Evaluation of the Higher Modes of Vibration on Rigid Response Coefficient and Earthquake Resistance Design of Structures

Structures need to be more flexible in order to withstand significant earthquakes, which cause undesired vibrations, deformations and accelerations. When structures are subjected to excessive vibration, it can lead to a variety of negative effects on people and the environment, including human discomfort, energy waste, structural part failure, & transmission of harmful forces and, in the worst case scenario, a collapse. Structures susceptible to dynamic loads, such as earthquakes must be studied in detail in order to eliminate the harmful consequences of vibrations. There's a lot of work to be done by structural engineers today to develop creative design concepts that protect civil engineering structures and their contents. Inelastic cyclic deformations at structural elements' intricately detailed plastic hinge areas may be a source of energy loss in such systems. Recent years have seen a rise in the use of novel methods to improve structural functionality and safety against dynamic loads. A mode of the structure in an oscillating system, in which all parts of the structure are oscillating with the same frequency. The sum of mass participation in structure vibrations calculated in the modal analysis increases as the number of modes increases. However, to obtain 90-percent mass participation required by codes, it is necessary to consider contribution of large number of eigenmodes, which lengthen calculations and analysis results. The limit of total missing mass could be 10%. The seismic response of vibration consists of two combinations of two different motions, which are referred in structural dynamics as the damped-periodic motion (also called as simply ―periodic‖) response and the ―rigid‖ response. (In the theory of vibrations, these two parts of motion are referred to as ―transient response‖ and ―steady-state response‖ respectively.) The periodic responses have the frequencies of the oscillators (or individual modes), and the rigid responses have the frequencies of the input motion.

Between the end of the region of amplified spectral acceleration, D, and the beginning of the rigid region E, the modal responses consist of both the periodic and rigid components. Appropriate methods, should be used to separate the two components in this transition region. The Gupta method and Lindley-Yow method are considered acceptable. Gupta separated the periodic and rigid components of a response by a rigid response coefficient αi. The periodic response component of Rpican then be expressed as follows: The rigid response component of a modal response, Rri, is defined as follows:

αi = rigid response coefficient With proper selection of key frequencies rigid response coefficient can be idealized as follows: Finally, after calculating the total periodic response, total rigid response, and residual rigid response, an appropriate combination method, should be used to obtain the total response. Periodic Response of ONE Mode =

Rigid Response of ONE Mode = Periodic Response of ALL Modes = for n = number of modes below fzpa Rigid Response of ALL Modes = RRI = + RMissing mass OR RRI = RStatic ZPA Complete Response of ALL Modes =

RI = [R2PI +R2RI]1/2

Combining Effects Caused by Three Spatial Components of an Earthquake:

R= [R2IX + R2IY + R2IZ]1/2

OR

R= RIX + 0.33 RIY + 0.33 RIZ

The present methodologies used, and recommendations specified by seismic building codes of practice for the seismic analyses of structures, when high frequency modal responses are involved are reviewed. The 90% modal mass criteria defined by design codes in the number of modes measured for analysis are studied with the help of model studies. The effect of the damping ratio on the rigid response coefficient is studied. Regression analysis is conducted for determining rigid frequency and rigid response coefficient. The expressions are validated by comparing output of the model analysis. The variation of mass throughout the structure when higher modes are truncated is studied. Intense studies are conducted with the help of statistical patterns to study the contributions of the rigid and periodic parts of the response and modal mass corresponding to the residual mode, by varying the frequencies and mass contribution of the truncated higher modes. The approximate residual mode contribution is determined to the contributions of truncated higher modes. The modified residual mode method is used for developing a simplified procedure for the design of structures with vertical mass and stiffness irregularity. The method is validated using different modes. Selection of Structures& data collection The design model of real-life structures are considered for the study. Total twenty-three structures were modelled using the software. Out of these twenty models are of real-life projects with different seismic zones, different soil type, different materials. These structures include residential, commercial, and industrial. Three models are sample models used for validation purpose with standard database. The summery table on the software models is as below. Table 1: Model Summary Methods & tools for analysis Dynamic analysis is carried out by developing programme in MathCAD and validating it with professional structural analysis software STAAD.Pro and ETABS. Seismic analysis is carried out for each model in following Steps-

• Structural configuration

• Section and Material Properties

• Member Specification

Dynamic analysis of all the models were carried out for different variable factors. An earthquake force is applied in both (X and Z) horizontal directions. The output of the results isanalyzed on the basis of following parameters. Results of analysis are investigated for following Parameters:

• RC & Steel Structures

• Damping

• Results at 90% modal participation

• Results upto 99% - 100 % modal participation

• Number of Mode Shapes at 90% & 100 % modal participation

• Natural Period (Natural Frequency) at 90% & 100 % modal participation

• Rigid Response coefficient

• Residue Rigid Response

• Base Shear at 90% &upto 99% - 100 % modal participation

• Drift at 90% &upto 99% - 100 % modal participation

• Effect of Soil Type at 90% &upto 99% - 100 % modal participation

Similarly results of all models are evaluated and summarizedas below.

 Performance of steel structures and concrete structures are very well differentiated.  Steel structures required higher frequency to reach to desired mass participation.

 Structures with irregular configuration need to consider modes with higher natural frequency.  Response of all structures converges at higher frequencies.  Convergence both steel structures and concrete structures happens at higher modal frequencies.  Residue Rigid Response of steel and concrete structures is similar.

 Structural Rigid Response achieved at frequency 33Hz for 5 % damping ratio, which is consistent with CQC Method defined in IS 1893

 Rigid response frequency decreases with increase in damping ratio.  Rigid Response coefficient for different damping values converges at frequency 33 Hz and above this frequency, the periodic component of the modal response is essentially zero.  Rigid Response coefficient becomes 1 at 33 Hz frequency for different damping ratios.

 As the structural irregularity increases, modal mass participation of modes at higher frequency increases.

performance.

 In most of structures 90% modal mass participation is achieved below frequency 33Hz. To get Total response one need to consider response of frequencies higher than 33Hz up to 80Hz. Most of times response at higher modes is rigid.  Frequency ratio of frequency at 100% mass participation to frequency at 90% mass participation lies between 2 to 10.

 Error in Base Shear at 90% mass participation and 100% mass participation lies between 3% to 20%. Effect of this error may get amplify in moment at base. response will be in the range of2.5% to 25%.  The error at higher modal frequencies can be eliminated by considering effect of Residue (Missing) Mass.

 Error in Base Shear at 90% mass participation and 100% mass participation lies between 3% to 20%. Effect of this error may get amplify in moment at base.  Error in storey shear at 90% Modal mass increases with storey height. The Error can be reduced by adding effect of Residual Rigid Response at higher frequencies.  SRSS method underestimate Base Shear by 3 to 20% as compared to CQC Method.  CQC Method is more acceptable method as compared to SRSS Method for modal combination.

 Very little difference is observed between drift at 90% mass participation and drift at 100% mass participation

For regular buildings, building rules of practice specify shear is underestimated at both the top and bottom levels of the building, according to instances cited in the paper as part of a novel strategy, truncated upper modes are taken into consideration by employing a modified residual mode to calculate the response. This method produces reasonable results in response calculations, as can be seen from the provided examples. Using this method, the pushover analysis of structures can include contributions from higher modes other than first mode, making it elastic. For regular buildings, first few modes (Modes with lower frequency) have major contribution in overall vibration of the structure and contribution of higher modes (Modes with higher frequency) is considerably very less in overall vibration of the structure. For Irregular buildings, contribution of first few modes (Modes with lower frequency) is decreases in overall vibration of the structure and contribution of higher modes (Modes with higher frequency) goes on increasing with increase in irregularity of the structure. Hence particularly for structures with large amount of irregularity involved in it, then it is suggested to consider contribution of higher modes in earthquake resistant design of the structure. It is also recommended to perform nonlinear dynamic analysis for high rise structures with large amount of irregularity.  At lower frequencies response of structure is mainly periodic. Higher frequencies consist of ‗periodic‘ response and the ‗rigid‘ response.  Gupta Method separates the periodic and rigid components of a response by a rigid response coefficient.  Rigid response coefficient converges about 90% modal participation.  No effect of Soil type on frequency and modes shape; but there is effect on base shear. Base shear increases with soft soil.  SRSS Method underestimate forces by 3% to 20% as compared to CQC Method in combination of closely spaced modes. Complete Quadratic Combination (CQC) Method is more accurate for combination of closely spaced modes.  Most of Structures reaches to Rigid Response up to frequency 33Hz. At higher frequencies (beyond 33Hz) rigid response is predominate (ZPA) having frequencies that of the input motion.  Ignoring contribution of Residue (Missing) Mass of 1% to 10% at higher frequencies  Rigid Response Error between Modes with 90% modal participation to total response lies between 3% to 20% . This error can be eliminated by considering Rigid Response of Residue Modes of higher modes.  Structures with irregular configuration, participation of higher frequency modes are more significant. Effect of higher modes could not be ignored for irregular structural configuration.  To obtain 90-percent mass participation required by codes, it is necessary to calculate a large number of eigenmodes, lengthening calculations. Residue Mode consideration is useful without increasing the number modes.  For Flexible Structures (Ductile, Tall, Irregular, Steel) to attain 90% participation factor, large number of modes with higher frequency need to be considered. Hence for these structures it is important to consider higher modes with residue mass participation.

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Research Scholar, Shridhar University