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100 7.1 Overview The advantages, capabilities, limitations, and uses of several levels of load and response analysis techniques have been outlined in previous sections of this report. In the sections below, application of both simplified analysis techniques and detailed finite element modeling for the case of blast-loaded bridge columns are illustrated. Although the procedures described below provide a starting point for engineers who wish to learn the intricacies of modeling blasts acting on bridge columns, they should not be taken as a comprehensive lesson in analytical modeling of structures subjected to blast loads, and it is recommended that only analysts experienced in both blast phenomenology and analytical modeling predict struc- tural response to blast loads. 7.2 Simplified Analyses Guidelines Design procedures based on the results of a single-degree- of-freedom analysis are those most commonly used for blast- resistant design. Although engineers increasingly feel the need to use highly complex finite element analyses for design, the majority of design cases do not warrant such highly detailed analyses due to the many uncertainties associated with blast loads, such as the exact charge weight, shape of the charge, type of explosive material, and standoff distance. Additionally, blast-resistant designs typically allow significant nonlinear behavior to expend the energy associated with dynamic blast loads, and SDOF analyses have been shown to provide acceptable results when structural components undergo large inelastic deformations (Department of the Army, 1990). The theory behind and justification for the SDOF design pro- cedure is presented in Section 4.2 of this report and several publicly available texts, such as Structural Dynamics: Theory and Application by Joseph Tedesco (1999) and Introduction to Structural Dynamics by John Biggs (1964). Additionally, the U.S. Army Technical Manual TM 5-1300, Structures to Resist the Effects of Accidental Explosions (Department of the Army, 1990), contains the procedure most commonly applied to the blast- resistant design of building components. The design procedure for reinforced concrete bridge columns outlined in this section closely adheres to the widely accepted and trusted procedure from TM 5-1300 with special considerations for bridge columns derived from the experimental portion of this research. Bridge columns should be designed using an SDOF analysis procedure, which initially requires the computation of a few basic column properties such as the plastic moment capacity, Mp, the total mass of the member, M, and the moment of inertia, Iz. The material strengths used in the calculation of the plastic moment capacity should include a strength increase factor to account for the presumed actual material strengths and dynamic increase factors to account for strength increases due to strain-rate effects. The dynamic increase factors are material specific, and they should be applied to the material properties prior to calculating the section capacity. Table 16 shows common strength increase factors for concrete, and Table 17 shows common dynamic increase factors for con- crete and steel. Increasing material strengths for design may seem counter-intuitive relative to normal design procedures; however, a typical bridge structure is not expected to experience repeated blast loadings over its lifespan, and typical design objectives are to sacrifice the structure to protect life, provide egress, and, in the case of bridge columns, allow motorists and pedestrians to flee the bridge. Therefore, the blast-resistant design of bridge columns should use the entire available capacity of a column to resist these extreme loads. The SDOF-based design procedure also requires an engineer to predict the magnitude, distribution, and time-history of the blast load applied to a member. Available load prediction techniques, such as the charts or empirical equations provided in the U.S. Armyâs TM 5-1300 or software developed to pre- dict loads on structures based on first-principal or empirical data, such as ConWep, BlastX, or BEL, are valuable tools for this purpose. Although load cases for typical design C H A P T E R 7 Analysis Guidelines
101 scenarios include load factors that conservatively account for uncertainties in the assumed loads, these factors should not be used when designing bridge columns to resist blast loads. Regardless of the load prediction technique selected, the result should be a distributed load that varies only with time and approximates the predicted load as it varies in both time and position. Figure 86 shows an example load shape for a hypo- thetical bridge column and blast scenario. This assumed load distribution is necessary to conduct a plastic collapse analysis of a member to calculate the maximum resistances and hinge locations for each stage of response. Figure 19 shows the three stages of response for a beam with two fixed ends and a uniformly distributed blast load. The hinge locations will provide the boundary conditions needed to determine the normalized deflected shapes that result during each stage of response from the application of the assumed load shape. These normalized deflected shapes allow the computation of the equivalent load, equivalent mass, and system stiffness that equate the single-degree-of-freedom system to the real system for each stage of response. The visual relationship between the single-degree-of-freedom system and the real system is shown for the elastic stage of response for a simply supported beam in Figure 87. The calculation for the equivalent load is based on the work done by the assumed distributed load as the member undergoes a unit deflection at the point of maximum deflection, and the equivalent mass represents the inertia contributed to the system by the mass distributed along the length of the member as the member undergoes a unit deflec- tion at the point of maximum deflection. While the intent of this section is to describe the blast-resistant design process for bridge columns, it is not meant to provide a tutorial on developing the SDOF analysis procedure, and Biggâs Intro- duction to Structural Dynamics (1964) and the U.S. Armyâs TM 5-1300 Structures to Resist the Effects of Accidental Explosions (Department of the Army, 1990) provide charts that contain factors to quickly convert the total load and total mass into SDOF-equivalent values for common loading scenarios and boundary conditions and the equations needed to calculate these equivalent properties for other cases. The determination of equivalent system properties for this analysis and design procedure should assume a single load distribution that does not vary with position along the member. Real structural systems do not experience loads in this manner, as the shock wave will propagate along the length of a member, applying the load at different locations along the member at different points in time. While it technically may be possible to vary the load distribution along the length of the member during an SDOF analysis by updating the equivalent load and mass Material SIF Structural (f y 50 ksi) 1.10 Reinforcing Steel (f y 60 ksi) 1.10 Cold-Formed Steel 1.21 Concrete* 1.00 * The results of compression tests are usually well above the specified concrete strengths and may be used in lieu of the above factor. Some conservatism may be warranted because concrete strengths have more influence on shear design than bending capacity. TM 5-1300 specifies a SIF of 1.10. Source: American Society of Civil Engineers, 1997 Table 16. Strength increase factors. Concrete Concrete f dy /f y f du /f u f' dc /f' c f dy /f y f du /f u f' dc /f' c Flexure 1.17 1.05 1.19 1.23 1.05 1.25 Diagonal Tension 1.00 1.00 1.00 1.10 1.00 1.00 Direct Shear 1.10 1.00 1.10 1.10 1.00 1.10 Bond 1.17 1.05 1.00 1.23 1.05 1.00 Compression 1.10 1.00 1.12 1.13 1.00 1.16 *Far Design Range: Z 2.5 ft/lb1/3 â Close-in Design Range: Z < 1.0 ft/lb1/3 Source: Department of the Army, 1990 Stress Type Far Design Range* Close-in Design Rangeâ Reinforcing Bars Reinforcing Bars Table 17. Dynamic increase factors. wo Figure 86. Example of hypothetical assumed load shape for an SDOF analysis.
102 factors with each time step, most analyses should not include this additional complexity because the uncertainty in the loads does not justify this effort, and to the knowledge of the authors, no currently available SDOF software has this capability. Additionally, as mentioned above, the current procedure for building components has shown good correlation to experi- mental results with the selection of appropriate assumed load distributions. Once the equivalent SDOF system properties are known, an SDOF analysis should be used to determine the peak deflec- tion and maximum support rotations of the bridge column being analyzed. User-friendly software packages are available for this purpose, or the bridge designer may wish to create an SDOF analysis tool using the basic mechanics of structural dynamics. In either case, the results of an SDOF analysis should be compared to allowable ductility ratios and rotation limits. Based on the results of the experimental portion of this research program, acceptability criteria for blast-resistant concrete bridge columns are a maximum support rotation of 1.0 degree and a maximum ductility ratio of 15.0, and the U.S. Army TM 5-1300 Structures to Resist the Effects of Acci- dental Explosions (Department of the Army, 1990) contains similar values for building components. A designer should resize a member and repeat the analysis process if the SDOF analysis results in a ductility ratio or support rotation that exceeds the allowable limits. Both direct shear and diagonal shear are important for blast-resistant design, and a bridge column design 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. An important and notable difference between the design of blast-loaded bridge columns and blast-loaded build- ing columns that is captured in the design procedure developed under this research project is the manner in which diagonal shear and direct shear at the base of a bridge column are handled. In the design procedure proposed for bridge columns, transverse steel detailing requirements and flexural response limits have been selected to ensure that corresponding shear deformations remain within acceptable limits. Accordingly, design for direct shear at the base of a blast-loaded bridge column is inherently included in the SDOF design procedure and is not checked using the same procedure as is used for blast-loaded building columns. Furthermore, the design guidelines presented in the previous chapter outline prescrip- tive requirements for the minimum allowable volumetric reinforcement ratio. Additional details and a demonstration of how the design procedure is implemented can be found in Chapters 6 and 8, which provide specification-ready design guidelines and detailed design examples, respectively. As described above for the calculations for flexural capacity, the shear design should consider strength and dynamic increase factors and no load factors. The simplified design and analysis procedure described in this section is based on the widely accepted and trusted procedure presented in the U.S. Armyâs TM 5-1300 Structures to Resist the Effects of Accidental Explosions (Department of the Army, 1990) with specific modifications for bridge columns based on the results of the experimental portion of this research. As such, this procedure should provide a reasonable yet conservative design at a minimal cost, and it is recommended for bridge columns when a simplified analysis procedure is appropriate, as described in Chapter 5. Additional informa- tion regarding this procedure and its applicability to building components can be found in various sources, including Struc- tural Dynamics: Theory and Application (Tedesco, 1999) and Introduction to Structural Dynamics (Biggs, 1964). 7.3 Airblast Modeling Using Computation Fluid Dynamics Finite element codes employing computational fluid dynamics are valuable tools for predicting blast loads for situations involving complex designs, research problems, and post-event evaluations because they can include phenomena not captured by simplified analysis procedures, such as multi- ple reflections off complicated geometries, localized member failure, and blast loads coupled with structural response. wo(x) assumed distributed load mo(x) mass per unit length δ maximum deflection of member Fe equivalent load Me equivalent mass ke equivalent stiffness δ maximum deflection of member Fe Me ke δ δ mo(x) wo Figure 87. Relationship between real system and equivalent SDOF system.
While some finite element implementations of high-explosive modeling have user-friendly graphical user interfaces (GUIs), many general-purpose finite element codes require a user to manually specify a wide range of parameters, including the geometry, material properties, and mesh characteristics. While these codes can produce reliable results for a wide range of com- plex airblast scenarios, they also can generate highly misleading and erroneous solutions. These models are highly mesh- dependent, and several variables greatly influence the result- ing load predictions, such as the explosive and air material properties, the size and shape of the elements in the explosive region, and the density and shape of the air mesh. Successful airblast modeling using computational fluid dynamics codes requires an analyst experienced in theoretical shock physics, realistic physical behavior of high-explosive detonations, and the capabilities and limitations of the selected finite element code. Various general-purpose finite element codes with com- putational fluid dynamics capabilities implement the detona- tion of high explosives differently, and multiple methods exist even within the same finite element code. It is important to refer to the userâs manual of the selected software to deter- mine the most appropriate method for the scenario under consideration. Most implementations require separate defi- nitions for the explosive and the air, and this separation can be defined in at least two ways. One way to separate the explosive and air is to have regions of elements that represent both materials. In this case, the elements in each region have the property definitions of the material they represent, and the regions share common nodes on their boundaries. The other approach to separate the explosive and the air is a volume fraction technique, which requires the user to create a single mesh and then define the volumes within that mesh that contain the individual materials. Both of these methods will produce similar results if implemented correctly because the Arbitrary-Lagrangian-Eulerian (ALE) elements that define both the explosive and the air allow multiple materials within each element over the duration of an analysis. Using this analy- sis technique, general-purpose finite element codes generally require the user to allow the ALE mesh to contain multiple fluid definitions within the same elements. Regardless of the approach selected to separate the materials, efficient and reliable analyses have different mesh requirements for the explosive and air regions. These requirements are described further below. As with any finite element analysis, the mesh shape and size greatly influence the fidelity of the results. The literature documents the influence of mesh size on analytical results, and the work completed for this research reinforces its importance. Pressure and impulse predictions improve significantly as the air mesh becomes progressively finer (Cendon et al., 2004; Knight et al., 2004; Luccioni et al., 2006), and airblast models lacking sufficiently small elements throughout the mesh will generate inaccurately low pressures and impulses. Therefore, the analyst should always conduct a thorough sensitivity study comparing analytical results with known empirical results to determine the minimum mesh density required for the scenario under consideration. Additionally, the mesh size specifically in the explosive region is very important. The explosive mesh should be as uniform as possible, with elements of roughly the same dimensions, and no less than 16 elements along each edge of the explosive region is sufficient to allow the explosive to burn completely (Alia and Souli, 2006). Accordingly, the analyst should avoid using cylindrical or spherical meshes in the explosive region (i.e., a combination of wedge or conical elements with quadrilateral elements). The elements closest to the origin in these meshes will have disproportional edge lengths, and the analyses will produce lower peak pressures and impulses at a much larger computational cost than a model with only quadrilateral elements in the explosive region. Computational fluid dynamics analyses often employ âtracers,â which are the analytical equivalent of pressure gauges, to track variations in pressure over the duration of an analysis, and the mesh dimensions and fineness around these tracers greatly influence the computed results. Thus, in addition to mesh requirements for the explosive region and for the model as a whole, the region around a target demands a sufficiently fine mesh to capture clearing effects and reflections. If the mesh is too coarse in an area where the blast wave will reflect off a target, the tracers in this region may detect very inaccurate pressureâtime histories because the mesh will not be able to adequately capture pressure differentials from one location to another. For example, if the mesh is too coarse around tracers located on a target, two different tracers at dif- ferent locations on the same target with different clearing times and angles of incidence may record identical pressureâtime histories because they reside within the same element. In fact, a simulation may not be able to detect the difference between, for example, a square column and a circular column at the same standoff when the element edge length at the face of the columns is greater than the width of the columns (i.e., the front faces of the columns completely reside within the width of one element). Thus, an airblast model needs a very fine mesh in the target region when tracers are present, with several elements across the width and along the length of a target, to accurately capture clearing effects and the differences in pres- sure along the face of a target. While the mesh size is a well-established and documented variable that influences the accuracy of an explosion simula- tion, the mesh shape alone also plays an important role. A uniform mesh with the shape of the expected shock front propagation is preferable to other discretizations, as it allows the pressure perpendicular to the direction of propagation to remain in equilibrium. As such, a spherical mesh is most 103
104 appropriate for a spherical burst, and likewise, a cylindrical mesh is most appropriate for a cylindrical burst. A non-uniform mesh may not allow the pressure to equilibrate perpendicular to the shock front, and a pressure differential can occur between elements as shown in Figures 88 and 89. These two simula- tions represent similar explosions while using two different meshes. The mesh in Figure 88 is a symmetrical wedge from a uniform cylindrical mesh, and the mesh in Figure 89 is a symmetrical wedge with a mesh that has a transition to main- tain a finer mesh in the region around a target. (The target is not shown in the figure.) The different shades represent pres- sure levels, similar to a topographical map, and the pressure values in the scales in the upper right of the figures are in Mbar units. Figure 88 shows a uniform spherical blast wave, and the pressure in this model is in equilibrium perpendicular to the direction of flow. Figure 89 shows distortion in the blast wave perpendicular to the direction of flow, and this distortion is evident in the variation in shading seen in the shock front. The concentration of darker shading in the fine mesh region indicates a concentration of higher pressure in the denser element region. This pressure differential is an inherent result of a non-uniform mesh because the uniformly expanding shock front has to distribute fluid among uneven element widths, and as a result, coarse mesh regions lose pressure and do not have accurate results. Thus, as illustrated by these figures, distortion in the blast wave can occur in models that do not use uniform meshes, and a mesh with elements of varying sizes may lose accuracy in computing pressure and energy Figure 88. Isosurfaces of blast pressures in a uniform mesh. Figure 89. Isosurfaces of blast pressures in a non-uniform mesh.
relative to a model with a uniform mesh. At the very least, a model with elements of varying sizes will contain relative inaccuracies from one location of interest to another, and in some cases, the entire model will give deficient results. While the geometry of the air mesh outside the explosive region should conform to these rules, the elements within the explosive region should adhere to the guidelines presented above. Thus, the explosive region should employ only quadrilateral elements with similar proportions, even if the air mesh is cylindrical or spherical. The selected time step is another important parameter that the analyst must specify. While the results of this research do not necessarily show that continually decreasing the time step will always increase the accuracy of a model, the literature states that smaller time steps generally lead to more accurate results than models with larger time steps, albeit while com- promising computational efficiency (Knight et al., 2004). The analyst should always conduct a study to determine the sen- sitivity of an analysis to the time step, and the optimum time step should be used. Strategies do exist to achieve adequate mesh fineness while eliminating the need to construct cost-prohibitive model geometries. One such method is an adaptive mesh, which automatically contracts in regions of high pressure gradients, such as around a shock front. While this method can prevent costly mesh geometries by allowing a coarse mesh to auto- matically subdivide into a fine mesh in regions where cal- culations are important, the adaptive mesh algorithm can significantly increase the computational time of the analysis. Accordingly, there is a trade-off in efficiency between using an adequately fine mesh that is uniform with one that employs an initially coarse mesh and mesh refinement. The appropriate alternative will depend on the specifics of the model being solved. Symmetry is another strategy to achieve adequate mesh fineness without compromising cost. Symmetry in an airblast model can mean cubic, cylindrical, or spherical fractions as appropriate, and an example of cylindrical symmetry is shown in Figure 88. Regardless of the mesh shape and size chosen for a given application, the analyst should always consider symmetry as a useful resource to reduce computational cost. While employing symmetry is commonly used whenever possible, perfect reflecting surfaces do not exist in real scenarios, and an analysis that employs symmetry may produce results that do not compare well to high-explosive detonations on or near deformable surfaces (e.g., the earth). The term âexplosive burnâ commonly refers to the procedure of analytically simulating the burning of a high-explosive material to produce high-pressure gaseous detonation prod- ucts. While the explosive burn modeling technique varies depending on the finite element code of choice, a common implementation involves one or more mathematical equations that essentially expand the explosive material at a high rate to the pressure, volume, and density of the gaseous detonation products. The explosive burn is the most computationally demanding aspect of a free-field airblast simulation because the mesh in the explosive region needs to be very fine for suitably accurate results, and a strategy to reduce the computational demand of a model is to eliminate the need to repetitively burn the explosive material during an analysis. To that end, it is possible in some cases to burn the explosive with one model and then remap the shock front onto a separate mesh. While this approach may cost the analyst some additional preparation time, it can prevent the analyst from rerunning computation- ally demanding detonation simulations. While these guidelines will improve the efficiency and accuracy of modeling high-explosive detonations in finite element codes, obtaining reliable load predictions from high- level analyses is challenging. Reproducing experimental results can be especially challenging, as in many cases it is difficult to know the exact atmospheric conditions, explosive material properties, and locations of pressure gauges. Moreover, some codes may have inherent deficiencies and simplifying assump- tions built into their calculations, and these deficiencies can compound during repetitive calculations over numerous time steps. Thus, prior to endorsing any structural loads obtained from a finite element analysis, the analyst should always verify the accuracy of the computational fluid dynamics (CFD) capabilities within the code of choice by checking free-field overpressures at a few locations in a model against verifiable empirical results. The analyst should adjust model variables to reflect reality as needed, and as a last resort, one can overcome certain deficiencies and inaccuracies by artificially adjusting the explosive properties and specifically calibrating the model to match experimental results. Only analysts with a firm grasp of high-explosive theory and realistic behavior should attempt this approach, and those interpreting the results should exer- cise extreme caution. While adjusted explosive properties may reproduce accurately part or all aspects of a pressureâtime history at a given point in space and time, altering model input can compromise the results at other locations within a model, and the net load throughout the structure being modeled may not represent reality nor satisfy the intent of the analysis being performed. 7.4 Concrete Modeling Several options exist in most finite element codes for modeling concrete material behavior, and the performance of each varies depending on the accuracy of the material model, the implementation within the finite element code, and the experience of the analyst. Concrete constitutive models used in finite element analyses should employ at least a two-invariant formulation and a nonlinear equation-of-state to capture the nonlinear hardening and softening behavior characteristic of 105
106 concrete, and three-invariant formulations are preferable when available to correctly model volumetric expansion (Magallanes, 2008). Additionally, past research shows that constant-stress solid elements produce good results for both static and impact analyses (Schwer and Malvar, 2005; Gokani, 2006). Regard- less of the concrete model and element formulation chosen, however, an analyst should verify the fidelity of that model by comparing the results of a single element stress analysis to known theoretical or experimental behavior of the selected concrete (Magallanes, 2008), and the results from a model of a complete reinforced concrete structure should show good correlation with known experimental results prior to trusting analytical results. Most concrete structures contain reinforcement, and finite element models should include the concrete and reinforcing steel explicitly. Models should employ solid and beam ele- ments that have displacement functions of the same order. For example, truss elements are best for rebar when a model employs constant-stress solid elements for concrete because both element formulations have linear displacement functions, thereby guaranteeing displacement compatibility along the shared element edges. The simplest approach to modeling rebar is to connect the rebar elements to the concrete elements at common nodes. Although this method is a simplification of reality in which potential slip between rebar elements and concrete elements is ignored, analysts typically choose this approach because it provides reasonable results for the preparation time required. This simplification may not be appropriate in some cases, and models can incorporate addi- tional aspects of concrete behavior, such as bar slip and rebar buckling, by using beam element formulations for rebar or connecting rebar elements to concrete elements using springs. While including such features may be more representative of actual behavior, this increase in accuracy comes at a large preparation and computational cost. In some cases, however, the flexural resistance of rebar can contribute significantly to structural response, as when significant spall and breach de- grades a member cross-section. Finite element models should include rebar elements with a beam element formulation rather than a truss element formulation for these cases. In general, the analyst should first try an analysis with truss elements to determine if section loss and the flexural resistance of the reinforcing steel are important. As with concrete, all steel material property definitions should consider strain-rate dependent properties when pre- dicting response to airblast, and two procedures are available for this purpose. As outlined in Section 2.2.4, the first method to consider the effect of strain rate on reinforcement properties involves increasing the yield and ultimate strengths of the reinforcing steel by applying constant dynamic increase factors such as those shown in Table 17. While this method is often acceptable for design because it quickly and easily allows the computation of conservative material properties for most air- blast cases, it does not allow an accurate comparison between member stresses at a given time step and material proper- ties computed using the strain rate from the same time step. Therefore, simply applying constant dynamic increase factors may not be appropriate for some analysis cases. Experimental studies show that actual material properties vary significantly with strain rate and continue to increase as the strain rate increases (Department of the Army, 1990; Tedesco, 1999), and finite element analyses that include erosion (i.e., removal of failed elements) need accurate material properties during each time step to assess the adequacy of each element. Because both member stresses and actual material properties vary significantly during an analysis, the application of constant dynamic increase factors could result in premature or delayed erosion of some elements. Most general-purpose finite ele- ment software includes strain-rate dependent material defi- nitions that are applicable to reinforcing steel, and two of these are the Johnson-Cook and Cowper-Symonds relationships (LSTC, 2007). Several recent and past studies illustrate the use of these relationships for scenarios involving structures subjected to blast and impact loads (Karagiozova and Jones, 2000; Raftenberg, 1997; Rusinek, 2008; Rushton et al., 2008; Turhan et al., 2008). 7.5 Coupled Analyses Coupled analyses are very complex, involving the inter- action of an Eulerian mesh and a Lagrangian mesh, and the results can vary widely depending on the user-selected inputs. To run a coupled analysis, the user must build a model with a Lagrangian mesh (i.e., typically the structure) overlapping (but not connected to) an Eulerian mesh (i.e., typically the fluid) before evacuating the fluid from the Lagrangian mesh. When the fluid in the Eulerian mesh (i.e., the shock front as it travels through air in the case of airblast) penetrates the surface of a Lagrangian structure during a simulation, the finite element code will move the fluid back to the surface of the Lagrangian structure and apply a reaction force to the fluid normal to the surface of the Lagrangian structure. Because some codes allow modeling of porous surfaces, a user-defined penalty factor or penalty curve typically governs the amount of fluid that will leak through the surface of the Lagrangian mesh, and it also defines the relationship between the depth of fluid penetration and the resulting force applied to the fluid. This concept is similar to Hookeâs Law, where the force is equal to the stiffness multiplied by the displacement. The stiffness of the coupled surface can be constant or vary linearly, and a user can increase or decrease the âstiffnessâ of the coupled surface by adjusting the penalty factor or penalty curve; therefore, the solution can vary widely depending on the user selected inputs. For example, if the analyst sets the penalty
factor too high, the coupled surface can be too âstiff,â and the shock front will bounce off the Lagrangian surface with too much momentum (Knight et al., 2004). A penalty factor set too âlowâ can result in a large amount of fluid leaking through the coupled surface. The analyst bears the responsibility of defining an appropriate penalty factor or penalty curve, and many models may require calibration with experimental data. As a result, these problems can be very complex, requiring a very experienced analyst and reliable empirical data. Thus, using these models is not advisable unless structural response will significantly change the flow of the shock front. Detailed information on coupled analyses is beyond the scope of this report, as the intention of this chapter is not to provide a tuto- rial on finite element modeling for such cases. If more detailed information is desired, the reader should consult the research literature and the userâs manual of the software being used. 7.6 Summary This chapter outlines available methods to predict blast loads on and the resulting response of bridge columns. Methods that employ single-degree-of-freedom analyses, such as the one described in this chapter, are valuable for design scenarios because they yield acceptable accuracy given the uncertainty of the loads. As a result, they are commonly used by blast engineers. While these simplified methods are often useful and desirable, they may not always be appropriate, and the analyst may wish to use a more advanced procedure. As described above, several general-purpose finite element soft- ware packages have the ability to conduct high-level airblast and structural response analyses, and the implementation of explosive modeling, the availability of material constitutive models, and the analysis procedure employed are different for each. Thus, while the above sections give general principles for modeling high-explosive detonation, shock propagation, and structural response using these codes, the analyst should be familiar with the capabilities, requirements, and limitations of the selected software. With proper knowledge and experi- ence, general-purpose finite element software is a valuable tool for predicting blast loads and response in advanced design or research environments. Such load scenarios include cases with complex geometries, structures for which venting due to localized failure may drastically reduce loads, and situations for which the charge weight, explosive properties, and struc- ture parameters are known or can be reasonably well estimated. Applicable response scenarios include cases in which localized failure (i.e., spall or breach) may affect the loading or the global response of a structure. 107
