1 Introduction

The nonlinear static analysis (NSA) of structures for seismic evaluation, known also as the pushover analysis method, has been introduced as an affordable substitute for the costly but rigorous nonlinear dynamic analysis. The main components of an NSA are the pattern of increasing lateral loads and a maximum (target) displacement for the roof to attain. Introduction of NSA goes back to the studies of Gulkan and Sozen (1974). They developed an equivalent nonlinear single-degree-of-freedom (SDF) system substituting its associated multi-degree-of-freedom (MDF) multi story building. In the same line, simplified nonlinear analysis methods for MDF systems were proposed by Saiidi and Sozen (1981) and Fajfar and Fischinger (1988). However, nonlinear analysis was not endorsed by the practice of structural engineering until mid 90’s. Release of the documents ATC-40 (1996) and FEMA-273 (1997) was a milestone in the process of evolution of engineering analysis with nonlinear static/dynamic methods. A set of nonlinear procedures was also promoted by SEAOC in 1995. The conventional pushover analysis method (CPA) has its own shortcomings. For instance, Kim and D’Amor implemented a series of comparative analysis with NSA and nonlinear dynamic analysis and concluded that use of NSA for irregular or tall buildings could result in large errors (Kim and D’Amore 1999). An adaptive pushover method, with the essence of using current dynamic properties of yielding structures for determination of the lateral load pattern, was suggested by Bracci et al. (1997). They reported an increased accuracy in response prediction, compared with that of CPA. Antoniou and Pinho introduced a number of alternatives for the adaptive pushover procedure (Antoniou and Pinho 2004), based on story shears and displacements.

Afterwards, efforts were focused on developing multi-modal pushover methods to account for the higher modes and predict the response of taller buildings with acceptable accuracy. Meanwhile, work of Chopra and Goel (2002) succeeded in becoming more widely accepted (Chopra et al. 2004). In their modal pushover analysis (MPA), a pushover analysis is implemented each time for a certain mode of vibration and the total nonlinear response is calculated by combination of modal responses. Different alternatives for combination of the adaptive and modal pushover procedures have also been proposed (Gupta and Kunnath 2000; Kalkan and Kunnath 2006). Because NSA in essence has been proposed as a simple substitute for the nonlinear dynamic analysis, use of newer versions of NSA with increasing complexity might be looked upon as a contradictory move. For the same reason, some researchers have tried to introduce methods that while retain the simplicity of CPA, enjoy greater accuracy. In the mentioned methods, use of the adaptive approaches has been refrained from, but the effects of higher modes were taken into account in one way or another. The works of Sahraei and Behnamfar (2014) using the height-wise drift pattern as a basis for NSA, Behnamfar and Tavakoli based on story shears (Tavakoli 2012) and Poursha et al. (2009) introducing a consecutive modal pushover (CMP) with use of the mass participation factor for combination of modal responses can be mentioned in this regard.

This paper is an extension of the work of Sahraei and Behnamfar (2014). It aims to extend the pushover analysis to the cases of high-rise buildings while retaining its simplicity. Similar to the mentioned reference again the story drifts are taken to develop the pattern of lateral loads, but in contrast, instead of using the square-root-of-sum-of-the-squares (SRSS) procedure to combine the modal drifts, a new combination rule is proposed that retains the sign of the response quantities. Values of the relative displacements of stories are calculated simply using the elastic modal analysis and the theoretical basis of the method is developed utilizing the spectral analysis method. Different approaches are examined for deriving a combination rule and the modal participation factor. Within various alternatives, the one with more accuracy is explored in detail. Also, a procedure for correction of responses in upper stories is presented. Accuracy of the proposed method is compared with CPA, MPA and nonlinear time history analysis (NLTH), in estimation of story drift, shear and plastic hinge rotations of medium to tall buildings.

2 The extended drift pushover analysis (EDPA)

As mentioned above, the lateral load pattern at the floor levels and the target displacement roof have to be known before implementing a pushover analysis. In the method proposed in this research, new equations are presented for the lateral load pattern but the target displacement is calculated either as the average of the maximum roof displacements under different earthquakes, or as the value determined using the prescribed code-based formula. Therefore, the focus is on the pattern of the lateral load. Value of the lateral displacement of an MDF structure in the j-th mode at the i-th level, uij, is calculated using Eq. (1) (Chopra 2007):

$$u_{ij} = \varPhi_{ij} \cdot y_{j}$$
(1)

in which \(\varPhi_{ij}\) is the i-th component of the j-th column vector of the mode shapes of structure. \(y_{j}\) is the j-th modal response from Eq. (2):

$$y_{j} = \varGamma_{j} \cdot S_{dj}$$
(2)

where \({\varGamma}_{j}\) and \(S_{dj}\) are the j-th mode participation factor and spectral displacement of the j-th mode, respectively, and are determined from Eqs. 3 and 4:

$$\varGamma_{j} = \frac{{L_{j} }}{{M_{j} }}$$
(3)
$$S_{{d_{j} }} = \left( {\frac{{T_{j}^{2} }}{{4\pi^{2} }}} \right) \cdot S_{{a_{j} }}$$
(4)

In the above equations, \(L_{j}\), \(M_{j}\), \(T_{j}\), and \(S_{{a_{j} }}\) are the influence factor, modal mass, period, and spectral acceleration of the j-th mode, respectively. \(L_{j}\) and \(M_{j}\) are calculated as:

$$L_{j} = \phi_{j}^{T} \cdot \left[ M \right] \cdot r$$
(5)
$$M_{j} = \phi_{j}^{T} \cdot \left[ M \right] \cdot \phi_{j}$$
(6)

where \(\phi_{j}\) is the j-th mode shape column vector, \(\left[ M \right]\) is the mass matrix of structure, and r is the influence column vector with its components being unity for the degrees of freedom in the direction of ground motion, and null elsewhere.

Substitution of Eqs. 2–4 in Eq. 1 results for \({\text{u}}_{\text{ij}}\) in:

$$u_{ij} = \varPhi_{ij} \cdot \varGamma_{j} \cdot S_{{d_{j} }} = \varPhi_{ij} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right) \cdot S_{{a_{j} }}$$
(7)

Equation (7) can be rewritten in two different forms:

$$u_{ij} = \bar{u}_{ij} \cdot S_{{d_{j} }} , \quad \bar{u}_{ij} = \varPhi_{ij} \cdot \varGamma_{j}$$
(8)
$$u_{ij} = \bar{u}_{ij} \cdot S_{{a_{j} }} ,\quad \bar{u}_{ij} = \varPhi_{ij} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right)$$
(9)

The drift at the i-th story in the j-th mode, \({\text{D}}_{\text{ij}}\), can be calculated by deducing the lateral displacement of the roof of the i-th story from that of its floor. This results in:

$$D_{ij} = u_{ij} - u_{i - 1 j}$$
(10)
$$D_{ij} = \varPhi_{ij} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right) \cdot S_{{a_{j} }} - \varPhi_{i - 1 j} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right) \cdot S_{{a_{j} }}$$
(11)

or:

$$D_{ij} = \bar{\varPhi }_{ij} \cdot \varGamma_{j} \cdot S_{{d_{j}}}= \bar{\varPhi }_{ij} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right) \cdot S_{{a_{j}}}$$
(12)

where:

$$\bar{\varPhi }_{ij} = \varPhi_{ij} - \varPhi_{i - 1 j}$$
(13)

in which \(\bar{\Phi }_{\text{ij}}\) is rate of change of the j-th mode shape in the i-th story.

Equation (12) can be re-written in two different forms:

$$D_{ij} = \bar{D}_{ij} \cdot S_{{d_{j} }} ,\quad \bar{D}_{ij} = \bar{\varPhi }_{ij} \cdot \varGamma_{j}$$
(14)
$$D_{ij} = \bar{D}_{ij} \cdot S_{{a_{j} }} ,\quad \bar{D}_{ij} = \bar{\varPhi }_{ij} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right)$$
(15)

The values of the story drifts in each mode, \(d_{ij}\), have to be combined one way or another to account for all of the desired modes. In such a combination, non-concurrency of the maximum modal responses must be considered. A well-known classical method for combining the modal responses is computing the SRSS of the modal responses. This method lacks the ability to retain the signs of the responses. In this research keeping the signs of quantities is a prime concern. Then, the question is how to calculate the maximum story drift including: the desired number of modes, accounting for the fact that the maximum drifts do not occur at the same time, and retaining the signs of story drifts in each mode.

Two options are evaluated in this study for tackling the above problem:

  1. 1.

    Direct adding of the modal maximum drifts after modifying each one by a modal participation factor \({\upalpha}_{\text{ij}}\) corresponding to the i-th story in the j-th mode. This results in Eq. (16):

    $$d_{i} = \mathop \sum \limits_{j = 1}^{n} \alpha_{ij} .d_{ij}$$
    (16)

    in which \({\text{d}}_{\text{i}}\) is the maximum drift of story i, and n is the desired number of modes.

  1. 2.

    Direct addition of the modal maximum drifts, and then applying a correction factor to the sum. The correction factor is considered to be the sum of the individual modal participation factors. This approach is illustrated by Eq. 17:

$$d_{i} = \mathop \sum \limits_{j = 1}^{n} \alpha_{ij} \cdot \mathop \sum \limits_{j = 1}^{n} d_{ij}$$
(17)

In other words, in the first approach, the responses are first modified and then added, while in the second approach the response are added and then modified.

In addition to how the modal combination is implemented using participation factors, the participation factor itself has also to be introduced. Several options are possible. For each option, the accuracy of results can be evaluated. In this research the following alternative equations have been assessed for accuracy for the modal participation factor:

$$\alpha_{ij} = \left| {\frac{{\bar{d}_{ij} }}{{\mathop \sum \nolimits_{j = 1}^{n} \bar{d}_{ij} }}} \right|,\quad \bar{d}_{ij} = \bar{\varPhi }_{ij} \cdot \varGamma_{j}$$
(18)
$$\alpha_{ij} = \left| {\frac{{\bar{d}_{ij} }}{{\mathop \sum \nolimits_{j = 1}^{n} \bar{d}_{ij} }}} \right|, \quad \bar{d}_{ij} = \bar{\varPhi }_{ij} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right)$$
(19)
$$\alpha_{ij} = \left| {\frac{{\bar{d}_{ij} }}{{\sqrt {\mathop \sum \nolimits_{j = 1}^{n} \bar{d}_{ij}^{2} } }}} \right|,\quad \bar{d}_{ij} = \bar{\varPhi }_{ij} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right) \cdot S_{{a_{j} }}$$
(20)
$$\alpha_{j} = \left| {\frac{{\bar{u}_{Nj} }}{{\mathop \sum \nolimits_{j = 1}^{n} \bar{u}_{Nj} }}} \right|,\quad \bar{u}_{Nj} = \varPhi_{Nj} \cdot \varGamma_{j}$$
(21)
$$\alpha_{j} = \left| {\frac{{\bar{u}_{Nj} }}{{\mathop \sum \nolimits_{j = 1}^{n} \bar{u}_{Nj} }}} \right|,\quad \bar{u}_{Nj} = \varPhi_{Nj} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right)$$
(22)
$$\alpha_{j} = \left| {\frac{{u_{Nj} }}{{\sqrt {\mathop \sum \nolimits_{j = 1}^{n} u_{Nj}^{2} } }}} \right|,\quad u_{Nj} = \varPhi_{Nj} \cdot \varGamma_{j} \cdot \left( {\frac{{T^{2} }}{{4\pi^{2} }}} \right) \cdot S_{{a_{j} }}$$
(23)

Equations 18, 19 and 20 are conjoined with Eqs. 14, 15 and 13, respectively. Also, Eqs. 21, 22 and 23 are associated with Eqs. 8, 9 and 7, respectively. In Eqs. 21–23, \({\upalpha}_{\text{j}}\) is the j-th mode participation factor calculated based on the lateral displacement of the roof (N-th story) of building, where N is number of stories. It is used in Eqs. 16 and 17 instead of \({\upalpha}_{\text{ij}}\) to calculate the total i-th story drift, \({\text{d}}_{\text{i}}\). In fact, in Eqs. 18–20, drifts are used as the basis for calculation of \(\alpha_{ij}\) while in Eqs. 21–23, lateral displacement of the top story is used for calculating a unique j-th mode participation factor for all stories.

After calculation of \({\upalpha}_{\text{ij}}\) or \({\upalpha}_{\text{j}}\), the story drifts are calculated using Eq. 15 or 16. Then, the load pattern of the equivalent lateral forces, necessary for pushover analysis, is determined from Eq. 24:

$$\bar{f}_{i} = \frac{{u_{t} }}{{\mathop \sum \nolimits_{i = 1}^{N} d_{i} }}\frac{{k_{i} d_{i} - k_{i + 1} d_{i + 1} }}{{k_{1} d_{1} }}$$
(24)

In Eq. 24, \(\bar{f}_{i}\) is the load pattern at floor i (i = 1,…, N), \({\text{u}}_{\text{t}}\) is the target displacement (at the roof), and \(k_{i}\) is the interstory lateral stiffness of story i. In fact, the first fraction on the right side of Eq. 24 is meant to correct all story drifts such that their sum, i.e. the lateral displacement at the top of building, equals the known target displacement. The second fraction is simply deduction of the adjacent story shears normalized to the base shear to result in the lateral load pattern.

The EDPA analysis is implemented using Eq. 24 and the prescribed code-based target displacement. The numerical calculations of this research have been employed using four medium to tall buildings (10, 15, 20 and 30 stories) and 10 scaled earthquake records. Responses including lateral displacements, shear forces and absolute sum of plastic hinge rotations in each story have been calculated and compared with the averages of the maximum response values from nonlinear dynamic analysis. In all, about 5000 analysis cases have been implemented (totally 75 stories, 3 response quantities in each story, 10 earthquake records: 75 × 3 × 10 = 2250, 6 different formulas for the participation factor, 2 approaches for modal combination: 75 × 3 × 6 × 2 = 2700). Out of the above analysis, it was concluded that determining the participation factor using the third approach (Eq. 20) and combining the modal drifts using Eq. 17 results in the most accurate results (Taherian 2014). Because of the huge volume of the analysis, in this paper only the results of the selected method consisting of Eqs. 17 and 20 are presented.

3 The buildings studied

Since the method developed herein is meant to enhance the accuracy of the pushover analysis for taller buildings where the affect of higher modes is more important, a range of stories between 10 and 30 is selected. To cover this interval, four regular residential buildings having 10, 15, 20 and 30 stories are selected. In each building, the structural system consists of special steel moment frames having three bays in each direction. Each bay spans 5 m and the floor to floor story height is uniformly selected to be 3.20 m. The seismicity of the region is assumed to be very high. The ground consists of a medium soil, e.g. soil type C of NERHP09 (2009). The dead (weight of floors plus partitions) and live loads on the floors are 650 and 200 daN/m2, respectively, and the diaphragms are rigid.

Before implementing a nonlinear static/dynamic analysis, the structural frames are designed according to AISC-ASD (2010), with the above assumptions. Use is made of I-sections for beams and box sections for columns with the dimensions shown parametrically in Fig. 1 and with specific dimensions in Tables 1 and 2.

Fig. 1
figure 1

Cross sections of beams (a) and columns (b)

Full size image
Table 1 Types and dimensions of beams and columns
Full size table
Table 2 Beam and column types of the buildings
Full size table

Table 3 shows the periods and Fig. 2 exhibits the mode shapes of the first three modes of the designed buildings. It is to be noted that with the use of three modes, at least 90 % of the buildings weight is excited for all the buildings studied.

Table 3 The first three modal periods of the studied buildings
Full size table
Fig. 2
figure 2

The first three mode shapes of the buildings under study

Full size image

4 The earthquake records

While use of a minimum of seven earthquake records is enough for when utilizing average of values is targeted, ten strong ground motions are selected for less discrepancy of results. The PEER ground motions database (PEER 2014) is used for selection of earthquake records. The selected records have all been recorded on the soil type C with their magnitudes being in the range of 6–7.5. Table 4 shows the characteristics of the earthquakes selected.

Table 4 Characteristics of the selected records
Full size table

According to ASCE7-10, the earthquake records must be scaled before being suitable for time history analysis such that their average spectrum is not lower than the design spectrum in the period range of 0.2T–1.5T where T is the fundamental period of the building under study (see Table 3). Table 5 gives the scale factors for each building-earthquake case. Figure 3 shows the mean spectrum of the 10 records before and after scaling for each building along with the design spectrum.

Table 5 Scale factors of the earthquakes
Full size table
Fig. 3
figure 3

Mean spectra of the earthquakes before and after scaling

Full size image

5 Nonlinear modeling and analysis issues

Modeling of the structures for nonlinear static/dynamic analysis is accomplished in this study using the OpenSEES software (Mazzoni et al. 2006). The structures are modeled as plane frames. In OpenSEES, the nonlinear behavior of beams and columns of a moment frame can be incorporated either by assuming the plastic deformations all being concentrated at a certain section at the end of the member, or being distributed along the member. While the second approach is obviously more accurate, it is much more common to adopt the first approach, called the concentrated plasticity, for computational efficiency. The same practice is followed in this study. It results in introducing zero-length nonlinear \(M - \theta\) springs at both ends of an elsewhere elastic member. The properties of the \(M - \theta\) spring are calculated using the resultant moment of axial stresses of the several longitudinal fibers the section is assumed to be composed of. For each steel fiber, a normal stress-normal strain relation must be defined. Several backbone curves are currently available for the steel material in OpenSEES to select. Among the possible options, the Steel02 material has proved to be accurate and efficient enough for nonlinear analysis of plane frames (Mazzoni et al. 2006). This material type is able to follow the gradual elastic–plastic behavior, strain-hardening, the Bauschinger effect, and the isotropic/kinematic plastic behavior of steel in cyclic loading/unloading.

The Steel02 material is used in this study with its properties being as shown in Table 6.

Table 6 Properties of the material Steel02
Full size table

6 Numerical results

6.1 Introduction

In this section, results of the nonlinear static/dynamic analysis are presented. The purpose is to see how accurate is the EDPA method presented in this research in comparison to the CPA and MPA methods with regard to the accurate responses determined as the average of maximum responses corresponding to each earthquake. The relative difference of results of the EDPA, CPA and MPA methods with regard to those of the exact nonlinear time-history analysis is calculated using Eq. 25:

$${\text{R}}.{\text{M}}.{\text{S}}\left( \% \right) = 100\sqrt {\frac{1}{\text{N}}\sum\limits_{{{\text{i}} = 1}}^{\text{N}} {\left( {\frac{{{\text{X}}_{\text{iD}} - {\text{X}}_{\text{iP}} }}{{{\text{X}}_{\text{iD}} }}} \right)^{2} } }$$
(25)

in which RMS is the error percentage, \({\text{X}}_{\text{iD}}\) the dynamic response calculated at the i-th story and \(X_{iP}\) is the response at the same story due to the pushover analysis.

6.2 The analysis methods

In the CPA method, a certain pattern for distribution of the lateral loads is adopted. Often, this pattern is taken to be proportional to the fundamental mode shape of structure. The lateral loads are increased to the above pattern until the displacement of roof equals a predefined target displacement. At this point, the structural responses are established.

In the MPA procedure, a number of lateral load patterns, equal to the number of important modes, are selected. Each load pattern is consistent to the corresponding mode shape of structure. The force–displacement (capacity) curve corresponding to each pattern is drawn. Then the characteristics of the equivalent SDF system associated with each mode are calculated using the corresponding capacity curve. Then the target displacement in each mode can be determined. Then the building is pushed in each mode using the modal pattern of lateral loads up to the target displacement of the mode. The desired responses are established in each mode at the target displacement. Finally, the total responses are calculated using a conventional mode combination rule, such as SRSS.

In EDPA, first the story drifts are calculated using one of the approaches introduced as Eqs. 16–23. Then distribution of lateral forces is calculated using Eq. 24. The structure is pushed upon until the roof displacement becomes equal to the target displacement. The story responses are calculated at this displacement. In the nonlinear time history analysis (NLTH), in this study, maxima of lateral displacements and shear forces are calculated at each story under each earthquake. Then the averages of the maxima are calculated for the story displacements and shear forces separately and used as a basis of comparison for accuracy analysis of the mentioned pushover procedures. In addition, maxima of the sum of the absolute values of the plastic hinges of beams and columns of each story are processed in the same way as a measure of ductility demand or seismic damage of each story.

6.3 The target displacement

In all of the pushover procedures evaluated in this study, the lateral loads on the buildings are increased up to when the lateral displacement of the roof of each building reaches a predefined value called the target displacement. The target displacement of each building has to be identical for all pushover procedures to make the comparison with the NLTH procedure possible. A customary approach is determining the target displacement as being equal to the maximum roof displacement averaged between NLTH results under the selected earthquakes. This is not valid for two reasons. First, the pushover analysis is meant to evaluate structures without necessarily resorting to NLTH analysis. Then, the maximum roof displacements under earthquakes are not known beforehand in practical cases. Second, taking the maximum displacements from the NLTH analysis and then performing the pushover analysis up to such a displacement for comparison with the NLTH results inherently makes a prejudice as the pushover analysis will not be independent from NLTH procedure in this case. Then, in this research, a more rational approach is adopted with scaling the earthquakes based on a design spectrum and determining the target displacement using the same spectrum. For the latter calculation, the method of displacement coefficient of ASCE41-13 is followed [21]. The target displacements of the buildings under study are calculated to be 24.60, 33.28, 44.17 and 55.19 cm for 10, 15, 20 and 30-story buildings, respectively.

6.4 The analysis results

6.4.1 Modal participation factors and lateral load patterns

The modal participation factor,\({\upalpha}_{\text{ij}}\), is calculated using Eq. (20). The results are mentioned in Table 7 for each mode and collectively for each story of the buildings under study. As seen, values of \({\upalpha}_{\text{ij}}\) in the first mode are larger at the lower floors than those of the top floors. In the higher modes a reverse trend is observed. Also, \({\upalpha}_{\text{ij}}\) generally decreases with the mode number.

Table 7 Values of \({\varvec{\upalpha}}_{{{\mathbf{ij}}}}\)
Full size table

For all the buildings studied, \(\mathop \sum \nolimits_{{{\text{j}} = 1}}^{\text{n}} {\upalpha}_{\text{ij}}\) varies between 1 and 2, and its variation along height (changing i) for a certain building is smooth. The maximum variation of \(\mathop \sum \nolimits_{{{\text{j}} = 1}}^{\text{n}} {\upalpha}_{\text{ij}}\) as the relative difference between its minimum and maximum value for a specific building, is observed to be about 30 %.

The distribution of lateral forces corresponding to the utilized pushover methods is shown in Fig. 4.

Fig. 4
figure 4

Distribution of lateral loads in pushover analyses

Full size image

In Fig. 4, distribution of lateral story forces in CPA is proportional to the fundamental mode shape of each building. In MPA, the distribution is consistent with the mode shape (multiplied by distribution of structural mass) of each mode. In the figure, this distribution is given for the first three modes. In EDPA, the distribution is calculated with Eq. 24. Therefore, in EDPA contrary to MPA, a combination of modal drifts is used to determine the distribution instead of each individual mode. In the figure, number of modes is three for EDPA. According to Fig. 4, distribution of lateral forces in EDPA is similar to the first mode in the lower stories. For taller buildings, the lateral distribution in EDPA distorts in the higher stories due to larger participation of the higher modes.

6.4.2 Lateral displacements

Figure 5 shows distribution of lateral displacements of stories for the CPA, MPA and EDPA procedures along with the averaged maxima of the NLTH analysis method. The RMS’s of errors with regard to the NLTH results are mentioned in Table 8.

Fig. 5
figure 5

Lateral displacements of stories

Full size image
Table 8 RMS errors of pushover displacements with regard to those of the NLTH analysis
Full size table

According to Fig. 5, prediction of displacements is generally good in all of the pushover procedure studied. In most cases, error is larger in the upper half of a building. As observed in Table 8, while the RMS value of MPA is less than CPA in all cases, the smallest RMS in all buildings belongs to EDPA. Therefore, EDPA has been successful to predict the maximum floor displacements of the studied buildings with a better accuracy with regard to the other pushover methods. While the displacement prediction error increases among the above pushover method for the taller buildings, it remains below 17 % for EDPA and MPA but reaches 19 % for CPA.

6.4.3 Story shear forces

Figure 6 illustrates the story shears using the above analysis methods. Also, Table 9 gathers the RMS errors of the pushover methods in estimation of the story shears. The overall error in prediction of story shears in all of the above pushover methods has been increased relative to displacement predictions. At the same time, EDPA possesses the smallest shear RMS errors between the others, being at most 29 % for the 20-story building. MPA comes next with 30 % and the largest error belongs to CPA, which is 35 %. EDPA is most successful in the lower half stories. It tends to an error similar to other methods in the upper stories.

Fig. 6
figure 6

Total shear forces of stories

Full size image
Table 9 RMS errors of story shear forces relative to the NLTH responses
Full size table

6.4.4 Plastic hinge rotations

The sum of the absolute values of the plastic hinge rotation in all plastic hinges of the beams and columns of each story, called the story plastic hinge rotation, are calculated in this section. The ability of each pushover method to predict these values is quantified with its RMS error with regard to NLTH response. Figure 7 shows the story plastic hinge rotations. The RMS values of errors of the pushover methods are mentioned in Table 10.

Fig. 7
figure 7

Total plastic hinge rotations of stories

Full size image
Table 10 RMS errors of story plastic hinge rotations relative to the NLTH responses
Full size table

Figure 7 shows that generally EDPA has been successful in prediction of plastic hinge rotations with good accuracy in the lower half of structures (perhaps except of the 30-story building). The prevailing trend is that the prediction errors reach to their largest values for plastic hinge rotations compared to shear forces and displacements. The error is larger for taller buildings. Meanwhile, the largest error belongs again to CPA with 55 %; next is MPA with 48 %, and the least value belongs to EDPA with 45 %.

6.4.4.1 Enhancing accuracy of EDPA

While in prediction of the nonlinear maximum seismic responses of the tall buildings studied, EDPA has been marginally more successful than MPA, it is much easier to implement because it is accomplished only in one stage. This is in contrast to MPA that must be repeated for all of the important modes.

Number of modes necessary to be included in EDPA is simply number of modes with a total effective modal mass of at least 90 % of the seismic mass of building. For all of the studied structures the above criterion results in inclusion of only 3 modes. To evaluate the accuracy, the story hinge rotations of the 30-story building are illustrated in Fig. 8 using 1, 2, 3 and 4 modes for comparison. This figure and other figures alike for other responses and other buildings confirms the above criterion that only modes with a total relative effective mass of over 90 % are enough for the EDPA procedure.

Fig. 8
figure 8

Story hinge rotations as predicted by EDPA including different numbers of modes

Full size image

On the other hand, if distribution of RMS errors along height of each building is investigated, a correction for EDPA can be derived that results in a considerably enhanced accuracy for this method.

Table 11 shows the values of EDPA’s RMS errors for different responses of each building averaged between the lower and upper half stories separately. It shows that EDPA is much less successful in estimation of shear and hinge rotations of the upper stories than it is in the lower ones.

Table 11 Average of RMS errors of EDPA in the lower and upper half stories
Full size table

Because the average error in estimation of displacement is small it is not discussed further.

Table 12 mentions the average correction factors that if are multiplied by the corresponding responses of the upper half stories, will result in accurate average values of NLTH.

Table 12 Correction factors of average EDPA responses in the upper half stories to be resembled to those of NLTH
Full size table

The correction factors of Table 12 all happen to be placed between 1 and 2. Moreover, they are very similar to \(\sum\nolimits_{j = 1}^{n} {\alpha_{ij} }\) values of the upper half stories mentioned in Table 7. Therefore, while it is possible to derive regression equations for correction factors of responses of the upper half stories, it is easier and more consistent to assign \(\sum\nolimits_{j = 1}^{n} {\alpha_{ij} }\) as the response correction factors as follows:

$${\bar{\text{X}}}_{{\text{i}}} = \left\{ {\begin{array}{*{20}l} {{\text{X}}_{{\text{i}}} } \hfill & {{\text{if}}\; {\text{i}} < \left[ {\frac{{\text{N}}}{2}} \right]} \hfill \\ {{\text{X}}_{{\text{i}}} \cdot \mathop \sum \nolimits_{{{\text{j}} = 1}}^{{\text{N}}} \alpha _{{{\text{ij}}}} } \hfill & {{\text{if}}\; {\text{i}} \ge \left[ {\frac{{\text{N}}}{2}} \right]} \hfill \\ \end{array} } \right.$$
(26)

in which […] shows the integer part, \(X_{i}\) is the uncorrected shear or plastic hinge rotation of story i, and \(\bar{X}_{i}\) is the associated corrected value.

To evaluate the above correction procedure, the RMS errors are again calculated for the studied buildings but this time with EDPA with correction. The results are mentioned in Table 13 in comparison to those of MPA and EDPA without correction. As observed, when corrected, the accuracy of EDPA will be much superior to MPA and is located within the acceptable margin of error for design applications.

Table 13 RMS errors of EDPA with/without correction and other procedures relative to NLTH
Full size table

7 Conclusions

In this study a drift based pushover for estimation of nonlinear seismic responses of structures was presented. The theory of the method was derived using the basics of the spectral analysis method using story drifts. Spectral drifts in each mode were combined to calculate the total story drifts and the distribution of lateral forces for pushover analysis. With adhering to the condition of preserving the signs of response values, two different approaches for a combination rule and six various equations for the modal participation factor were examined. The procedure with the least errors was explored in detail. The selected procedure was superior in accuracy and easier in implementation compared with the existing well-known pushover methods. Finally, the sum of the modal participation factors in each story was used as a correction factor for maximum story shears and plastic hinge rotations. It enhanced the accuracy of the proposed method to a level much superior to other available pushover methods.