RESPONSE SPECTRUM ANALYSIS : AN ADVANCE ANALYSIS

Response Spectrum Analysis :  An Advance Analysis

Its a very complex and challenging analysis according to International Building Code (IBC). Response Spectrum Analysis(RSA) is a linear-dynamic method which does not consider non-linear assignments formulation as well as which measures the contribution from each natural mode of vibration to indicate the likely maximum seismic response of an essentially elastic structure.Response Spectrum Analysis provides insight into dynamic behavior by measuring pseudo-spectral shaking,velocity or displacement as a function of structural period for a given time history and level of damping.

It is practical to envelop response spectrum such that a smooth curve represents the peak response for each realization of structural period. RSA is useful for design decision making as it relates structural type selection to dynamic performance.Structures of shorter period experience greater acceleration whereas those of longer period experience greater displacements.Structural performance objectives should be taken into account during preliminary design and response spectrum analysis(RSA).

Factors influencing & applicable for RSA: The Response Spectrum values depends upon the following parameters:

1) Skyscraper/ Tall Building

2) Energy release mechanism/Energy dissipation mechanism which depends on loading and unloading type.

3) Epicentral distance

4) Focal depth

5) Soil condition

6) Ricter magnitude

7) Damping in the system.

8) Time period of the system.

9) Type of irregularity of building.

10) Inherent torsion & Accidental torsion.

11) Type of Analysis.

Damping &  RSA: 

A) RSA is significantly affected by damping.If damping increases response spectra shifts downward.

B) According to IBC damping should be taken not exceding 5%.

C) Viscous dampers do not affect structural stiffness,are not modeled during RSA and are not accounted for in the IBC provision for 5% damping.

Consideration in RSA:

1) All Response quantities are positives therefore RSA is not suitable for extreme torsional irregularity.A static lateral-load procedure is best for measuring accidental torsion.

2) Tension /Uplift and Compression should be considered to design foundation i.e foundation should be designed for resonant force also.

3) Response may be combined using SRSS, CQC,ABS or GMC methods. CQC method is best when periods are closely spaced with cross-correlation between mode shapes. SRSS is suitable when periods differ by more than 10%.

4) Ritz vectors are recommended for RSA because this formulation is computationally efficient.Only pertinent mode shapes which occurs in the horizontal plane are identified.

5) Eigen vectors use the full stiffness and mass matrices,which also account for vertical modes. Eigen formulation is useful when considering floor vibration,out of plane vibration & shear wall system etc.This application also useful for locating modeling errors.

Irregular Structures: A well designed structure should be capable of resisting earthquake motions equally from all possible directions.The theory is based on the assumption that motions which take place during an earthquake have one principle direction,Or that during finite period of time around the time of the occurrence of the maximum acceleration,a principle direction exist.For most structures this direction is not known and for most geographical locations,cannot be estimated.Therefore,the only rational earthquake design criterion is that 

        ".....the structure must resist an earthquake of a given magnitude in any possible direction,there is a probability that motion normal to that direction will occur simultaneously.Also,it is valid to assume that these normal motions are statistically independent because of the complex nature of these three dimensional wave propagation....." 

Common mistakes in approximation of Response Spectrum: The curve plotted between maximum response of SDOF system subjected to specified earthquake ground motion and its time period( or frequency) is called Response Spectra curve.Response Spectrum can be interpreted as the locus of maximum response of a SDOF for given damping ratio.Response Spectra thus helps in obtaing the peak structural responses under linear range,which can be obtained.Usually,response of a SDOF system is determined by time domain or frequency domain analysis,and for a given time period of system,maximum response is peaked.This process is continued  for all range of possible time periods of SDOF system.Finally,plot with system time period on x-axis and response quantity on y-axis is the required response spectra pertaining to specified damping ratio and input ground motion.Same process is carried out with different damping ratios to obtain overall response spectra.

1) Straight line Approximation: In the digital computation of spectra,the actual earthquake record  is replaced by linear segments between the points of digitization.This is a minor  approximation provides that the length of the time interval is much shorter than the periods of interest.

2) Trancation error : In general,a trancation error exists in numerical methods for integrating differential  equations.

3) Error due to rounding the time record: For earthquake records digitized at irregular time intervals,the integration technique proposed in this report requiresrounding of the time record and the attendant error depends on the way the rounding is done.For round-off to 0.05 sec,the average error in spectrum values is expected to be less than 2%.

4) Errors due to discretization: In any numerical method of computing the spectra,the response is obtained at a set of discrete points.Since,spectral values represent maximum values of response parameters which may not occur at these discrete points, discretization introduces an error which gives spectrum values lower than the true values.The error will be a maximum if the maximum response occurs exactly midway between two discrete points.

Torsion is major issue here to be discussed as it plays a vital role to select Analysis type,time of analysis,buildings effective cost etc..There is two types of torsion associated with types of irregularity of structures: 

A) Inherent torsion: This torsion includes the effects of the difference in location of the center of  rigidity from center of mass.

B) Accidental torsion: Accidental torsion includes unaccounted effects from non-uniform ground motion input,variations in floor masses over the life of the building,strength and stiffness degradation during non-linear response of lateral force resisting elements.The uncertainty in the location of the center of mass is normally the largest influence  to accidental torsion.

Load Combination in RSA: Response Spectrum Analysis is a useful tool to calculate resonant behavior of structure.As, in this analysis procedure ground shaking force is come into consideration,so load combination (specially for foundation) should be modified.According to various codes it is:  1) RSx+ 0.3*RSy+T       2)  RSy+0.3*RSx+T

Irregularity of Building: In analysis of Response Spectrum,Irregularity of building should meet the criteria of International Building Code(IBC),most likely Building Design shear & total base shear,Building total drift, Storey drift,Vertical Deflection,Lateral displacement and Drift due to irregularity should lies beyond level.

Whereas,

                The following Controlling value varies from Code to Code wrt geographical site condition                  & regional seismic sensitivity.
             
Such as,
                Boundary limit for total drift is H/250
                According to Indian Code & UBC'97
                Boundary limit for drift due to irregularity is Dmax/Davg > 1.2
                Boundary limit for drift due to extensive irregularity is  Dmax/Davg> 1.4
According to Canadain Code,
                Boundary limit for drift due to irregularity is Dmax/Davg > 1.4 
                Boundary limit for drift due to extensive irregularity is  Dmax/Davg> 1.7 

                                             

                                    

                      (THANKS TO ALL OF YOU)