-resistant design of precast concrete systems.

Acknowledgments

This analytical study was funded by the PCI Daniel P. Jenny Fellowship and Lehigh University. The experimental program was supported by the contributions of the Portland Cement Associa-tion, Metromont Inc., and PCI. Technical support was provided by Jeff Fisher and Robert Dinan of Tyndall Air Force Base, Jason Krohn of PCI, Metromont Inc., members of the PCI Blast and Structural Integrity Committee, and High Concrete Structures. The authors acknowledge these groups and individuals for their support. The findings and conclusions presented in this report are those of the authors and do not necessarily reflect the views of the sponsors.

rapidly assess whether the blast resistance is adequate.

It is important to note that the extension of the model to double tees was purely analytical. Blast experiments verifying the analytical model results would be beneficial. Issues that may contribute to deviation from the presented results include, but are not limited to, the following:

Brisance, or local shattering, of concrete is possible for close-in •explosions.

Adjacent double-tee panels may contribute to the flexural •resistance.

Two-way flexural action could occur between the double-tee •webs.

Higher-order dynamic modes could contribute substantially to the •behavior causing the analytical model to break down.

Conclusions

From the results and discussion presented, the following conclusions can be made:

Single degree of freedom modeling methods provide a conserva-•tive approximation of deformation response for precast concrete wall panels.

To be conservative, damping should be ignored when determining •the maximum expected displacement.

If the displacement history is to be approximated, a damping ratio •of 10% should be included for the wall panels in this study.

Modeling the sandwich panels as fully composite provided a rea-•sonable approximation of peak wall displacement. Modeling the panels as non-composite provided overly conservative predictions.

Due to the flexibility of the wall panels examined in this study, •the peak displacement occurs during or after the negative pressure phase. Therefore, the negative impulse must be included for ac-curate prediction of deformation. Discounting the negative phase provides an overly conservative estimation of expected deformation.

Accurate isodamage assessment curves can be developed to •account for the negative pressure phase by applying realistic pressure-time histories.

Isodamage curves can be developed for double-tee floor dia-•phragm members of various size, cross section, and reinforcement. These models, however, are only theoretical and must be experi-mentally verified before they are used in practice.


Ip = positive impulse

KLM = load mass factor

L = live load

M = mass

Me = equivalent mass for one degree of freedom

M-Φ = moment-curvature relationship

p(t) = applied uniform blast load as a function of time

p(x,t) = applied uniform blast load as a function of location and time

Pmax = maximum peak reflected pressure

Pmin = negative reflected pressure

R = resistance

R(y) = component dynamic resistance

t0 = time at which positive pressure occurs

ta = time of pressure arrival after the detonation

td = duration corresponding to Ip and Pmax

Tn = natural period

V(t) = dynamic support reaction

x = critical damping ratio

X = standoff distance greater than Xcloser

Xcloser = standoff distance at which panels will fail in flexural mode

Xfarther = standoff distance greater than X

ycr = critical displacement

yf,max 1 = peak displacement in free vibration

yf,max 2 = peak displacement in free vibration

ymax = maximum peak positive displacement

yperm = permanent displacement (deformation)

yreb = peak rebound displacement

yult = theoretical ultimate displacement

y· = velocity

ÿ = acceleration

Φ1 =mode shape corresponding to elastic response

Φ2 = mode shape corresponding to plastic response

Φ1(x) = mode shape corresponding to elastic response as a function of distance

Φ2(x) = mode shape corresponding to plastic response as a function of distance

ωn = natural circular frequency

References

U.S. Army Corps of Engineers. 1998. 1. Pro-tective Structures Automated Design System (PSADS). Omaha, NE: U.S. Army Corps of Engineers.

American Society of Civil Engineers (ASCE) 2. Committee 7. 2005. Minimum Design Loads for Buildings and Other Structures. Reston, VA: ASCE.

Biggs, J. M. 1964. 3. Introduction to Structural Dynamics. New York, NY: McGraw-Hill Book Company Inc.

Naito, C. J, and K. P. Wheaton. 2006. Blast 4. Assessment of Load Bearing Reinforced Con-crete Shear Walls. ASCE Practical Periodical on Structural Design and Construction, V. 11, No. 2 (May): pp. 112–121.

Park, R., and T. Paulay. 1975. 5. Reinforced Concrete Structures. Hoboken, NJ: John Wiley and Sons.

Chopra, A. K. 2001. 6. Dynamics of Structures. 2nd ed. Upper Saddle River, NJ: Prentice Hall.

American Society of Testing and Materials 7. (ASTM) Committee A706. 1993. Standard Specification for Low-Alloy Steel Deformed Bars for Concrete Reinforcement. West Con-shohocken, PA: ASTM.

ASTM Committee A185. 2006. 8. Standard Specification for Steel Welded Wire Reinforce-ment, Plain, for Concrete. V. e1. West Consho-hocken, PA: ASTM.

Baker, W. E., P. A. Cox, P. S. Westine, J. J. 9. Kulesz, and R. A. Strehlow. 1983. Explosion Hazards and Evaluation. New York, NY: Elsevier Scientific Publishing Co.

Unified Facilities Criteria (UFC). 2005. 10. Design of Buildings to Resist Progressive Collapse. UFC 4-023-03. www.wbdg.org/ccb/DOD/UFC/ufc_4_023_03.pdf.

Notation

2 f

c

' = static shear strength of concrete

C = damping

D = dead load

F = applied load demand

F(t) = load demand as it varies with time

F-y = load-displacement relationship

hDT = double-tee height

In = negative impulse

J

About the authors

Nicholas Cramsey is a graduate research assistant for ATLSS Research Center at Lehigh Univer-sity in Bethlehem, Pa.

Clay Naito, Ph.D., P.E., is an assistant professor of structural engineering in the Department of Civil and Environmental Engineer-ing at Lehigh University in Bethle-hem, Pa.

Synopsis

Four full-scale explosive experiments were performed on 30-ft-tall (9 m) precast, prestressed concrete wall panels. Two conventional sandwich panels and a control panel were examined and shown to perform adequately under the blast demands. An approximate dynamic analysis technique for finding the blast resistance of the wall pan-els was developed and validated using the experimental

results. For the wall panels in this study, it was found that an undamped analytical model provided reasonable peak displacement approximations. Using the results from the undamped analytical model, conservative isodamage curves were developed that characterize wall-panel be-havior when the panels are subjected to varying levels of blast demands. It was found that accuracy of the isodam-age curve can be increased by accounting for the nega-tive impulse regime. The analytical model is extended to assess the capacity of double tees subjected to a close-in explosion.

Keywords

Blast loading, double tee, dynamic analysis, experimental results, isodamage curve, sandwich panels.

Review policy

This paper was reviewed in accordance with the PCI’s peer-review process.

Reader comments

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


analytical assessment of blast resistance of precast

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