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NONLINEAR SEISMIC ANALYSIS REPORT

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Project Description

The project structure is a seven story, reinforced concrete moment frame. The lateral loadresisting system consists of two parallel frames in the north-south direction and four parallelframes, two of which consist of a single bay, in the east-west direction. In addition to themoment frame, gravity frames are distributed throughout the structure and run primarily in thenorth-south direction with several transfer frames in the east-west direction.

Analysis Basis

Governing Documents: The design is governed by the 2001 California Building Code (CBC).However, the code provides little guidance for nonlinear analysis. Therefore, extensive use ismade of FEMA-356, Prestandard and Commentary for Seismic Rehabilitation of Buildings.FEMA-356 provides guidelines for selecting component properties, including nonlinear force-deformation characteristics and acceptance criteria. In addition, a nonlinear static analysisprocedure is detailed in the FEMA document, along with details of the nonlinear dynamicanalysis procedure specified by the CBC.

Analysis Procedures: The analysis was performed using two different procedures, the nonlinearstatic procedure (NSP) and the nonlinear dynamic procedure (NDP). The NSP is a pushoveranalysis wherein the basic lateral load-deformation curve is determined from the modelconsidering the nonlinear behavior, including yielding, cracking, strength loss (if any), and P-∆effects. A target displacement is then calculated based on the ground motion spectra for the site.This displacement is meant to represent the maximum displacement that will be experienced bythe structure for the design basis earthquake. The acceptance criteria are compared to thestructure response at the target displacement level. If the structure response quantities (memberforces, nonlinear deformations, drifts, etc.) are below the acceptable values then the structure isconsidered to have adequate lateral capacity. Earthquake motion in two orthogonal directions,including combined motions, must be analyzed.

The second procedure (NDP) involves running full nonlinear dynamic analyses of the structurefor at least three ground motions. Again, bi-direction motion must be considered. Themaximum value of each response quantity from all of the analyses is determined and comparedto the acceptance criteria.

Model Details

The parking structure is modeled as a collection of beams and columns. The floor slabs areassumed to be rigid, as is the foundation. This section outlines the choices made in modeling ofthe structure.

Component Modeling: The beams and columns are modeled using the chord-rotation modeloutlined in FEMA-356. This model assumes a plastic hinge can form at each end of the elementand that there is an inflection point at midspan. The hinges can form due to pure moment(beams) or due to the interaction of axial force and biaxial bending (columns). Although a

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somewhat simplified representation of the beam or column behavior, the FEMA-type model isgenerally sufficiently accurate for most structures and loading conditions, and has the greatadvantage of having recommendations for the strength, stiffness, and failure properties. Therecommendations require that for concrete frames the panel zones are assumed rigid.

Component Properties: The component properties are based on recommendations in FEMA-356. The moment resisting frame beams and columns, gravity frame beams and columns, andtransfer frame girders and columns are explicitly modeled as nonlinear elements. Eachcomponent is modeled using the chord-rotation model outlined in FEMA-356, with stiffness andstrength properties based on the material properties and section geometry. Section strengths aretaken equal to the ACI-318 specified values and section stiffness properties are taken as outlinedin Table 1.

Table 1. Effective Stiffness Values (from FEMA-356).

Component Flexural Rigidity Shear Rigidity Axial RigidityBeams gc IE5.0 wc AE4.0 -Columns, cg fAP ′> 5.0 gc IE7.0 wc AE4.0 gc AEColumns, cg fAP ′< 3.0 gc IE5.0 wc AE4.0 ss AE

The nonlinear behavior is assumed to be ductile. The basic force-deformation curve is shown inFigure 1. Since no element is allowed to deform beyond the ductile limit and still meet theacceptance criteria, no strength loss is modeled and all moment-rotation relationships areassumed to be elastic-perfectly plastic. A summary of the component properties is given inAppendix A.

Figure 1. Generalized Force-Deformation Relationship for Concrete Components.

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Strength Sections: Axial compression in columns is controlled by strength rather than ductility.In order to monitor the axial loads on the columns and to flag compression in excess ofallowable, axial strength sections were added to the column definitions. There is only onedamage level for strength-controlled components.

Structure Sections: The story shear can be obtained using structure sections. All of the columnsat a story are included in the section, and the shear force at the base of each column is summed togive the total story shear. Structure sections were defined for shear in each direction at everystory.

Mass and Gravity Loads: The mass and dead load were obtained from the RAMFrame modelsupplied by the Structural Engineer of Record. The loads included the self weight of themembers plus the additional dead load due to the members that were not modeled. Bothdistributed loads on the elements and concentrated loads at the nodes were used to completelymodel the dead load. All mass was assumed to be lumped at the center of mass, offset by codedefined distances to account for accidental torsion. No live load information was provided andhence live loads were not included for this analysis, but typically 25% of the live load is appliedprior to performing the lateral load analyses.

Seismic Loads: The project was placed on hold at the point where seismic analyses were to beperformed. Therefore, site-specific earthquake loads were not obtained. However, in order todemonstrate how the analysis would be completed, seismic loads were assumed using aprocedure similar to that required by the code and guidelines. Two types of seismic loads arerequired - spectral and time history.

A design response spectrum is required for the Target Displacement method described in FEMA-356 and a general spectrum for seismic loads is specified in the CBC and will be used for theNSP. Values for Ca and Cv, acceleration and velocity seismic coefficients respectively, arerequired. Assuming Seismic Zone 4, Soil Profile Type SD, and Seismic Source Type A located 5km from the project site we obtain

024.1528.0

==

v

a

CC

.

The resulting design response spectrum is shown in Figure 2.

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Figure 2. Design Response Spectrum.

In addition, static pushover load patterns must be defined in order to calculate the lateral load-drift relationship. Several load patterns are required by code including a uniform (proportional tomass) vertical distribution and a distribution more closely approximating the first mode shape(essentially triangular). Only the uniform load distribution was used for this analysis.

Earthquake time histories are required for the NDP. Normally, these would be obtained from theGeotechnical Engineer and would be either generated specifically for the project site or derivedfrom existing earthquake records by scaling both the acceleration and time axes to match thedesign response spectrum. For this sample calculation, the north-south and east-west recordsfrom the El Centro earthquake were chosen with east-west record scaled to 30% of the originalaccelerations.

Acceptance Criteria:

Three levels of earthquake protection are outlined in the FEMA guidelines, immediateoccupancy (IO), life safety (LS), and collapse prevention (CP). All three levels are included inthis analysis model, but it is likely that the LS level would be required for the project. Theacceptance criteria for all beam and column components are based on the plastic rotation at theends of the members. The allowable rotations are based on the transverse reinforcement andlevel of shear in both the beams and columns. In addition, the beam allowable rotations considerthe amount of longitudinal reinforcement and the column allowable rotation is dependent uponthe axial load. A summary of the allowable end rotations, assuming all elements are primary forgravity loads, is given in Table 2.

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Table 2. Acceptance Criteria per FEMA-356.

Plastic Hinge Rotation (radians)Performance Level

Component IO LS CPMoment Resisting Frame Beams .010 .020 .025Gravity Frame Beams .010 .020 .025Transfer Frame Girders .010 .020 .025Moment Resisting Frame Columns, Levels Ground-5 .005 .012 .016Moment Resisting Frame Columns, Levels 6-Roof .005 .015 .020Gravity Frame Columns, Levels Ground-4 .005 .012 .016Gravity Frame Columns, Levels 5-Roof .005 .015 .020Transfer Frame Columns, Levels Ground-5 .005 .012 .016Transfer Frame Columns, Levels 6-Roof .005 .015 .020

Limit States: Defining limit states can mean the difference between an analysis whose resultsare easy to interpret and having just a series of numbers that must be further investigated. Assuch, multiple limit states were defined to allow quick identification of the critical elements andthese states were grouped together to give an immediate overview of the response in relation tothe acceptance criteria. Of particular importance in this analysis were the life safety level limitstates.

Deformation-based limit states were defined for the immediate occupancy, life safety, andcollapse prevention damage levels for the moment resisting beams and columns, gravity beamsand columns, and transfer girders and columns. Strength-based limit states were defined for eachof the column types, giving a total of 21 basic limit states. Since the life safety level limit statesare of primary importance they were grouped together for easy reference.

Analysis Results

The analysis results for both the nonlinear static and dynamic procedures are presented in thissection.

Nonlinear Static Procedure: The Target Displacement method is used to evaluate the push-overanalysis results. In this procedure a “target displacement”, meant to approximate the maximumdisplacement expected during an actual earthquake, is calculated using the site response spectraand some information about the structure. In this case Type 2 framing was assumed (betterstructural performance – appropriate for moment frames) along with a life safety performancelevel. The actual target displacement calculation is an iterative process. A preliminary targetdisplacement is chosen and a bilinear approximation of the push-over curve is generated. Basedon the approximate curve the target displacement is calculated. If the calculated and preliminarydisplacements are not equal a new bilinear curve is generated and the process continues.

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The area above and below the approximate curve should be equal and the bilinear curve shouldintersect the actual curve at a strength equal to 60% of the effective yield strength, as defined bythe break in the bilinear curve. In practice it is often impossible to meet both of these criteria andconsiderable judgment must be applied. Figure 3 illustrates one possible solution wherein theareas are approximately equal but the strength at the intersection point is 80% of the effectiveyield. Similarly, Figure 4 shows a solution where the 60% strength guideline is met, but theareas above and below the curve are not equal.

Figure 3. Target displacement plot based on approximately equal areas.

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Figure 4. Target displacement plot based on matching the initial secant stiffness at 60% of theeffective yield strength.

In both of the target displacement plots the structure has failed to meet the acceptance criteria.The maximum expected displacement is larger than the displacements at which the limit statesare met as indicated by the vertical red lines on the pushover curves. All of the limit states areexceeded in Figure 3 while all limit states except collapse prevention for the beams are exceededin Figure 4.

Nonlinear Dynamic Procedure: A large number of response quantities are calculated at eachstep for all of the elements in the model. Making use of limit states allows us to easily interpretthe results relative to the acceptance criteria using only a few basic plots. The Usage Ratio plotshows the fraction of the allowable value of each limit state that is obtained at each step of theanalysis. Any usage ratio that exceeds 1.0 has failed to meet the criteria. This plot presents asimple pass/fail representation of the analysis results and lets the user determine which limitstates are of concern. In order to obtain detailed information about the specific elements thathave failed the displaced shape plot is used. This plot shows, on the deflected shape of thestructure, exactly which elements have exceeded the limit states that have been chosen fordisplay.

The usage ratio plots showing the ratios for all limit states, only life safety level limit states, andonly strength-based limit states are shown in Figures 5 through 7 respectively. The resultsindicate that the immediate occupancy limit state are greatly exceeded (usage ratioapproximately 2.5 for the moment resisting columns), the life safety limit state is just exceeded(1.04 for the moment resisting columns), and the strength limit state is not exceeded.

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Figure 5. Usage ratio plot for nonlinear dynamic procedure showing all limit states.

Figure 6. Usage ratio plot for nonlinear dynamic analysis showing only life safety-level limitstates.

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Figure 7. Usage ratio plot for nonlinear dynamic analysis showing only strength-based limitstates.

While the usage ratios give a quick overall snapshot of the structure performance they do notindicate if the damage is widespread or localized. Figure 8 shows the maximum usage ratio ineach element for all limit states. The usage ratio is color coded with red indicating a valuegreater than 1.0. Although the maximum usage ratio is large, only four elements, all at the southend of the structure, have exceeded any limit state. A handful of other elements have reachedbetween 70 and 100% of the allowable deformations.

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Figure 8. Deflected shape plot at end of nonlinear dynamic analysis showing the element-by-element maximum usage ratio for all limit states.

Figure 9. Deflected shape plot at end of nonlinear dynamic analysis showing the element-by-element maximum usage ratio for the life safety limit state.

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The situation is similar for the life safety level limit state as shown in Figure 9. In this case onlya single element has exceeded the allowable end rotation with one other element above 70%utilization of the capacity. It is likely that, with only minor revision, the structure would beacceptable for the applied earthquake load.

Conclusions

The structure was modeled for nonlinear seismic analysis according to the California BuildingCode requirements and guidelines from FEMA-356. All significant nonlinear modes of behaviorwere modeled using appropriate elements and member properties as required. Limit states forimmediate occupancy, life safety, and collapse prevention damage levels were chosen inaccordance with FEMA guidelines.

The results presented in this report are those that are most useful for determining the adequacy ofa design. Much additional information is available in RAM Perform including time histories ofdisplacements and element forces and hysteresis loops. However, while these plots are ofinterest to researchers and can help provide insight into the actual behavior, their usefulness indesign is limited.

Although the project was halted before final results could be obtained, seismic loadscorresponding roughly to those expected at the site were generated and analysis results wereproduced. The nonlinear static procedure indicated that the structure was not adequate for theseismic loads. The target displacement method used in the NSP does not allow for determiningthe extent of damage that exceeds the acceptance criteria, but merely the presence of at least onemember that has not met the requirements. The nonlinear dynamic procedure also indicated thatthe structure was not acceptable, but further examination of the results showed that only a singlemember failed to meet the life safety level acceptance criteria and that minor revisions to thestructure would allow it to pass the code requirements for the applied load.

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Appendix A – Component Properties

Beam Properties

Component Ag (in2) Imajor (in4) Ec (ksi) My (k-in)Grid 6 – Ground Level 1152 110,590 3281 ±13,572Grid 6 – Level 2 1152 110,590 3281 ±20,340Grid 6 – Level 3 1152 110,590 3281 ±18,336Grid 6 – Level 4 1152 110,590 3281 ±15,168Grid 6 – Level 5 1152 110,590 3281 ±12,216Grid 6 – Level 6 and Roof 1152 110,590 3281 ±8772Grid 7, 15, 16 – Ground Level 1274 127,460 3605 ±12,240Grid 7, 15, 16 –Level 2 1274 127,460 3605 ±18,240Grid 7, 15, 16 –Level 3 1274 127,460 3605 ±16,560Grid 7, 15, 16 –Level 4 1274 127,460 3605 ±13,920Grid 7, 15, 16 –Level 5 1274 127,460 3605 ±11,280Grid 7, 15, 16 –Level 6 and Roof 1274 127,460 3605 ±8160Grid E, H – Ground Level 980 98,040 3605 ±9240Grid E, H –Level 2 980 98,040 3605 ±14,280Grid E, H –Level 3 980 98,040 3605 ±12,600Grid E, H –Level 4 980 98,040 3605 ±10,080Grid E, H –Levels 5, 6, and Roof 980 98,040 3605 ±8160

Gravity Beams 525 26,797 3281 +3948-13,872

Transfer Girders 840 42,875 3281 +11,592-14,892

Column Stiffness Properties

Component Ag (in2) Imajor (in4) Iminor (in4) Ec (ksi)Grid 6, 7, 15, 16 1260 92,610 47,250 3605Grid E, H 864 46,656 20,736 3605Gravity 576 13,824 13,824 4031Transfer 720 27,000 17,280 4031

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Column Strength Properties

Axial Only Balance Point Bending OnlyComponent C

(k)T

(k)P

(k)M2

(k-in)M3

(k-in)M2

(k-in)M3

(k-in)Grid 6, 7, 15, 16 – Ground Level 4536 1440 1800 16,794 23,515 26,448 37,032Grid 6, 7, 15, 16 – Levels 2 and 3 4536 1210 1800 13,861 19,398 24,840 34,764Grid 6, 7, 15, 16 – Level 4 and 5 4536 910 1800 11,771 16,461 22,680 31,716Grid 6, 7, 15, 16 – Level 6 and Roof 4536 756 1800 12,442 17,398 21,600 30,204Grid E, H –Levels Ground, 2, and 3 3110 1037 1234 9669 14504 14,808 22,212Grid E, H –Levels 4 and 5 3110 726 1234 6907 10,360 13,032 19,548Grid E, H –Levels 6, and Roof 3110 518 1234 5294 7944 11,844 17,772Gravity 2592 346 988 3525 3525 9084 9084Transfer 3240 433 1234 4409 5508 11,364 14,196