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The detailed investigation of blast phenomena and their catastrophic effects on existing structures are the main objectives of the present paper. It is well known that blast phenomena may be characterized by significant complexity, often involving complicated wave propagation effects as well as distinguishable material behaviors. Considering the above and in an attempt to provide a simplified modelling approach for the simulation of blast effects, a novel procedure is presented herein based on well-established methodologies and common engineering practices. In the above framework, firstly, the “predominant” deformation shape of the structure is estimated based on elastic finite element simulations under blast loads and then the structural response of the system is evaluated as a result of common computational beam-element tools such as displacement-based pushover analysis. The proposed methodology provides an immediate first estimation of the structural behavior under blast loads, based on familiar engineering procedures. A two-span reinforced concrete bridge was thoroughly investigated and the results provide insightful information regarding the damage patterns and localization.

Although blasts are considered phenomena of significant severity and potential socioeconomic impact, only recently did the authorities realize the necessity for the formulation of an integrated design and assessment protective framework. Modeling the effects of these phenomena on structures is very demanding, requiring highly sophisticated simulations including advanced constitutive material models. These procedures are resource- and time-consuming. On the other hand, there is a lack of simplifying procedures that could be implemented by practicing engineers through the utilization of common computational tools. Herein, a simplifying procedure is proposed based on common analytical and computational tools that would provide a preliminary but yet reliable estimation of the blast impact to the structural integrity of bridges.

Blasts are short duration dynamic events that generate dynamic pressure waves which propagate radially from the source in space, exciting dynamic response in the structures that are encountered in their path. The pressures acting on the affected surfaces are impulsive loads that impart significant amount of potential energy which sets damage-causing vibrations in the structure.

The various loads that act on a structure during its lifetime (natural or man-made) are characterized [

(a) Frequency range of different dynamic loads (according to [

Schematic representation of the evolution of a surface blast wave in space; reflected and main pressure blast wave on an adjacent structure.

The effects of explosions are studied in this paper based on a simplifying procedure, with particular emphasis on one type of structure whose operation is vital in emergencies, namely, reinforced concrete highway overpasses. In the simplifying framework, the structure is approximated using concepts from generalized single-degree-of-freedom systems (Clough and Penzien [

The first section of the study deals with the mathematical definition of the excitation function exerted by the explosion and its numerical representation in the framework of established structural analysis software, considering both the temporal and spatial definition of the evolving pressure profiles acting on the exposed structural surfaces. The investigation uses a selected bridge case study subjected to a passing blast wave for illustration of concepts. A linear 3D finite element model of the bridge structure was combined with a consistent time history simulation of the explosion pulse, placing particular emphasis on the sequence of contact of the pressure wave with the structure, which depends on the physical distance and the location of the source relative to the structure. An essential ingredient for assessment of deformation demands and damage potential caused by the event is the deformed shape assumed by the structure through the time history of the event; this is extracted from the calculated response results. Next, using this pattern of deformations, the structure was analyzed based on the implementation of a simplified ordinary beam-element structural model so as to enable the use of the results of a nonlinear pushover analysis of the bridge under transient pressure profiles that simulate the explosive loads. This simplification alleviated partly the complexity of the problem associated with the time dependency of the constitutive properties of the materials, whereas it was possible to identify the tendency for damage localization throughout the structure. In this way, it is possible to take advantage of the nonlinear modelling technology that is stable and convergent when dealing with linear prismatic elements while avoiding convergence problems that would be owing to brittle failures in the continuous 3D finite element model.

At any single point, the pressure wave has a time history of the type shown in Figure

Diagram of pressure variation as a function of time.

An event is classified as blast if, compared with other catastrophic events such as wind and earthquake, the pressure momentum is several orders of magnitude greater than that of other phenomena (FEMA 426 [

The size and distribution of the applied pressures on the structure depend on the amount and type of released energy (which depend on the explosives used), the position of the source relative to the structure, and the magnitude and possible amplification of the resulting blast pressure owing to its interactions with objects encountered during dilation and propagation of the wave (Birhane [^{1/3} (obtained after pertinent normalizing; Ngo et al. [

For hemispherical explosions, that is, explosions that occur on ground surface, it is recommended that the variable term in (

High strain rates affect the mechanical properties of the materials and thus the mechanisms of degradation of the various elements of the structure. The effect of high deformation rate on material strengths is quantified by the Dynamic Increase Factor (DIF), which is defined as the ratio of the dynamic to static strength (Javier and Allen [^{2}/s–10^{7}/s, whereas for usual loads the rate of deformation ranges between 10^{−7}/s and 10^{0}/s (for reference, note that the rates associated with Creep and Relaxation phenomena are in the range of 10^{−8}/s to 10^{−5}/s; pseudostatic loads occur at strain rates from 10^{−8}/s to 10^{−4}/s; earthquake loads occur at strain rates from 10^{−5}/s to 10/s. Explosions and collisions occur at strain rates from 10/s to 10^{7}/s, whereas even higher rates correspond to astrophysical phenomena, CEB-FIP, Bulletin 56 [

For reinforced concrete structures subjected to blast loading, the strength of concrete and reinforcing materials may experience a significant increase due to the rate effects. The increase may exceed 50% for the reinforcing steel, whereas it may exceed 100% for concrete in compression and more than 600% for concrete in tension (Javier and Allen [

Consider^{−6} s^{−1} to 3 × 10^{2} s^{−1}, and ^{−6} s^{−1} is the rate of deformation under pseudostatic compression loading.

Consider^{−6} s^{−1} to 3 × 10^{2} s^{−1}, and ^{−6} s^{−1} (the rate of deformation at pseudostatic tensile loading conditions). Apart from material strength, the high deformation rate affects all other mechanical properties. For example, by definition, the modulus of elasticity of concrete under high deformation rate may be estimated from the following equation:

Javier and John [

This model applies to reinforcing steel with yield stress ranging between 290 and 710 MPa and for strain rates ranging between 10^{−4} and 225 s^{−1}. The strength increase estimated according to (

Blast effects on bridges are studied on a model bridge highway overpass, as illustrated in Figure

(a) Schematic representation of the bridge. (b) Geometric characteristics (transverse cross section). (c) Lengthwise cross section of the bridge.

The central bent columns have a circular cross section of 1.50 m diameter supported by separate pile caps resting on pile groups; each foundation block was a 3.75 m × 4.75 m × 1.00 m rectangular block. Clear height of the columns was 5.50 m. The portal frame bent was centered at the midpoint of the deck width with no eccentricity, whereas the clear transverse distance between columns was 3.88 m. Columns were monolithically connected with a bent cap beam (transverse beam) which in turn was connected monolithically with the adjacent deck superstructure. In the analysis of the bridge case study, the following parameter values were assumed: C30/37 concrete class (Eurocode 2, 2004) with a characteristic compressive strength

The problem considered in this study concerns the dynamic response of the bridge described above, owing to a surface explosion at near distance. A key issue to resolve first is the pressure profile occurring throughout the bridge structure and how the pressure wave propagates in space and time.

At the moment of the explosion the main pressure wave is transmitted uniformly in all directions. Over the duration of the transmission process the pressure varies with time: the peak pressure that occurs in each point in space depends on the distance from the source and the amount of the explosive material (Figure

Attenuation of peak overpressure with distance from source: blast wave propagation.

Furthermore, the pressure of the blast wave does not attenuate at the same rate in all points in space since it depends on the coefficient of degradation

The effect of the pressure on the structure begins at the instant when the blast wave arrives at a specific point on its exposed surface, denoted henceforth as the “arrival time.” Pressure value is maximum for any given spatial point in consideration at the arrival time. The pressure magnitude attenuates from that peak value according to (

The duration of the time period over which the pressure values exceed the atmospheric pressure also depends on

Regarding the bridge under consideration, the source of the explosion was assumed to be located on ground surface. Thermal radiation released upon the explosion and its effects on the material response were ignored in the present study. Surfaces that lie on the path of the wave front receive the reflected pressure, whereas the others receive the incidental pressure (Figure ^{2} s^{−1}) the DIF may be estimated from (

Schematic representation of contact of the blast wave with the bridge superstructure.

In the present section, a two-step simplifying methodology for the evaluation of blast effects on structures is proposed and implemented. The main scope is to provide an approximate yet reliable tool for instant estimations of the above effects based on well-established engineering design and assessment strategies. According to the methodology, a corresponding

Therefore, based on the deformation shape pattern obtained in the initial step, a displacement-based pushover analysis is conducted on a detailed beam-element computational simulation. Lumped plasticity or even brittle failure [

Let time be defined with the reference starting point set at the instant of the explosion (

During time

The same thing holds at time instances,

The set of contact points which are loaded simultaneously is evaluated automatically by calculating their normalized distance from the source based on the assumption that the blast wave evolves at constant speed radially (the geometric distance and the amount of the explosive matter are combined to calculate the normalized distance; this is then introduced in the corresponding relationships for calculation of pressure at every point in a given time). As illustrated in Figures

The speed of the wave front is calculated from the following equation (Ngo et al. [

The magnitude of the pressure felt by each surface depends on its position relative to the direction of evolution of the wave front. Surfaces located at the forefront of the wave transmission receive the reflected pressure, whereas all other surfaces receive the incidental pressure (Figure

Calculating in this manner the pressure that acts on the various points of the structure in time, the problem is reduced to a classical problem of structural dynamics which may be solved numerically using established procedures (e.g., with finite elements); the boundary of each finite element lying on the perimeter of the structure is loaded by a time-varying pressure function depending on its distance from the source. The volume of data that must be calculated is proportional to the size and complexity of the structure, and thus significant computing capacity is required for a usual structure. To deal with this problem, in the present investigation, a collection of geometric loci representing the intersection of the advancing hemispherical wave front with the structure were determined at no loss of accuracy. Based on the consecutive, time-dependent contact patterns with the bridge depicted in Figure

Separation of the loading surfaces of column and bridge deck to model the spherical propagation of the blast wave.

FE elastic analysis for a case with monolithic connections in the abutments and central pier. (a and b) Loading surfaces on the structure to model the wave transition. (c and d) Snapshots of the deformed state of the structure at the deck and around the central bent at peak displacement response.

In the present section, a simplifying methodology is proposed for a fast evaluation of blast effects based on the utilization of common analytical tools. In general, such complicated phenomena as blast effects would demand modelling and analytical approaches with significant computational cost. The dynamic nature of the problem, in conjunction with the advanced material constitutive properties, would drive to extremely complicated FE models with very large number of nodes, significant computational cost, and questionable results (e.g., convergence problems during inelastic finite element analysis). Based on the above, a simplifying, easy to use methodology was developed herein in order to reduce the computational cost and at the same time provide reliable solutions based on widely used software packages. The core of the proposed methodology is to define the predominant deformation shape of the structure under the blast wave and then to proceed with ordinary pushover analysis based on simplified, ordinary structural beam-element models with lumped properties. The entire method may be summarized in the following steps:

Based on the theoretical background presented herein (Section

Evaluate the structural behavior of all critical components (columns and beams) through the implementation of established analytical methodologies (i.e., moment-curvature section diagrams, shear and axial tension/compression ultimate strength values). The above-mentioned values will be used as an input to the lumped plasticity beam-element model and at the same time will form the failure/yielding criteria for the selection of a valid deformation pattern on the FE model of the next step. Strain rate phenomena should be also considered herein for concrete and steel, based on the provisions of Section

Generate a finite element model (simplified or detailed) and, through the implementation of elastic analytical tools, find the predominant deformation shape of the structure due to blast wave effects at peak dynamic displacement response. The detailed 3D finite element modelling approach presented in Section

Using the evaluated predominant deformation shape of the previous step, proceed with the conventional displacement-based pushover analysis in a simplified beam-element structural model with lumped properties. Detailed section properties and interactions are necessary to be incorporated in the model, based on ordinary structural modelling strategies (Step (b)). In any case, the resulting total force should be equal to or less than the maximum estimated applied force extracted from the FE model.

A key issue for the implementation of the proposed methodology is the estimation of a representative, predominant deformation shape of the structure, under the induced blast loads. For this reason the “predominant” deformation shape is defined herein as the elastic deformation state of the structural system just before brittle failure or yielding of the selected, important for the stability, structural components. Considering the severity of the induced loads and the corresponding instant brittle failures in critical structural components, the above-mentioned deformation shape approximation usually provides representative patterns of force distribution along the structure. Note that local yielding of structural elements under blast loads is rare, while brittle failure modes are usually the dominant pattern of damage. According to the preceding, based on the anticipated failure characteristics of each critical structural component of the bridge (e.g., shear failure, axial force, or moment capacity), the elastic deformation shape is extracted from the implemented FE computational study. For the structural system under examination (post-tensioned concrete bridge), the critical components that control the entire response are the post-tensioned beams as well as the central bent columns and the dominant deformation shape of the system is depicted in Figures

As it has been already stated in Section

Considering that the problem under investigation was reduced into a model comprising typical frame elements, appropriate inelastic hinge properties were evaluated for each structural component of the model. Based on well-known structural engineering evaluations, a wide range of possible constitutive behaviors (brittle or ductile) were modelled analytically for each existing critical component (shear and axial compression/tension ultimate forces, moment-axial load interaction diagrams, moment-rotation envelopes for lumped plastic hinges, and plastic hinge lengths and moment-curvature relationships for member sections where a mode detailed idealization was needed). Objective of this approach was to quantify the extent of damage in terms of values of established stress and strain resultants in the critical elements of the bridge (Figure

Analysis results for a case with fixed supports: monolithic connections in the abutments and central pier (obtained using nonlinear static analysis).

From the pushover analysis results it was found that plastic hinges form quickly in the structure, the most immediate being those at the top and the base of the two pier columns. Upon further examination of the estimated values it was concluded that columns would fail by a combination of shear and axial tension (as a result of the upwards pressure applied in the underside of the deck) after flexural yielding in the ends of the members. Considering the brittleness of the estimated mode of failure, the anticipated damage in the column could be extensive depending on the intensity of the blast wave. Beyond that point, a sequential formation of plastic hinges at various points in the structure is observed, owing to the vertical component of the blast wave throughout the span. The deck beams near the abutments and the central pier attain shear failure over an extensive portion of the span; no flexural failure is observed at mid-span suggesting that shear failure precedes flexural modes leading directly to nonproportional damage and collapse.

For the case under consideration, it is evident that, in contrast with ordinary seismic events, blast actions induce excessive loads which are distributed based on significantly different patterns along the superstructure and the substructure elements. Early axial tension and shear brittle failures dominate the response of the entire system, while flexural yielding phenomena, which are commonly considered to be critical in earthquake design practices, seem to be of minor importance.

Based on the above, it is evident that common earthquake design and detailing practices do not cover at all blast actions, as the developed mechanisms are significantly different from those considered in conventional load combinations including earthquake. Therefore, it should be underscored that existing infrastructure elements which are designed based on ordinary structural/earthquake design practices and provisions are vulnerable to significant blast loads and that additional measures need to be taken (active or passive protection) in order to secure the structural integrity of those bridge systems that would be considered critical for continuous functionality in an emergency situation.

A simplified modelling procedure for the fast assessment of the effects of blast loads on bridges is proposed. The methodology is a versatile tool for first-order estimation of the effects of blast explosions on structures, which would otherwise require an extensive and complicated nonlinear time history analysis. First the evolving pressure wave front was described mathematically in space and time so as to define its loci with the exposed structural surfaces. This enabled definition of the pressure forcing functions on the structure. This was applied using linear dynamic analysis on a detailed elastic finite element model of the structure in order to identify the predominant displacement profile experienced by the structure. Deformations were normalized to the peak response so as to develop a shape function for the bridge structure at the extreme displaced response. The identified displacement pattern was then applied on a nonlinear frame model of the structure. Through this modelling option, nonlinear static analysis was possible at a much reduced computational cost as compared to 3D solid nonlinear finite element modelling which is still, today, prohibitively time-consuming and computationally inaccessible when used to conduct time history investigation of the complete structure; a reason for this is lack of pertinent nonlinear cyclic brick FE models appropriate for modelling 3D solid reinforced concrete structures of realistic complexity, whereas brittle nonlinearity causes insurmountable convergence problems in 3D solid FE models. The frame model was subjected to the identified displacement pattern, with the intensity being increased gradually to the displacement levels identified by the elastic analysis. Note that the concepts used are extended from earthquake engineering where performance is established from a displacement-based pushover analysis up to the anticipated level of displacement demands. Nonproportional damage is the characteristic consequence of surface, near field explosions such as the example considered in the present study for a two-span typical highway overcrossing. It was found that axial tension as well as shear failure in the columns and deck owing to the excessive displacements caused by the blast having the intensity examined would lead to catastrophic collapse of the bridge. Clearly, blast actions induce severe loads on the structure which are distributed based on different patterns along the superstructure and the substructure elements. Considering that conventional structural and earthquake design fails to provide a solid framework against blast actions, additional measures are necessary to be implemented (e.g., active or passive protection of the system). Based on the simplicity and the reduced computational cost, the proposed method could form the basis for future investigations of blast phenomena as it is intended for a fast assessment procedure of structures subjected to accidental explosions.

The authors declare that they have no competing interests.