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Nonlinear analysis is widely applied in the studies of seismic response and progressive collapse of structures. A sound nonlinear analysis must consider inelastic material and geometric nonlinear behavior, damping, element type selection, acceptance criteria and properly scaled ground motions. That is, nonlinear analyses based on incorrect modeling/calibration methods will lead to unreliable results.
In this Chapter, the element-level modeling, structure-level modeling and selection of ground motions will be presented in reference to FEMA P695 (2009). The computer programs used in this study will be discussed as well. Then the PBPD frames and the code compliant baseline frames will be subjected to extensive inelastic pushover and time-history analyses with the same software (PERFORM 3D) for comparison of response and performance evaluation purposes.
A beam-column element was created in order to simulate the behavior of reinforced concrete beams and columns ot the study frames. The beam-column element was idealized using an elastic element and two zero-length lumped flexural plastic hinges at the ends of the element as shown in Figure 5-1.
Figure 5-1. Hybrid RC beam and column models in PERFORM 3D program
The monotonic backbone of basic plasticity model developed by Ibarra et al. (2005) was selected for this study as was done in the FEMA P695 (2009) study. To facilitate accurate structural modeling, 255 tests of RC columns were calibrated by Haselton et al., for this proposed RC element model. It is noted that this basic element model was implemented in PEER's open-source structural analysis and simulation software tool, OPENSEEs.
This model is capable of capturing the important modes of deterioration that lead to global sideway collapse. Using these calibration data, a full set of equations were also developed for the parameters (mean and uncertainty) of lumped plasticity element model as shown in Figure 5-2.
Figure 5-2. Monotonic moment-rotation model
Those parameters include: initial stiffness, post-yield hardening stiffness, plastic rotation capacity, post-capping rotation capacity, and cyclic energy dissipation capacity. The equations, which are applicable to any rectangular RC section element that fails in flexure or flexure-shear will be discussed in more detail in the next section. This portion of the study showed that the median plastic rotation capacity of modern RC elements is larger than reflected in documented such as FEMA 356 (2000).
Monotonic backbone plot which defines characteristic force deformation relationship of nonlinear structural element model can be initially formatted in FEMA 273/356 project (1997, 2003). However, due to its high degree of idealization and conservatism (Haselton, 2007), the backbone curves in FMMA 356 may not be accurate enough for realistice modeling of reinforced concrete beam-column components.
As mentioned earlier, the proposed equations for backbone characteristics of plasticity model as shown in Figure 5-2 were calibrated by Haselton et al (2007) based on an element model developed by Ibarra, Medina, and Krawinkler (2005, 2003), as implemented in OPENSEEs. The calibration work was based on the data from 255 reinforced concrete column tests assembled by Berry et al. (Berry et al. 2004, PEER 2006a). For ease and consistency of comparison, all the test configurations and force-deflection data were reduced to the case of an equivalent cantilever column as shown in Figure 5-3.
Figure 5-3. Converting (a) double-curvature and (b) double-ended column into equivalent column
For each test, the element model parameters (eg. plastic rotation capacity, cyclic deterioration parameters, etc.) were systematically calibrated such that the analysis results closely matched the experimental results. The mean modeling parameters and the uncertainty were also quantified and detailed discussion can be found in FEMA P695 (2009). The summary of accuracy of proposed equations used in this study is briefly presented in Table 5-1.
Table 5-1. Accuracy of proposed equations used in this study
logarithmic standard deviation
Plastic rotation capacity,
Post-capping rotation capacity,
Post-yield hardening stiffness,
Cyclic energy dissipation capacity,
The empirical equations proposed in FEMA P695 (2009) are briefly presented and discussed in the following.
The effective initial stiffness is defined by the secant stiffness at 40% of yield force since it was observed that the stiffness changes noticeably in most tests. The equation for effective stiffness is given as follows:
, where is the axial load ratio and is the column aspect ratio. It is noted that in FEMA 356, is permitted to be simply set as or while or , respectively, which is generally 2.5 times higher than that calculated by Equation 6-1. Elwood and Eberhard (2006) showed that most of this difference can be accounted for by significant bond-slip and shear deformations, which were not incorporated in FEMA 356.
Plastic rotation capacity,
The plastic rotation capacity, , is mainly affected by the axial load ratio () and confinement ratio (), while other parameters, such as concrete strength ( unit: MPa), rebar buckling coefficient ( ) and longitudinal reinforcement ratio () also have statistically significant influence. The equation for is as follows:
, where is bond-slip indicator variable and can be assumed as 1 or 0 depending on whether slip is possible or not.
It is also worth mentioning that Wight and Sozen (1975) indicated that the transverse reinforcement must be proportioned to carry the total shear required to develop the ultimate moment capacity of the column. That is, confinement ratio (), has a significant effect on the plastic rotation capacity. Table 5-2 shows the effects of on plastic rotation capacity, .
Table 5-2. Effects of on plastic rotation capacity,
Plastic rotation capacity, *
*: =30 MPa; = 0.1; =1; = 12.7; = 0.02
Post-capping rotation capacity,
In the proposed equation for post-capping response shown below, axial load ratio () and transverse steel ratio () are considered as key parameters.
Post-yield hardening stiffness,
Post-yield hardening stiffness () is defined by the ratio of the maximum moment capacity and the yield moment capacity. According to regression analysis, two major factors in determining are concrete strength ( unit: MPa). The predictive equation for is given as follows.
Cyclic energy dissipation capacity,
Cyclic energy dissipation capacity () was most closely related to both the axial load level () and the degree of confinement of the concrete core. In terms of quantifying the effect of confinement, the ratio of stirrup spacing to column depth () was found as a better predictor than transverse steel ratio () by Haselton (2007). A more detailed discussion of cyclic energy dissipation capacity is given in the next section. The simplified predictive equation of cyclic energy dissipation capacity is given in the following.
In summary, all these empirical equations developed in FEMA P695 (2009) give element modeling parameters which are based on design section parameters of RC column. Even though there are still limitations of these equations due to limited availability of test data, for fair comparison purpose, the input modeling parameters of all RC SMF, including code compliant base frames and PBPD frames, were mainly calculated by these equations.
Based on the studies of Comité Euro-International du Béton (1996), it is noted that cyclic degradation is most closely related to both the axial load level and the degree of confinement of the concrete core. That is, the cyclic energy dissipation capacity decreases with increasing axial load and decreasing confinement. As mentioned earlier, certain key parameters can be calculated based on the equations presented in FEMA P695 (2009), including cyclic energy dissipation capacity,, developed by Ibarra. By considering ratio of stirrup spacing to column depth () and axial loading ratio, Ibarra presented a good equation for cyclic energy dissipation factor,.
The hysteretic model captures four modes of cyclic deterioration: basic strength deterioration, post-cap strength deterioration, unloading stiffness deterioration, and accelerated reloading stiffness deterioration. This cyclic energy dissipation factor was also implemented in OPENSEEs by Altoontash (2005).
However, there is no exact cyclic energy dissipation factor, , available in PERFORM 3D. Only cyclic degradation energy factor, e, defined as the ratio of the area of degrading hysteretic loop to the area of elastic perfectly-plastic hysteretic loop, can be set in PERFORM 3D. Proper transformation of and e based on respective definitions was needed for this study. The factor e can be obtained from when , and backbone shapes are known as shown in Figure 5-2 and 5-4. Figure 5-5 shows a comparison of hysteretic loops obtained by this transformation procedure. The value of e in this study is found to be around 0.15 to 0.25 depending on element types.
Figure 5-4. Transformation of Î» and e
Figure 5-5. Comparison of hysteretic loops obtained by transformation procedure
It is noted that OPENSEEs was used in FEMA P695 (2009) instead of PERFORM 3D. Although OPENSEEs is considered to be more accurate to model the hysteretic characteristics as seen in Figure 5-5, the nonlinear axial-flexural interaction was not considered in the plastic hinge models used in FEMA P695 (2009). In contrast, axial-flexural interaction has been quite accurately modeled in the formulation of column elements in this study.
As mentioned in Chapter 4, a three-bay frame was selected as main archetype structure model in this study as more realistic representative capture of frame design and behavior, as shown in Figure 5-5. A three-bay model contains both interior and exterior columns. The interior and exterior columns are important for capturing the effects of strong-column weak-beam design provisions as well. Furthermore, the three-bay frame can capture the additional axial loads due to overturning, which influences both the column design and behavior.
Figure 5-5. Archetype analysis model for RC SMF
It should noted that the P-Delta effect is captured by applying the story gravity loads on a "P-Delta column" element (columns not part of the lateral force resisting frame), which is connected to the main frame by rigid links. The seismic mass for each floor is equal to corresponding story weight since every study frame is designed as space frame.
The damping ratio is set as 6.5% for RC SMF as suggested in the research by Chopra (1995) and Miranda (2005).
In this study, PERFORM 3D (CSI, 2007) was selected as the main analysis program. PERFORM 3D is a highly focused nonlinear software tool for seismic analysis and design. Complex structures and element models can be analyzed nonlinearly using a wide variety of deformation-based and strength-based limit states. Nonlinear analysis can be static and/or dynamic, and can be run on the same model in PERFORM 3D. Loads can be applied in any sequence, such as a dynamic earthquake loading followed by a static pushover. The output includes usage ratio plots, pushover diagrams, energy balance displays, as well as mode shapes, deflected shapes, and time history records of displacements and forces.
As mentioned earlier, PCA-COLUMN was used for designing the columns. Once the section size and reinforcement layouts were obtained, XTRACT (Chadwell and Imbsen, 2002) was applied to get P-M interaction diagrams as input parameters for P-M-M column lumped plastic hinge properties in PERFORM 3D as shown in Figure 5-1. The application sequence of software for simulation of column P-M-M plastic hinges is shown in Figure 5-6.
Figure 5-6. Sequence of software use for simulation of column P-M-M plastic hinges
XTRACT (Chadwell and Imbsen, 2002) was developed originally at the University of California at Berkeley by Dr. Charles Chadwell. XTRACT is a general cross section analysis software for analysis of any section shape and material subject to any force based loading as shown in Figure 5-7.
Figure 5-7. Column section input window in XTRACT
Analysis of XTRACT begins with the specification of nonlinear material models, and the cross section will be cut into fibers so that the moment curvature and axial force-moment interactions can be generated. For reinforced concrete, three typical material models must be defined: steel, unconfined concrete and confined concrete (Figure 5-8). Confined concrete mathematical models incorporate effects of increased compressive strain capacity in addition to an increased compressive strength as a function of passive confinement from transverse reinforcing steel. An axial force-moment interaction surface for a typical rectangular reinforced concrete cross section is given in Figure 5-9.
Figure 5-8. Confined concrete (left) and unconfined concrete (right) material models
Figure 5-9. Axial force-moment interaction surface
The computer program PERFORM-3D (CSI, 2007) is a highly focused nonlinear software tool for earthquake resistant design. It supports a wide variety of element types, including beams, columns, braces, shear wall, floor slabs, dampers and isolators. Nonlinear analyses can be performed in static and/or dynamic mode. Different types of loading can be applied in any sequence. PERFORM 3D output includes usage ratio plots, pushover diagrams, energy balance displays, as well as mode shapes, deflected shapes, and time history records of displacements and forces.
Figure 5-10 and 5-11 show that the backbone curve of all plastic hinge models in PERFORM 3D can be determined by parameters, such as basic force-deformation relationship, strength loss, deformation capacity and cyclic degradation, which are calculated according to the equations presented in Section 5.2.2.
Figure 5-10. Backbone curve of moment hinge of beam
Figure 5-11. Backbone curve of P-M-M moment hinge of column
As mentioned earlier, in terms of column element, there is no nonlinear axial-flexural interaction considered in the plastic hinge models in FEMA P695, while it was indeed engaged in this study by using PERFORM 3D. Certain characteristics of P-M diagram obtained from XTRACT can be well implemented in PERFORM 3D as shown in Figure 5-12. The maximum axial forces in compression and tension, moment and axial force at balance point, flexural moment without axial loading are required for column section model in PERFORM 3D.
The window interface is one of the advantages of PERFORM 3D. Occurrences, locations and sequences of plastic hinges can be easily monitored and tracked by color change of elements during static pushover or dynamic time history analyses. Animation is also available to provide better picture about how a structure behaves subjected to earthquakes. Thus, all nonlinear analyses in this study were performed by mainly using the PERFORM 3D program.
Figure 5-12. Determination of P-M interaction of column sections
Structures generally deform far beyond the elastic range while subjected to strong earthquakes. In the building codes, additional capacity after first yielding is accounted for through some factors, which are mostly judgment based. For properly accounting for the nonlinear behavior, a more sophisticated analysis is required.
Nonlinear analyses can offer greater insight into the behavior of the structure and to determine if the structures satisfy performance requirements. Two types of nonlinear analysis, static pushover and dynamic time history analyses, are the most comprehensive and common tools to be used in accordance with several guidelines.
Nonlinear static pushover analyses
Nonlinear static pushover analyses are carried out by applying increasing monotonic lateral forces and pushing the structure models to large displacements. The lateral loads which are statically applied to the model should be properly distributed as defined by the design standard. In this study, the pushover analyses were performed using a static lateral force distribution derived from the equivalent lateral force procedure in the PBPD method (Chao et al., 2007) as mentioned in Chapter 3 instead of that given by the seismic design provisions (ASCE 2005).
The lateral loads are incrementally increased and the resulting force-displacement plot for the structure is obtained. This plot can be assumed to represent the inelastic structural response to earthquake ground motions. In general, displacement control instead of force control is used to study the formation of mechanisms and structural behavior characteristics after mechanism formation.
Nonlinear dynamic time history analyses
Despite the complexity and computational effort, dynamic time history is generally deemed as the most accurate analysis method. By applying series of base acceleration records to the study structures, the response can be directly determined. Multiple, representative and properly scaled earthquake records must be used for dynamic analyses in order to ensure that the range of possible responses is properly captured instead of only single loading case considered in static pushover analysis.
It is worth noting that the most advanced nonlinear dynamic procedure in FEMA 356 (2000) requires only three to seven earthquake ground motions scaled to a single design hazard level. However, neglect of variations between selected records, the variability in the structural modeling and in the limit state criteria may cause variability in response. In FEMA P695 (2009), a more general method was proposed and used in this study as well. That is, to select 10-30 earthquake ground motions, scale the ground motions to different hazard levels, and estimate both the mean and the variability in response due to the variability between different earthquake ground motions. The detailed description of how this method accounts for the effects of uncertainties in structural design and structural modeling can be found in FEMA P695.
Site hazard and ground motions
MCE and DE Demand (ASCE/SEI 7-05)
Maximum considered earthquake (MCE) ground motion is the most severe earthquake effect considered by ASCE /SEI 7-05. The site specific MCE response spectra for ASCE 7-05 design evaluations should be determined in accordance with Chapter 21 of ASCE 7-05. MCE ground motions are generally described with the probabilistic criteria specified corresponding to the risk of a 2 percent probability of exceedance within a 50-year period, which is equivalent to a return period of 2,475 years.
On the other hand, design earthquake (DE or 2/3MCE) ground motion is defined as the earthquake ground motion that is two-thirds of the corresponding MCE ground motion. The site specific DE response spectra and the DE design acceleration parameters SDS and SD1 should be determined in accordance with Sections 21.3 and 21.4 of ASCE 7-05. The DE (2/3MCE) ground motions are adopted for practical purposes which correspond to the risk of a 10 percent probability of a 50-year period, also meaning a return period of 475 years.
Record Selection Criteria
As mentioned earlier, selection of proper ground motions is essential for reliable dynamic time history analysis. For considering variety of earthquake records, the PEER NGA database is an update and extension to the PEER Strong Motion Database and provides a larger set of records, more extensive meta-data, with some corrections made to information in the original database. It is noted that the NGA site includes only acceleration time history files so far.
In FEMA P695 (2009), by considering unique spectral shapes of some rare earthquakes which are much different from that the shape of a typical building code spectrum, a set of far field strong ground motions were selected. This typically occurs at rather extreme levels of ground motion. So this ground motion set was selected to represent these extreme motions to the extent possible.
Minimum limits on event magnitude, as well as peak ground velocity and acceleration were imposed to ensure that all records represent strong motions. The selection criteria are summarized in Table 5-3.
Table 5-3. Far field ground motion records selection criteria (FEMA P695)
Magnitude > 6.5
Distance from source to site > 10 km
Peak ground acceleration > 0.2g and peak ground velocity > 15 cm/sec
Soil shear wave velocity, in upper 30m of soil, greater than 180 m/s
(NEHRP soil types A-D; note that all selected records happened to be on C/D sites)
Limit of six records from a single seismic event
Lowest useable frequency < 0.25 Hz
Strike-slip and thrust faults
No consideration of spectral shape
No consideration of station housing
Far-Field Record Set
According to the selection criteria described in the previous section, a set of far-field ground motion records was selected by Haselton (2007); it contains 44 records composed by 22 horizontal motions in both perpendicular direction components (x and y). The pseudo acceleration elastic spectra of this ground motion set (only x-direction) is shown in Figure 5- 13.
Figure 5-13. Pseudo acceleration elastic spectrum of 22 selected ground motion records
For this study, 11 ground motions were selected from the 44 far-field ground motion records. These 11 ground motions were picked by anchoring the corresponding periods of 4, 8, 12 and 20-story RC frame (0.86,1.80, 2.14 and 2.36 second) and then the 3, 4, and 3 ground motions which representatively highest, closest, and lowest from the median curves were selected. The set of ground motions used in this study is shown in Table 5-4.
Table 5-4. Far field ground motion records used in this study
Earthquake records used in this study
To scale the records to the 2/3 MCE and MCE levels, all normalized records were multiplied by the same scale factor. The scaling factor was obtained by the ratios of 2/3 MCE and MCE pseudo acceleration elastic spectrum to the median of all 44 normalized ground motion set. The detailed calculation is given in Table 5-5. The median and mean of pseudo acceleration elastic spectrum curves of these 22 selected ground motions as well as the code design spectrum for 2/3 MCE and MCE hazard level are shown in Figure 5-14.
Table 5-5. Scaling factors of selected ground motion set
Normalized Factor (1)
Anchor MCE factor (2)
Anchor 2/3 MCE factor (3)
Scaling factor MCE (1)*(2)
Scaling factor 2/3MCE (1)*(3)
Figure 5-14. Pseudo acceleration elastic spectrum of mean and median of selected 22 ground motions as well as 2/3 MCE and MCE design spectrum
Structural modeling documentation for the study RC SMF
For this study, the code compliant frames and PBPD frames were subjected to extensive inelastic pushover and time-history analyses by using PERFORM 3D. This section provides the documentation of the modeling parameters used for the structural models of each study frame. The modeling parameters of 4, 8, 12 and 20-story code compliant and PBPD RC SMF are shown in Figure 5-15 to 5-18 respectively. The units are in US system (kip and in).
Figure 5-15. Modeling documentation of 4-story code compliant and PBPD frames
Figure 5-16 Modeling documentation of 8story code compliant and PBPD frames
Figure 5-17 Modeling documentation of 12tory code compliant and PBPD frames
Figure 5-18 Modeling documentation of 20tory code compliant and PBPD frames
Summary and conclusions
The element-level modeling, structure-level modeling and selection of ground motions were presented in reference to FEMA P695 (2009). The computer programs used in this study was discussed as well. It is noted that OPENSEEs was used in FEMA P695 (2009) instead of PERFORM 3D. Although OPENSEEs is considered to be more accurate to model the hysteretic characteristics, the nonlinear axial-flexural interaction was not considered in the plastic hinge models used in FEMA P695 (2009). In contrast, axial-flexural interaction has been quite accurately modeled in the formulation of column elements in this study. Thus, all nonlinear analyses in this study were performed by mainly using the PERFORM 3D program.