What inspired me to write this article was a query I received via e-mail concerning the difference between Linear Dynamic Analysis and Non-linear Dynamic analysis. To those familiar with the Indian seismic code of IS:1893-2002 [1], Dynamic analysis is mostly Response Spectrum Analysis (RSA), and thus the former one in the query; and to those familiar with the research and academic field, Non-linear Dynamic analysis is Non-linear Time History Analysis (THA) and thus the latter. But RSA and THA are totally different things – the second one is not the nonlinear counterpart of the first. This was where the core of the confusion on how to respond was. The linear procedure that corresponds to the Nonlinear THA is obviously the Linear THA. The question that remained was: Which non-linear procedure corresponds to RSA? An effort is being made in the following sections to suggest an answer by explaining the various approaches to seismic analysis as applied to multi-storied RC buildings, trying to elaborate and justify an acceptable answer to the above question.

Before venturing into the topic, a basic classification of the different methods of approach to performing a seismic analysis of a structure need to be introduced. In general, the methods of seismic analysis can be classified as (A) Static and (B) Dynamic. Dynamic analysis can further be classified as (i) Frequency domain analysis (which is also Time period domain analysis — since Time period is simply the reciprocal of frequency — though such terminology is never used) and (ii) Time domain analysis. Under the Frequency domain analyses falls (a) the Response Spectrum Analysis (RSA) which, in my definition, is a Dynamic Characteristics based (static) Analysis and (b) the Power Spectrum Density (PSD) analysis, which though can be utilized for seismic applications, doesn’t seem to have as much advantage considering to the effort one puts in to do it, especially when RSA is easy and good enough. The seismic codes, including IS:1893-2002, obviously don’t suggest this. Under Time domain analysis falls the Time History Analysis (THA). These are represented as a tree (Fig. 1) and appropriately ‘numbered’ as referenced above. This numbering is used consistently throughout the article for easy referring its position in the tree.

In addition, all of the above have their (1) Linear and (2) Non-linear counterparts. Please note that while linear analysis is an approach where the results as obtained from the analysis — namely, Bending moment (BM), Shear force (SF), Axial Force (AF), etc. — are directly considered for the design of the members. In case of nonlinear analysis, it is done by running a non-linear analysis on a non-linear building model. Non-linearity is incorporated in the building’s analysis model in form of non-linear hinges inserted into an otherwise linear elastic model which one generates using a common analysis-design software package like SAP2000 or STAAD.Pro. Since the determination of the hinge properties require the details of reinforcement to be provided in the members, which, in turn, are obtained only by designing the structure, a nonlinear analysis can only be done on a designed building. (Of course, one can fall back on the approximate hinge property tables, which is based on probable reinforcement requirement, available in the appendix of both FEMA-356 [2] and ATC-40 [3], but the procedure will give only approximate results, which is not the intention of doing such an analysis in the first place). Thus while the linear methods are used to obtain forces (BM, SF, AF, etc.) which the building is designed for, the nonlinear methods are usually used to evaluate and ‘fine tune’ the design of a ‘design completed but not finalised’ building.

Now each of the categories shall be discussed in detail:

**A) Static Analysis**

A.1) Linear Static analysis: This is basically the Seismic Coefficient Method (SCM)(IS:1893-2002, Cl.7.5.3 & 7.7.1): Here the total base shear for the building under seismic loading is determined by using an empharically calculated Time period, and distributed over the stories as lateral load, proportional to an assumed ‘mode shape’, which is parabolic in the Indian code [1]. The base shear is obtained from the Spectral acceleration corresponding to the empherical time period using the same Response Spectrum curve (to be covered later) used for the RSA. Here determination of the building’s Time period, as well as the lateral load distribution is all formula based, no modal analysis is required, and the method is therefore considered a ‘static’ one.

A.2) Non-linear static analysis: This is basically the NSP (Non-linear Static Procedure), also known commonly as the Pushover analysis (PoA) (FEMA-356, Sec.3.3.3; ATC-40, Ch. 8), as mentioned in the introduction to the classification of analysis methods, is done by running a non-linear analysis on a non-linear building model. Nonlinearity is incorporated in the form of non-linear hinges inserted into an otherwise linear elastic model which one generates using a common structural analysis & design software package such as SAP2000 or MIDAS/Gen, having facilities for PoA. An earlier article by the author explains the method in detail [4].

Unlike as in SCM (where the lateral load of a calculated intensity is applied in whole – in one shot), in PoA , the building model is gently ‘pushed over’ by a monotonically increasing lateral load applied in steps up to a predetermined value (Displacement Coefficient Method, FEMA-356, also ATC-40). The seismic base shear for the building is distributed over the stories as lateral load which is either uniform (FEMA-356, Sec.3.3.3.2.3-2.1), or proportional to an assumed mode shape which is a power distribution (FEMA-356, Sec.3.3.2.3-1.1). For each floor, the base shear is distributed based on a factor of h (the height of each floor from the foundation), raised to the power of k.The value of k is determined by an empharical method to have a value between 1 (inverted triangular distribution) and 2 (parabolic distribution). In the SCM method explained in the last section, in IS:1893-2002, Cl.7.7.1, this k is having a value of 2. The option of distributing lateral load according to the first mode shape, also supported by FEMA-356, is not included here and is deliberately reserved for the next section. Thus the version of PoA described here requires no modal analysis and thus fits the ‘static’ category.

**B) Dynamic Analysis**

**B.i) Frequency Domain Analysis**

**B.i.a.1) Response Spectrum Analysis – Linear:** Before going into the procedure, a little note on what the Response Spectrum curve is: A response spectrum is simply a plot of the peak acceleration of each of a series of hypothetical oscillators of varying natural frequency, that are forced into vibration at its base by the same seismic ground motion record. Although in practice, this is done using software simulation – a software converter that takes the ground motion record as input and outputs the corresponding response spectrum curve, by simulating a series of oscillators – imagine such a setup in real, whose schematic representation is shown in Fig. 2a to obtain the response spectrum curve from a given ground motion record. It consists of an ensemble of oscillators, each having different time periods of oscillation. Although the conversion software simulates numerous oscillators in order to build the response spectrum curve, here only seven oscillators (with time periods marked T1 to T7) are shown for simplicity. Each of the oscillators – shown separately in Fig. 2b, which is one of the seven oscillators in Fig. 2a – is equipped with a facility to record its back-and-forth response, and plot its acceleration versus time graph (marked in Fig. 2b as ‘Response oscillation’, also in Fig. 2c, shown against each of the oscillators). Each also has facility to note down the peak acceleration (marked in Fig. 2a as ‘Peak response’, and noted in Fig. 2c as A1 to A7) from the response oscillations. A curve is drawn connecting these values (Fig. 2d) to obtain the response spectrum curve corresponding to the given ground motion record.

Fig.2 Schematic representation of a setup to obtain the response spectrum curve – (a) series of oscillators, (b) a typical oscillator subjected to ground motion, with acceleration response graph and peak response shown, (c) The series of oscillators subjected to a ground motion record, with acceleration response graph and peak response of each oscillator shown, and (d) the peak responses used to plot the response spectrum curve corresponding to the ground motion record.

Shown in Fig.3 is a more realistic example of Response spectrum conversion. Figure shows three ground motion records (Fig.3, a to c), converted to each one’s corresponding Response spectrum and overlaid (Fig.3d) – note the designations on top right corner of each ground motion record (Fig.3, a to c, marked EQ-1, EQ-2 and EQ-3, the same designations the corresponding response spectrum curves in Fig.3d are marked with).

Since earthquake motions are random and dependent on factors such as local soil conditions and distance from the source of the earthquake, the practical thing to do was to develop a smooth envelope of spectra for a range of expected earthquakes in a region. Moreover, it is interesting to note that the obtained curves of most of the seismic data when converted to response spectrum (Fig.3d) shows more or less the same pattern – starts from very low values at zero second of time period (T), quickly rises and fluctuates in high values up to around 0.5 second of T, and then gently descents to very low values up to around 3 to 4 seconds of T. Noticing this trend inspired the researchers to come up with an “idealized” smooth seismic response spectrum curve which is an average of response spectrum curves generated from several earthquake ground motion records. This was then further developed into a design response spectrum for use in structural design, and this basic form of which is now the basis for structural design in seismic regions throughout the world. The Response Spectrum curve is typically plotted with Spectral seismic acceleration (Sa, in units of g, acceleration due to gravity, written as Sa/g) in x-axis against Time Period T of the structure in y-axis, usually for a damping ratio that matches that of RCC buildings – ie., 5%. The standard response spectrum currently prescribed for design by the IS code is given in Fig.4.

Every building can vibrate in different deflection patterns (ie., the shape of the deflected building during its vibration), and each pattern has a different frequency (or time period) of vibration. These patterns are known as Mode Shapes and the frequencies are known as Mode (or Natural) Frequencies. Fig.5b to 5d shows the three typical mode shapes of a schematic building (Fig.5a), which are the oscillatory deflection of the building, ie., the pattern with which the building oscillates back and forth.

The seismic force acting on a building depends on its site characteristics (seismic zone of its location and soil type), and the dynamic characteristics of the structure (ie., its natural frequencies and other mode related parameters). The steps in an RSA as applied to buildings are: 1) Perform a modal analysis to identify the modes, which software packages like STAAD.Pro or SAP2000. 2) Calculating the structural loading corresponding to each mode, based on the spectral acceleration Sa/g read off from the codal Response Spectrum curve, corresponding to the Time period T of that mode (Fig. 6). The base shear acting on the building is calculated using the the obtained Sa/g, the location based zonal parameters and ‘importance’ factor of the building, and distributed to the different storey levels based on the mode shape factored using a parameter known as the mode participation factor calculated for the mode. This is repeated for each mode. 3) Static analysis is then done separately for each mode to obtain the responses (deflections and member forces such as BM, SF, AF, etc). 4) These responses are then combined to estimate the total response of the structure, for which the beams, columns and shear walls of the structure are designed. All the above steps are done automatically by the software package used for analysis.

The method of ‘combining’ the responses mentioned above requires a little consideration. When a structure oscillates, the modal peaks for different modal patterns of oscillations, oscillating in the different frequencies, do not occur simultaneously. Thus a simple summation of the absolute values of responses from different modes ends up in an over estimation of combined responses. Combination methods such as the Square Root of the Sum of the Squares (SRSS) are therefore used. The IS code, apart from SRSS method, prescribes CQC (Complete Quadratic Combination) method as most ideal to be adopted for such combinations (IS:1893-2002, Cl.7.8.4.4).

Thus in RSA (IS:1893-2002, Cl. 7.8.4), a ‘dynamic’ (modal) analysis is done to get the dynamic characteristics of the building (natural frequencies and mode shapes) from which the lateral loads corresponding to each mode shape is calculated, corresponding lateral loads applied and ‘static’ analyses are performed for each mode, and the results (BM, SF, AF, etc.) of each are then combined to get the design forces for each and every member, which are then used for design. It is this requirement of a modal analysis, which is a dynamic one, that has resulted in the classification of the whole procedure of RSA as a ‘Dynamic’ one.

B.i.a.2) Response Spectrum Analysis – Nonlinear: This is the same PoA mentioned above (section A.2), where the analysis model is gently ‘pushed over’ by a monotonically increasing lateral load applied in steps up to a predetermined value (Displacement Coefficient Method, FEMA-356 and ATC-40) or until a specified condition is met (Capacity Spectrum Method, ATC-40, Ch. 8, the method later improved in FEMA-440 [5]), but with the lateral loads proportional to the 1st mode or (FEMA 356, Sec.3.3.3.2.3-1.2 and ATC-40, Sec 8.2.1) or proportional to a combination (SRSS) of the first few modes (FEMA 356, Sec.3.3.3.2.3-1.3). As in case of the computer model for the PoA with lateral load pattern proportional to an empherically determined distribution, explained in section A.2, this analysis is also done by running a non-linear analysis on a non-linear model of the building [4,8]. Here, unlike the RSA, it’s not the results corresponding to each mode shape that is SRSS’ed, but the loads themselves. Then there are alternative methods developed — found almost only in published technical papers — for i) doing PoA for each mode separately and then combining the results from each (Fig. 7), as well as ii) the method where each set of separate lateral loads corresponding to the different modes are applied consecutively on the model in the same analysis, and the forces at the end of the last step considered for ‘fine tuning’ the design. These are yet to be standardised and included in the seismic codes.

**B.ii) Time Domain Analysis**

**B.ii.1) Time History Analysis – Linear:** This is the method referred to in IS:1893-2002, Cl. 7.8.3. Here the support points of the building model are oscillated back and forth in accordance to a ground motion record (Fig. 8), usually of an actually occurred earthquake (as recorded by a seismograph, and available in tabular form of time vs. acceleration). This is known as base excitation.

The most commonly implemented linear THA method is the Mode superposition method. The steps for a building model subjected to a base excitation with an earthquake ground motion record is as follows : 1) A modal analysis is performed for obtaining the Natural frequencies and mode shapes. For the easiness of explaining further steps, let us assume that only three modes are considered for the analysis — which is not sufficient in a real analysis, though — and thus we start with three modal time periods T1, T2 and T3, and three mode shapes {1}, {2} and { 3}, where the curly brackets indicate a column matrix, usually termed as a vector, where each value in it represents the modal displacement of each floor (Fig. 9a). In mode superposition, it is assumed that throughout the event of the earthquake, at any point of time, the instantaneous displaced shape of the structure can be represented by 1 * {1} + 1 * {1} + 1 * {1}, where the values of ’s vary with time. Thus the analysis has actually reduced to finding the values of ’s for all points of time, right from start to the time at the end of the seismic event (Fig 9b). 3) Therefore at any point of time during the event of the earthquake, the instantaneous displaced shape of the structure can be found, from which the member forces (BM, SF, AF, etc. ) also can be found. As in case of RSA, here also the above steps are automatically done by the software package for analysis.

Once the structural responses are obtained, the results (BM, SF, AF, etc.) are usually taken as the maximum enveloped over time (ie., for example, the max. BM for a particular beam is the maximum among all the BMs, considering the values corresponding to each time point over the duration of earthquake) for that ground motion record.

However, the question of how to chose a ground motion record for application in a THA for it to be considered conforming to the Response Spectrum of any of the design codes, like IS:1893-2002, is an issue.

Since any ground motion record is the recording of a particular seismic event, no single record conforms to any code. As we have already seen, for a given ground motion, a Response spectrum curve specific to that record can be generated (Fig. 3d). This ground motion record specific Response spectrum curve generated is not the same as the smooth Response spectrum curve of the code (as in Fig. 4), and therefore a THA with any given ground motion does not comply to the code. The two methods of tackling this discrepancy are as follows:

One is to use a generated ground motion record: A ground motion record can be tweaked such that the Response spectrum curve specific to that ‘tweaked ground motion record’ will come close to resemble the smooth idealized Response Spectrum curve (a kind of jagged idealised curve, Fig. 10). This tweaked data is termed Generated ground motion record (or artificial earthquake). There are such ‘tweaking’ programs available. In this method one should have the THA done for at least four such generated ground motion records for the procedure to be acceptable for design. The analysis should be repeated for each ground motion record and the maximum of the results taken as the forces for the design of the structure.

But recent trends discourage such ‘code compliant time histories’ generated, and recommend what is known as an ‘ensemble of selected ground motion records’ that together comply to the idealized codal Response spectrum curve; ie., if one generates the the ground motion record specific Response spectrum curve for each record of the ensemble (ie., the set) and take their mean, it will be more or less the Response spectrum curve given in the code that the ensemble conforms to (Fig. 11). The ensemble should have at least seven selected ground motion records. The method of selection from a vast no. of recorded ground motion records is quite an extensive subject in itself. There are internet sites that accepts the codal Response Spectrum curve provided by the user and returns the analytically selected set of ground motion records. The analysis should be repeated for each ground motion record and the mean of the results (BM , SF , etc.) taken as the forces for design of the structure.

**B.ii.2) Time History Analysis (THA) – Nonlinear:** Nonlinear THA (FEMA356, Sec.3.3.4.2.3) is the non-linear version of the linear THA we have already seen earlier (section B.ii.1). Here the intension remains the same: to oscillate the support points of the structure back and forth in accordance to a ground motion record. But the method implemented for analysis is different. This is because there is non-linearity in the structure. As already mentioned, these non-linearity’s are incorporated in the analysis model in form of non-linear hinges inserted into an otherwise linear elastic model. But that’s not all; since there are repeated cycles of loading, unloading and force reversals that these inserted hinges are subjected to, hysteresis parameters too are required to be provided.

Also, for non-linear structures, the method of Modal superposition method explained above (section B.ii.1) is no longer applicable. It’s the Time Integration method that is mostly used. In this method, which is also known as time stepping, the total duration of analysis – or rather the duration of the earthquake – is divided into very small time steps. Starting from zero time when the structure is un-displaced and at rest, the support displacements at the end of the first time step is applied and the analysis progresses time step by time step. At each time step (or point of time), having known the displacement, velocity and acceleration at every node in the structure, as well as the direction and intensity of base excitation, the displacement, velocity and acceleration at every node for the next time point is determined. This is repeated for the full duration of the earthquake through each time step.

**Conclusion**

Of all the methods touched upon in this article — namely static analysis and two dynamic analyses, plus the nonlinear counterparts of the three — the method most popular among structural engineers is the RSA. As of now it is not known whether any other method will gain enough acceptance to replace it. If, or rather when, it happens, one such candidate is the linear THA, due to the affordable computer processing power available, compared to those of the yesteryears, plus the availability of internet resources to select or generate and provide designers with the ground motion record data for the analysis. But the popular structural analysis packages are yet to provide the facility that automates, or rather eases, the process to accepting multiple ground motion record data, analysing for each, and calculating the design forces from those results to be presented to the designer.

Another possible route of development is the use of RSA for design of the structure, followed by a PoA for the ‘fine tuning’ of the design done. Probably this is what is meant by the latest edition of IS:13920 [7] when it says “Buildings with any of the listed irregularities perform poorly during earthquake shaking … When any such irregularities are adopted, detailed nonlinear analysis shall be performed … “. Thus it seems the statement in the author’s earlier article that, “Seismic design is slowly transforming … to a stage where a specifically dedicated nonlinear procedure is to be done, which finally influences the seismic design as a whole” [4] is slowly being realised.

Here again the popular structural analysis packages are yet to provide the facility that automates the nonlinear hinge insertion in the structural members: After analysis and design, the user should be able to right click on each beam (or a set of continues beams) and input the reinforcements to be provided, with which the program should generate the hinge with appropriate parameters automatically and insert into the beams at appropriate distances from the ends. And similar facilities should be available for columns, and also for shear walls and column strips of flat slabs.

With more and more earthquakes making headlines — the latest being a series of intense tremors at central Italy (24th Aug 2016 onwards) as I was starting to scribble the bits and parts of this article — seismic design requirements are only going to get complicated and stringent. But whatever be the methods and requirements for seismic design in the future, we, as structural engineers, thankfully with the help of the analysis packages (and, of course, those not being blindly used as black boxes), will equip ourselves for it.

**Reference**

1. IS:1893-2002 Part 1, “Indian Standard Criteria for Earthquake Resistant Design of Structures”, 2002, Bureau of Indian Standards, New Delhi

2. FEMA-356, “Prestandard and Commentary for the Seismic Rehabilitation of Buildings”, 2000, Building Seismic Safety Council, Federal Emergency Management Agency, Washington DC, USA

3. ATC-40, “Seismic Evaluation and Retrofit of Concrete Buildings”, 1997, Applied Technology Council, Redwood City, CA, USARahul Leslie, “The Pushover Analysis in its Simplicity”, Jun 2012, Civil Engineering and Construction Review magazine, India, pp. 118-126

4. Rahul Leslie, “The Pushover Analysis in its Simplicity”, Jun 2012, Civil Engineering and Construction Review magazine, India, pp. 118-126

5. FEMA-440, “Improvement of Inelastic Seismic Analysis Procedures”, 2003, Applied Technology Council, Redwood City, CA, USA

6. C.V.R.Murty et al., “Some Concepts in Earthquake Behaviour of Buildings”, Gujarat SDMA, 2012

7. IS:13920-2016, “Indian Standard Code of Practice for Ductile Detailing of Reinforced Concrete Structure Subjected to Seismic Forces”, 2016, Bureau of Indian Standards, New Delhi

8. Blog: http://rahulleslie.blogspot.in/p/blog-page.html

Rahul Leslie

Deputy Director, Buildings Design,

DRIQ, Kerala PWD, Trivandrum