Pushover Analysis of RC Building Frames With Symmetrical Setback

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Pushover Analysis of RC Building Frames With Symmetrical Setback

1Nayer A. El-Esnawy, ²Bahaa E. H. Mahmoud, ³Ahmed G. Fouad

1

Professor of Structural Engineering and Vice-Dean, Faculty of Engineering and Technology, Badr University in Cairo (BUC), Egypt (on leave from Faculty of

Engineering, Cairo University, Egypt)

2

Associate Professor, Structural Engineering Department, Faculty of Engineering, Cairo University, Egypt

3

Structural Engineer, Egyptian Projects Operation and Maintenance (EPROM), Egypt

.

Abstract

Due to architectural needs, the geometrical irregularities in buildings, either in plan or elevation, have become among the common challenges that face structural engineers nowadays. Irregularities in elevation are usually associated by the presence of setbacks in the building. In this paper, the seismic responses of several R/C multi-storey moment-resisting frame buildings with single symmetrical setback are examined and evaluated via pushover analysis, which is a non-linear static analysis technique used to estimate the seismic capacity of buildings for design purposes. Four pushover analysis methods are applied for different vertical configurations of setbacks: the mass proportional uniform pushover analysis, the mass proportional triangular pushover analysis, the method of modal combinations, and the improved upper bound method. Also, the nonlinear time-history analysis using seven scaled earthquakes is applied to determine the mean values and standard deviations for the seismic responses.

The results of the inter-story drifts for damage and the base shear for strength determined by the pushover analysis methods are compared with the mean results determined by the nonlinear time-history analysis. This comparative study with the nonlinear time-history analysis shows that the improved upper bound pushover method is the most suitable to estimate damage based on inter- story drifts for R/C multi-storey moment-resisting frame buildings with setbacks, whereas the mass proportional uniform pushover method is better for estimating the base shear capacities of buildings.

Keywords: pushover analysis; symmetrical setback; RC frames; seismic damage; nonlinear time-history analysis; inter-story drifts; improved upper bound pushover method; modal combinations pushover method

1. Introduction

Modern architecture designs involve geometrical irregularities in buildings due to aesthetics and/or functionality purposes. These irregularities in elevation and/or plan can alter the distribution of building mass, stiffness, and strength along the total height. The common reason for irregularities in elevation is the presence of setbacks. Thus, the seismic responses of the building are affected by the irregularities [1-3].

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The seismic responses and damage of RC building frames with irregularities in elevation can be determined by the nonlinear time-history analysis [1]. Yet, this analysis method is not practical for everyday earthquake-resistant design because of its complexity, time, and effort. A recent practical approach for estimating the seismic responses and damage in RC frames with irregularities is the pushover analysis [2-5]. The pushover analysis is a nonlinear static analysis that has been applied to estimate the seismic response for regular RC and steel frames [6-10]. In the pushover, the frame is subjected to lateral load patterns along the total height, which represent the inertia forces induced by the design earthquake at each floor level. Accordingly, the frame is pushed laterally in an incremental manner using the selected lateral load patterns, starting from the initial state under the action of the seismic gravity loads till the frame yields and then collapses. Thus, the nonlinear relation between the total base shear and the top total drift of the frame is plotted to provide the global capacity curve of the frame. Due to its practicality, the standard pushover analysis for regular frames is recommended by modern seismic design codes such as Eurocode 8 [11] and FEMA-356 [12].

Many seismic design codes allow buildings with setbacks to be analyzed using methods of regular buildings if the setback configuration and size satisfy a prescribed criteria. The criteria in the Egyptian code of Practice for loads and forces ECP-201 [13] is similar to the criteria of Eurocode 8 [11]. Accordingly, buildings with symmetrical single setback can be considered regular in elevation if they satisfy the following criteria, see figure 1:

 Criterion “c” states that if the base height of the building, the part below the setback, is less than 15% of the total height of the building, then the building may be considered regular in elevation on condition that the total size of the setback does not exceed 50% of the total size of the base and the base shear at the level of the setback connection with the frame base is larger than 75% of the total base shear of the tower only at the foundation level.

 Criterion “b” states that if the base height of the building exceeds 15% of the total height of the building, then the building may be considered regular in elevation on condition that the total size of the setback does not exceed 20% of the total size of the base.

Figure 1: ECP-201 ccriteria for vertical regularity of RC buildings with setbacks.

In this paper, the seismic responses of several RC moment-resisting frames with symmetrical setbacks of different sizes are determined by the nonlinear

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time-history analysis using seven earthquakes. The mean values and the standard deviations of the base shear for strength and the inter-story drifts for damage are evaluated. Then, four pushover analysis methods are applied to estimate the seismic responses, and a comparison with the mean values of the nonlinear time- history analysis is presented.

2. RC Moment-Resisting Frames with Symmetrical Setbacks

Several RC moment-resisting frames (MRF) with single symmetrical setbacks of different sizes are studied. Each frame consists of 16 stories with typical story height of h = 3 m. They are designed according to the Egyptian code of Practice for RC structures ECP-203 [14], considering the following grades: 30 MPa for concrete and 360 MPa for steel reinforcing bars. The plan of ground floor has dimensions of 40-by-16 m, as shown in figure 2. All columns have constant size along the total height of the frame and are considered fixed to the foundation. The concrete dimensions and details of the reinforcing bars of the beams and columns are listed in Table 1.

Figure 2. Plan of the ground floor for RC MRF.

Table 1. Concrete dimensions and details of reinforcing bars of RC MRF

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Earthquake-resistant design specifications of Egyptian code of Practice for loads and forces ECP-201 are considered for the third seismic zone, which is characterized by ground acceleration of 0.15 g. The design earthquake for the RC MRF is defined using response spectrum Type (1), soil factor S = 1.35 of class B, and importance factor I = 1 for normal-risk occupancy. Hence, the design peak ground acceleration is about 0.20 g. The seismic weight (Ws) of the RC MRF includes 25% of the live loads. The RC frames are considered to have limited ductility, where the response modification factor is R = 5.0.

A single symmetrical setback exists in each RC MRF after the second story, so the ratio between the base height and the total height of the frame (H) is 12.5% (less than 15%). Different sizes for the symmetrical setback are considered as follows: 10% (MRF-1), 20% (MRF-2), 30% (MRF-3), 45%

(MRF-4), and 60% (MRF-5) of the total length of the base, as shown in figure 4.

According to ECP-201 and Eurocode 8, RC frames MRF-1 to MRF-4 meet the regularity criterion (c). Also, a reference RC moment-resisting frame with no setbacks (MRF-0) is analyzed to illustrate the effect of the setback on the seismic responses.

All RC MRF are modeled and analyzed via SAP2000 program [15]. Beams and columns are modeled using inelastic frame elements with plastic hinges placed at the ends of each member to detect the events of damage in the RC MRF and their progression during the analysis. The inelastic force-deformation relationships of the plastic hinges follow the standards of FEMA-356, as shown in figure 3. For beams, the bending moments control the development of plastic hinges. For columns, the P-M interaction curves following ECP-203 are used to control the development of plastic hinges during various stages of the nonlinear analysis. The vibration periods of the RC MRF determined from the SAP2000 structural models are listed in Table 2, and are compared with the design value estimated by the empirical formula of ECP-201 which does not account for the presence of setbacks in regular buildings, as well as with the classical formula of 0.1N, where N is the number of stories of the regular building.

Figure 3. Moment-rotation relation of FEMA-365 used by the plastic hinges.

Table 2. Periods of vibration of RC MRF

MRF- 0

MRF- 1

MRF- 2

MRF- 3

MRF- 4

MRF- 5

Empirical Formula of ECP-

201

Classical Formula of

0.1N

1.27 1.27 1.33 1.34 1.40 1.48 1.37 1.60

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(a) MRF-0 (b) MRF-1

(c) MRF-2 (d) MRF-3

(e) MRF-4 (f) MRF-5

Figure 4. RC MRF with single symmetrical setbacks.

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3. Seismic Responses of RC MRF by Nonlinear Time-History Analysis

Seven earthquakes from the Peer Strong Ground Motion Database of Berkley are used to perform the nonlinear time-history analyses for the RC MRF with single symmetrical setbacks. The acceleration-time histories of the selected earthquakes are shown in figure 5, and the acceleration repsonse spectra for 5%

damping are shown in figure 6. The mean response spectrum of these seven earthquakes is plotted by the black solid line. These earthquakes provide an adequate range of seismic excitations with different frequency content for assessing the seismic demands via ECP-201, which has two main requirements.

The first requirement requires no collapse under the design-basis earthquake having return period of 475 years, i.e., the probability of exceedance in 50 years is 10%. The second requirement requires no/limited damage under service conditions considering that the probability of exceedance in 50 years is 50%.

(a) EQ1, Imperial Valley 1940, El-Centro Valley station

(b)EQ2, Coyote Lake 1979, Gilroy Array 2station

(c)EQ3, Landers 1992, Barstow station

(d)EQ4, Landers 1992, Yermo station (e)EQ5, Loma Prieta 1989, Gilroy station

(f)EQ6, Whittier 1987, Puente Hills station (g)EQ7, North Palm Springs 1986, Morongo station

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Figure 5.Acceleration-time histories of the selected earthquakes.

Figure 6. The elastic acceleration response spectra of the selected earthquakes.

The nonlinear time-history analyses are performed using the direct integration method of Hilber-Hughes-Taylor with α = 0 [15] and 5% damping.

All seven earthquakes are scaled to have PGA = 0.2 g, as required by the third seismic zone of ECP-201. The results for the peak base shear (Vbase) of each RC MRF due to individual earthquakes are listed in Table 3, normalized as a percentage of the seiemic weight (Ws) of the MRF. Also, the mean value and the standard deviation (SD) of the seven earthquakes are listed. Table 4 lists the results for the peak top drift () of each RC MRF due to individual earthquakes, normalized as a percentage of the total height (H) of the MRF. Also, the mean value and the standard deviation (SD) of the seven earthquakes are listed. Figure 7 shows the inter-story drift ratios (IDR) expressed as percentages of the story height at the state of peak top drift of each RC MRF. The mean values of IDR are shown by the black solid line.

Table 3. Normalized peak base shear determined by nonlinear time- history analysis (Vbase/Ws)

EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7 Mean SD MRF-0 7.3% 11.4

% 10.0% 7.2% 4.7% 10.0

% 6.0% 8.1% 2.4%

MRF-1 7.4% 11.4

% 10.0% 7.4% 4.9% 10.0

% 6.1% 8.2% 2.4%

MRF-2 7.1% 11.4

% 9.6% 7.6% 4.9% 10.0

% 6.2% 8.1% 2.3%

MRF-3 7.0% 11.3

% 9.3% 7.8% 5.0% 9.9% 6.3% 8.1% 2.2%

MRF-4 7.4% 11.3

% 8.9% 8.3% 5.2% 9.8% 6.5% 8.2% 2.1%

MRF-5 7.9% 11.1

% 8.6% 8.9% 6.3% 9.5% 7.7% 8.6% 1.5%

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Table 4. Normalized peak top drift determined by nonlinear time-history analysis (/H) EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 EQ7 Mean SD MRF-0 0.10% 0.14% 0.20% 0.11% 0.06% 0.16% 0.08% 0.12% 0.05%

MRF-1 0.10% 0.14% 0.20% 0.11% 0.06% 0.16% 0.08% 0.12% 0.05%

MRF-2 0.11% 0.13% 0.20% 0.11% 0.07% 0.07% 0.05% 0.11% 0.05%

MRF-3 0.11% 0.13% 0.20% 0.11% 0.07% 0.17% 0.07% 0.12% 0.05%

MRF-4 0.11% 0.13% 0.20% 0.11% 0.07% 0.17% 0.07% 0.12% 0.05%

MRF-5 0.06% 0.14% 0.20% 0.12% 0.02% 0.16% 0.03% 0.11% 0.07%

(a) MRF-0 (b) MRF-1

(c) MRF-2 (d) MRF-3

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(e) MRF-4 (f) MRF-5

Figure 7. Inter-story drift ratios determined by nonlinear time-history analysis.

4. Evaluation of Seismic Responses Using Pushover Analysis Methods

The seismic responses of RC MRF with symmetrical setbacks are estimated by applying the pushover analysis. The standard pushover analysis methods are based on using either a mass proportional triangular lateral load pattern (TLP) or uniform lateral load pattern (ULP). The seismic responses estimated by using these two lateral load patterns can bound the expected seismic responses during the event of design earthquake [10]. Recent pushover analysis methods have shown that other lateral load patterns can give better estimates for the seismic responses [7]. Another drawback is that the standard pushover analysis considers only the first mode of vibration for the multistory MRF [10]. Among the recent pushover analysis methods are the method of modal combinations (MMC) [8,9]

and the improved upper bound method (IUBM) [6].

The following four pushover analysis methods are applied to estimate the seismic responses of the RC MRF for design purposes: TLP, ULP, MMC, and IUBM. For the MMC pushover method, the first mode (S1) and the second mode (S2) of vibration are combined to define the lateral load patterns as follows: S1+S2 and S1–S2 [8]. The design seismic responses are given by the envelope of results estimated by these two combined load patterns. Figure 8 shows the lateral load patterns used for each pushover analysis method, where all the load patterns are normalized to have a unit maximum value.

Each RC MRF is subjected to each of the four selected lateral load patterns till collapse. The capacity curves showing the nonlinear relation between the normalized base shear (Vbase/Ws) and the normalized top drift (/H) are determined as shown in figure 9. As shown, the largest base shear for all MRF with symmetrical setbacks is provided by the method of ULP, whereas the smallest base shear is provided by the method of MMC based on adding the first two modes of vibration (S1+S2).

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4.1 Comparison of base shear results

The base shear is the main response for strength-based seismic design. For the pushover analysis, the normalized design base shear (Vbase/Ws) is estimated at two drift levels. The first level is the Mean value of peak top drifts determined by the nonlinear time-history analyses, as shown in Table 5. The second level is the Mean+SD value of peak top drifts determined by the nonlinear time-history analyses, as shown in Table 6. The estimated values of the design base shear are compared with the mean value of the peak base shear determined by the nonlinear time-history analyses. The percentages deviation between the estimated values by each of the pushover analysis methods and the mean base shear determined by the nonlinear time-history analysis are listed in Tables 5 and 6 between parentheses.

The results in Tables 5 and 6 show that the ULP provides the best estimate of the normalized design base shear (Vbase/Ws) for all the RC MRF, compared with the mean value of the nonlinear time-history analysis. The estimates of the normalized base shear (Vbase/Ws) by the other pushover analysis methods are ordered as follows: IUBM, MMC (S1–S2), TLP, and MMC (S1+S2) as the least accurate method. The deviation in base shear between the ULP and IUBM estimates and the mean value of nonlinear time-history analysis increases as the size of setback increases, where this deviation is less than 20% and 30%, respectively, for MRF-1 to MRF-4 that satisfy the ECP-201 regularity criterion in elevation since the setback size <50%. For, MRF-5 with setback size of 60%, and thus irregular in elevation, the deviation percentages increase to 30% and 45%, respectively. Besides, the deviation in base shear values estimated at the Mean+SD of the top drift decreases to less than 10% using ULP and 25% using IUBM for MRF-1 to MRF-4.

(a) MRF-0 (b) MRF-1

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(c) MRF-2 (d) MRF-3

(e) MRF-4 (f) MRF-5

Figure 8. Normalized lateral load patterns used for the pushover analysis methods.

(a) MRF-0 (b) MRF-1

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(c) MRF-2 (d) MRF-3

(e) MRF-4 (f) MRF-5

Figure 9. Capacity curves of RC MRF using four methods of pushover analysis.

Table 5. Normalized design base shear (Vbase/Ws) estimated bythe pushover analysis methods at Mean value of peak top drifts determined by

the noninear time-history analyses.

Nonlinear Time-

History Analysis Pushover Analysis Methods

Mean SD TLP ULP MMC

S1+S2

MMC

S1–S2 IUBM MRF-0 8.1% 2.4% 6.6%

(18.5%)

7.7%

(4.9%)

5.9%

(27.2%)

6.9%

(14.8%)

7.2%

(11.1%) MRF-1 8.2% 2.4% 6.3%

(23.2%)

7.8%

(4.9%)

5.5%

(32.9%)

6.9%

(15.9%)

7.2%

(12.2%) MRF-2 8.1% 2.3% 5.5%

(32.1%)

7.2%

(11.1%)

4.7%

(42.0%)

6.0%

(25.9%)

6.3%

(22.2%) MRF-3 8.1% 2.2% 5.3% 7.2% 4.6% 5.8% 6.1%

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(34.6%) (11.1%) (43.2%) (28.4%) (24.7%) MRF-4 8.2% 2.1% 5.1%

(37.8%)

6.9%

(15.9%)

4.1%

(50.0%)

5.6%

(31.7%)

6.0%

(26.8%) MRF-5 8.6% 1.5% 4.0%

(53.5%)

6.2%

(27.9%)

3.3%

(61.6%)

4.3%

(50.0%)

4.8%

(44.2%)

Table 6. Normalized design base shear (Vbase/Ws) estimated bythe pushover analysis methods at Mean+SD value of peak top drifts determined

by the noninear time-history analyses.

Nonlinear Time-

History Analysis Pushover Analysis Methods

Mean SD TLP ULP MMC

S1+S2

MMC

S1–S2 IUBM MRF-0 8.1% 2.4% 6.9%

(14.8%)

8.2%

(-1.2%)

6.3%

(22.2%)

7.2%

(11.1%)

7.7%

(4.9%) MRF-1 8.2% 2.4% 6.7%

(18.3%)

8.2%

(0.0%)

6.0%

(26.8%)

7.2%

(12.2%)

7.4%

(9.8%) MRF-2 8.1% 2.3% 6.0%

(25.9%)

8.0%

(1.2%)

5.2%

(35.8%)

6.6%

(18.5%)

6.9%

(14.8%) MRF-3 8.1% 2.2% 5.8%

(28.4%)

7.7%

(4.9%)

5.0%

(38.3%)

6.4%

(21.0%)

6.8%

(16.0%) MRF-4 8.2% 2.1% 5.4%

(34.1%)

7.5%

(8.5%)

4.6%

(43.9%)

6.0%

(26.8%)

6.3%

(23.2%) MRF-5 8.6% 1.5% 4.8%

(44.2%)

6.9%

(19.8%)

3.8%

(55.8%)

5.0%

(41.9%)

5.7%

(33.7%)

4.2 Comparison of inter-story drift results

The inter-story drifts are practical measures for the expected seismic damage in performance-based seismic design. For the pushover analysis, the inter-story drifts are estimated at the stage of the Mean value of peak top drifts determined by the nonlinear time-history analyses. Figure 10 shows the estimated inter-story drift ratios using the pushover analysis methods. For comparison purposes, the Mean inter-story drift ratios determined by the nonlinear time-history analyses are plotted using the heavy solid line. Also, the Mean ± ½SD inter-story drift ratios determined by the nonlinear time-history analyses are plotted using the light solid lines.

Figure 11 shows the percentages deviation between the estimated values by each of the pushover analysis methods and the mean inter-story drifts determined by the nonlinear time-history analysis. In this figure, (+ve) deviations indicate that the inter-story drifts are overestimated by the pushover analysis, while (–ve) deviations indicate that the inter-story drifts are underestimated. The results show that the inter-story drifts by the MMC are overestimated along the height of frame. But, the MMC estimates for the bottom stories are significantly overestimated, where the deviations have reached 60%. The ULP also significantly overestimates the inter-story drifts of the bottom stories, and it

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considerably underestimates the inter-story drifts of the upper stories. On the other hand, the IUBM and TLP produced better estimates of the inter-story drifts along the height of the frame, where values of the bottom story are overestimated by less than 30%, and values of the upper stories are underestimated by less than 20%, in general. The IUBM results are slightly better than the TLP results.

(a) MRF-0 (b) MRF-1

(c) MRF-2 (d) MRF-3

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(e) MRF-4 (f) MRF-5

Figure 10. Inter-story drift ratios determined bythe pushover analysis methods.

(a) MRF-0 (b) MRF-1

(c) MRF-2 (d) MRF-3

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(e) MRF-4 (f) MRF-5

Figure 11.Perctage deviation of inter-story drifts by pushover analysis from the mean values by the nonlinear time-history analysis.

5. Conclusions

The seismic responses of RC moment-resisting frames with single symmetrical setbacks have been evaluated by nonlinear time-history analyses and the following pushover analysis methods: the uniform pushover analysis (ULP), the triangular pushover analysis (TLP), the method of modal combinations (MMC), and the improved upper bound method (IUBM). The results of the base shear and the inter-story drifts have been compared with the mean values and the standard deviation of the peak results determined by the nonlinear time history analysis.

The study shows that the pushover analysis can produce adequate estimates for RC moment-resisting frames with vertical irregularities due the presence of setbacks, in particular if the frame satisfies the code criteria for vertical regularity. The results indicate that the ULP and IUBM can provide reasonable estimates for the base shear. On the other hand, the TLP and IUBM can provide adequate estimates for the inter-story drifts of the bottom stories. For the upper stories, the MMC can provide better estimates for the drifts that are higher than the mean values determined by the nonlinear time-history analysis.

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