Essay: Response spectrum method

To perform earthquake analysis and design of structure to be fabricated at a specific area, the actual time history record is needed. On the other hand, it is impractical to have such records at each and every location. Further, the earthquake analysis of the structure can’t be done basically taking into account the peak value of the ground acceleration as the response of the structure rely upon frequency content of the ground movement and its own dynamic characteristics. To conquer the above problems, seismic response spectrum is the most prevalent technique in the seismic investigation of the structure. There are computational benefits in using the response spectrum method of earthquake analysis for the evaluation of displacements and member forces in the structural system. The method includes the computation of only the maximum values of the displacement and member forces in each mode of oscillation using smooth design spectra which are the average of several earthquake motions.
Maximum response of the bridge is determine form the modal spectral analysis with the use of the spectrum of a given ground motion. Analysis model utilized for modal spectral analysis are linear elastic model based on effective stiffness characteristics and on assumed equivalent viscous damping ratios. With these requirement response spectrum analysis can be performed on bridges.
1) For bridges systems which are expected to perform essentially in the linear elastic range based on cracked or effective stiffness properties.
2) For inelastic response of bridge systems where the equivalent response is linearized to the initial effective stiffness and subsequently modified by means of equal energy or equal displacement.
Results from linear elastic model spectral analysis are most applicable to displacement response estimates since bridge structure respond typically with a fundamental period T greater than the period T0 at maximum intensity in the response spectrum, where equal displacement principle apply. Equivalent member forces need to be adjusted with the use of force reduction factor.
Furthermore, it should be recognized that only maximum modal response values are determined which do not occur at the same time during the earthquake time history. The modal combination techniques outlined in the IS: 1893 part 3(draft code) accounts for this effects of linear elastic bridge system. However since modal analysis techniques rely on the principle of superposition, these techniques are valid only as long as in addition to the linear elastic response, only small displacements occur. During the inelastic seismic response of bridge structure, displacements can exceed the small displacement as stipulated in small displacement theory, and frequencies and mode shapes can no longer be considered simple harmonics since they depends on the displacement amplitude.
Many modal analysis program provide an effective mass or mass participation factor for each mode. This mass participation factor, which is similar in physical meaning to the modal participation factor and has the same form except that the numerator is squared, can be used to determined how many modes should be considered in a given response direction. The sum of the effective masses for all modes in a given response direction must equal to the total mass of the bridge. Effective mass participation of 80% to 90% of the total bridge mass in any given response direction can be considered sufficient to capture the dominant dynamic response of the bridge structure.
Some bridge seismic codes specify the use of inelastic response spectra, which implies that the acceleration response spectrum has been modified to account for ductile member actions that result in reliable structural displacement ductilities. Member forces obtained from these inelastic acceleration response spectra can then be directly considered as actual maximum member demand. Indian standard code IS: 1893-part 3(draft version) considered the elastic response spectra in the response spectrum analysis of bridges.
Response spectra:
Response spectra are curves plotted between maximum response of SDOF system subjected to specified earthquake ground motion and its time period (or frequency). Response spectrum can be interpreted as the locus of maximum response of a SDOF system for given damping ratio. Response spectra thus helps in obtaining the peak structural responses under linear range, which can be used for obtaining lateral forces developed in structure due to earthquake thus facilitates in earthquake-resistant design of structures.
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 picked. This process is continued for all range of possible time periods of SDOF system. Final 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. The following figure shows that the response spectra used in the IS: 1893 part-3 (draft version).
Steps involved in response spectrum method:
Following steps required in response spectrum analysis as specified in IS: 1893 part-3:
1. An appropriate mathematical model with lumped mass system is formulated with the use of 2D or 3D finite element beam element. Mathematical model ought to suitably represent dynamic properties of the different element of the bridge such as superstructure (deck), bearing properties, substructure, and foundation with soil/rock spring. In medium or soft soil flexible base and in the rocky and very stiff soil fixed base can be considered.
2. Calculation of fundamental natural frequency and the mode shapes with the use of dynamic stiffness and mass matrix method or other standard approaches. Fundamental time period of the bridge member is to be calculated by any rational method of analysis adopting the Modulus of Elasticity of Concrete as per IRC:112, and considering moment of inertia of cracked section which can be taken as 0.75 times the moment of inertia of gross uncracked section, in the absence of rigorous calculation or The fundamental natural period T (in seconds) of pier/abutment of the bridge along a horizontal direction may be estimated by the following expression:
T= 2 * sqrt(D/1000F)
D = Appropriate dead load of the superstructure and live load in kN
F = Horizontal force in kN required to be applied at the centre of mass of superstructure for one mm horizontal deflection at the top of the pier/ abutment for the earthquake in the transverse direction; and the force to be applied at the top of the bearings for the earthquake in the longitudinal direction.
3. Determine total response by combining responses in various modes by
i. by mode combination procedure such as
‘ Absolute Sum (ABSSUM) Method,
‘ Square root of sum of squares (SRSS) method, and
‘ Complete quadratic combination (CQC) method
ii. Time-wise superposition of responses using ground motion time history(s).
In ABSSUM method, the peak responses of all the modes are added algebraically, assuming that all modal peaks occur at same time. The maximum response is given by
The ABSSUM method provides a much conservative estimate of resulting response quantity and thus provides an upper bound to peak value of total response. (Chopra, 2007)
In the SRSS method, the maximum response is obtained by square root of sum of square of response in each mode of vibration and is expressed by
The SRSS method of combining maximum modal responses is fundamentally sound where the modal frequencies are well separated. However, this method yield poor results where frequencies of major contributing modes are very close together.
In Complete Quadratic Combination (CQC) method the maximum response from all the modes is calculated as
Where ri and rj are maximum responses in the ith and jth modes, respectively and ??ij is correlation coefficient given by
Where ??i and ??j are damping ratio in ith and jth modes of vibration, respectively and
The range of coefficient, is 0 < ??ij < 1 and ??ii = ??jj =1. For the system having the same damping ratio in two modes i.e. ??i= ??j= ??, then 4. Calculation of horizontal seismic coefficient (Ah). It is calculated by the following expression: Provided that for any structure with T< 0.1 sec, the value of Ah will not be taken less than Z/2 whatever be the value of I/R Where, Z = Zone factor I = Importance factor, Table 2 R = Response reduction factor, Table 3 Sa/g= Average Acceleration coefficient for rock or soil sites as given in Fig.1. 5. Calculation of base shear. Base shear is calculated with the following expression. Fah= Ah*W Where W= seismic weight of the bridge.