1 Introduction

Nowadays, the performance-based engineering concepts constitute the main basis for compilation of most of the seismic analysis and design codes of the day (FEMA-356 2000; FEMA-440 2005; ATC40 1996; FEMA-P695 2009; ATC-58 2009). In low performance levels, Life Safety and Collapse Prevention, and under the effect of high intensity quakes, structures enter into the nonlinear phase, so that the structural damage index is generally controlled by the non-elastic deformation capacity of the structural and non-structural elements. Hence, displacement-based nonlinear analysis would be necessary to control and design the performance-based structures. Regarding the nature of the seismic loads, categorized as base excitation and imposed as accelerations to the upper masses, the dynamic nonlinear time-history analysis, NTHA has been widely welcomed as the most accurate method of analysis for the structures. However, the method entails complexities and application problems as follows:

  1. (a)

    Earthquake record selection procedures, appropriate for each region and their scaling, for which currently extensive researches are being accomplished;

  2. (b)

    Lack of reliable commercial software and time-consuming nature of the NTHA;

  3. (c)

    Abundant data output of such analyses, requiring too much time for analyzing and post-processing the results;

  4. (d)

    Problem associated with the construction of damping matrix (Aydinoğlu and Onem 2010).

Taking notice of the above issues, different researchers around the world have tried to propose simple and applicable methods for seismic evaluation of structures based on the deformation. The outcome of such studies has been the “Nonlinear Static Procedure using the pushover concept.”

In this regard, adaptive load pattern was first used by Reinhorn (1997). In this method, the initial load distribution pattern was often triangular. The load pattern is adapted at each step based on the story’s internal resisting force. The eigen values and spectral analysis are used only as rudimentary and to provide the demand spectrum. Satyarno et al. (1998), presented a pushover method with adaptive load pattern based on the modal changes of structure in each step of loading in which the Rayleigh method is used to modify, based on incremental displacement of floors. Requena and Ayala (2000) proposed two different methods of adaptive pushover analysis called 2-A and 2-B method. According to the method 2-A, the imposed forces in different floor levels in each step of loading is gained from a combination of modal forces using the square root of sum of squares (SRSS) method. On the other hand, the presence of an equivalent fundamental mode shape has been accepted in the 2-B method. This equivalent mode is a combination of the mode shapes using the SRSS method in each step of loading.

A method which was conceptually similar to the 2-A method was presented by Elnashai (2001), with the difference that in his new method, the fiber model analysis was used instead of using discrete model; also load pattern is updated continually. Albanesi et al. (2002), proposed an adaptive energy-based pushover analysis method (AEPOA) in which the natural attributes of the structure, such as the kinetic energy resulting from the seismic effect, are considered in the adaptive load pattern.

Antoniou and Pinho (2004) proposed a force-based adaptive pushover (FAP) analysis method. The proposed method was multi-modal, and the decrease in the structural stiffness, period elongation and modification of inertial forces based on the spectral values are considered. The load vector in each step of the analysis is determined from the structure’s stiffness at the end of the previous step. Then the lateral forces resulting from each vibration mode are combined using the SRSS method or complete quadratic combination method (CQC), depending on the interdependency of the modes. The load vector can be updated in each step of the analysis, using any of the two, total or incremental adaptive methods.

Shakeri et al. (2010) presented an advanced story shear-based adaptive pushover (SSAP) analysis method in which the following three key concepts have been used: (A) adapting the employed load pattern based on the modal story shear in each step. (B) Exploiting the assumed fundamental mode shape of the structure from the load profile employed during loading. (C) Converting the pushover curve coordinates of MDOF structure into the coordinates of the equivalent SDOF system capacity curve, using assumed fundamental mode shape of the structure and the energy concept (Hernandez-Montes et al. 2004). This method attempts to consider the higher-mode effects and the structural stiffness changes during the analysis, as well as interaction between the modes. The method exploits the target displacement utilizing the energy concept. In this line, Shakeri et al. (2012) suggested the story-shear-and-torque-based adaptive for asymmetric-plan buildings. In this method, the single modal load pattern with regard to effects of the higher modes and the interaction between the modes was considered. Abbasnia et al. (2013) proposed a displacement-based adaptive modal pushover procedure based on effective modal mass combination rule for considering the sign reversals in the load vectors. This combination rule was defined in order to obtain a modification factor associated to each mode and this modified factor applied to the corresponding load vector. Finally, these modified load vectors are algebraically subtracted and added and result in a range of load pattern.

Also some other researchers used the adaptive load application as displacement instead of force (Kalkan and Kunnath 2006; Ferracuti et al. 2009; Pinho et al. 2006).

2 Producing an Adaptive Load Pattern

Single-run modal pushover analysis is a method in which the modal shape vectors are combined using modal combination rules and are applied as force vector to the structure through a pushover process, and then, the results are extracted in a target displacement. In fact, such methods use modal combination rules to determine the load pattern for considering the higher-mode effects.

Also, some single-run methods utilize variable load pattern to consider the effects resulted from structural nonlinearity (yielding) during the analysis.

In the SSAP, in each analysis step, based on the instantaneous dynamic parameters, the story shear associated with each mode is calculated by Eqs. (1) and (2) (Fig. 1a, b). The story shears associated with each of the modes considered, are combined using the SRSS rule (Eq. (3), Fig. 1c). In calculation of the story shears for each mode, the sign reversal of the modal forces is also considered.

Fig. 1
figure 1

The process of defining the applied load pattern at one step of the proposed SSAP and FAP procedures. a Modal story forces, b story shears profiles for each mode, c combined modal story shears profiles, d load pattern in the SSAP method, d′ load pattern in the FAP method

Full size image
$$F_{ij} = \varGamma_{j} \emptyset_{ij} m_{i} Sa_{j}$$
(1)
$${\text{SS}}_{ij} = \sum\limits_{k = i}^{n} {F_{kj} }$$
(2)
$${\text{SS}}_{i} = \sqrt {\sum\limits_{j = 1}^{m} {{\text{SS}}_{ij}^{2} } }$$
(3)

where i is the story number, j—the mode number, Øij—the ith component of the jth eigenvector (mode shape), mi—the mass of the ith story, Saj—the spectral acceleration corresponding to the jth mode, Γj—modal participation factor for the jth mode, \({\text{SS}}_{ij}\)—the story shear in level i associated with mode j, \({\text{SS}}_{i}\)—the modal story shear in level i associated with all the considered modes.

2.1 Different Methods for Producing Adaptive Load Pattern

There are two main procedures for adapting load patterns in each step of loading, namely total adaptive method and incremental adaptive method. In the former method, after each step of loading, modal forces are extracted based on the current stiffness of structure, and then, modal forces are combined and the load pattern is provided. Finally, previous load pattern will totally be replaced by the provided new load pattern (Fig. 2).

Fig. 2
figure 2

Process for defining total adaptive load pattern

Full size image

where λt is the base shear at step t, Pt is lateral load distribution at step t in total adaptive load pattern. The total adaptive method is more conceptual, because the force is distributed based on the structural stiffness; in spite of that, usually there would be some convergence problem in the analysis due to sudden changes in load pattern in the inelastic phase and also sometimes there would be reversal displacement of roof story because of the variable load distribution (Fig. 3).

Fig. 3
figure 3

Lateral load pattern in different stage of total updating load pattern

Full size image

Hence, the incremental adaptive load patterns, which are conceptually a better procedure for displacement-based methods, were emerged. In this method, the normalized lateral load pattern is amplified by incremental base shear and so provide incremental lateral load for each floor. Then at this step incremental load will be added to the previous lateral load pattern and after normalizing, it would be implemented as the new load pattern. So there would not be a big sudden change in incremental adaptive load patterns (Antoniou and Pinho 2004, Fig. 4).

Fig. 4
figure 4

Process for defining incremental adaptive load pattern

Full size image

3 Validation of the Adaptive Modal Pushover Procedure

In this study, in order to evaluate the efficiency of adaptive load patterns, the equivalent non-adaptive load patterns of SSAP and FAP methods, namely story shear-based pushover, “SSP”, and force-based pushover, “FP”, are introduced.

FAP method utilizes the maximum roof displacement resulted from NTHA, as the target displacement and does not calculate the target displacement by itself. Because the main concern of this study is to evaluate the efficiency of adaptive load patterns, assumed fundamental mode shape of the structure was calculated instantaneously based on the load profile. Then energy concept was implemented to convert pushover curve of multi degree of freedom system “MDOF”, into capacity curve of single degree of freedom system “SDOF”. In this stage, first the SDOF capacity curve is idealized to a bilinear graph, and then for each ground motion record, a NTHA was carried out to achieve the target displacement. Incremental adaptive load pattern has been used in both SSAP and FAP methods in this study. For more information about the process of SSAP and FAP methods, refer to the original papers.

3.1 Structural Model

In this paper, three steel moment-resistant frame (SMRF) of the Phase II of SAC group buildings has been observed (FEMA-355 2000). Accordingly, SAC-3, a 3 story building as low-rise structure, SAC-9, a ten-story construction as the representative of mid-rise constructions and SAC-20, a twenty-two-story construction as a high-rise structure, have been used to evaluate efficiency and precision of the mentioned method in comparison with NTHA. These structures have been specially designed according to the UBC1994 code for Los Angeles region, the USA. Given that in each direction only two moment frames resist against the lateral forces, only one of the northward–southward moment frames has been modeled in two-dimensional state. A half of each story mass is considered as its seismic lumped mass. The gravity load has been ignored in modeling. In this study, in order to define the Rayleigh damping matrix, a damping ratio of 5% was considered for the first and third modes of vibration. The details related to the 3-, 9- and 20- story buildings are shown in Figs. 5, 6 and 7. The levels of the 3-, 9- and 20-story building are numbered with respect to the ground level (see Fig. 5, 6, 7). Further details of these buildings are presented in Gupta and Krawinkler (1999).

Fig. 5
figure 5

3-Story building

Full size image
Fig. 6
figure 6

9-Story building

Full size image
Fig. 7
figure 7

20-Story building

Full size image

The first three modal frequencies of the 3-, 9- and 20-story buildings are presented in Table 1. Also, the first three modal masses of the buildings are presented in Table 2.

Table 1 Modal frequency (Hz)
Full size table
Table 2 Modal mass % age
Full size table

3.2 Software

Open System for Earthquake Engineering Simulation, OpenSEES, has been used to model and perform nonlinear static and dynamic analyses. Using the software capabilities, the material specifications have been modeled considering nonlinearity using “uniaxialMaterial Steel02” command. Element sections are modeled in fiber state. The structural elements have been modeled in the “Force-Based Beam-Column Element” command, which consider distributed plasticity along the element length (Mazzoni et al. 2007). The first three modes of vibration were considered to extract modal forces in all of the NSP methods.

3.3 Ground Motions

For performing nonlinear dynamic analyses of MDOF and SDOF systems, forty SAC Los Angeles region ground motions records (Somerville et al. 1997), twenty at 10% (design basis earthquake “DBE”) and twenty at 2% (maximum considered earthquake “MCE”) probability of exceedance in 50 years, were used. The characteristics of the selected SAC ground motions are given in Table 3. In order to achieve modal spectral accelerations, the elastic spectrum with 5% of damping was calculated. The elastic pseudo-acceleration for DBE and MCE records, together with the corresponding the mean spectra, is presented, for 5% damping ratio in Figs. 8 and 9.

Table 3 Specifications of the earthquake records
Full size table
Fig. 8
figure 8

Pseudo-acceleration spectra of the DBE records of ground motions, damping ratio = 5%. The mean spectra are shown by a thicker line

Full size image
Fig. 9
figure 9

Pseudo-acceleration spectra of the MCE records of ground motions, damping ratio = 5%. The mean spectra are shown by a thicker line

Full size image

3.4 Surveyed Pushover Methods

The following advanced pushover procedures have been investigated:

  1. 1.

    Force-based adaptive pushover (FAP) analysis method, presented by Antoniou and Pinho (2004);

  2. 2.

    Force-based pushover (FP) method, equivalent non-adaptive load pattern procedure of FAP methodology;

  3. 3.

    Story shear-based adaptive pushover (SSAP) analysis method, presented by Shakeri et al. (2010);

  4. 4.

    Story shear-based pushover (SSP), equivalent non-adaptive load pattern procedure of SSAP methodology;

3.5 Comparison Criteria

Since the structural damage is mainly controlled first by the story drift and then by the displacements, it will be evaluated and compared with NTHA counterpart.

Maximum responses gained from NTHA have been considered as benchmark. The error index is defined by Eq. (4):

$${\text{Error}}_{\Delta } (\% ) = 100 \times \left( {\frac{{\Delta_{{i - {\text{NSP}}}} - \Delta_{{i - {\text{NTHA}}}} }}{{\Delta_{{i - {\text{NTHA}}}} }}} \right)$$
(4)

where \(\Delta_{{i - {\text{NTHA}}}}\) is the peak inter-story drift or displacement at a given level i, resulting from the NTHA and \(\Delta_{{i - {\text{NSP}}}}\) is the corresponding inter-story drift or displacement from the nonlinear static procedure.

4 Analyses Results

The average inter-story drift and displacement profiles resulted from the NTHAs are plotted in Figs. 10, 11, 12, 13, 14 and 15.

Fig. 10
figure 10

Average displacement and inter-story drift profile and their corresponding error diagram resulted from NTHA, FAP, FP, SSAP and SP method, subjected to twenty 10% probability of exceedance ground motions of Los Angeles region for 3-story building

Full size image
Fig. 11
figure 11

Average displacement and inter-story drift profile and their corresponding error diagram resulted from NTHA, FAP, FP, SSAP and SP method, subjected to twenty 2% probability of exceedance ground motions of Los Angeles region for 3-story building

Full size image
Fig. 12
figure 12

Average displacement and inter-story drift profile and their corresponding error diagram resulted from NTHA, FAP, FP, SSAP and SP method, subjected to twenty 10% probability of exceedance ground motions of Los Angeles region for 9-story building

Full size image
Fig. 13
figure 13

Average displacement and inter-story drift profile and their corresponding error diagram resulted from NTHA, FAP, FP, SSAP and SP method, subjected to twenty 2% probability of exceedance ground motions of Los Angeles region for 9-story building

Full size image
Fig. 14
figure 14

Average displacement and inter-story drift profile and their corresponding error diagram resulted from NTHA, FAP, FP, SSAP and SP method, subjected to twenty 10% probability of exceedance ground motions of Los Angeles region for 20-story building

Full size image
Fig. 15
figure 15

Average displacement and inter-story drift profile and their corresponding error diagram resulted from NTHA, FAP, FP, SSAP and SP method, subjected to twenty 2% probability of exceedance ground motions of Los Angeles region for 20-story building

Full size image

Average seismic demand predicted by pushover methods and NTHA for SAC-3 can be observed in Figs. 10 and 11. It can be seen that the average of FAP and FP seismic demands is more accurate. On the other hand, the average of SSAP and SSP responses indicates that these two methods predict upper stories drift and displacement values better. In addition, there is no remarkable difference in the responses of FAP and SSAP in comparison with their equivalent non-adaptive procedures, FP and SSP, respectively. It means adaptive load patterns cannot significantly increase the accuracy of NSP methods in low-rise buildings.

Regarding Fig. 12, responses of the FAP and the FP procedures seem closer to NTHA than SSAP and SSP in lower stories for DBE records. Conversely, for the upper stories the average of SSAP and SSP responses is achieved better for the SAC-9. Similar to the SAC-3 model, the difference between adaptive and non-adaptive procedures is not considerable for DBE records.

Figure 13 illustrates the results for MCE set of records. Interpretations are approximately the same as they were for DBE set of records. It is noticeable here to state that for MCE records, the gap between adaptive and non-adaptive procedures is clearer than it for DBE. SSAP led to better result in upper stories than SP, while FAP led to better responses in lower stories than its non-adaptive counterpart.

Referring Fig. 14, the SSAP is much more accurate than all other procedures in DBE set of records.

It can be inferred from Fig. 15 that when SAC-20 is subjected to MCE records, SSAP is better for upper stories than all other procedures and FAP and FP are better for lower stories.

5 Conclusion

The equivalent non-adaptive load patterns of story shear-based adaptive pushover “SSAP”, and force-based adaptive pushover “FAP” methodologies are introduced: story shear-based pushover, “SSP”, and force-based pushover, “FP”. In order to evaluate the efficiency of adaptive load patterns, three well-known steel frames that are SAC groups, covering wide range of periods, have been subjected to two sets of earthquake records, including twenty design basis earthquakes and twenty maximum considered earthquake records. The inter-story drift and displacement have been extracted to compare the efficiency of the methods.

However, the improvement due to the adaptive characteristic is not very considerable. In this respect, SSAP led to more precise drift responses in comparison with SSP in all structures, especially in SAC-20’s upper stories. On the other hand, there is just a very negligible difference between FAP and FP drift responses. It can be said that FAP predicts drift demand more in lower stories, and less in upper stories, than FP. Furthermore, adaptive load patterns are more efficient under the effects of high intensity records, the MCE set.

All in all it can be inferred that adaptive load patterns’ efficiency and accuracy are higher for intense earthquakes and also in high-rise buildings. Furthermore, results showed that the SSAP and SSP procedures result in more accurate drift responses than FAP and FP procedures.