Blast-Resistant Highway Bridges: Design and Detailing Guidelines (2010)

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46 4.1 Overview While bridge engineers can adapt many currently available methods to predict blast loads and the response of building components for use with bridges, there is a need for bridge- specific, simple models that provide quick and accurate results because such models facilitate an efficient design process. In addition, the most sophisticated techniques are not always necessary because these methods often require large compu- tational demands and time commitments, while providing a level of accuracy that may not be warranted given the uncer- tainty associated with identifying threat scenarios and cor- responding blast loads. Simple models, however, may not be appropriate for every case due to the complexity of blast loadings and associated responses. The following sections outline characteristics and capabilities of various analysis methods and software, along with guidelines for the selection of the appropriate analysis technique for a given scenario. Load determination programs and response determination programs are typically independent software packages, and thus the information in this chapter is divided accordingly. Because designers benefit from analytical methods that con- tribute simplicity, efficiency, and accuracy to the design process, information is provided to aid in the selection of the sim- plest and most appropriate analysis techniques for a particu- lar case. Therefore, characteristics, advantages, limitations, and current uses are provided for each analysis level, and a brief description of commonly used analysis software is given. While the capabilities of some of the software discussed have been verified by past research found in the literature, the assertions of other programs have not been verified. More- over, the authors do not endorse any particular software or program, and the codes listed below are simply examples of applications one could use to predict loading and response. In all cases, predicting blast loads and response requires the dis- cretion of an engineer experienced in the field of blast analy- sis, and analysis methods that are based on or corrected by empirical data are preferable when available due to their accu- racy and efficiency. These semi-empirical programs, how- ever, are often not available due to a lack of existing data or because they have limited distributions and are available only to government contractors. 4.2 Current State-of-Practice: SDOF A single-degree-of-freedom analysis is standard practice for most blast-resistant designs. While an analyst could employ complex three-dimensional finite element analyses, the uncer- tainty associated with the threat size and location in blast sce- narios typically does not justify such a detailed analysis. “It is a waste of time to employ methods having precision much greater than that of the input of the analysis” (Biggs, 1964). Furthermore, unlike traditional design loads, most blast designs allow significant nonlinear behavior to dissipate the energy associated with dynamic blast loads, and SDOF results compare well with experimental test data when members experience considerable plastic deformation (Department of the Army, 1990). Additionally, consideration of multi- member coupling is not necessary if the natural frequencies of connected elements differ by a factor of two or more (Biggs, 1964). Therefore, a member-by-member SDOF analysis ap- proach is applicable for most blast design scenarios and repre- sents the current state-of-practice for blast-resistant structural design. Structural Dynamics: Theory and Approach by Joseph Tedesco (1999) and Introduction to Structural Dynamics by John Biggs (1964) both include a basic introduction to SDOF systems and the application of structural dynamics to blast-loaded structures, and this section summarizes SDOF analysis concepts pertaining to blast-loaded structures. Fig- ure 41 illustrates an SDOF system comprised of a simply sup- ported beam with a uniformly distributed load on half the beam and a corresponding deformed shape, Δ(x). Shown in C H A P T E R 4 Analytical Research Program

47 the figure is the idealized spring-mass system with an equiv- alent mass, Me, that goes through a displacement, δ, against the resistance of an equivalent stiffness, ke, under the appli- cation of an equivalent force, Fe. An engineer can transform the real system into the idealized system and obtain equiva- lent system properties by applying work and energy princi- ples to the real beam using an assumed normalized displaced shape. In the SDOF method, each stage of response has a dif- ferent mode or characteristic displaced shape that requires the computation of unique equivalent beam properties. Fig- ure 42 illustrates the deflected shapes and resistance diagrams of a fixed–fixed beam that undergoes three stages of deforma- tion: purely elastic, a combination of elastic and plastic, and purely plastic. The accuracy of the SDOF method depends on how well the assumed deflected shapes represent each stage of response of the real structure in both space and time (ASCE, 1997). While the total response of a continuous member the- oretically includes contributions from an infinite number of individual modes, the response of an SDOF system relies only on a single mode of deformation for each stage of response. This simplification is possible when one mode dominates the response, as is often the case for blast-loaded components. Most designs of structures to resist blast allow significant in- elastic deformation, and the displaced shape associated with the plastic mechanism ultimately dominates the response. While the analysis could use one of the modal shapes that cor- respond to a vibration mode during each stage of response of the structural component, the SDOF method typically uses the displaced shape resulting from the application of a static load that has the same form as the assumed blast load. The value of δ for the real beam in Figure 41 is equal to the displace- ment of the equivalent SDOF system, both in magnitude and in variation with time, at the point of maximum deflection. The use of a static displaced shape as the only mode shape simplifies calculations because for many structural members it is too difficult to determine mode shapes exactly, and for most practical applications, this assumption provides more accurate results than using only the displaced shape that cor- responds to the first mode of vibration (Biggs, 1964). Using the assumed normalized displaced shapes for each stage of response, an analyst may choose to derive equivalent SDOF system transformation factors based on work-equivalency Fe Me ke Δ(x) = δ⋅φ(x) δ δ Figure 41. Idealized SDOF system. R es ist an ce Deflection Elastic Elastic-Plastic PlasticElastic Elastic-Plastic Plastic Plastic hinge Plastic hinges Figure 42. Stages of response for fixed–fixed beam (Biggs, 1964).

48 principles or use factors for known scenarios. Transforma- tion factors for known scenarios can be found in Introduction to Structural Dynamics by Biggs (1964) and the U.S. Army’s TM 5-1300 Structures to Resist the Effects of Accidental Explo- sions (Department of the Army, 1990). Blast-resistant designs typically allow inelastic (i.e., perma- nent) deformations to dissipate energy, and SDOF analyses also include this behavior. “Although plastic behavior is not generally permissible under continuous operating condi- tions, it is quite appropriate for design when the structure is subjected to a severe dynamic loading only once or at most a few times during its life. Among other examples which might be cited, plastic behavior is normally anticipated in the design of blast-resistant structures and at least implied in the design of structures for earthquake” (Biggs, 1964). Plastic hinges and mechanisms can form during a blast scenario, allowing a struc- ture to dissipate energy through large plastic deformations and rotations. This plastic deformation is important in structural design for dynamic loads because allowing a structure to dis- sipate energy creates the most economical design possible (ASCE, 1997). “The primary method for evaluation of structure response is evaluation of the ductility ratio and hinge rotations” (ASCE, 1997). Structures to Resist the Effects of Accidental Explosions (Department of the Army, 1990) and Design of Blast Resis- tant Buildings in Petrochemical Facilities (ASCE, 1997) pro- vide allowable rotation limits for various types of structural members. For the case of blast design, hinge rotation refers to a support rotation and indicates the degree of deformation present in critical areas of a member. Equation 12 defines a ductility ratio that measures the overall inelastic response of a structural component. where: µ = ductility ratio Δmax = maximum displacement of a member (in.) Δelastic = displacement at the elastic limit (in.) The ductility ratio and hinge rotations should not exceed allowable response limits from current blast guidelines (ASCE, 1997). Existing guidelines, however, are applicable to building components, and the limiting values will likely require adjustment as test data on the response of bridges subjected to blast loads become available (see Chapters 6 and 8). Hinge rotations are separated into three levels of response: low, medium, and high. A low level of response includes localized structure or component damage. The main structural components do not need repair, while non- structural components require moderate repairs. A medium μ = Δ Δ max elastic ( )12 level of response entails extensive structure or component dam- age. In the case of a medium level of response, the structure cannot sustain normal loads until repairs are complete, and the total cost of those repairs is significant. A high level of re- sponse includes a structure or component that does not re- tain structural integrity, and this structure or member may collapse under minor loads from environmental conditions (e.g., wind, snow, rain). The total cost of repairs for a high level of response approaches the replacement cost of the structure (ASCE, 1997). In general, the shear reinforcement detailing and the con- trolling stress states influence the response limits for reinforced concrete members. The allowable deformations are very low for elements with significant shear or compression demands, while large deflections are permitted when a member has ade- quate shear capacity. As shown in Table 9, hinge rotations alone are specified for concrete elements in flexure because the relatively stiff nature of concrete members produces very high ductility ratios for members with low maximum deformations, indicating that the use of a ductility ratio is inappropriate for these conditions. Elements responding primarily in shear are subject to brittle failures at low support rotations; therefore, the ductility ratio is the main design criterion for these elements (ASCE, 1997). For blast-resistant structural design, shear demand is typ- ically calculated based on the flexural capacity associated with an SDOF analysis. Both direct shear and diagonal shear are important for blast-loaded components, and structural mem- bers should contain enough capacity through a combination of transverse reinforcement and longitudinal reinforcement dowel action to prevent a shear failure and force a flexurally dominated response. While a member-by-member analysis uses the loads that result from the reaction forces and applied load of an SDOF analysis to load a supporting member, the shear design should not consider these loads because the deflected shapes assumed for those analyses do not match the actual deflected shape of a component early in time when shear dominates the response. Rather, designing a member to have enough shear capacity to prevent shear failure and allow a flexurally dominated response will provide satisfactory behavior. Thus, a designer should consider the available shear capacity of a member only after it has sufficient flexural capac- ity to resist the assumed blast loads, and a plastic analysis using the assumed load distribution and the plastic moment of the final section provides the maximum distributed load the member must sustain prior to forming a mechanism leading to flexural failure. The shear demand resulting from this analysis is the theoretical maximum shear a member will experience during an event that loads a component with the assumed load shape, and thus it is the shear demand a mem- ber must resist. The U.S. Army Technical Manual 5-1300 Structures to Resist the Effects of Accidental Explosions contains

information on computing the shear demand and capacity for structural components, and the ACI and AISC design manuals contain additional information needed to calculate the shear capacity of concrete and steel members, respectively. 4.3 Simplified Modeling and Software Blast design specialists have a large number of options when choosing a technique or software to compute blast loads and structural response, and these methods vary widely in both cost and accuracy. While an analyst may desire results from high-level finite element analyses, the exact location and mag- nitude of a threat typically remains unknown, and the uncer- tainty surrounding load prediction typically does not justify the cost of such high level analyses. As a result, several simpli- fied methods for both blast load and response prediction are available to bridge engineers. This section describes the capa- bilities and limitations of several such procedures. 4.3.1 Load Prediction Techniques Simplified load prediction techniques fall into two groups. The first group contains the methods that utilize blast phe- nomenology from basic equations and curves developed em- pirically from “ideal explosions” in free air. The second group of load determination methods, which is described in the fol- lowing subsection, includes those that consider multiple reflec- tions and pressure magnification, while not strictly employing computational fluid mechanics techniques. The following two subsections describe both types of load prediction techniques. 4.3.1.1 Free-Field Load Prediction Techniques Ideal explosions include bare spherical charges in air or hemispherical charges on the ground. Many researchers have conducted controlled experiments using ideal explosives over many years, and several curve fits allow engineers to quickly determine basic blast load parameters, such as peak over- pressure, peak reflected pressure, positive phase duration, negative phase duration, time of arrival, and impulse. From these parameters, a designer can construct an elementary pressure–time history. Several texts, military manuals, and computer programs include various forms of these equations and curves, including the U.S. Army TM 5-1300 Structures to Resist the Effects of Accidental Explosions (Department of the Army, 1990), the Unified Facilities Criteria (2002), and Con- Wep (U.S. Army Corps of Engineers, 2001), which is a widely used software application that provides an automated version of these equations and curves. These basic techniques are very useful for preliminary designs and other basic problems. As they are based on explosions in free air, these methods cannot Low Medium High Flexure N/ A Shear 1 : Concrete Only 1.3 Concrete + Stirrups 1.6 Stirrups Only 3.0 Com pression 1.3 Flexure N/ A Shear 1 : Concrete Only 1.3 Concrete + Stirrups 1.6 Stirrups Only 3.0 Com pression 1.3 Flexure: Com pression ( C ) 1 .3 Tension ( T ) -- ‡ Between C & T 1 0 Shear 1 1.3 Flexure 3 Shear 1 1.5 Source: American Society of Civil Engineers, 1997 ‡Ductility Ratio = 0.05 (ρ - ρ') < 10 4 1 1 .5 2 1Shear controls when shear resistance is less than 120% of flexural resistance †Stirrups are required for support rotations greater than 2 degrees Beam -C olum ns Shear Walls, Diaphrag ms 1 2 4 1 2 Slabs Support Rotatio n † , a (degrees ) Element Type Controlling Stres s Du ctility Ra tio, µ a Beams 2 4 8 Table 9. Response criteria for reinforced concrete. 49

50 represent reflections off and interactions with structures, and any user must clearly understand the limitations of the selected method before applying it to a specific loading case. While simplified analyses using available equations and curves are useful for relatively large-standoff problems (i.e., where the center of detonation is greater than a scaled range of three or more from the structure under study), they typi- cally do not provide good results for small-standoff problems because the accuracy of the empirical data decreases at close ranges. The definition of scaled range (i.e., scaled standoff) is the actual distance from the center of an explosive to the tar- get, divided by the equivalent TNT charge weight raised to the 1⁄3 power (i.e., Z = R/W1/3). The analyst must remember that the data included in these simplified load prediction methods are from actual field measurements during detonation events, and very close-in data have been, and still remain, very diffi- cult to measure due to extremely high temperatures and pres- sures in the immediate vicinity of an explosive source. Because most critical threats associated with bridge-related blast prob- lems will likely involve very close-in detonations with scaled standoffs much less than 1 ft/lbs1/3, analysts should consider the limitations in using simplified load prediction techniques for computing blast-load parameters for such scenarios. Additionally, these analysis methods do not account for reflections and confinement, which can be significant for several scenarios involving explosives acting against bridges, such as a detonation inside a box girder and a detonation beneath a bridge overpass. Pressures from confined and par- tially confined explosions cannot dissipate as quickly as un- confined explosions, and as a result, they can produce much higher impulses on a structure (Ray et al., 2003). Magnifica- tion factors exist, though, to adjust incident blast pressures for varying levels of confinement, and one can choose values conservatively with success (Gannon, 2004). Because of the associated limitations, methods that utilize blast phenomenology from basic equations and curves developed empirically from “ideal explosions” in free air may not be ade- quate for determining design blast-load parameters for many applications. They may be useful and adequate, however, for cases when only one reflection is significant (e.g., above-deck blasts). These techniques also provide a useful and expedient “sanity check” for high-level blast prediction software pack- ages (i.e., nonlinear finite element analyses employing compu- tational fluid dynamics). While such advanced programs can produce impressive graphical results, the analyst should always study the results to ensure against the “garbage-in-garbage- out” possibilities of large, data-intensive calculations. 4.3.1.2 Load Prediction Techniques That Consider Confinement and Reflections The most widely used application in this group is BlastX (SAIC, 1994). The BEL code (U.S. Army Corps of Engineers, 2000) is an open-distribution version of BlastX that includes a bridge-specific graphical user interface. Although some of these load determination techniques are intended to model blast propagation through enclosed areas such as buildings, they can be adapted for use with bridges. These methods track pressure values as they radiate from an explosion source and as they reflect off surfaces through basic first-order principles of wave reflection. In addition, they can include correction factors to account for differing wave magnitudes reflecting off various surfaces. As a result, they provide higher accuracy than those load determination methods that do not consider confinement and reflections. In addition, a recent study inves- tigated the use of BlastX for modeling a blast below a bridge deck, and the results showed that BlastX has the ability to pro- vide conservative but reasonably accurate results for this case (Ray et al., 2003). The authors of this report caution, however, that results from these codes require additional adjustments for some scenarios. For example, software such as BlastX and BEL are not capable of modeling round geometries such as columns of bridge piers, and experimental and analytical research described in this report show that these codes sig- nificantly overestimate loads on slender square and circular members (i.e., bridge columns) subjected to blast loads. This second group of load determination analysis methods is recommended for use in bridge design because they con- sider reflections and pressure magnification. For most cases, the result is an acceptably accurate but conservative loading obtained at a modest computational cost, although inaccura- cies incurred when modeling explosions against slender flat or curved members (i.e., bridge columns) were overly conser- vative in some cases. This method should not be employed for contact charges as it does not predict internal detonation pres- sures (i.e., pressures that have not gone through the explosive detonation product–air interface). Its use for very complex envi- ronments with many varied reflecting surfaces should also be carefully considered, and, if possible, a fluid-mechanics or empir- ically based model should be used instead. The capabilities and limitations of load prediction methods based on fluid mechan- ics are discussed later in this document. 4.3.2 Response Prediction Engineers may choose from a wide variety of methods when analyzing the responses of structures subjected to blast loads, ranging from simple and easy-to-use SDOF analyses to highly complex, 3D nonlinear finite element analyses. While ad- vanced finite element codes are advantageous for specialized design, research, and post-event scenarios, the uncertainty in the load prediction does not usually justify this level of effort for most design problems. Additionally, simplified methods for response prediction are extremely useful for providing a baseline by which to validate the results of high-level finite element analyses. Thus, all engineers analyzing structures

subjected to blast should be familiar with the advantages, uses, and inadequacies of simplified analysis methods. This section describes two levels of analyses for predicting structural re- sponse due to blast and includes examples of commonly ac- cepted and used software. 4.3.2.1 Single-Degree-of-Freedom Systems The simplest response determination methods are those based on SDOF mass-spring-damper systems. The vast major- ity of currently available design procedures for blasts utilize these approximate systems (Conrath et al., 1999) because they provide reasonably accurate estimates of response while minimizing time and cost. Accordingly, uncoupled SDOF analyses are the most widely used methods for determining response to blast loads throughout the structural engineer- ing community. For example, engineers frequently use these methods in the design of structural members for sensitive control rooms (Barker and Whitney, 1992), and investigators used an SDOF method to analyze the response of the blast- loaded columns and slabs in the Alfred P. Murrah federal build- ing (Mlakar et al., 1998) that was attacked in Oklahoma City in 1995. In addition, the U.S. Army TM 5-1300 Structures to Resist the Effects of Accidental Explosions (Department of the Army, 1990), which is widely considered to be one of the leading references for blast-resistant design, recommends SDOF analyses for most cases. Although it may seem that higher resolution techniques are preferable because blast loadings and the resulting re- sponses are often very complicated, such accuracy is not necessarily warranted. Many uncertainties exist in blast loadings, including the location of the explosive, the mag- nitude of the explosive, and the type of explosive. Unless a very specific threat is expected, a general understanding of structural response to an assumed blast source, which can be provided by SDOF methods, is sufficient for design. John Biggs, a former MIT professor who pioneered SDOF analy- ses for response to blast loads and whose work provides the basis for most blast-resistant design and analysis procedures currently used (including the Army’s TM 5-1300), wrote that SDOF analysis techniques “should not be regarded as merely crude approximations, to be used for rough or preliminary analysis, nor should they be regarded as methods to be used only by engineers who lack the training or intellect to employ more sophisticated techniques. Problems in structural dynam- ics typically involve significant uncertainties, particularly with regard to loading characteristics. It is a waste of time to employ methods having precision much greater than that of the input of the analysis” (Biggs, 1964). If desired, an analyst can combine an SDOF analysis with a continuously updated sectional analysis when varied local- ized changes in material behavior through the depth of a cross- section may significantly affect member resistance. Because this approach considers composite material behavior and changes in constitutive properties through the depth of a cross-section, it may be useful when a member does not have an easily identifiable single value for the yield moment, such as with prestressed concrete girders, or when the assumption of idealized hinging may not be appropriate, such as when loads vary considerably over the length of a member. Examples of sectional analysis programs include RCCOLA (Farahany, 1983), RESPONSE-2000 (Bentz, 2001), and RECONASANCE (Alaoui, 2004). Although this procedure will provide more accurate representations of cross-sectional behavior, little is known about the accuracy of the resulting response compared to responses determined by ordinary SDOF analyses, and en- gineers should study the results carefully before implementing this procedure as common design practice. It is possible to use hand calculations with simple formu- las for SDOF response determination, and charts are also available in Bigg’s Introduction to Structural Dynamics (Biggs, 1964) and the U.S. Army’s TM 5-1300 Structures to Resist the Effects of Accidental Explosions (Department of the Army, 1990) to establish the SDOF nonlinear response of structural ele- ments. In addition, many easy-to-use programs, such as SBEDS (U.S. Army Corps of Engineers, 2007) and SPAn32 (U.S. Army Corps of Engineers, 2002), exist for this purpose. Some SDOF response determination techniques, such as the “Israeli Method,” incorporate empirical data to improve the accuracy of SDOF analysis results (Eytan, 1992). In the “Israeli Method,” a load determination program called Car Bomb (Eytan, 1992) first independently produces the load on a structure, and then SDOF models determine the response of structural members. The models, however, include input coefficients that relate to the specific design scenario. These values come from extensive data and adjust the response obtained by the SDOF analysis to provide more accurate results than might be computed without such adjustments. As indicated earlier, any model that is accurately adjusted or has been accurately val- idated using empirical data should be used when available. Even if an engineer feels it is necessary to use higher level analysis techniques to compute structural response to blast loads, SDOF analyses are extremely useful for the preliminary design and sizing of members (American Society of Civil Engi- neers, 1997; Biggs, 1964; Conrath et al., 1999; Department of the Army et al., 2002). Parameter studies with high-level mod- els, such as with MDOF models, may not be practical due to the large number of input requirements. Moreover, 3D non- linear general-purpose finite element analyses are not con- ducive to parameter studies due to the large amount of time needed to build, mesh, refine, and run a model. The limited number of input values needed to define an SDOF model, however, facilitates extensive parameter studies. Therefore, if an analyst desires results from high-level analyses, SDOF meth- 51

52 ods may be useful to size individual members before using a high-level model to verify the global response of the structural system. Practicing blast engineers sometimes use this type of procedure involving a progression of analyses (Hinman, 1998). SDOF models are valuable because they can provide rea- sonably accurate predictions of response for a wide range of structural components. This accuracy is in large part attributed to the use of complex spring resistance (i.e., load-deformation) functions that are based on empirical data. Such resistance functions can account for complex modes of behavior, in- cluding tensile and compression membrane effects and mem- ber instabilities. Because blast scenarios are unpredictable, detailed response analyses of attack scenarios are often not necessary. Rather, understanding the response of a member due to a general threat is sufficient for designing blast-resistant structural components. Many components, whose end re- straints can be readily determined or conservatively estimated, can be analyzed by such methods. Examples include beams, slabs, columns, and walls. Bridge applications include the analysis of individual girders, piers, and truss members as well as individual wall sections of towers and box girders. In some cases, as is often done for the analysis of bridges subjected to seismic events (BERGER/ABAM Engineers Inc., 1996), com- posite sections can be modeled as responding as a single ele- ment so that bridge decks and entire superstructures may also be analyzed using SDOF techniques. Such an approach may not always be appropriate, however, and guidance concern- ing the limitations of SDOF analyses is provided below. When appropriate assumptions for end restraints are un- clear, boundary conditions can be assumed and then varied to maximize the response quantity of interest. For example, a propped-cantilever or two fixed ends can be conservatively assumed for shear calculations because design criteria are based on maximum forces at the supports, and the stiff end restraints associated with the fixed support condition will lead to con- servative estimates of the required shear capacity. For flexural calculations, because maximum deformation controls the design, simply supported ends can be conservatively assumed (the cantilever case should also be verified not to control the design). Moment capacities can be adjusted to include the effects of axial loads by using interaction diagrams, and resis- tance diagrams can be approximated by straight lines. For the reasons mentioned above, use of an SDOF analysis is the most appropriate choice for the design of most individ- ual bridge components and systems, including those often designed for non-blast loads using MDOF frame analysis procedures. Although SDOF analyses can only consider one component at a time, the interaction between two structural components can be considered by determining the reaction forces as a function of time that must be transferred from one component to another (American Society of Civil Engineers, 1997; Conrath et al., 1999; Department of the Army et al., 2002). This approach has been shown to provide acceptably accurate predictions of structural performance when the ratio of the natural period of the connected components is larger than or equal to two (American Society of Civil Engi- neers, 1997; Conrath et al., 1999; Department of the Army et al., 2002). Although SDOF methods can be used for most structural members, discretion should be used when choosing the most appropriate analysis method. For example, the designs of some bridges, including suspension and cable-stayed bridges, do not ordinarily receive guidance from AASHTO specifica- tions because they require extensive experience or high-level analyses even for “ordinary” loadings, such as dead loads, live loads, wind loads, and seismic loads. The designs of the sys- tems and components of these bridges often require spe- cialized engineering firms that use high-level finite element analysis software such as that described in Section 4.3.3. For those bridges, the choice of analysis methods when designing for blast loads should reflect the same considerations that re- quire advanced analysis procedures or experience to design for “ordinary” loads. As mentioned above, it is important to remember that, although some structures require sophisti- cated analysis procedures, SDOF methods are well-suited for the preliminary sizing of most components and the complete design of many bridge members, including those bridges often designed using MDOF frame analysis codes. Because SDOF analyses provide acceptable results while min- imizing costs, they should be used whenever possible. Some lim- itations for their use do exist, however, and such methods should not be used for all purposes. In general, SDOF analyses can and should be used for any and all structural components that are not included in the following cases: • Cases when conservative load values are not acceptable. Because SDOF procedures are uncoupled, blast loads com- puted from load determination methods may be inaccurate if significant localized failure and venting occurs, such as when trying to use only an SDOF analysis to model an en- tire bridge superstructure that may experience load relief if part of the deck fails and vents pressure. If the use of con- servative loads that do not account for pressure relief due to localized structural failure within the component being analyzed is not acceptable, the analyst should use an alternate modeling approach in which connected components (e.g., bridge deck and girders) are treated as individual SDOF sys- tems where the reactions from the directly loaded compo- nent act as the load input to the supporting component. The vast majority of cases do not require coupled analyses. SDOF analyses typically can account for the effects of localized fail- ure and venting through modification of system properties much more readily than performing a coupled analysis. • Cases when localized failure may occur. Examples of such localized failures include spall and breach for concrete

members and local buckling and fracture for steel mem- bers. These failures often result from close-in or contact blasts, but they are not necessarily limited to such attacks. Although it is possible to use reduced cross-sections, espe- cially for concrete members, to alter existing SDOF methods to account for such events, it requires extensive experience and knowledge of the failure mode. Some empirical methods exist for concrete members to determine response involv- ing localized failure, but they are often not publicly avail- able and are applicable only to specific cases. Because such failures often can be unpredictable, varying widely accord- ing to both blast and component characteristics, scenarios with the potential for localized failure should be analyzed with care, and SDOF methods should not be used for all members and all standoffs. • Cases when the failure mode is uncertain. Although SDOF methods may be able to determine structural re- sponse up to failure for individual members within a sys- tem, they cannot predict global response when the failure mode is uncertain, such as with highly confined blasts, when potential structural instabilities exist, or when local- ized material or member failure may occur. Examples po- tentially include blasts at abutments, within double-decked bridges, within or close to hollow piers, within or close to segmental box sections, in contact with a structure, and with significant fragmentation. • Cases when P- effects may contribute to failure. Although axial-flexural interaction diagrams can be used to alter member resistance to account for changes in moment capacity due to axial loads, typical SDOF analyses generally do not consider P-Δ effects. It may be possible, however, to incorporate an iterative technique that accounts for geo- metric nonlinearity within SDOF calculations. Such tech- niques are not common, and inexperienced analysts should not use SDOF methods to analyze long, slender, axially loaded members that may experience P-Δ effects. Tall bridge piers loaded by large surface blasts are examples of such members. The terms “tall bridge piers” and “large surface blasts” are intentionally vague because each design scenario will be unique, and an experienced engineer will need to de- termine whether or not P-Δ effects will be significant. • Cases when the member under consideration is not suf- ficiently slender. Most currently available SDOF methods neglect rotational inertia and shear effects by default. There- fore, analysts should be careful when using SDOF methods to analyze deep members. Examples of such non-slender members include concrete anchors for suspension bridge cables, short columns, most pier caps, and deep beams. It is important to point out, however, that the resistance function and stiffness can be modified to account for shear defor- mation and rotational inertia if these effects are considered to be important for the problem under consideration. • Cases when loading exists on more than one principal axis. Loading on more than one principal axis may result in biaxial bending or torsion. Current SDOF methods cannot determine such responses, and constructing a new code for such purposes would be difficult due to the essen- tially infinite number of various structural members that can respond in any direction. An example is a bridge super- structure loaded by a vehicle bomb detonating at an angle to the superstructure. • Cases when the loading is not centered on the primary axis (i.e., not centered along the width) of the mem- ber considered. Off-centered loading can create biaxial bending and torsion, and, as mentioned previously, SDOF methods cannot easily determine these types of responses. An example is a bridge superstructure loaded by a vehicle bomb at a widthwise edge of a bridge (i.e., near the railing). • Cases when more than one mode shape may contribute to the response. Although a distributed-mass modal analy- sis can accurately determine such responses, pure SDOF methods cannot because they assume the response to be characterized by a single mode shape, which is approxi- mated by a characteristic deflected shape. Examples in which SDOF methods may not be accurate include cables and very long superstructures as well as structures in which the response is primarily elastic. • Cases when the blast load cannot be easily approximated over the entire affected area as a function of time. If signif- icant load variations occur over the length of the structure being analyzed (i.e., blast pressures arrive at significantly different times or structural geometry prevents easy approx- imation of the pressure distribution), the load will not be adequately represented by a simple time function, and a more substantial analysis may be required. Examples may include close-in blasts and blasts against unusual structural geometries involving multiple corners and steps. • Cases when the rebound response will be significant, pos- sibly causing failure in the reverse direction. Although SDOF analyses can predict rebound of members, the accu- racy of these calculations is uncertain. Damage incurred during the initial response may alter the resistance capac- ity for the rebound response. No specific examples of these cases are available because the significance of the rebound phase depends solely on the characteristics of each mem- ber and loading considered. • Cases when more than one member contributes to the re- sponse. SDOF methods intrinsically lend themselves to the analysis of only one member or component. Thus, any component that receives significant structural contribu- tions from an attached system should not be analyzed using SDOF models. Examples include entire superstructures when the determination of individual member failure is desired and piers when knowledge of the contribution of 53

54 the superstructure is desired. In most cases, however, as- sumptions can be made for the end restraints in order to use SDOF analyses to obtain conservative results. • Cases when changes in material behavior that varies through the depth of a cross-section may influence re- sponse, preventing easy approximation of an effective resistance diagram based on static loading. Examples of such localized material behavior that may affect response include local plastic deformation, failure of extreme fibers, strain-hardening, localized strain-rate effects, or strain- hardening induced material changes. Because typical SDOF analyses use approximate resistance diagrams based on a static load-deflection curve, they typically do not include the effects of localized changes in material behavior. As- suming a direct transition from linear response to a plastic hinge for cross-sectional behavior may not adequately rep- resent material behavior. In addition, when the loading varies significantly across the length of a member, idealized hinging may not be appropriate. In these cases, MDOF frame analyses may be necessary 4.3.2.2 Multiple-Degree-of-Freedom Systems In some cases, engineers may need to use response tech- niques consisting of 2D or 3D MDOF structural analysis soft- ware when simplified response analysis procedures cannot adequately analyze a structural component. Examples of these programs include ETABS (Computers and Structures Inc., 2006a), SAP 2000 (Computers and Structures Inc., 2006b), and RISA 3-D (RISA Technologies, 2005). These techniques have several advantages that, for many cases, provide more accurate and reliable results than simplified response determi- nation methods such as SDOF analyses. For example, MDOF 2D and 3D structural analysis software can determine the interaction among the individual responses of multiple mem- bers within a structural system, providing a better understand- ing of global behavior than can a combination of individual SDOF analyses. In addition, MDOF frame analysis methods can investigate member response that is governed by more than one mode shape, and they can also predict response due to P-Δ effects. Furthermore, while MDOF systems provide these advantages over SDOF analyses, MDOF frame analy- sis software packages are easier to use than general-purpose finite element analysis procedures. As with any analysis method that provides an increase in accuracy, however, MDOF systems have additional costs over SDOF models due to the additional time needed to build and analyze models. Additionally, MDOF analysis methods ex- change increased accuracy of response calculations with more complex loading input demands. Because a blast may load multiple members within a structural system, the analyst may need to calculate and output several complex load histories with differing pressure magnitudes and time variations for a large number of targets and then input these values into the analysis model. Because of the given structural geometry and the need to account for interaction among various structural components that lead to the need for an MDOF analysis, it is expected that at least a load determination method that con- siders reflections and confinement, which may be difficult to obtain without proper authorization, would be necessary in order to track and record blast pressures at this level of detail. Even with such methods, the initial geometry of the structure would provide the framework for the loading, and, because the loading would not reflect localized failure of members that may occur within a model, an inaccurate prediction of behavior may still result. On the contrary, when SDOF analy- sis methods are used individually on members within a struc- tural system, the reaction forces of the first member loaded become the loading on the members supporting the first member. Thus, the reaction forces that pass from one mem- ber to another reflect the load reduction observed in some members due to the occurrence of individual member failure. Although the increased accuracy provided by MDOF frame analysis methods can be very useful for some design situa- tions, it is important to understand that this level of accuracy is not always warranted. For example, the interaction of struc- tural members is only important when the natural periods of the interacting members differ by a factor less than two (American Society of Civil Engineers, 1997; Biggs, 1964; Con- rath et al.; Department of the Army et al., 2002). For all other cases, SDOF systems yield sufficiently accurate results, and the reaction forces of one member can be used to accurately load another member. Also, in comparison to SDOF analytical methods, which have minimal input requirements, MDOF frame analysis methods require greater time and effort for conducting parameter studies. Thus, the analyst may desire SDOF methods for the initial sizing of members. It is important to note that MDOF analyses may not be suf- ficient for all purposes. While SDOF analyses can provide ac- curate representations of blast-loaded components because spring resistance functions can account for complex modes of deformation, incorporating such features as large deforma- tions and membrane effects, MDOF frame analysis models cannot typically rely on the same empirical data to compute results with the same degree of accuracy as the SDOF analy- ses. The ability of MDOF models to provide reasonable pre- dictions of structural response to blast loads will depend on the ability of the software to account for nonlinear dynamic behavior, including both large deformations and inelastic material response. Furthermore, the results will also depend on the methods used to model a structure, including the num- ber of elements selected to represent a particular structural component, location of the nodes, and so on. As mentioned above, SDOF analyses may reveal the effects of individual

member failure on response that MDOF frame analysis methods may fail to capture. Moreover, because MDOF analy- sis methods are not coupled (at least in the context in which they are being described in this section), they cannot deter- mine load changes due to structural response, and, accord- ingly, they incorporate overestimated blast loads, which can result in structural members being overdesigned. In addition, MDOF methods cannot predict spall and breach without sig- nificant adjustments that require a very experienced engineer. Thus, a high-level model and a specialist are required for such predictions (Conrath et al., 1999). Although MDOF frame analysis methods can be very useful for determining the response of complex structural systems in- volving the interaction of multiple members, contributions from multiple modes, and the influence of P-Δ effects, they are not ap- propriate for the following cases: • Cases when localized failure or large localized deforma- tions may affect response. Although MDOF frame analy- sis methods can determine the effect of individual member response on the global response of structural systems, they cannot take into account the effects of member material loss due to localized failure such as breaching or spalling. If such damage is likely, the problem will require a more sophisticated analysis and a very experienced analyst. Ex- amples of such cases include blasts against hollow piers and close-in or contact blasts against any member. • Cases when localized failure is expected to affect loading. Because MDOF frame analysis methods are uncoupled, they cannot determine changes in loading due to localized failure or large deformations, which can result in venting. In some cases, such as with blast pressure on a bridge deck, venting can reduce loads on other structural members (e.g., the girders), affecting global response and reducing the required resistance of structural components in the system. In other cases, such as with blast pressure against hollow piers, failure of a wall can cause blast pressures to vent into the interior of the pier. Neglecting the effects of venting by using MDOF frame analysis methods may greatly affect both loading and response. Examples of cases when the design may be significantly influenced by localized failure include blasts at abutments, within or close to hollow piers, within or close to box-like sections (e.g., box girders, towers of cable-stayed bridges), in contact with a structure, and with significant fragmentation. 4.3.3 Advanced Modeling Methods and Software While simplified methods for load and response prediction are appropriate for most design cases, some situations require an advanced understanding of blast loading and structural re- sponse that SDOF and MDOF frame analysis methods do not provide. Examples include blasts against slender or curved members, explosions within confined regions, venting due to localized failure, and cases in which localized large deforma- tion or member failure may affect response. These analysis problems require 3D computational fluid dynamics models to determine loads and 3D nonlinear general-purpose finite element analyses to predict response. While these methods can produce more accurate results than simplified models and depict behavior not captured by SDOF and MDOF sys- tems, they can also generate highly misleading and erroneous results, and only analysts experienced with both blast phe- nomena and advanced modeling techniques should conduct and interpret the results of these simulations. This section describes the characteristics, limitations, and uses of various advanced modeling techniques. 4.3.3.1 Explicit versus Implicit Analysis Methods All computationally based analysis methods employ either an implicit solution technique or an explicit solution tech- nique to solve the governing equations of motion for the system under consideration. Both procedures have unique advantages and disadvantages that make each appropriate for different cases. An implicit method calculates a solution at a given time step based on the equilibrium of the external, internal, and inertial forces in the system at that time step. As a result, depending on the constants used, implicit solutions can be unconditionally stable and can provide accurate pre- dictions of structural response at time steps that are larger than those used with explicit methods. A drawback with im- plicit methods, however, is that they require the factorization of the stiffness matrix (i.e., inverting the stiffness matrix) for each time step, and this requirement greatly increases com- putation time for problems in which the stiffness of elements in the structural model change due to nonlinear response, which is often the case with blast loads. The increased com- putation time is the result of a costly iteration at each time step to determine equilibrium, and each iteration step requires the reformulation and refactorization of a new stiffness matrix that reflects adjustments based on the nonlinearity of the sys- tem at that iteration step. An explicit solution technique, unlike an implicit method, determines response values at each time step based on equi- librium at the previous time step. One outcome of such a for- mulation is that explicit methods do not require factorization of the stiffness matrix for each time step. Because equilibrium is not satisfied precisely at each time step, however, explicit solutions can become numerically unstable if appropriately small time steps are not chosen. As a result, explicit methods require more time steps than implicit methods to converge to the same solution. Depending on the case being considered, 55

56 this fact may make explicit solutions less efficient than im- plicit solutions. It is important to note that the solution technique employed by an analysis method varies with each individual program, even within the levels of analysis defined within this document. Thus, it is the responsibility of the analyst to understand how the solution procedure employed by the selected software will influence the computed results. Although many factors must be considered when choosing an appropriate solution tech- nique, most analysts choose an explicit method for complex analyses of structural response to blast because the benefit gained by not having to factorize the stiffness matrix during each time step typically outweighs having to use small time steps. Furthermore, allowing for failure and removal of ele- ments from an analysis model is readily achieved with an explicit code, while such a scenario poses a significant com- putational challenge for implicit codes. Alternatively, for simple analyses, involving only one member or a system ex- periencing only elastic response, an implicit analysis may be appropriate. 4.3.3.2 Load Prediction The most sophisticated level of load determination is defined as one that employs fluid mechanics computations. Examples include SHAMRC (Applied Research Associates Inc., 2005), which is available only to government contrac- tors, and LS-DYNA (LSTC, 2007). These load determination methods are “high resolution” models that use the mechan- ics and characteristics of fluids (i.e., air in the case of blast) and fluid flow to calculate variations in pressure, density, velocity, and so on as a function of time and position. As a result, they have the ability to consider multiple reflections, pressure magnification, turbulence, pressure buildups, and “hot spots” (i.e., localized areas of large pressure buildup). Thus, very accurate predictions of blast loads can be obtained using these methods. Due to the increased resolution, how- ever, these methods are very computationally intensive, and significantly more personnel time and experience is needed to correctly input required analysis parameters over that re- quired for lower level analyses. In addition, close-in blasts require response calculation (i.e., coupling) to determine the influence of material failure and breach on load variation. Because lower resolution tools have the ability to provide reasonable and conservative results (Ray et al., 2003) and because the specifics of future attacks are generally uncertain, the level of accuracy provided by high resolution models is not warranted for most design cases. The decision to use a highly accurate and likewise expensive computational pro- cedure must be on the basis of a cost–benefit analysis. The expense may be well warranted on a large project where the removal of even small degrees of conservatism could poten- tially save large amounts of money. In general, however, analy- ses employing computational fluid dynamics are primarily suggested for research environments and for analyzing past events when a specific attack scenario is known. 4.3.3.3 Response Prediction Advanced design scenarios, research problems, and post- event evaluations often require general-purpose, uncoupled, nonlinear finite element software packages. The primary dis- tinction that differentiates these analyses from MDOF frame analyses is that the higher level of analysis allows for the dis- cretization of general-shaped geometries using a large num- ber of small elements consisting of solids, plates, shells, beams, and so on. Examples of analysis software with these capabili- ties include ABAQUS (ABAQUS Inc., 2004), ANSYS (ANSYS Inc., 2004), and LS-DYNA3D (FEA Information Inc., 2006). Among these software packages, LS-DYNA3D is frequently used for predicting the response of structures subjected to blast loads. Finite element analyses make use of a large num- ber of degrees of freedom that interact together, and the model is then analyzed using structural analysis techniques similar to those employed for the MDOF frame analysis methods. Thus, increased accuracy is possible by dividing the system into small, discrete “finite elements.” These models are often very complex, however, involving large personnel time re- quirements to construct a model and input system parameters. In addition, significantly more time is often necessary to ana- lyze a model and to ensure that the results converge to a stable and accurate solution. Because engineers with limited experi- ence can easily input and analyze details of a physical system, unsubstantiated confidence can often be obtained in the valid- ity of the results. Some programs are so complex that they are accurately used only by the developing organization (National Research Council, 1995). Hence, only highly trained person- nel should use 3D nonlinear general-purpose finite element methods when absolutely necessary. Additionally, as analysis techniques become more sophisti- cated, they move further away from the quantities desired for design. For example, rather than providing bending moment acting at a given cross-section in a beam, nonlinear general- purpose finite element methods typically only give stress, strain, and displacement values at discrete locations. The analyst must take this information and determine bending moment through post-processing of the results. Furthermore, the input and time requirements of this level of analysis are not conducive to the parameter studies often used to size members. Therefore, these methods have limited usefulness for the design of most members. If nonlinear general-purpose finite element techniques are desired for design, the process may be expedited by using lower level analysis methods to size members initially before using these higher level response determination techniques to verify the final design.

Despite the increase in cost, nonlinear general-purpose finite element methods can be valuable because they can pro- vide detailed information regarding failure modes and the effects of localized failure. “Meshes,” or grids containing the “finite elements,” model the physical system and allow com- putation of localized effects. Accordingly, these models can be very useful for predicting structural response for scenarios that lower resolution techniques do not easily model. Exam- ples include examining local deformation and failure of walls in hollow piers, finding the onset of deck failure for blast- loaded superstructures, calculating the global response of post-tensioned segmental box girders due to localized failure, and predicting response to blasts at abutments. Although nonlinear general-purpose finite element methods do have concrete constitutive models that can provide spall and breach predictions, the results should be used with caution. These detailed analyses are very complex, and existing concrete constitutive models struggle to capture complex dynamic material behavior and failure due to uncertainties involving rebar bond, rebar buckling, variable strength and modulus, strain-rate effects, complex crack propagations, and variations in mix design and workmanship (Department of the Army et al., 2002). Such predictions require very sophisticated models developed by a specialist with significant experience to correctly build a model and interpret the results of such an analysis (Conrath et al., 1999). It is also important to note that uncoupled nonlinear general- purpose finite element methods may be very useful for pre- dicting a response that involves localized failure and large deformations, but they cannot include the effect local failure has on loading, nor can they determine how the resulting load- ing and response interact over time. This “coupling” of the re- sponse is usually not necessary because neglecting venting and pressure redistribution results in conservative load values, and the uncertainties that exist in the determination of design blast loads do not warrant such accuracy. Thus, for most prac- tical design applications, uncoupled nonlinear general-purpose finite element methods provide the highest level of resolution needed. In some situations, however, it may be appropriate to consider coupling, and those cases are as follows: • Cases when localized failure is expected to affect load- ing. Uncoupled nonlinear general-purpose finite element method response techniques cannot determine changes in loading resulting from venting due to localized failures or large deformations. In some cases, such as blast pressure on a bridge deck, venting can reduce loads on other struc- tural members, affecting global response and reducing the required resistance of some components. In other cases, such as with blast pressure against a hollow pier, failure of a wall can allow pressure to vent into the interior of the pier, complicating both loading and response. For an optimum design, the analysis assumptions should not neglect the effects of venting, thus requiring a coupled analysis. 4.3.3.4 Coupled Analyses The highest level of response computation techniques con- sists of coupled 3D nonlinear general-purpose finite element procedures. These programs can provide the most accurate predictions of both loading and response because, unlike lower-level methods, they account for the interaction of load- ing and response over time. Thus, this highest level of analy- sis can consider pressure release and redistribution resulting from localized failure or large deformations, providing the most accurate representations of actual structural performance. Although any level of analysis has limitations concerning ap- plicability, coupled 3D nonlinear general-purpose finite ele- ment techniques are valid for any purpose because they are the highest resolution procedures currently available. It is important to remember, however, that use of such methods is not typically necessary because lower resolution methods usually provide acceptable results, and it is important to note applicable limitations for proper use of these programs. Although some programs may advertise the ability to couple loads and response, the coupling is often very limited, and AUTODYN (Century Dynamics, 2004), DYSMAS (U.S. Army Corps of Engineers, 2006), and LS-DYNA (LSTC, 2007) are three of the few programs that are truly capable of coupling blast pressures with structural response. In addition, ensur- ing the accuracy of results can be very difficult due to the large number of inputs needed. Successful operation requires exten- sive experience and time to properly model structural systems, achieve the correct failure modes, and interpret the results. Response predictions are usually applicable only for the spe- cific case under consideration. Furthermore, as with uncou- pled general-purpose finite element analysis methods, coupled general-purpose finite element methods do not directly gen- erate the quantities needed for design, and the input and time requirements of this level of analysis are not conducive to the parameter studies often used to size members. It is also impor- tant to note that, as with uncoupled general-purpose finite element methods, coupled general-purpose finite element methods do have concrete constitutive material models that can provide spall and breach predictions. An analyst should review the results with discretion, however, because such mod- els often struggle to accurately predict dynamic behavior and failure due to the many uncertainties intrinsically character- istic to concrete (Department of the Army et al., 2002). Only a specialist with significant experience should perform these simulations (Conrath et al., 1999). Because of the extensive issues associated with such methods and because uncoupled analyses provide conservative results, most research and design problems only need uncoupled 57

58 analyses (National Research Council, 1995). Thus, coupled 3D nonlinear general-purpose finite element analysis methods are useful only for specialized cases to conduct research, to ana- lyze past attacks when the specifics of the blast and structural response are known, and to investigate specific design concerns for which uncoupled methods have been deemed unsuitable. Design scenarios that may benefit from this highest level of response prediction include optimizing walls of hollow tow- ers for blast loads and sizing superstructure members to allow for the failure of bridge decks to vent blast loads. In general, as stated earlier, the decision to use a highly accurate and like- wise expensive computational procedure must be on the basis of a cost–benefit analysis. The expense may be well warranted on a large project where the removal of even small degrees of conservatism could potentially save large amounts of money. Nonetheless, increasingly complex analyses often can pro- duce highly erroneous results, and only analysts experienced with predicting blast loading and response should undertake this option. 4.3.3.5 Combining Load Prediction Techniques with Response Prediction Techniques Most response determination programs are uncoupled, meaning that load determination is separate from response determination. As such, an engineer must first predict struc- tural loads due to blast, output the results, and then input the load data into the response analysis model. The combination of load determination and response determination techniques for a given scenario is the choice of the designer, and prac- tical limitations for the combinations of load and response methods are important factors in that decision. Low-level load analyses do not easily provide information needed for high-level response analyses, and low-level response determi- nation methods do not warrant the increased accuracy and ex- pense associated with high-level load determination methods. In order to maximize efficiency and accuracy, a designer must choose appropriate analysis combinations based on the output limitations of load determination techniques and the input requirements of response determination techniques. Although the outputs obtained from different load analy- ses vary even within the analysis levels described in this doc- ument, the analyst should note some general observations. Typically, load prediction methods based on blast phenome- nology from basic equations and curves developed empiri- cally from “ideal explosions” in free air are able to provide, for example, a time–pressure history acting over a given tar- get area or a time-varying equivalent uniform load acting on a single structural component. As noted earlier, however, the loads computed by those methods do not account for the ef- fects of multiple reflections, confinement, and other impor- tant factors that influence the actual blast loads that can act on bridges. The load prediction methods account for con- finement and reflections typically produce a time-dependent load history for a user-specified number of “targets.” The targets are defined within a three-dimensional space and are defined as individual points on planar surfaces. Load analy- ses that employ computational fluid dynamics have the capa- bility to calculate time-varying pressure histories on various planes in three dimensions. The input requirements of response determination tech- niques also vary within a given level of analysis, but, as with load analyses, some general observations can provide guid- ance when choosing an appropriate combination of load and response analyses. SDOF response analyses need one- dimensional loading data for a single member, such as an equivalent uniform load, a pressure distribution along a member, or a pressure history for selected points. In contrast, MDOF frame analysis methods and 3D nonlinear general- purpose finite element methods require that the analyst input load histories for a variety of points on a structure. This load- ing data must usually be specified in three dimensions. For example, to utilize a load analysis that adjusts blast phenom- enology based on empirical data to consider reflections and confinement in conjunction with an MDOF frame analysis, the analyst must input loading data that acts over various faces and members of an individual bridge. Thus, consid- ering the case of a below-deck blast scenario against a girder bridge, the analyst would need to specify individual loads acting at a suitable number of points to accurately capture its variation in time and space on the columns in the pier, the girders, and the deck. The discussion above does not address all possible scenarios, but it illustrates the important fact that, although it may be possible to combine all levels of load analyses with all levels of response analyses, it is not appropriate to do so. Based on the characteristics of available analysis methods, some general guidance can be developed to aid in the selection of appropri- ate load and response analysis combinations. For example, depending on the level of accuracy desired and the geometric properties of the structure under consideration, load data required for an SDOF analysis is easily extractable from the output of a load analysis that uses blast phenomenology based on empirical data with or without consideration of confine- ment and reflections, and it is appropriate to use at least a load analysis that considers reflections and confinement to predict loads for an MDOF frame analysis or a 3D nonlinear general- purpose finite element analysis. 4.3.3.6 Summary This chapter outlines the capabilities and limitations of sev- eral levels of analysis programs available to blast engineers. Not all levels of load and response prediction techniques

are appropriate for all cases, and similarly not all levels of load prediction techniques are compatible with all levels of response prediction methods. An engineer should take care to select analysis methods that are suitable for the resources available, the goals of the project team, and the specific sce- nario being considered. Use of inappropriate methods may produce results that are not conservative enough or are overly conservative, or may be a waste of resources. The informa- tion provided in the previous subsections provides guide- lines to assist engineers in selecting the most appropriate combination of load and response analyses for the scenario under consideration. 59