Professor Anil K. Chopra Johnson Chair in Engineering University of California Department of Civil and Environmental Engineering Berkeley, CA 94720-1710 USA chopra@berkeley.edu Dynamics of structures, earthquake engineering

Although I had known Luis Esteva for many years, the opportunity to work closely with

him came only a few years ago when he become President of the International Association for Earthquake Engineering (IAEE). I interacted with Luis in my role as Editor of Earthquake Engineering and Structural Dynamics, the official journal of IAEE. During the period 2002-2004, the journal underwent major changes, which involved difficult negotiations with other organizations. I went to him several times asking for advice; in fact, so many times that I am sure that he grew to dread my emails. Every time I asked for his help, I admired his ability to identify immediately the essence of the problem, how he was able present his views calmly and effectively in the midst of disagreement, and was steadfast in searching for solutions that were fair to all parties.

Anil K. Chopra August 12, 2005

ESTIMATING SEISMIC DEMANDS FOR STRUCTURES IN ENGINEERING

PRACTICE: CONSTRAINED BY COMPUTER SOFTWARE, BUILDING CODES, AND BUILDING EVALUATION GUIDELINES

ANIL K. CHOPRA

ABSTRACT The main thesis of this paper is that adoption of improved analytical procedures for estimating seismic demands for structures in engineering practice is constrained by the lack of computer software that implement these techniques and by prevailing codes of practice. Two classes of structures, concrete dams and buildings, are chosen to illustrate different aspects of this contention.

Part A: Arch Dams Analysis Procedures The dynamic analysis of arch dams is especially complicated because they must be treated as three-dimensional systems that recognize the semi-unbounded size of the reservoir and foundation rock domains, and the following factors should be considered: dam-water interaction, wave absorption at the reservoir boundary, water compressibility, and dam-foundation rock interaction.

A series of Ph.D. theses [Chakrabarti and Chopra 1973; Hall and Chopra 1980; Fenves and Chopra 1984 (a); Fok and Chopra 1985; Zhang and Chopra 1990; Tan and Chopra, 1995] at the University of California, Berkeley, during 1970 to 1995, culminated in the substructure method formulated in the frequency domain, and its implementation in computer programs EAGD-84 for two-dimensional analysis of concrete gravity dams [Fenves and Chopra 1984 (b)] and EACD-3D-96 for three-dimensional analysis of arch dams [Tan and Chopra 1996] both of which were distributed pro bono by the National Information Service for Earthquake Engineering. Extensive parametric studies of the earthquake response of actual dams led to an understanding of how each of the preceding factors influences the response, and the practical range of conditions where each factor is significant. Commonly used finite element analysis techniques for dams, however, ignore most of these factors. Typically, hydrodynamic effects are represented by an added water mass moving with the dam, implying water compressibility is neglected. Also ignored in these analyses is the partial absorption of hydrodynamic pressure waves by the sediments invariably deposited at the reservoir bottom and sides, or even by rock underlying the reservoir. The foundation rock is usually assumed to be massless and a portion is included in a finite-element idealization of the system. This extremely simple idealization of the foundation rock, in which only its flexibility is considered but inertial and damping effects are ignored, is popular because the foundation impedance matrix (or frequency-dependent stiffness matrix) is very difficult to determine for unbounded domains.

Although all of these factors known to be significant in the response of concrete dams were included in computer software developed before desktop computer became standard by graduate students at the University of California, Berkeley, they made no effort to develop input and output interfaces that were user friendly. Thus, they remained primarily as research programs with limited application to actual projects. Easier to use, the EAGD-84 program was chosen as the analytical tool for the seismic safety evaluation of several concrete gravity dams, but the less convenient EACD-3D-96 program was used only for a few arch dam projects. However, starting in 1996, the U.S. Bureau of Reclamation (USBR) embarked upon a major program to evaluate the seismic safety of dams and found it necessary to consider water compressibility, reservoir boundary absorption, and dam-foundation rock interaction effects. Among the 12 dams investigated were: (1) Deadwood Dam, a 50-m high single curvature dam; (2) Monticello Dam, a 100-m high single curvature dam; (3) Morrow Point Dam, a 150-m high double curvature dam; and (4) Hoover Dam, a 221-m high thick arch dam. The results of EACD-3D analyses of these dams, conducted by L. Nuss and R. Munoz (USBR staff) are presented next. Implications of Neglecting Water Compressibility We compare the earthquake-induced stresses in two of the four dams computed under two conditions: water compressibility considered or neglected (Figs. 1 and 2). These figures permit the following observations: the effects of water compressibility, which are generally significant, vary with the location on the dam surface. By neglecting water compressibility, the stresses may be significantly underestimated, as in the case of Monticello Dam (Fig. 1), or significantly overestimated as in the case of Morrow Point Dam (Fig. 2). Thus, water compressibility should be included in the analysis of arch dams; however, most standard finite element analysis software neglects water compressibility. Implications of Neglecting Foundation Rock Inertia and Damping We compare the earthquake-induced stresses in each of the four dams, computed under two conditions considering: (1) dam-foundation rock interaction; and (2) foundation rock flexibility only (Figs. 3 through 6). For these two conditions, the largest arch stress on the upstream or downstream face of the dam is 476 psi versus 844 psi for Deadwood Dam; 730 psi versus 1410 psi for Monticello Dam; 665 psi versus 1336 psi for Morrow Point Dam; and 758 psi versus 2204 psi for Hoover Dam. If only foundation rock flexibility is considered (i.e., foundation rock is assumed to be massless), the stresses are overestimated by a factor of 2 (approximately) for the first three dams and by a factor of 3 (approximately) for Hoover Dam. As demonstrated in the above examples, stresses are overestimated as a result of ignoring the foundation rock material and radiation damping. Because such overestimation of stresses may lead to overconservative designs of new dams and to the erroneous conclusion that an existing dam is unsafe, and, hence, requires remediation, it is imperative that dam-foundation rock interaction effects should be included in earthquake analysis of concrete dams. However, dynamic analyses for most seismic safety evaluation projects assume massless foundation rock,

thus ignoring dam-foundation rock interaction effects, because most commercial and proprietary computer software is based, erroneously, on these simplifying assumptions. Advancement of Engineering Practice Most of the research developments to include the effects of dam-water interaction, water compressibility, reservoir boundary absorption, and dam-foundation rock interaction in analysis of dams were reported over twenty years ago for gravity dams and over ten years ago for arch dams. However, the profession has resisted using these analytical advances, in part because they have not been incorporated in widely used commercial software. The state of earthquake engineering practice for dams is not likely to advance unless user-friendly software is developed for desktop computers that includes these interaction effects in linear and nonlinear analysis of dams considering spatial variations in ground motion.

Part B: Buildings Building Code Development Most buildings are designed according to procedures specified in building codes, which define the state of practice. In the United States, for example, the most recent code is the International Building Code, first developed in 2000 and subsequently revised in 2003. The structural analysis procedures in this code are based on the NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (1997), which, in turn, are based on the ATC-3-06 document (1978). ATC-3-06 represented a major departure from traditional U.S. seismic codes. The new features relevant to structural analysis included:

• Response modification factors (R factors) to reduce elastic response of the structure due to realistic ground motions to obtain design forces; the R factors were dependent on ductility, damping, and past performance of a given structural system.

• Strength-based design instead of allowable stress design. • Provisions for dynamic (modal) analysis. • Provisions to consider soil-structure interaction effects.

The ATC-3-06 document was the result of a massive effort during 1974 through 1976 by a team of nearly 70 experts from around the country, and it was reviewed by various overview panels and committees. Although all prominent engineers and academics were involved, its acceptance into formal code documents was a slow process. Evolution of ATC-3-06 to the 1997 NEHRP provisions took twenty years. Until the analysis procedures in the latter were adopted into the IBC (2000), they did not become a part of structural engineering practice in the United States. In contrast, many of the ideas in ATC-3-06 were incorporated very quickly into the 1976 Mexico Federal District Code by Emilio Rosenblueth, who participated in the ATC-3-06 project.

Building Evaluation Guidelines In contrast to the slow progress in building codes, the guidelines for seismic evaluation of buildings evolved rapidly after the Loma Prieta (1989) and Northridge (1994) earthquakes in California. Two major guideline documents appeared: ATC-40 (1996) and FEMA 273 (1997) [and its successor FEMA 356 (2000)]. According to these guidelines, seismic demands are computed by nonlinear static analysis of the structure subjected to monotonically increasing lateral forces with an invariant heightwise distribution until reaching a roof displacement determined from the deformation of an inelastic SDF system derived from the pushover curve. According to the ATC-40 guidelines, the deformation of the inelastic SDF system is estimated by the capacity-spectrum method, an iterative method requiring analysis of a sequence of equivalent linear system; the method is typically implemented graphically. Unfortunately, the ATC-40 iterative procedure does not always converge; when it does converge it does not lead to the exact deformation (Chopra and Goel, 2000). Because convergence traditionally implies accuracy, the user could be left with the impression that the calculated deformation is accurate, but the ATC-40 estimate errs considerably. This is demonstrated in Fig. 7a, where the deformation estimated by the ATC-40 method is compared with the value determined from inelastic design spectrum theory and three different y nR Tµ− − equations. Both the approximate and theoretical results are presented for systems covering a wide range of period values and ductility factors subjected to ground motions characterized by an elastic design spectrum. The discrepancy in the approximate result presented in Fig. 7b shows that the ATC-40 method underestimates by 40-50% the deformation over a wide range of periods (Chopra and Goel, 2000). Prompted by research results, such as those reported above, the ATC-55 (or FEMA-440) project was launched in 2002 to develop improved procedures to estimate seismic demands. The two flaws in the ATC-40 capacity spectrum method—lack of convergence in some cases and large errors in many cases—appear to have been rectified in the FEMA-440 report (Comartin et al., 2004). The optimal vibration period and damping ratio parameters for the equivalent linear system are now derived by minimizing the differences between its response and that of the actual inelastic system. Such an equivalent linear method would obviously give essentially the correct deformation. However, the benefit in making the equivalent linearization detour is unclear when the deformation of an inelastic system can be readily determined using available equations for the inelastic deformation ratio [e.g., Ruiz-Garcia and Miranda, 2003; Chopra and Chintanapakdee, 2004) or from the inelastic design spectrum (e.g., Chopra, 2001). The nonlinear static procedure (NSP) in FEMA-356 requires development of a pushover curve, a plot of base shear versus roof displacement, by nonlinear static analysis of the structure subjected first to gravity loads, followed by monotonically increasing lateral forces with a specified invariant height-wise distribution. At least two force distributions must be considered. The first is to be selected from among the following: first mode distribution, equivalent lateral force (ELF) distribution, and response spectrum analysis (RSA) distribution. The second distribution is either the “uniform” distribution or an adaptive distribution; several options are mentioned for the latter, which varies with change in deflected shape of the structure as it yields. The other four force distributions mentioned above are defined as follows:

1. First-mode distribution: * 1,j j js m φ= where jm is the mass and 1jφ is the mode shape

value at the jth floor.

2. ELF distribution: * ,kj j js m h= where jh is the height of the jth floor above the base, and k

varies between 1 and 2, depending on the fundamental vibration period. 3. RSA distribution: *s is defined by the lateral forces back-calculated from the story shears

determined by response spectrum analysis of the structure, assumed to be linearly elastic. 4. “Uniform” distribution: *j js m= .

Each of these force distributions pushes the building in the same direction over the height of the building (Fig. 8). The limitations of the FEMA-356 force distributions are demonstrated in Figs. 9 and 10, where the resulting estimates of the median story drift and plastic hinge rotation demands imposed on the SAC−Los Angeles buildings by the ensemble of 20 SAC ground motions are compared with the “exact” median value determined by nonlinear response history analysis (RHA). The FEMA-356 lateral force distributions provide a good estimate of story drifts for the 3-story building. However, the first-mode force distribution grossly underestimates the story drifts in the upper stories of the 9- and 20-story buildings, showing that higher-mode contributions are especially significant in the seismic demands for upper stories. Although the ELF and RSA force distributions are intended to account for higher-mode responses, they do not provide satisfactory estimates of seismic demands for buildings that remain essentially elastic (SAC Boston buildings) or buildings that are deformed far into the inelastic range (SAC Los Angeles buildings) (Goel and Chopra, 2004). The “uniform” force distribution seems unnecessary because it grossly underestimates drifts in upper stories and grossly overestimates them in lower stories. Because FEMA-356 requires that seismic demands be estimated as the larger of results from at least two lateral force distributions, it is useful to examine the upper bound of results from the four force distributions considered. This upper bound also significantly underestimates drifts in upper stories of taller buildings but overestimates them in lower stories (Fig. 9). The FEMA-356 lateral force distributions provide a good estimate of plastic hinge rotations for the 3-story building, but either fail to identify, or significantly underestimate, plastic hinge rotations in beams at the upper floors of 9- and 20-story buildings (Fig. 10). Many discussions of the potential and limitations pushover analysis are available in the literature (e.g., Naeim and Lobo, 1998; Krawinkler and Seneviratna, 1998; Elnashai, 2001; Fafar, 2002). Despite the limitations of the ATC-40 procedure for estimating the target roof displacement and the FEMA 273/356 nonlinear static procedure for estimating seismic demands, these procedures have had profound influence on engineering practice not only in the United States, where these procedures were developed, but in several other countries, and they continue to be used. Eventually, they will be replaced by improved procedures that are emerging, but their acceptance into codes and guidelines is likely to be a slow process. Because it is so difficult to make major modifications to such documents, it is imperative that in the future any analytical procedure should be thoroughly researched and tested before it is included.

Improved Nonlinear Static Procedures It is clear from the preceding discussion that the seismic demand estimated by NSP using the first-mode force distribution (or others in FEMA-356) should be improved. One approach to reducing the discrepancy in this approximate procedure relative to nonlinear RHA is to include the contributions of higher modes of vibration to seismic demands. It is well known that when higher-mode responses are included in the RSA procedure, improved results are obtained for linearly elastic systems. Although modal analysis theory is strictly not valid for inelastic systems, the fact that elastic modes are coupled only weakly in the response of inelastic systems (Chopra and Goel 2002, 2004) permitted development of the modal pushover analysis procedure (MPA). In this MPA procedure, the peak “modal” demand nr (n denotes mode number) is determined by a nonlinear static or pushover analysis using the modal force distribution *n n=s mφ (m is the mass

matrix and nφ is the nth vibration mode) up to a target displacement determined from the

deformation of the nth-“mode” inelastic SDF system; and the peak modal demands nr are then combined by an appropriate modal combination rule.

Figures 11 and 12 show the median values of story drift and beam plastic rotation demands, respectively, for SAC—Boston, Seattle, and Los Angeles buildings—including a variable number of “modes” in MPA superimposed with the “exact” result from nonlinear RHA. The first “mode” alone is inadequate in estimating story drifts, but with a few “modes” included, story drifts estimated by MPA are generally similar to the upper floors of all buildings and also in the lower floors of the Seattle 20-story building. Including higher-“mode” contributions also improves significantly the estimate of plastic hinge rotations. In particular, plastic hinging in upper stories is now identified, and the MPA estimate of plastic rotation is much closer—compared to the first-“mode” result—to the “exact” results of nonlinear RHA. Based on structural dynamics theory, the MPA procedure retains the conceptual simplicity and computational attractiveness of the standard pushover procedures with invariant lateral force distribution. Because higher-mode pushover analyses are similar to the first-mode analysis, MPA is conceptually no more difficult than procedures now standard in structural engineering practice. Because pushover analyses for the first two or three modal force distributions are typically sufficient in MPA, it requires computational effort that is comparable to the FEMA-356 procedure, which requires pushover analysis for at least two force distributions. Without additional conceptual complexity or computational effort, MPA estimates seismic demands much more accurately than FEMA-356 procedures (Goel and Chopra 2004). Advancement of Engineering Practice Building codes should explicitly state the underlying basis for each provision, so that it can be improved as we develop a better understanding of structural dynamics and earthquake performance of structures. Such a transparent format should be combined with a mechanism to incorporate improvements more quickly than has been feasible in the past.

Design codes and evaluation guidelines for buildings should be consistent with research results and actual performance of structures during earthquakes. They should be tested extensively before promulgation. The ATC-40 and FEMA-273 documents do not seem to have satisfied this requirement. Presently, developers of commercial software are followers, in the sense that they implement analysis procedures specified in building design codes and evaluation guidelines. To advance the state of engineering practice, they should become leaders in incorporating research results—for example, improved nonlinear static procedures—into their software so that the profession can use these procedures during the many years they are being considered by committees for possible adoption in building evaluation guidelines.

Closing Comments This paper has attempted to demonstrate that adoption of improved analytical techniques for estimating seismic demands for structures in engineering practice is constrained by the lack of computer software that implement these techniques and by prevailing codes of practice. The structural types and analytical procedures chosen to illustrate this contention were obviously based on my research experience. However, the overall issue is of broader concern and other researchers may be able to identify supporting examples from their own experience.

References Applied Technology Council (1978). Tentative provisions for the development of seismic regulations for buildings, Report ATC-3-06, NBS Special Publication 510, NSF Special Publication 78-08. Applied Technology Council (1996). Seismic evaluation and retrofit of concrete buildings, Report ATC-40, ATC, Redwood City, Calif. Building Seismic Safety Council (1997). NEHRP guidelines for the seismic rehabilitation of buildings, FEMA-273, Federal Emergency Management Agency, Washington, D.C. Chakrabarti, P., and A. K. Chopra (1972). Earthquake Response of Gravity Dams Including Reservoir Interaction Effects, Report No. EERC 72-6, Earthquake Engineering Research Center, University of California, Berkeley, 161 pp. Chopra, A. K. (2001). Dynamics of Structures: Theory and Applications to Earthquake Engineering, 2nd ed., Prentice Hall, Upper Saddle River, N.J., 844 pp. Chopra, A. K., and C. Chintanapakdee (2004). Inelastic deformation ratios for design and evaluation of structures: single-degree-of-freedom bilinear systems, ASCE, J. Struct. Engrg., 130 (9), 1309−1319.

Chopra, A. K, and R. K. Goel (2000). Evaluation of NSP to estimate seismic deformation: SDF systems, ASCE, J. Struc. Engrg., 26, 482−490. Chopra, A. K., and R. K. Goel (2002). A modal pushover analysis procedure for estimating seismic demands for buildings, Earthq. Engrg. Struc. Dyn., 31, 561−582. Chopra, A. K., and R. K. Goel (2004). A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings, Earthq. Engrg. Struc. Dyn., 33 (8), 903−927. Comartin, C. D., et al. (2004). A summary of FEMA-440: improvement of nonlinear static seismic analysis procedures, Proceedings, 13th World Conf. Earthq. Engrg., Paper 1476, Vancouver, British Columbia, Canada. Fenves, G., and A. K. Chopra (a) (1984). Earthquake analysis and response of concrete gravity dams, Report No. UCB/EERC-84/10, Earthquake Engineering Research Center, University of California, Berkeley, 213 pp. Fenves, G., and A. K. Chopra (b) (1984). EAGD-84: A computer program for earthquake analysis of concrete gravity dams, Report No. UCB/EERC-84/11, Earthquake Engineering Research Center, University of California, Berkeley, 78 pp. Elnashai, A. S. (2001). Advanced inelastic static (pushover) analysis for earthquake applications, Struc. Engrg. Mech., 12 (1), 51−69. Fajfar, P. (2002). Structural analysis in earthquake engineering—a breakthrough of simplified nonlinear methods, Proceedings, 12th Eur. Conf. Earthq. Engrg., London, Paper Ref. 843. Fok, K.-L., and A. K. Chopra (1985). Earthquake analysis and response of concrete arch dams, Report No. UCB/EERC-85/07, Earthquake Engineering Research Center, University of California, Berkeley, 207 pp. Goel, R. K., and A. K. Chopra (2004). Evaluation of modal and FEMA pushover analyses: SAC buildings, Earthq. Spectra, 20, (1), 225−254. Hall, J. F., and A. K. Chopra (1980). Dynamic response of embankment, concrete-gravity and arch dams including hydrodynamic interaction, Report No. UCB/EERC-80/39, Earthquake Engineering Research Center, University of California, Berkeley, 220 pp. Krawinkler, H., and G. D. P. K. Seneviratna (1998). Pros and cons of a pushover analysis of seismic performance evaluation, Engrg. Struc., 20 (4-6), 452−464. Naeim, F., and R. M. Lobo (1998). Common pitfalls in pushover analysis, Proceedings, SEAOC Annu. Conv., Reno, Nev. Ruiz-Garcia, J., and E. Miranda (2003). Inelastic displacement ratios for evaluation of existing structures, Earthq. Engrg, Struc. Dyn., 32 (8),1237−1250.

Tan, H. C., and A. K. Chopra (1995). Earthquake analysis and response of concrete arch dams, Report No. UCB/EERC-95/07, Earthquake Engineering Research Center, University of California, Berkeley,168 pp. Tan, H. C., and A. K. Chopra (1996). EACD-3D-96: A computer program for three-dimensional earthquake analysis of concrete dams, Report No. UCB/SEMM-96/06, University of California, Berkeley, 131 pp Zhang, L., and A. K. Chopra (1991). Computation of spatially-varying ground motion and foundation-rock impedance matrices for seismic analysis of arch dams, Report No. UCB/EERC-91/06, Earthquake Engineering Research Center, University of California, 121 pp.

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Figure 1. Peak values of earthquake-induced stresses in Monticello Dam computed under two conditions: (a) water compressibility considered, and (b) water compressibility neglected.

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Figure 2. Peak values of earthquake-induced stresses in Morrow Point Dam computed under two conditions: (a) water compressibility considered, and (b) water compressibility neglected.

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Figure 3. Peak values of earthquake-induced stresses in Deadwood Dam, considering (a) dam-foundation interaction, and (b) foundation rock flexibility only.

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Figure 4. Peak values of earthquake-induced stresses in Monticello Dam, considering (a) dam-foundation interaction, and (b) foundation rock flexibility only.

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Figure 5. Peak values of earthquake-induced stresses in Morrow Point Dam, considering (a) dam-foundation interaction, and (b) foundation rock flexibility only.

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Figure 6. Peak values of earthquake-induced stresses in Hoover Dam, considering (a) dam-foundation interaction, and (b) foundation rock flexibility only.

0.05 0.1 0.2 0.5 1 2 5 10 20 500.1

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Figure 7. Deformations computed by ATC-40 and from inelastic design spectrum using three different y nR Tµ− − equations (identified by NH, KN, and VFF; part (a) compares deformations and part (b) shows discrepancy in ATC-40 method. [Adapted from Chopra and Goel, 2000.]

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Figure 8. FEMA-356 force distributions for SAC-Los Angeles 9-story building: (a) first mode,

(b) ELF, (c) RSA, and (d) “uniform.

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Figure 9. Median story drifts for SAC-Los Angeles buildings: (a) 3-story; (b) 9-story; (c) 20-story. Determined by nonlinear RHA and four FEMA-356 force distributions: first Mode, ELF, RSA, and “uniform” (Goel and Chopra 2004).

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Figure 10. Median plastic rotations in interior beams of SAC−Los Angeles buildings: (a) 3-story; (b) 9-story; (c) 20-story. Determined by nonlinear RHA and four FEMA-356 force distributions: first Mode, ELF, RSA, and “uniform” (Goel and Chopra 2004).

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Figure 11. Median story drifts determined by nonlinear RHA and MPA with variable number of “modes”; P-∆ effects due to gravity loads are included (Goel and Chopra 2004).

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Figure 12. Median plastic rotations in interior beams determined by nonlinear RHA and MPA

with variable number of “modes”; P-∆ effects due to gravity loads are included (Goel and Chopra 2004).