Hysteretic MDOF Model to Quantify Damage for RC Shear Frames Subject to Earthquakes Köylüoglu, H. Ugur; Nielsen, Søren R. K.; Cakmak, Ahmet S.

Transcription

1 Aalborg Universitet Hysteretic MDOF Model to Quantify Damage for RC Shear Frames Subject to Earthquakes Köylüoglu, H. Ugur; Nielsen, Søren R. K.; Cakmak, Ahmet S. Publication date: 1996 Document Version Publisher's PDF, also known as Version of record Link to publication from Aalborg University Citation for published version (APA): Köylüoglu, H. U., Nielsen, S. R. K., & Cakmak, A. S. (1996). Hysteretic MDOF Model to Quantify Damage for RC Shear Frames Subject to Earthquakes. Aalborg: Dept. of Building Technology and Structural Engineering. Structural Reliability Theory, No. 148, Vol.. R9601 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.? You may not further distribute the material or use it for any profit-making activity or commercial gain? You may freely distribute the URL identifying the publication in the public portal? Take down policy f you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from vbn.aau.dk on: juli 05, 2018

2 NSTTUTTET FOR BYGNNGSTEKNK DEPT. OF BULDNG TECHNOLOGY AND STRUCTURAL ENGNEERNG AALBORG UNVERSTET AUC AALBORG DANMARK STRUCTURAL RELABLTY THEORY PAPER NO. 148 ~ Submitted to ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, Worcester, USA, August 1996 H. U. KOYLUOGLU, S. R. K. NELSEN & A. ~. QAKMAK HYSTERETC MDOF MODEL TO QUANTFY DAMAGE FOR RC SHEAR FRAMES SUBJECT TO EARTHQUAKES JANUARY 1996 SSN R9601

3 The STRUCTURAL RELABLTY THEORY papers are issued for early disserr{ination of research results from the Structural Reliability Group at the Department of Building Technology and Structural Engineering, University of Aalborg. These papers are generally submitted to scientific meetings, conferences or journals and should therefore not be widely distributed. Whenever possible reference should be given to the final publications (pni>ceedings, journa:ls, etc. ) and not to the Structural Reliability 'Theory papers. Printed at Aalborg University

4 NSTTUTTET FOR BYGNNGSTEKNK DEPT. OF BULDNG TECHNOLOGY AND STRUCTURAL ENGNEERN.G AALBORG UNVERSTET AUC AALBORG DANMARK STRUCTURAL RELABLTY THEORY PAPER NO. 148 Submitted to ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, Worcester, USA, August 1996 H. U. KOYLUOGLU, S. R. K. NELSEN & A. S QAKMAK HYSTERETC MDOF MODEL TO QUANTFY DAMAGE FOR RC SHEAR FRAMES SUBJECT TO EARTHQUAKES JANUARY 1996 SSN R9601

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6 Hysteretic MDOF Model to Quantify Damage for RC Shear Frames Subject to Earthquakes H. Ugur Koyliioglu 1, S!lren R. K. Nielsen 2 and Ahmet ~. <;akmak 3 ABSTRACT A hysteretic mechanical formulation is derived to quantify local, modal and overall damage in reinforced concrete (RC) shear frames subject to seismic excitation. Each interstorey is represented by a Clough and Johnston (1966) hysteretic constitutive relation with degrading elastic fraction of the restoring force. The local maximum softening damage indicators are based on the variation of the local stiffness and strength deterioration. The modal damage indicators are calculated from the variation of the eigenfrequencie~ of the structure. A statistical analysis is performed where a sample 5 storey shear frame is subject to simulated earthquake excitations, which are modelled as a stationary Gaussian stochastic process with Kanai-Tajimi spectrum, multiplied by an envelope function. The relationship between local, modal and overall damage indices is investigated statistically. 1. NTRODUCTON For RC structures modelled by non-linear mechanical theories, local damage can be quantified by the degradation of local stiffness and strength. Damage indicators are quantities characterizing the damage state of the structure after an earthquake excitation, and such can be used in decision-making during design, or in case of postearthquake reliability and repair problems. The maximum softening damage indicators (MSD) used here measure the maximum relative reduction of the vibrational frequencies for an equivalent linear system with slowly varying stiffness during a seismic event, hence, display the combined damaging effects of the maximum displacement ductility of the structure during extreme plastic deformations and the stiffness deterioration in the elastic regime. The authors have previously studied different merits of the MSD concept, including applications to available real data, testing the Markov property, prediction of future performance and reliability of damaged structures in a series of papers, the latter being Koyliioglu et al. (1995). 2. HYSTERETC MODEL FOR MDOF SHEAR FRAMES Consider ann storey RC shear frame. The relative displacement between the ith and (i + l)th storeys is designated Xi, and x1 signifies the displacement of the first storey relative to ground surface excited by the horizontal acceleration u 9 With a shear force of the magnitude Qimi where mi is the storey mass, the equations of motion in terms of the relative displacements are : 1 Ko<; University, TR istinye, istanbul, Turkey. 2 Aalborg University, DK-9000, Aalborg, Denmark. 3 Princeton University, Princeton, NJ 08544, USA.

7 2 Figm;e 1. MDOF shear frame. iig( t) Figure 2. Clough-Johnston hysteretic model. X = J.l2Q2 - xi = J.li+ Qi+l - Ql - Ug (J.Li + 1 )Qi + Qi-1 Xn = -(J.Ln + l)qn + Qn-1 t>o t>o t>o i = 2, 3,, n - } {) ffij J.li = - ffii-1 (2) (3) (4) (5) 2zo i ( - 2zo + D 1 1 )no; a - ', 1 (6) k(xi, zi, Di; zo,i) = H(zi){ AiH(ii)(l- H(zi- zo,i)) + H( -xi)}+ H( -zi){ AiH( -ii)(l- H( -zi- zo,i)) + H(xi)} (7)

8 3 H(x)= { ~ x~o x<o (8) (9) (10) ai ( Di) = elastic fraction of the restoring force which is a function of damage. The Clough-Johnston model deals with the stiffness degradation by changing the slope Ai of the elastic hranches as the accumulated plastic deformations Dt and Di at positive and negative yielding increase, see Fig.2. Di = Dt + Di = total accumulated plastic deformations. A novelty is the modelling of ai(di) as a non-increasing function of the damage parameter Di. Since, ai(di) measures the fraction of the restoring force from linear elastic behaviour, this fraction must decrease from 1 as more and more parts of the structure become plastic. 3. MODAL, LOCAL AND OVERALL MSD The modal MSD for the jth mode, DM,j, is defined as To DM,j = max {1- TiC:)} (11) To,j = jth and Tj(t) = jth period of the linear and equivalent linear structure. Locally, a hysteretic loop-averaged softening value Si(t) is defined using the average slope mi. 2zo,i ffij=--~!.,;,_- 2zo,i + Di(t) (12) Si(t) = 1- -Jmi(1- ai) + ai (13) Si(t) is non-decreasing during a seismic event and fully correlated to Di(t). The ith local MSD SM,i is defined as the maximum of Si(t). Si(t) = 0 denotes no local damage in the columns and Si(t) = 1 means total collapse of columns under the ith storey. A scalar numerical overall damage indicator is defined with weights of modal participation factors for modal SM and with equal weights for local MSD SM.

9 4 4. NUMER CAL NVEST GATONS Consider a five storey RC shear frame. All storeys have the same mass, stiffness and damping characteristics. The parameters are : f-li = 1, wo,i = 77r sec- 1, (o,i = 0.03, zo,i = 24 mm and no,i = 0.8 fori= 1, 2,, 5. Then, the first two eigenfrequencies are 1.00 Hz and 2.90 Hz. The ground excitation, ii. 9 (t) is taken as ii. 9 (t) = E(t)V(t), with. { c 1 t Et - c 2 e ( ) - -ca(t-to) t <to t >to (14) (15) For all cases, to = 7 sec, ( 9 = 0.3, S 0 = 1. A match in w 9 to the jth frequency of the structure denotes an earthquake exciting the jth mode the most. Two different types of ground motion exciting different modes and with statistically equivalent energy contents named Type A and B are utilized. For Type A, (cl, c2, c3) = (0.005, 0.035, 0.2). For Type B, (ell c2, c3) = ( , , 0.2). The simulation of the stationary Gaussian stochastic processes is performed using the procedure of Shinozuka et al. (1991). A statistical analysis based on Monte Carlo simulations is performed. 30 realizations are generated and the results are tabulated below in Tables 1, 2 and 3. The results are consistent with the mode shapes. Type A excitation would cause the most damage. The coefficient of variation in the Type A excitation is observed to be relatively small compared to Type B. This shows that the reliability models can estimate small and sharper confidence intervals for severe damage compared to light damage. 5. REFERENCES [1] Clough, W. and Johnston, S.B. (1966) Effect of Stiffness Degradation on Earthquake Ductility Requirements, Proc. 2nd Japan Earthquake Symposium, [2] Koyliioglu, H.U., Nielsen, S.R.K., Abbott,J. and Qakmak, A.S. (1995) Local and Modal Damage ndicators for Reinforced Concrete Shear Frames subject to Earthquakes, University of Aalborg, Structural Reliability Theory Paper No. 145, SSN R9521. Submitted to J. Engineering Mechanics, ASCE. [3] Shinozuka, M. and Deodatis, G. (1991) Simulation of Stochastic Processes by Spectral Representation, Applied Mechanics Reviews, 40,

10 5 Table 1. Mean and coefficient of variation of the local and overall MSD Earthquake type SMl SM2 SMa SM4 SMs Type A ( w 9 = 6.6) 0.449, , , , ,- Type B (w 9 = 19.2) 0.167, , , , ,- SM 0.208, , Table 2. Mean and coefficient of variation of modal and overall MSD Earthquake type 8M1 8M2 Type A ( w 9 = 6.6) 0.354, , Type B (w 9 = 19.2) 0.109, , DM 3 DM4 8Ms 0.240, , , , , , M 0.301, , g~ N ~~ n! 1 o 1 ~ 20 :-l=:===; t ( ~. ( ~. ;! { ~~ x ~..~,_ (!!. 11!> :.:o t ~-~~ t (!o 0 ~ :!C ~-:~ X ;;,_ l,. 1'\ t (1!. l ( ~ :.!h t AJ~r"w, lt~e~ A...A f-... Mrvvv--- :~ : ~, )h l!o.,:l o ~] u]".. " "] Ul.. ~:~ u]" " + U lu " ' U "] r( Ul " ' U ~ ~ i u:l ' U t '.. 1'. ~ : 11 J', : ~ 11 '.:11 ' :: 1! Figure 3. Type A excitation and x;(t) in mm. Figure 4. Corresponding S;(t).

11 6 Table 3. Correlation between the local, modal and overall MSD for Type A excitation SM,l SM,2 SM,3 SM,4 SM,5 SM t5m,l t5m,2 t5m,3 t5m,4 t5m,5 8M SM,l SM, SM, SM, SM, SM t5m,l t5m, :997 t5m, t5m, t5m, M 1.000

12 STRUCTURAL RELABLTY THEORY SERES PAPER NO. 121: H. U. Koyliioglu, S. R. K. Nielsen & A. S. Qakmak: Applications of nterval Mapping for Structural Uncertainties and Pattern Loadings. SSN R9411. PAPER NO. 122: H. U. Koyli.ioglu, S. R. K. Nielstm & A. S. Qakmak: Fast Cellto- Cell Mapping (Path ntegration) with Probability Tails for the Random Vibration of Nonlinear and Hysteretic Systems. SSN R9410. PAPER NO. 123: A. A kar, H. U. Koyli.ioglu, S. R. K. Nielsen & A. S. Qakmak: Faster Simulation Methods for the Nonstationary Random Vibrations of Nonlinear MDOF Systems. SSN R9405. PAPER NO. 125: H.. Hansen, P. H. Kirkegaard & S. R. K. Nielsen: Modelling of Deteriorating RC-Structures under Stochastic Dynamic Loading by Neural Networks. SSN R9409. PAPER NO. 1_26: H. U. Koyli.ioglu, S. R. K. Nielsen & A. S. Qakmak: Reliability Approximations for MDOF Structures with Random Properties subject to Random Dynamic Excitation in Modal Subspaces. SSN R9440. PAPER NO. 127: H. U. K0yli.ioglu, S. R. K. Nielsen and A. ~- Qakmak: A Faster Simulation Method for the Stochastic Response of Hysteretic Structures subject to Earthquakes. SSN R PAPER NO. 128: H. U. Koyli.ioglu, S. R. K. Nie'lsen, A. S. Qakmak & P. H. Kirkegaard: Prediction of Global and Loealized Damage and Jlutu1'e Reliability for RC Structures subject to Earthquakes. SSN R9426. PAPER NO. 129: C. Pedersen & P. Thoft-Christensen: nteractive Structural Optimization with Quasi-Newton Algorithms. SSN 09(i) R9436. PAPER NO. 130:. Enevoldsen & J. D. S!Z!rensen: Decomposition Techniques and Effective Algorithms in Reliability-Based Optimization. SSN R~412. PAPER NO. 131: H. U. Koyli.ioglu, S. R. K. Nielsen & A. S- Qakmak: Approximate Forward Difference Equations for the Lower Order Non-Stationary Statistics of Geometrically Non-Linear Systems subject to Random Excitation. SSN R9422. PAPER NO. 132:. B. Kroon: Decision Theory applied to Structural Engineering Problems. Ph.D.-Thesis. SSN R9421. PAPER 133: H. U. Koyli.ioglu, S. R. K. Nielsen & A. S. Qakmak: Stoehas tic Dynamics of N onlinear Structures with Random Properties subject to Random.Stationary Excitation. SSN R9520. PAPER NO. 134' H. U. K0yli.ioglu, S. R. K. Nielsen & A. S. Qakmak Solution of Random Structural System subject to Non-Stationary Excitation: Transforming the Equation with Random Coefficients to One with Deterministic Coefficients and Random nitial Conditions. SSN R94!29. PAPER NO. 135: S. Engelund, J. D. S!Z!rensen & S. Krenk: Estimation of the Time to nitiation of Corrosion in Existing Uncracked Concrete Structures. lssn R9438.

13 STRUCTURAL RELABLTY THEORY SERES PAPER NO. 136: H. U. Koyliioglu, S. R. K. Nielsen & A. S. Qakmak: Solution Methods for Structures with Random Properties subject to Random Excitation. SSN R9444. PAPER NO. 137: J. D. S!Zlrensen, M. H. Faber &. B. Kroon: Optimal Reliability-Based Planning of Experiments for POD Curves. SSN R9455. PAPER NO. 138: S.R.K. Nielsen & P.S. Skjrerbrek, H.U. Koyliioglu & A.S. Qakmak: Prediction of Global Damage and Reliability based upon Sequential dentification and Updating of RC Structures subject to Earthqv,akes. SSN R9505. PAPER NO. 139: R. wankiewicz, S. R. K. Nielsen & P. S. Skjrerbrek: Sensitivity of Reliability Estimates in Partially Damaged RC Structures subject to Earthquakes, using Reduced Hysteretic Models. SSN R9507. " PAPER NO 141: H. U. Koyliioglu, S. R. K. Nielsen & A. S. Qakmak: Uncertain Buckling Laad and Reliability of Columns with Uncertain Properties. SSN R9524. PAPER NO. 142: S. R. K. Nielsen & R. wankiewicz: Response of Non-Linear Systems to Renewal mpulses by Path ntegration. SSN R9512. PAPER NO. 145: H. U. Koyli.ioglu, S. R. K. Nielsen, Jamison Abbott and A. S. Qakmak: Local and Modal Damage ndicators for Reinforced Concrete Shear Frames subject to Earthquakes. SSN R9521 PAPER NO. 146: P. H. Kirkegaard, S. R. K. Nielsen, R. C. Micaletti and A. S. Qakmak: dentification of a Maximum Softening Damage ndicator of RC-Strv.ctures using Time Frequency Techniques. SSN R9522. PAPER NO. 147: R. C. Micaletti, A. S. Qakmak, "8. R. K. Nielsen & P. H. Kirkegaard: Construction of Time-Dependent Spectra using Wavelet Analysis for Determination of Global Damage. SSN R9517. PAPER NO. 148: H. U. Koyliioglu, S. R. K. Nielsen & A. S. Qakmak: Hysteretic MDOF Model to Quantify Damage for TC Shear Frame.s subject to Earthquakes. SSN R9601. PAPER NO. 149: P. S. Skjrerbrek, S. R. K. Nielsen & A. S. Qakmak: Damage Location of Severely Damaged RC-Structures based on Measured Eigenperiods from a Single Response. SSN R9518. PAPER 151: H. U. Koyliioglu & S. R. K. Nielsen: System Dynamics and Modified Cumulant Neglect Closure Schemes. SSN R9603. Department of Building Technology an~ Structural Engineering Aalborg University, Sohngaardsholmsvej 57, DK 9000 Aalborg Telephone: Telefaoc: