SAMPLE NONLINEAR SEISMIC ANALYSIS REPORT

SAMPLE NONLINEAR SEISMIC ANALYSIS REPORT Project Description The project structure is a seven story, reinforced concrete moment frame. The lateral l...
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SAMPLE NONLINEAR SEISMIC ANALYSIS REPORT

Project Description The project structure is a seven story, reinforced concrete moment frame. The lateral load resisting system consists of two parallel frames in the north-south direction and four parallel frames, two of which consist of a single bay, in the east-west direction. In addition to the moment frame, gravity frames are distributed throughout the structure and run primarily in the north-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 is made of FEMA-356, Prestandard and Commentary for Seismic Rehabilitation of Buildings. FEMA-356 provides guidelines for selecting component properties, including nonlinear forcedeformation characteristics and acceptance criteria. In addition, a nonlinear static analysis procedure is detailed in the FEMA document, along with details of the nonlinear dynamic analysis procedure specified by the CBC. Analysis Procedures: The analysis was performed using two different procedures, the nonlinear static procedure (NSP) and the nonlinear dynamic procedure (NDP). The NSP is a pushover analysis wherein the basic lateral load-deformation curve is determined from the model considering 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 by the structure for the design basis earthquake. The acceptance criteria are compared to the structure response at the target displacement level. If the structure response quantities (member forces, nonlinear deformations, drifts, etc.) are below the acceptable values then the structure is considered 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 structure for at least three ground motions. Again, bi-direction motion must be considered. The maximum value of each response quantity from all of the analyses is determined and compared to the acceptance criteria. Model Details The parking structure is modeled as a collection of beams and columns. The floor slabs are assumed to be rigid, as is the foundation. This section outlines the choices made in modeling of the structure. Component Modeling: The beams and columns are modeled using the chord-rotation model outlined in FEMA-356. This model assumes a plastic hinge can form at each end of the element and 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 Page 1 of 13

somewhat simplified representation of the beam or column behavior, the FEMA-type model is generally sufficiently accurate for most structures and loading conditions, and has the great advantage of having recommendations for the strength, stiffness, and failure properties. The recommendations require that for concrete frames the panel zones are assumed rigid. Component Properties: The component properties are based on recommendations in FEMA356. The moment resisting frame beams and columns, gravity frame beams and columns, and transfer frame girders and columns are explicitly modeled as nonlinear elements. Each component is modeled using the chord-rotation model outlined in FEMA-356, with stiffness and strength properties based on the material properties and section geometry. Section strengths are taken equal to the ACI-318 specified values and section stiffness properties are taken as outlined in Table 1. Table 1. Effective Stiffness Values (from FEMA-356). Component Beams Columns, P > 0.5 Ag f c′

Flexural Rigidity 0.5 E c I g

Shear Rigidity 0.4 Ec Aw

0.7 E c I g

0.4 Ec Aw

Axial Rigidity E c Ag

Columns, P < 0.3 Ag f c′

0.5 E c I g

0.4 Ec Aw

E s As

The nonlinear behavior is assumed to be ductile. The basic force-deformation curve is shown in Figure 1. Since no element is allowed to deform beyond the ductile limit and still meet the acceptance criteria, no strength loss is modeled and all moment-rotation relationships are assumed to be elastic-perfectly plastic. A summary of the component properties is given in Appendix 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 of allowable, axial strength sections were added to the column definitions. There is only one damage level for strength-controlled components. Structure Sections: The story shear can be obtained using structure sections. All of the columns at a story are included in the section, and the shear force at the base of each column is summed to give the total story shear. Structure sections were defined for shear in each direction at every story. Mass and Gravity Loads: The mass and dead load were obtained from the RAMFrame model supplied by the Structural Engineer of Record. The loads included the self weight of the members plus the additional dead load due to the members that were not modeled. Both distributed loads on the elements and concentrated loads at the nodes were used to completely model the dead load. All mass was assumed to be lumped at the center of mass, offset by code defined distances to account for accidental torsion. No live load information was provided and hence live loads were not included for this analysis, but typically 25% of the live load is applied prior to performing the lateral load analyses. Seismic Loads: The project was placed on hold at the point where seismic analyses were to be performed. Therefore, site-specific earthquake loads were not obtained. However, in order to demonstrate how the analysis would be completed, seismic loads were assumed using a procedure similar to that required by the code and guidelines. Two types of seismic loads are required - spectral and time history. A design response spectrum is required for the Target Displacement method described in FEMA356 and a general spectrum for seismic loads is specified in the CBC and will be used for the NSP. Values for Ca and Cv, acceleration and velocity seismic coefficients respectively, are required. Assuming Seismic Zone 4, Soil Profile Type SD, and Seismic Source Type A located 5 km from the project site we obtain C a = 0.528 . C v = 1.024 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 loaddrift relationship. Several load patterns are required by code including a uniform (proportional to mass) 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 the Geotechnical Engineer and would be either generated specifically for the project site or derived from existing earthquake records by scaling both the acceleration and time axes to match the design response spectrum. For this sample calculation, the north-south and east-west records from the El Centro earthquake were chosen with east-west record scaled to 30% of the original accelerations. Acceptance Criteria: Three levels of earthquake protection are outlined in the FEMA guidelines, immediate occupancy (IO), life safety (LS), and collapse prevention (CP). All three levels are included in this analysis model, but it is likely that the LS level would be required for the project. The acceptance criteria for all beam and column components are based on the plastic rotation at the ends of the members. The allowable rotations are based on the transverse reinforcement and level of shear in both the beams and columns. In addition, the beam allowable rotations consider the amount of longitudinal reinforcement and the column allowable rotation is dependent upon the axial load. A summary of the allowable end rotations, assuming all elements are primary for gravity loads, is given in Table 2.

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Table 2. Acceptance Criteria per FEMA-356.

Component Moment Resisting Frame Beams Gravity Frame Beams Transfer Frame Girders Moment Resisting Frame Columns, Levels Ground-5 Moment Resisting Frame Columns, Levels 6-Roof Gravity Frame Columns, Levels Ground-4 Gravity Frame Columns, Levels 5-Roof Transfer Frame Columns, Levels Ground-5 Transfer Frame Columns, Levels 6-Roof

Plastic Hinge Rotation (radians) Performance Level IO LS CP .010 .020 .025 .010 .020 .025 .010 .020 .025 .005 .012 .016 .005 .015 .020 .005 .012 .016 .005 .015 .020 .005 .012 .016 .005 .015 .020

Limit States: Defining limit states can mean the difference between an analysis whose results are easy to interpret and having just a series of numbers that must be further investigated. As such, multiple limit states were defined to allow quick identification of the critical elements and these states were grouped together to give an immediate overview of the response in relation to the acceptance criteria. Of particular importance in this analysis were the life safety level limit states. Deformation-based limit states were defined for the immediate occupancy, life safety, and collapse prevention damage levels for the moment resisting beams and columns, gravity beams and columns, and transfer girders and columns. Strength-based limit states were defined for each of the column types, giving a total of 21 basic limit states. Since the life safety level limit states are 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 this section. Nonlinear Static Procedure: The Target Displacement method is used to evaluate the push-over analysis results. In this procedure a “target displacement”, meant to approximate the maximum displacement expected during an actual earthquake, is calculated using the site response spectra and some information about the structure. In this case Type 2 framing was assumed (better structural performance – appropriate for moment frames) along with a life safety performance level. The actual target displacement calculation is an iterative process. A preliminary target displacement is chosen and a bilinear approximation of the push-over curve is generated. Based on the approximate curve the target displacement is calculated. If the calculated and preliminary displacements 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 should intersect the actual curve at a strength equal to 60% of the effective yield strength, as defined by the break in the bilinear curve. In practice it is often impossible to meet both of these criteria and considerable judgment must be applied. Figure 3 illustrates one possible solution wherein the areas are approximately equal but the strength at the intersection point is 80% of the effective yield. Similarly, Figure 4 shows a solution where the 60% strength guideline is met, but the areas 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 the effective 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 states are met as indicated by the vertical red lines on the pushover curves. All of the limit states are exceeded in Figure 3 while all limit states except collapse prevention for the beams are exceeded in Figure 4. Nonlinear Dynamic Procedure: A large number of response quantities are calculated at each step for all of the elements in the model. Making use of limit states allows us to easily interpret the results relative to the acceptance criteria using only a few basic plots. The Usage Ratio plot shows the fraction of the allowable value of each limit state that is obtained at each step of the analysis. Any usage ratio that exceeds 1.0 has failed to meet the criteria. This plot presents a simple pass/fail representation of the analysis results and lets the user determine which limit states are of concern. In order to obtain detailed information about the specific elements that have failed the displaced shape plot is used. This plot shows, on the deflected shape of the structure, exactly which elements have exceeded the limit states that have been chosen for display. The usage ratio plots showing the ratios for all limit states, only life safety level limit states, and only strength-based limit states are shown in Figures 5 through 7 respectively. The results indicate that the immediate occupancy limit state are greatly exceeded (usage ratio approximately 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 limit states.

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Figure 7. Usage ratio plot for nonlinear dynamic analysis showing only strength-based limit states. While the usage ratios give a quick overall snapshot of the structure performance they do not indicate if the damage is widespread or localized. Figure 8 shows the maximum usage ratio in each element for all limit states. The usage ratio is color coded with red indicating a value greater than 1.0. Although the maximum usage ratio is large, only four elements, all at the south end of the structure, have exceeded any limit state. A handful of other elements have reached between 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-byelement maximum usage ratio for all limit states.

Figure 9. Deflected shape plot at end of nonlinear dynamic analysis showing the element-byelement 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 only a 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 be acceptable for the applied earthquake load. Conclusions The structure was modeled for nonlinear seismic analysis according to the California Building Code requirements and guidelines from FEMA-356. All significant nonlinear modes of behavior were modeled using appropriate elements and member properties as required. Limit states for immediate occupancy, life safety, and collapse prevention damage levels were chosen in accordance with FEMA guidelines. The results presented in this report are those that are most useful for determining the adequacy of a design. Much additional information is available in RAM Perform including time histories of displacements and element forces and hysteresis loops. However, while these plots are of interest to researchers and can help provide insight into the actual behavior, their usefulness in design is limited. Although the project was halted before final results could be obtained, seismic loads corresponding roughly to those expected at the site were generated and analysis results were produced. The nonlinear static procedure indicated that the structure was not adequate for the seismic loads. The target displacement method used in the NSP does not allow for determining the extent of damage that exceeds the acceptance criteria, but merely the presence of at least one member that has not met the requirements. The nonlinear dynamic procedure also indicated that the structure was not acceptable, but further examination of the results showed that only a single member failed to meet the life safety level acceptance criteria and that minor revisions to the structure would allow it to pass the code requirements for the applied load.

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Appendix A – Component Properties Beam Properties Ag (in2) 1152 1152 1152 1152 1152 1152 1274 1274 1274 1274 1274 1274 980 980 980 980 980

Imajor (in4) 110,590 110,590 110,590 110,590 110,590 110,590 127,460 127,460 127,460 127,460 127,460 127,460 98,040 98,040 98,040 98,040 98,040

Ec (ksi) 3281 3281 3281 3281 3281 3281 3605 3605 3605 3605 3605 3605 3605 3605 3605 3605 3605

Gravity Beams

525

26,797

3281

Transfer Girders

840

42,875

3281

Ag (in2) 1260 864 576 720

Imajor (in4) 92,610 46,656 13,824 27,000

Iminor (in4) 47,250 20,736 13,824 17,280

Component Grid 6 – Ground Level Grid 6 – Level 2 Grid 6 – Level 3 Grid 6 – Level 4 Grid 6 – Level 5 Grid 6 – Level 6 and Roof Grid 7, 15, 16 – Ground Level Grid 7, 15, 16 –Level 2 Grid 7, 15, 16 –Level 3 Grid 7, 15, 16 –Level 4 Grid 7, 15, 16 –Level 5 Grid 7, 15, 16 –Level 6 and Roof Grid E, H – Ground Level Grid E, H –Level 2 Grid E, H –Level 3 Grid E, H –Level 4 Grid E, H –Levels 5, 6, and Roof

My (k-in) ±13,572 ±20,340 ±18,336 ±15,168 ±12,216 ±8772 ±12,240 ±18,240 ±16,560 ±13,920 ±11,280 ±8160 ±9240 ±14,280 ±12,600 ±10,080 ±8160 +3948 -13,872 +11,592 -14,892

Column Stiffness Properties Component Grid 6, 7, 15, 16 Grid E, H Gravity Transfer

Ec (ksi) 3605 3605 4031 4031

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Column Strength Properties

Component Grid 6, 7, 15, 16 – Ground Level Grid 6, 7, 15, 16 – Levels 2 and 3 Grid 6, 7, 15, 16 – Level 4 and 5 Grid 6, 7, 15, 16 – Level 6 and Roof Grid E, H –Levels Ground, 2, and 3 Grid E, H –Levels 4 and 5 Grid E, H –Levels 6, and Roof Gravity Transfer

Axial Only C T (k) (k) 4536 1440 4536 1210 4536 910 4536 756 3110 1037 3110 726 3110 518 2592 346 3240 433

P (k) 1800 1800 1800 1800 1234 1234 1234 988 1234

Balance Point M2 M3 (k-in) (k-in) 16,794 23,515 13,861 19,398 11,771 16,461 12,442 17,398 9669 14504 6907 10,360 5294 7944 3525 3525 4409 5508

Bending Only M2 M3 (k-in) (k-in) 26,448 37,032 24,840 34,764 22,680 31,716 21,600 30,204 14,808 22,212 13,032 19,548 11,844 17,772 9084 9084 11,364 14,196

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