Dynamic Response of SDOF System: A Comparative Study Between Newmark s Method and IS1893 Response Spectra

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1 Dynaic Response of SDOF Syste: A Coparative Study Between Newark s Method and IS893 Response Spectra [] P. Manoj Reddy [2] Dr. A. Viala [] Post Graduate Student, [2] Associate Professor [][2] Chaitanya Bharathi Institute of Technology, Hyderabad, India. Abstract: There are two basic approaches to nuerically evaluate the dynaic response of the structures. The first approach is nuerical interpolation of the excitation and the second is nuerical integration of the equation of otion. In the present study, the second approach (Newark s ethod) is used to find the response of a set of 40 structures idealised as SDOF syste. The 40 structures are chosen in such a way that their tie period varies fro 0. to 4 sec. The response of all 40 structures is evaluated for 8 past Indian earthquakes. A progra is developed in MATLAB to find the peak response of each structure for each earthquake using Newark s ethod. Average peak response of each structure under eight earthquakes is calculated. A response spectru is developed as average peak response versus tie period of 40 structures. As a second part of the work, the response of 40 structures is evaluated as per response spectra given in IS893: 2002 based on tie period of the structure. A coparative study is ade between Newark s response and the response as per IS893 response spectra. The study reveals that the responses calculated using IS893 response spectra and Newarks ethod are not siilar for all the earthquakes. Key word:-- Response spectru, Newarks ethod, Matlab. I. INTRODUCTION Response spectra provide a very handy tool for engineers to quantify the deands of earthquake ground otion on the capacity of buildings. Data on past earthquake ground otion is generally in the for of tie-history recordings obtained fro instruents placed at various sites that are activated by sensing the initial ground otion of an earthquake. The aplitudes of otion can be expressed in ters of acceleration, velocity and displaceent. The first data reported fro an earthquake record is generally the peak ground acceleration (PGA) which expresses the tip of the axiu spike of the acceleration ground otion (Fig ). Fig Typical accelerogra Although useful to express the relative intensity of the ground otion (i.e., sall, oderate or large), the PGA does not give any inforation regarding the frequency (or period) content that influences the aplification of building otion due to the cyclic ground otion. In other words, tall buildings with long fundaental periods of vibration will respond differently than short buildings with short periods of vibration. Response spectra provide these characteristics. Picture a field of lollipop-like structures of various heights and sizes stuck in the ground. The stick represents the stiffness (K*) of the structure and the lup at the top represents the ass (M*). The period of this idealized single-degree-of-freedo (SDOF) syste is calculated by the equation: T = 2π (M * / K*) /2 If the peak acceleration (S a ) of each of these SDOF systes, when subjected to an earthquake ground otion, is calculated and plotted with the corresponding period of vibration (T), the locus of points will for a response spectru for the subject ground otion. Thus, if the period of vibration is known, the axiu acceleration can be deterined fro the plotted curve. When calculating response spectra, a noinal percentage of critical daping is applied to represent viscous daping of a linear-elastic syste, typically five-percent. All Rights Reserved 207 IJERMCE 305

2 II. LITERATURE: Japanese Society of civil engineers. (2000) stated that Response spectra can also be used in assessing the response of linear systes with ultiple odes of oscillation (ulti-degree of freedo systes), although they are only accurate for low levels of daping. Modal analysis is perfored to identify the odes, and the response in that ode can be picked fro the response spectru. This peak response is then cobined to estiate a total response. The result is typically different fro that which would be calculated directly fro an input, since phase inforation is lost in the process of generating the response spectru. The ain liitation of response spectra is that they are only universally applicable for linear systes. Response spectra can be generated for non-linear systes, but are only applicable to systes with the sae non-linearity, although attepts have been ade to develop non-linear seisic design spectra with wider structural application. Bharat Bhushan Prasad, (200) stated that PGA, PGV, PGD are the ost coon and easily recognizable tie doain paraeters of the strong ground otion. Typically, the ground otion records tered seisograph or tie histories have recorded acceleration (these records are tered accelerogras) for any years in analogue for and ore recently, digitally. The axiu aplitude of recorded acceleration is tered as peak ground acceleration (PGA) peak ground velocity (PGV) and peak ground displaceent (PGD) are the axiu respective aplitudes of velocity and displaceent. The focus is usually on peak horizontal acceleration (PHA) due to its role in deterining the lateral inertial forces in structures. Pankaj Agarwal and Manish shrikande, (2006) stated that the seisic force generated in structures varies according to their dynaic properties even though they stand on the sae ground and are subjected to the sae seisic otion The paraeters representing the properties of structures and reflected directly in seisic otion, are the natural period and daping constant.the response spectru thus represents the relation between seisic wave shape and response value as a function of the natural period and daping constant of the structure. III. METHODOLOGY In 959, N.M. Newark developed a faily of tiestepping ethods based on the following equations: ú i+ = ú i + [ ( - ) Δt ] ü i + ( Δt) ü i+ (a) u i+ = u i + ( Δt ) ú i + [ (0.5 - β) (Δt) 2 ] ü i + [β (Δt) 2 ] ü i+ (b) Equation can be reforulated to avoid iteration and to use increental quantities: Δu i = u i+ u i Δú i = ú i+ ú i Δü i = ü i+ ü i (2) Δp i = p i+ p i While the increental for is not necessary for analysis of linear systes, Equation can be rewritten as Δú i = (Δt) ü i + ( Δt) Δü i Δu i = (Δt) ú i + Δt 2 i + β (Δt) 2 Δü 2 i (3) The second of these equations can be solved for Δü i = β Δt 2Δu i i ü i β Δt 2β (4) Substituting Equations (4) into Equation (3a) gives Δú i = Δu i ú i + Δt( - ) ü i (5) β Δt β 2β Next, Equations (4) and (5) are substituted into the increental equation of otion: Δü i + c Δú i + k Δu i = Δp i (6) Obtained by subtracting equation ( ü i + cú i + (fs) i = p i ) fro (ü i+ + cú i+ + (fs) i+ = p i+ ) both specialized to linear systes with (fs) i = ku i and (fs) i+ = ku i+. This substitution gives kcapδu i = Δ pcapi (7) Where, kcap = k + βδt And, Δ pcapi = Δp i + ( βδt + β c )ú i + [ 2β c + (8) β Δt 2 + Δt ( 2β ) c ] ü i (9) With kcap and Δ pcapi known fro the syste properties, k, and c, algorith paraeters and β, and the ú i and ü i at the beginning of the tie step, the increental displaceent is coputed fro Δu i = Δ pcapi kcap (0) Once Δu i is known, Δú i and Δü i can be coputed fro Equations (5) and (4), respectively, and u i+, ú i+, and ü i+ fro equation (2). The acceleration can also be obtained fro the equation of otion at tie i+, p i+ c ú i+ ku i+ ü i+ = () Below steps suarizes the tie-stepping solution using Newark s ethod as it ight be ipleented on the coputer.. Average acceleration ethod ( = ½, β = ¼) Initial calculations ü 0 = po cúo kuo. selectδt. k cap = k + c + βδt β Δt 2. a = + c: and b = + Δt ( )c βδt β 2β 2β 2. Calculations for each tie step, i All Rights Reserved 207 IJERMCE 306

3 2. Δpcapi = Δp i + aú i + bü i. 2.2 Δu i = Δpcapi. kcap 2.3 Δú i = Δu i ú i + Δt (- ) ü i. βδt β 2β 2.4 Δü i = β Δt 2Δu i ú i ü i. βδt 2β 2.5 u i+ = u i + Δu i, ú i+ = ú i + Δú i, ü i+ = ü i + Δü i. 3.0 Repetition for the next tie step. Replace i by i+ and ipleent steps 2. to 2.5 for the next tie step. By using the above steps a progra was developed in MATLAB software. This progra is very useful to find the spectral acceleration, velocity, and displaceent. A list of 40 structures was odeled insuch way that their tie periods varies fro 0. to 4 sec. IV. RESULTS AND DISCUSSION The peak responses of 40 structures idealized as SDOF is calculated using Newarks ethod for past 8 Indian Earthquakes. The 8 Earthquakes are of ediu to low agnitudes highest of 6.8 agnitudes. Earthquakes which are taken in this study are Uttarkashi, chaoli, Barkot, Dharshala, Northeast India, India-Bangladesh, Xizang-India, Chaba. Figure 4. shows peak response of 40 structures due to Uttarkashi Earthquake and copared with IS 893:2002 response spectru. This graph shows the clear dissiilarity between response spectru obtained by Newarks ethod and IS 893:2002 response spectru. The peak value in Newarks ethod is 4.8g where as in IS 893:2002 response spectru is 2.5g. Fig (4.) Uttarkashi response spectru. Figure 4.2 shows peak response of 40 structures due to Chaoli Earthquake and copared with IS 893:2002 response spectru. This graph shows the clear dissiilarity between response spectru obtained by Newarks ethod and IS 893:2002 response spectru. The peak value in Newarks ethod is 0.65g where as in IS 893:2002 response spectru is 2.5g. Figure 4.3 shows peak response of 40 structures due to Barkot Earthquake and copared with IS 893 Fig (4.2) Chaoli response spectru. response spectru. This graph shows the clear dissiilarity between response spectru obtained by Newarks ethod and IS 893:2002 response spectru. The peak value in Newarks ethod is 0.7g where as in IS 893:2002 response spectru is 2.5g. Fig (4.3) Barkot response spectru. Figure 4.4 shows peak response of 40 structures due to due to Dharashala Earthquake and copared with IS893:2002 response spectru. This graph shows the clear dissiilarity between response spectru obtained by Newarks ethod and IS 893:2002 response spectru. The peak value in Newarks ethod is.2g where as in IS 893:2002 response spectru is 2.5g. Figure 4.5 shows peak response of 40 structures due to due to Northeast India Earthquake and copared with IS893:2002 response spectru. The peak value in Newarks ethod is 0.3g where as in IS 893:2002 response spectru is 2.5g. Also fro 0.2sec to 4 sec the Sa/g value is alost 0.sec which is very less value when copared to IS 893:2002 response spectru, this graph shows the ost dissiilarity than any other graphs. Fig (4.4) Dharashala response spectru. All Rights Reserved 207 IJERMCE 307

4 893:2002 response spectru. The peak value in Newarks ethod is 0.2g where as in IS 893:2002 response spectru is 2.5g. Fig (4.5) NE India response spectru Figure 4.6 shows peak response of 40 structures due to due to India-Bangladesh Earthquake and copared with IS893:2002 response spectru. This graph shows the clear dissiilarity between response spectru obtained by Newarks ethod and IS 893:2002 response spectru. The peak value in Newarks ethod is 0.25g where as in IS 893:2002 response spectru is 2.5g. Figure 4.7 shows peak response of 40 structures due to due to Xizang-India Earthquake Earthquake and copared with IS893:2002 response spectru. This graph shows the clear dissiilarity between response spectru obtained by Newarks ethod and IS 893:2002 response spectru. The peak value in Newarks ethod is 0.2g where as in IS 893:2002 response spectru is 2.5g. Fig (4.8) Chaba response spectru Average peak response of 8 earthquakes for each structure is calculated and one standard deviation is added for each average peak response of structure. A response spectru is developed for average peak responses of 40 structures. The obtained response spectru is copared with IS893:2002 response spectru (Fig.4.9). The Sa/g values fro tie period 0.5 to sec are not identical. Fig (4.6) India Bangladesh response spectru. Fig (4.7) Xizang-India response spectru Figure 4.8 shows peak response of 40 structures due to due to Chaba Earthquake Earthquake and copared with IS893:2002 response spectru. This graph shows the clear dissiilarity between response spectru obtained by Newarks ethod and IS Fig (4.9) calculated response spectru and IS 893 response spectru. V. CONCLUSION ) A coparison between the response spectru developed by 8 Indian Earthquakes using Newark s ethod and the response spectru given in IS893 (part): 2002 have done. 2) Sa/g values of Uttarkashi has higher peak value than IS ) Sa/g values fro tie period 0.5 to sec are also not identical, this displays the response spectru provided in IS 893 are not so accurate for these Earthquakes. 4) Sa/g values dissiilarity is clearly presented in this paper. REFERENCES [] Anil K.Chopra, Dynaics of Structures, Pearson Education India publisher, third edition 2007 All Rights Reserved 207 IJERMCE 308

5 [2] Bharat Bhushan Prasad, Advanced Soil Dynaics and Earthquake engineering, published by PHI Learning Pvt. Ltd. 20. [3] Bijan Mohraj - A study of Earthquake response spectra for different geological condition, Journal: bulletin of the seisological society of Aerica, vol. 66, No. 3, pp, June 976 [4] Glen V. Berg, Eleents of Structural Dynaics, Prentice Hall publisher, 989 [5] H. Bolton Seed, Celso Ugas, And John Lyser - Site-Dependent Spectra for Earthquake-Resistant Design, journal: Bulletin of the Seisological Society of Aerica. Vol. 66, No., pp February 976 [6] Hideji Kawakai, Hidenori Mogi and Eric Augustus J. Tingatinga - A note on spatial variations in response spectra of earthquake ground otions,iset Journal of Earthquake Technology, Paper No. 478, Vol. 44, No., March 2007, pp [7] I. D. Gupta and R. G. Joshi - Response spectru superposition for structures with uncertain properties, Journal of Engineering Mechanics, Vol. 27, No. 3, March, 200. [8] IS 893 (part): 2002, Criteria for Earthquake Resistant Design of Structures, Bureau of Indian Standards (BIS), New Delhi. [9] Sudhir K Jain, - Strong Ground Motion and Concept of Response Spectru, IIT Gandhinagar, India.March 203. All Rights Reserved 207 IJERMCE 309