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27 CHAPTER FIVE STRUCTURAL ANALYSIS AND DESIGN pattern that represents the distribution of inertial forces imposed by earthquake shaking. This method is well suited to simple, regular structures that are dominated by their first mode response. The method predicts linear elastic response, and inelastic effects must be handled separately. For exam- ple, in design the performance and allowed inelasticity may be treated implicitly using R factors, which reduce the design forces as a function of the bridge importance and the structural systems ability to withstand plastic deformations. This method has been adopted into the AASHTO LRFD, but is limited in its treatment of inelasticity and structural complexity. A brief discussion of R-factors can be found in the Current AASHTO Practice section of chapter three. In general, the equivalent lateral force procedure loses accu- racy for structures where higher mode effects are significant, as in long-period structures, and where geometric irregu- larities or sharp discontinuities or asymmetry are present, as these breach the basic assumptions on which the method is founded. This analysis method is poorly suited for PBSD as it does not explicitly quantify bridge performance. LINEAR DYNAMIC PROCEDURES If a structure displays dynamically complex behavior that cannot be captured by the equivalent lateral force proce- dure, then modal response spectrum analysis (RSA) can be used. In this linear elastic procedure, a modal analysis of the structure is conducted to determine the deformed shape and natural frequency of all pertinent modes of vibration, typically including a sufficient number of modes to capture at least 90% mass participation in each orthogonal direc- tion of displacement. A design response (acceleration) spec- trum is then used to determine the magnitude of each modal response (e.g., displacements, shears, moments) based on the participation factor, response, and damping for each mode. The maximum modal responses are then determined using a modal combination rule such as the square root of the sum of the squares (SRSS) or the complete quadratic combination (CQC) rules, depending on the spacing of the natural frequencies. These combination methods are intended to provide estimates of response appropriately built up from individual elastic mode response. Modal RSAs are described extensively in structural dynamics texts and will not be further explained here. During the PBSD process, it is critical to accurately pre- dict the structural response to earthquake ground motions. Depending on the geometry of the system and the extent of inelastic behavior, various methods of increasing complexity and refinement have been developed. In general, structural analysis for earthquake forces can be broken into four catego- ries, as shown in Table 5. Fundamental to three of the analysis methods is some form of simplification from the most general and powerful, but time-consuming and complex methodol- ogyâfull nonlinear dynamic response history analysis. TABLE 5 TYPES OF STRUCTURAL ANALYSIS linear nonlinear Static ⢠Equivalent lateral force procedure ⢠Pushover analyses (e.g., coefficient method, equivalent linearization) dynaMic ⢠Modal response spectrum ⢠Linear response history ⢠Inelastic response spectrum ⢠Nonlinear response history analysis The differences between methods are contingent upon the explicit treatment of inelasticity and dynamic behavior. Many of the current PBSD procedures use a combination of techniques from more than one analytical category. For example, the AASHTO SGS uses a linear dynamic analy- sis adjusted by scalar multipliers to determine the estimated displacement demand that an earthquake might place upon the bridge, while a nonlinear static pushover analysis is used to determine the displacement capacity of either the indi- vidual piers or the bridge as a whole. Many analysis and design techniques exist in various stages of development and implementation, and detailed explanation of all possible options is beyond the scope of this synthesis. However, this chapter presents several of the more common techniques. There are excellent resources explaining in detail the dynamic response of structures to seismic excitation (Clough and Penzien 1975; Chopra 2007; Villaverde 2009), which pro- vide the theoretical basis for all of the analytical categories. LINEAR STATIC PROCEDURE The most basic analytical procedure uses equivalent static lateral forces imposed on the structure in a predefined load
28 structure remains essentially elastic under the selected ground motion. NONLINEAR STATIC PROCEDURES Whenever inelastic behavior is expected, linear static and linear dynamic analyses cannot reliably be used to precisely quantify structural performance; therefore, nonlinear meth- ods must be employed. The simplest methods are the nonlin- ear static procedures, which use what are known colloquially as âpushoverâ analyses, to evaluate the nonlinear deforma- tion capacity of the system and explicitly quantify the redis- tribution of internal forces as a result of yielding elements. In this technique, an assumed force or displacement field (typically corresponding to the first mode deformed shape) is applied to the structure. The magnitude of the applied force or displacement field is monotonically increased until failure or another applicable limit state is observed. A trace of the deformation versus base shear is recorded in forceâ displacement space. This response is known as the pushover or capacity curve. The maximum global displacement demands resulting from the seismic hazard are then predicted using simplified, approximate procedures that relate the inelastic response of a single degree of freedom (SDOF) oscillator to the elastic response of an infinitely strong elastic system. These proce- dures are ideally calibrated using nonlinear response history analyses. The global displacement demand is then compared with the global displacement capacity, as determined from the pushover curve, to determine the adequacy of the design. From the global displacement, member drifts, forces, and component actions can be determined, leading to the final design of the structure. Figure 8 is a schematic representa- tion of nonlinear static procedures. This method provides an accurate prediction of the elastic dynamic response of simple and complex structures and has the added benefit of being relatively easy and quick to per- form with modern software. Additionally, the method uses the same modeling assumptions that are typically employed for nonseismic loading, with which structural engineers are familiar. On their own, modal RSAs cannot predict inelas- tic displacement demands, plastic deformation capacity of the system, or accurate internal force fields if yielding is expected to develop. This method is therefore suitable for PBSD only if the structure is expected to remain nearly elas- tic under the ground motion of interest, or if the method is only used to determine displacement demands (as with the coefficient method described subsequently), or for analytical methods used to assess or check the inelastic response of a structure. This is how linear dynamic procedures are used in the AASHTO SGS. To further refine the accuracy of linear elastic analysis, step-by-step time integration or linear response history analysis methods can be used. These methods solve for the response of the structural system in the time domain by ana- lytically subjecting the structure to an earthquake accelera- tion record. This solution is accomplished by numerically using a series of small time intervals and integrating the incremental equations of motion for each time step based on the loading function for that time step and the state of the system from the previous time step. With small enough time steps, the solution will converge on the exact solution of the equation of motion. Because the response is solved explicitly, the assumption that the system mode shapes are orthogonal to the damping matrix, and the need for approxi- mate modal combination rules to determine maximum dis- placements, are unnecessary (Villaverde 2009). Despite these advantages, linear response history analyses are rarely performed in design practice and are only accurate if the FIGURE 8 Nonlinear static procedures (FEMA 440).
29 The benefits of this method are increased insight into inelastic behavior of a structure, including the location and formation sequence of plastic hinges, relative simplicity, and accuracy for simple structures, especially those that can be well represented as an SDOF system. This makes nonlinear static methods attractive for use in PBSD methodologies. The basic limitations of these procedures are that elastic modal properties are used to compute the inelastic system parameters, and it is assumed that the response of the struc- ture is controlled by a single mode (usually the first mode). Therefore, the contribution of the other modes (usually higher modes) may not be explicitly considered. In general, two primary manifestations of the nonlinear static procedure are commonly used in design practice: the coefficient method and the equivalent linearization method. If they are well calibrated against nonlinear response history analyses, these methods can provide reasonably accurate predictions of seismic behavior and are well suited to perfor- mance-based design of ordinary structures. A comparative evaluation of the two methods is provided in Miranda and Ruiz-Garcia (2002). Coefficient Method The coefficient, or displacement modification method, builds on the linear modal response spectrum procedure described previously. Because the modal response spectrum analysis is a linear-elastic method, inelastic behavior and damage cannot be captured explicitly. However, the inelastic displacement demand on a structure can be approximated if the so-called âequal displacement assumptionâ is imposed. This states that the maximum lateral displacement of a nonlinear structural system is approximately equal to the maximum displace- ment of the same system behaving elastically with unlimited strength, as shown in Figure 9. In other words, the yielding of the system does not affect the maximum displacement experi- enced during a ground motion. This assumption is only valid for medium- to long-period structures with minimal strength and stiffness degradation, and insignificant P-â effects. If these requirements are not met, scalar coefficients have been developed and calibrated to modify the predicted displace- ment. The coefficients are derived empirically from nonlinear response history analyses of SDOF oscillators with varying periods, strengths, and hysteretic shapes. The displacement capacity of the inelastic structural sys- tem is determined by the application of strain, rotation, drift, or displacement limits to the pushover response. The loca- tion of this displacement demand relative to the displacement capacity indicates whether the performance objective has been met. Simply stated, the displacement demand is deter- mined using a modified response spectrum analysis, while the displacement capacity is determined using a pushover analysis. This displacement is shown schematically in Figure 10. This method has been developed by Newmark and Hall (1982), Miranda (2000), and Chenouda and Ayooub (2008), among others, and has been implemented by the AASHTO SGS, FEMA 440, ASCE 7, ASCE 41, and FEMA 356. FIGURE 9 Linear and nonlinear approximations of structural response. Equivalent Linearization Method Another common nonlinear static analysis method is the equivalent linearization method, also known as the secant stiffness or substitute structure method. In this procedure, the nonlinear system is replaced with an equivalent linear system with an effective period defined by the secant stiff- ness of the nonlinear system at the displacement demand. Hysteretic energy dissipation is accounted for by equivalent viscous damping determined from the ductility and hyster- etic response of the nonlinear system. The global inelastic displacement of the system is calculated as the maximum displacement of the equivalent linear SDOF oscillator. This method has been implemented by Rosenblueth and Herrera (1964), Gulkan and Sozen (1974), Shibata and Sozen (1976), Kowalsky (1994), and Iwan (1980), among others. A convenient graphical representation of equivalent linearization is the capacity spectrum method of ATC-40, which has been further refined by FEMA 440 to incorporate various hysteretic properties, such as strength and stiffness degradation. In this method, the elastic response spectrum and the pushover capacity curve are converted into spectral ordinates (spectral displacement versus spectral accelera-
30 demand in relation to the displacement capacity. The sys- tem displacement capacity is determined using strain, rota- tion, drift, or displacement limits defined according to the required performance criteria. MULTIMODAL NONLINEAR STATIC PROCEDURES In some structures, an SDOF representation is inadequate, as neglecting higher mode effects will produce an inaccurate and often unconservative prediction of response. Because of this, several multimodal pushover procedures have been developed, in particular for tall buildings, but the concepts are potentially applicable to bridge systems sensitive to higher mode response. Modal Pushover Analysis In this procedure, pushover analyses are conducted indepen- dently in each mode, using lateral force profiles that represent the response in each of the first several modes. This procedure is performed even though the response in each mode may be nonlinear, whereby the mode shapes and lateral force profiles are assumed to be invariant as the inelastic mode shapes are only weakly coupled. Response values are determined at the tion). The demand spectrum is modified to account for the equivalent viscous damping generated by hysteretic energy dissipation. The intersection between the demand and capac- ity spectra when their equivalent viscous damping terms are equal represents the âperformance pointâ or the displace- ment demand, which is shown in Figure 11. Equivalent lin- earization methods are permitted in the AASHTO Guide Specifications for Isolation Design (2010a), the AASHTO SGS, and the provisions of ASCE 7 for seismically isolated structures and structures with damping systems. Another common use of equivalent linearization is the direct displacement based design (DDBD) procedure devel- oped and advocated by Priestley et al. (1996, 2007). This method uses the same fundamental equivalent linearization concepts but employs an elastic displacement spectrum, which is modified by the equivalent viscous damping to account for hysteretic energy dissipation. Specific equivalent viscous damping formulas have been calibrated for various hysteretic shapes (e.g., bilinear, Takeda, flag shaped), as well as various soil types for soil-foundation interaction of piles and drilled shafts (Priestley et al. 2007). As with the coefficient method, the system performance is determined based on the location of the displacement FIGURE 10 FEMA 356 Coefficient Method (FEMA 440).
31 target displacement associated with each modal pushover analysis. The target displacement values may be computed by applying the coefficient method or equivalent lineariza- tion procedures to an elastic spectrum for an equivalent SDOF system representative of each mode being considered. Response quantities obtained from each modal pushover are normally combined using the SRSS method. Modal pushover analyses are detailed in Chopra (2007). Limitations of this procedure are that (1) elastic modal properties are used to compute the inelastic system param- eters and (2) the displacements are approximated from the maximum deformation of an SDOF system for all the modes. Adaptive Modal Combination Procedure [This] methodology offers a direct multi-modal technique to estimate seismic demands and attempts to integrate concepts built into the capacity spectrum method recommended in ATC-40 (1996), the adaptive method originally proposed by Gupta and Kunnath (2000), and the modal pushover analysis advocated by Chopra and Goel (2002). The AMC [adaptive modal combination] procedure accounts for higher mode effects by combining the response of individual modal pushover analyses and incorporates the effects of varying dynamic characteristics during the inelastic response via its adaptive feature. The applied lateral forces used in the progressive pushover analysis are based on instantaneous inertial force distributions across the height of the building for each mode. A novel feature of the procedure is that the target displacement is estimated and updated dynamically during the analysis by incorporating energy-based modal capacity curves in conjunction with constant-ductility capacity spectra. Hence it eliminates the need to approximate the target displacement prior to commencing the pushover analysis (Naiem 2001). The primary feature of adaptive schemes is the updating of the applied story forces with respect to progressive changes in the modal properties at each step. This allows progressive system degradation resulting from inelastic deformations to be represented more realistically in a static framework. The method is described in Kalkan and Kunnath (2006). The use of these more elaborate modal analysis tech- niques may or may not be easily, or economically, adapted to bridge design, as the work in this area has been primarily directed toward building response. NONLINEAR DYNAMIC PROCEDURES Although the aforementioned nonlinear static methods aim to define inelastic displacement demands using a simplified framework, some situations warrant the use of nonlinear dynamic or nonlinear response history analytical procedures, often called the âtime-historyâ method. This method is an FIGURE 11 ATC-40 Capacity Spectrum Method (FEMA 440).
32 extension of linear response history analysis, with nonlinear material and geometric behavior explicitly accounted for in the equations of motion. In this most general of all procedures, it is necessary to (a) write the equations of motion in an incremental form, (b) make the assumption that the properties of the structure remain unchanged during any given time interval [or iterate the interval], (c) solve the equations of motion for such a time interval considering that the structure behaves linearly, and (d) reformulate its properties on the basis of the obtained solution at the end of the interval to conform to the state of stresses and deformations at that time (Villaverde 2009). The solution within each time step is iterated using vari- ous corrective solution algorithms, which reduce the error associated with (1) the assumption that system properties are constant during a time step and (2) the sudden change in system properties resulting from yielding or unloading dur- ing the time step. The details of nonlinear dynamic solution procedures, algorithms, and modeling guidelines are well documented elsewhere (Chopra 2006; Priestley et al. 2007; Villaverde 2009). These analysis procedures have the fewest limitations, as both the transient dynamic and inelastic responses are solved for explicitly. However, these are also the most time- consuming and difficult analyses to perform, troubleshoot, and interpret, requiring skilled analysts. Additionally, to obtain usable results, multiple ground motions must be selected. Typically, design forces and deformations are taken as the maximum structural actions if only three ground motions are used or taken as the average if seven or more ground motions are used. In most cases, each ground motion will consist of the two orthogonal horizontal components of shaking, and often the vertical component of shaking is incorporated as well. Some of the challenges associated with conducting non- linear response history analyses are the selection and scaling of the input ground motions (NEHRP 2011), the calibration and validation of element hysteretic response, the treatment of elastic damping (Charney 2008), and the computational hurdles of convergence, run time, and postprocessing. In many cases, the use of nonlinear response history analyses is simply too costly and time prohibitive for all but high-profile or critical structures. There is, however, a simplification of the nonlinear response history analyses, known as the constant ductility or inelastic response spectrum. This method is the nonlinear extension of an elastic response spectrum. Instead of record- ing the maximum response from an elastic SDOF oscillator to a suite of one or more ground motions, the constant ductil- ity spectrum records the maximum response of a nonlinear SDOF oscillator at specific system ductilities. The nonlinear SDOF oscillator has an assumed hysteretic behavior that sufficiently represents the nonlinear cyclic force-deforma- tion response of the system. This methodology is well docu- mented (Newmark and Hall 1982; Krawinkler and Nassar 1992; Han et al. 1999; Chopra and Goel 1999; Fajfar 1999; Chopra 2007) and can provide a direct and efficient analyti- cal method for structures that can be represented as SDOF systems. This method can also use a graphical solution pro- cedure similar to the capacity spectrum method described earlier (see Chopra 2006). MODELING OF NONLINEAR SYSTEMS The techniques and assumptions required to accurately model a nonlinear structural system are dependent on the system characteristics and configuration, the intended ana- lytical procedure, the computer software available, the level of accuracy needed, and the time available to perform the analysis. Therefore, detailed modeling recommendations are beyond the scope of this synthesis. Modeling guidelines are typically given in the literature describing particular analytical methodologies. It can be said, however, that for all structural modeling, the âanalysis should be as simple as possible, but no simpler,â according to Einsteinâs maxim. This requires the judicious use of modeling assumptions so unintended errors or gross oversimplifications are avoided. There are many excellent resources describing modeling for nonlinear structural analysis, including (Filippou and Issa 1988; Filippou et al. 1992; Priestley et al. 1996; Berry and Eberhard 2007; Priestley et al. 2007; Aviram et al. 2008; FIB 2008; Dierlein et al. 2010). The importance of accurate modeling of soil-structure interaction, abutment restraint, and movement joint effects must be emphasized here, as these effects can greatly change the predicted response. The details of the model- ing approaches depend on the type of foundation: typi- cally either shallow-spread footings or deep foundations (extended shafts, pile columns, pile bents, or pile groups). In most cases, spread footings and pile groups are capac- ity protected and are modeled using linear and/or nonlinear springs to capture footing rotation and translation. Ana- lytical methodologies are well documented (e.g., AASHTO SGS, FHWA Retrofitting Manual). However, in the case of extended shafts and pile bents/columns, inelastic action is often allowed to occur below ground. Including this inelas- tic response usually requires more sophisticated modeling and direct incorporation of the soil response into the solu- tion. These methods are well defined in the literature (Budek 1995; Priestley et al. 1996; Budek 1997; Boulanger et al. 1999; Chai 2004; Song 2005; Suarez and Kowalsky 2006a and 2006b; Blandon 2007; Priestley et al. 2007; Goel 2010), with several closed-form approximations being developed. As with foundations, abutment and movement joint type can also have a significant influence on the dynamic response of bridges. Modeling guidelines and recommenda-
33 tions can be found in ATC (1996), Priestley et al. (1996), FHWA (2006), Aviram et al. (2008), and Shamsabadi et al. (2010). Simplified models and analytical techniques for predicting movement joint response are difficult to define and have relatively high levels of inherent uncertainty. One such method that has gained considerable favor in the design community for its simplicity is to use a tension and a com- pression model of the structure to capture the response when the joint is fully open (the tension model) and fully closed (the compression model). The design actions are then taken as the envelope of member action from the tension and the compression model. While this appears to ensure adequate performance in strong ground motions movement, joint response is certainly an area warranting additional research. UNCERTAINTY IN NONLINEAR ANALYSIS METHODS Within the realm of nonlinear structural analysis for seis- mic design there is a constant struggle to balance simplicity and transparency of an analytical procedure with its ability to accurately predict often complex structural behavior. As with any simplification or approximation errors can be intro- duced. However, as long as the methods produce acceptable designs (i.e., not unconservative/unsafe or overly conserva- tive/unnecessarily expensive), then errors or bias can be tolerated. Ideally, structural analysis could be simplified without introducing significant error or bias into the results/ design; however, this is often not the case. Therefore, a con- siderable amount of research is currently being conducted to refine and simplify nonlinear analysis and design tech- niques. To be able to reliably apply PBSD in the full sense, the uncertainties in the structural analysis must be under- stood and quantified. Clearly delineating and considering such uncertainties is a significant challenge. It is of interest to examine the analytical methods described previously in relation to the relative uncertainty that is intro- duced into the solution solely by the assumptions inherent in an analytical technique. An increase in relative uncertainty is represented in Table 6 as a darker shade of grey. It should be noted, however, that uncertainty classified here as âhighâ is still within acceptable limits for usage within PBSD, espe- cially given the uncertainty inherent in the ground motion. The vertical columns represent the two primary ground motion input characterizations, elastic response spectra and individual ground motion records. The rows describe the refinement of the structural model; âDetailedâ represents a complex 2-D or 3-D model with refined element boundary conditions and associated nonlinear behavior. This level of refinement is typically reserved for academic research or critical structures. The equivalent multiple degree of freedom (MDOF) model defines the structural system as simply as possible while maintaining all critical degrees of freedom and modes of deformation. Structural analysis according to AAS- HTO SGS would fall into this category. Finally, the equivalent SDOF lumps all structural behavior into an SDOF oscillator with a nonlinear force-deformation response defined by the system pushover curve. TABLE 6 UNCERTAINTY IN NONLINEAR STRUCTURAL ANALYSIS METHODS (AFTER FEMA 440) ground Motion input Structural Model Response spectra Ground motion records Detailed Dynamic analysis Equivalent MDOF Multimode pushover analysis Simplified MDOF dynamic analysis Equivalent SDOF Nonlinear static procedures Simplified SDOF dynamic analysis HIGH LOW Relative Uncertainty SDOF = single degree of freedom. MDOF = multi degree of freedom. When the structural model or the ground motion input are simplified, the uncertainty increases. Because of this, the nonlinear static procedures typically have the largest inherent uncertainty owing to their modeling and analytical assumptions, and nonlinear response history analyses with detailed model definitions can often provide more accu- rate representations of actual structural behavior. However, increasing modeling refinement and complexity can intro- duce substantial uncertainty into the analytical solution by virtue of the difficulty in defining, verifying, and calibrat- ing nonlinear structural response. Furthermore, the detailed model category in Table 6 is broad and ranges from relatively simple predefined hysteretic models (e.g. bilinear or Takeda) to fiber and continuum type models, each of which carries its level of uncertainty, along with limitations and challenges regarding its implementation. Also, highly refined models and analysis may lull ana- lysts into believing that the results have a higher accuracy, when in fact the seismic input often carries nontrivial uncer- tainty that may greatly outweigh the uncertainty of the struc- tural model. Increased modeling complexity may then have a low benefit-to-cost ratio. In general, however, although uncertainty increases with analytical simplicity, time and cost decrease. There is typically a trade-off between cost and accuracy. PROBABILISTIC TREATMENT OF NONLINEAR ANALYSES One of the main goals of next-generation PBSD is the quan- tification of uncertainty throughout the analysis and design process. Therefore, how to characterize the amount of uncer-
34 tainty in the predicted structural demand is of vital impor- tance. Uncertainty can enter from two primary sources: (1) the input ground motion record-to-record variability and (2) the mathematical model used to predict structural response. As discussed in the previous section, simplified analytical procedures, such as nonlinear static analyses, increase the uncertainty in the estimation of structural demand. Therefore, there is an increasing trend to use nonlinear response history analyses in next-generation performance-based design proce- dures (ATC-58, PEER), as this provides a much more refined estimate of uncertainty and reduces the bias and uncertainty associated with simplified analytical techniques. A rigorous evaluation of uncertainty is accomplished by using what is known as a probabilistic seismic demand model (PSDM), which relates a selected IM such as peak ground acceleration, peak ground velocity, or first mode spectral acceleration, to an EDP, such as drift ratio or plas- tic rotation. In their most rigorous form, PSDMs quantify demand uncertainty by applying a suite of scaled ground motions representative of the site-specific hazard (fault mechanism, distance to fault, event magnitude, and local site conditions) to the structural model using nonlinear dynamic analyses. The use of suites of ground motions is necessary because each ground motion will produce a different struc- tural response, known as the record-to-record variability. ATC-58 (2011) recommends that at least 20 ground motions are necessary to truly represent the record-to-record vari- ability in nonlinear structural response. Perhaps the two most common rigorous treatments of record-to-record variability are the probabilistic seismic demand analysis (PSDA) and the incremental dynamic anal- ysis (IDA). PSDA uses a bin approach, where a portfolio of ground motions is chosen to represent the seismicity of an urban region. The intensities of the ground motions in the portfolio cover a range of seismic hazard Intensity Measures (IM), such as first mode spectral acceleration. Nonlinear time-history dynamic analyses are performed for each motion using a model of the structure to compute extreme values of structure-specific Engineering Demand Parameters (EDP) (Mackie and Stojadinovic 2003). PSDAs are explained in further detail in Mackie and Sto- jadinovic (2004) and Shome (1999). The IDA or dynamic pushover is the continuous exten- sion of a âsingle-point,â nonlinear response history analy- sis in much the same way as the nonlinear static pushover analysis is the continuous extension of a âsingle-pointâ static analysis (Vamvatsikos and Cornell 2001). IDA is done by conducting a series of nonlinear time- history analyses. The intensity of the ground motion, measured using an IM, is incrementally increased in each analysis. An EDP, such as global drift ratio, is monitored during each analysis. The extreme values of an EDP are plotted against the corresponding value of the ground motion IM for each intensity level to produce a dynamic pushover curve for the structure and the chosen earthquake record (Mackie and Stojadinovic 2003). Figure 12 shows example results from a PSDA and an IDA. Each demand model shows the structural response of a two-span ordinary California highway bridge. The only variation in the structural systems is the column diam- eter to superstructure depth ratio (Dc/Ds). In the case of PSDA, the relationship between first-mode spectral accel- eration and the longitudinal drift ratio is linear in log- log space, intuitively with higher spectral accelerations producing higher drift ratios. Each data point denotes a unique ground motion that represents a particular seismic hazard at the site. However, for the IDA only four ground motions were selected, but each record was scaled incre- mentally until a specific IM was reached. The trace of EDP in relation to increasing IM is shown by the IDA curve, which again shows that an increase in IM results in a gen- eral increase in EDP. However, the increase in EDP is not always proportional to the increase in IM, with some cases FIGURE 12 Example PSDA (left) and IDA (right) (Mackie and Stojadinovic 2003).
35 showing a higher IM generating a lower EDP. This is the result of the pattern and timing of response cycles, where âas the accelerogram is scaled up, weak response cycles in the early part of the response time-history become strong enough to inflict damage (yielding), thus altering the properties of the structure for the subsequent, stronger cyclesâ (Vamvatsikos and Cornell 2001). Characteristics of IDA response are further discussed in Vamvatsikos and Cornell (2001). PSDA and IDA can be used interchangeably to determine the PSDM as long as a sufficient number of appropriate ground motions are chosen. Mackie and Stojadinovic (2003) observed that the IDA method is sensitive to the choice of ground motions, because a small suite of ground motions are used, whereas the PSDA generally provides sufficient ground motion variation as a result of the bin approach of ground motion selection. Mackie and Stojadinovic (2003) provide recommendations on the accuracy and equivalency of PSDA and IDA for generating PSDMs. Regardless of the method of generating the PSDM, the uncertainty within the prediction of EDPs is quantitatively determined. As is obvious, the computational effort required to rigor- ously develop a PSDM is extremely high, and is likely only possible in academia and for critical or signature structures. To enable probabilistic treatment of demand uncertainty for noncritical structures, some assumptions must therefore be made. One such simplified method has been proposed by ATC-58 (2011). Although it still relies on nonlinear dynamic analyses, the number of permutations is greatly reduced by using best-estimate deterministic models and predetermined dispersion (β) factors for modeling uncertainty and ground motion record-to-record variability. The dispersion factors in this context represent the spread within the distribution of possible values; in other words it is the âfatnessâ or âwidthâ of the bell-shaped curve describing the distribution of pos- sible values. In many cases, the distribution within structural systems can be well represented using a lognormal probabil- ity density function, in which case the dispersion is the loga- rithmic standard deviation of the data set. The probability density function is then completely defined by the median response value and the dispersion. This method requires that the nonlinear dynamic analy- ses represent the median structural response, using expected material properties and ground motions scaled so that the spectral geometric mean of the selected records matches the target spectrum within a predefined range of first mode peri- ods. Typically, if there is strong correlation in spectral shape within the period range of interest, as few as three records can produce a reasonable prediction of the median response, whereas if there is high spectral fluctuation about the target spectrum, as many as 11 records may be needed. The selec- tion and scaling of ground motion records is discussed fur- ther in NEHRP (2011). ATC-58 (2011) provides ground motion record-to-record dispersion factors (βaâ, βaa, βgm) based on the effective fun- damental period, T , and the strength ratio, S, of the struc- ture. The strength ratio is effectively the ratio of the base shear demand on an infinitely strong elastic system to the base shear capacity of the yielding system, where Sa(T ) is the reactive weight, W is the first mode spectral acceleration, and Vy1 is the first mode yield strength. A strength ratio of unity or less means the system behaves elastically. Table 7 shows example dispersion values for drift, βaâ, and accel- eration, βaa. In general, the displacement and acceleration dispersion factors increase with fundamental period and a higher strength ratio (i.e., the yield strength is a smaller pro- portion of the elastic demand). These dispersion values are then used to determine the distribution of predicted struc- tural response about the best-estimate median response. Additionally, ground motion spectral demand dispersion values for scenario-based assessments specifically (βgm) are provided for western North America (WNA), CEUS, and the Pacific Northwest (PNW). The values for βm (described subsequently) are default values of modeling uncertainty for typical new construction in the design development phase of the project. TABLE 7 UNCERTAINTY DUE TO GROUND MOTION RECORD-TO- RECORD VARIATION (ATC 2011) Uncertainties will also develop from inaccuracies in component modeling (e.g., hysteretic behavior, material properties, imperfections in construction), damping, and mass assumptions. To account for this rigorously, mechani- cal properties can be treated as random variables with speci- fied distributions during parametric studies. Therefore, a fully probabilistic treatment of modeling uncertainties requires extensive permutations and significant amounts of time and effort. In many cases, this is entirely cost prohibi- tive, and modeling uncertainty is treated using simplified methods such as those prescribed in ATC-58 (2011), which are described here. The two primary sources of modeling uncertainty are (1) building definition and quality assurance, and (2) model quality and completeness. These are represented by disper-
36 sion values βc and βq respectively. βc represents the variation in as-built structural member properties, such as in mate- rial strengths, geometry, and reinforcement location, with respect to properties assumed in analysis or specified in the design drawings, or both. βq accounts for uncertainties in the modeling of actual structural behavior, such as the refine- ment and accuracy of the hysteretic model and the calibra- tion to large-scale experimental test results. Tables 8 and 9 show example values. βc and βq are combined using the SRSS method to form the modeling uncertainty, βm. TABLE 8 UNCERTAINTY DUE TO BUILDING DEFINITION AND CONSTRUCTION QUALITY (ATC 2011) TABLE 9 UNCERTAINTY DUE TO MODEL QUALITY AND COMPLETENESS (ATC 2011) For typical well-defined new construction of ordinary structures, the building definition and construction quality would fall into the first or second category with a βc value of 0.10â0.25, as the drawings are complete (or nearly com- plete) and the construction is monitored and of reasonable to high quality. However, the modeling uncertainty would be relatively high if a structural analysis is conducted with non- linear models of similar form and refinement to the ASCE 41 envelope curves. These curves have not been explicitly cali- brated to large-scale laboratory experiments. This results in a βq value of 0.40 if significant inelastic action is expected. This value may be reduced if the structure behaves nearly elastically or with low levels of ductility. Higher modeling sophistication is currently reserved for signature or critical structures and academic studies. Finally, the ground motion record-to-record variability is combined with the modeling uncertainty to generate the dispersion of the predicted structural response. This combination is again done using the SRSS method. At this point, the median structural response has been defined by the nonlinear response history structural analysis and the dispersion is quantified by the combination of the tabulated β factors. As seen in Tables 7, 8, and 9, there is no clear trend as to the dominant source of uncertainty for all applications, as the dispersion varies depending on structural characteristics (stiffness and strength), the desired output of the assessment (drift or acceleration), the building definition, the construc- tion quality control, and the modeling assumptions/calibra- tion used. Even with the ATC-58 simplifications, the uncertainty within seismic demand prediction still relies on complex, time-consuming, nonlinear dynamic analyses, which may prevent the complete methodology from being applied to ordinary or nonessential bridges. Continued emphasis should therefore be placed on the research and develop- ment of simple, robust, and accurate analytical procedures. It would appear that for ordinary and nonessential bridges, fully probabilistic analytical methods are simply too time and cost prohibitive. Even with the increase in computa- tional power and efficiency, detailed nonlinear dynamic analyses may not become the norm for ordinary structures, in which case simplified analytical techniques should be further refined and strengthened. Steps in this direction have recently been made with documents such as FEMA 440 (2005). Uncertainty can be classified with dispersion factors for the median response predicted by simplified methods that are adopted into next-generation performance- based codes and guidelines. However, there is still a long way to go before such methodologies can be implemented in practice, and this is just one link of a chain of calculations that must be completed for PBSD to be implemented in a probabilistic fashion.
