Seismic Fragility Assessment of Transmission Towers via Performance-based Analysis

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1 Seismi Fragility Assessment of Transmission Towers via Performane-based Analysis Liyu XIE, Jue TANG, Hesheng TANG & Qiang XIE Institute of Strutural Engineering and Disaster Redution, College of Civil Engineering, Tongji Universiy, Shanghai, China Songtao XUE Department of Arhiteture, Tohoku Institute of Tehnology, Sendai, Japan College of Civil Engineering, Tongji University, Shanghai, China SUMMARY: The knowledge and assessment of the seismi fragility urve is important to evaluate the integrity and reliability of transmission towers. In this paper, the seismi apaity assessment of the transmission tower is performed within a probabilisti frame, through a nonlinear bukling analysis and nonlinear dynami analysis, onsidering the internal unertainty of the tower and the randomness of ground motion. The performane limits of different damage states of transmission towers are determined. Finally, the seismi fragility urve of the transmission tower is evaluated by numerial Monte Carlo simulation. By the seismi fragility urve, the failure probability of the transmission tower under different magnitudes of earthquake an be visually predited. Keywords: Seismi, transmission towers, nonlinear bukling analysis, fragility urve, Monte Carlo simulations 1. INTRODUCTION Overhead transmission lines play an important role in the operation of a reliable eletri power system. Transmission towers are the vital omponents providing the supporters of high-voltage power lines. Many intensive earthquakes have happened in China reently, suh as Jiji earthquake in 1999 and Wenhuan earthquake in 2008, whih aused a great loss of eletri power system. Failure of transmission tower under extremely intense earthquake has been reported in the literature. Therefore, it s very imperative to evaluate the seismi risk of these towers for seismi retrofit and seismi mitigation planning. The aurate predition of tower failure is very important for the reliability and safety evaluation of the power transmission system (Li, 2009). The seismi risk analysis inludes three ontents: seismi hazard analysis, fragility analysis and earthquake-indued loss estimation. Among them, the fragility analysis is to study the probability of strutural failure for a given ground motion level, and an predit probabilities of the ourrene of different damage states indued by different magnitudes of earthquakes. One of the first appliations of seismi fragility analysis in ivil engineering was in the report ATC-13 submitted by Applied Tehnology Counil in USA (ATC, 1985). HAZUS developed under Federal Emergeny Management Ageny (FEMA) sponsorship, whih is the famous program for loss estimation, inorporates fragilities for 36 ategories of building and four damage states. But both of them are not entirely quantitative models, some qualitative evaluation relies on expert opinion to a onsiderable degree. Reently, the emerging methodologies depend more on omputation efforts, in other words, the trend of fragility analysis is shifting from qualitative paradigm toward quantitative paradigm. In this paper, the seismi apaity of transmission tower is evaluated by using nonlinear bukling analysis method and nonlinear dynami analysis, onsidering the inherent unertainty of the struture and ground motion. And the performane limits of different damage states indued by earthquake are determined. The objetive of this literature is to evaluate the fragility urve of transmission towers based on seismi performane analysis onsidering the inelasti strutural behaviour and the unertainties.

2 2. PROBABILITY-BASED SEISMIC PERFORMANCE ANALYSIS The inelasti behaviour of the transmission towers subjeted to the extreme earthquakes has been investigated extensively. The nonlinear stati pushover analysis and inremental dynami analysis (IDA) method (Vamvatsikos and Cornell, 2002) have been widely used in earthquake engineering for evaluating the strutural apaity urves onsidering seismi exitations. The towers might ollapse or be damaged when shaken by intensive earthquakes. However, the information relating the nonlinear inelasti responses of suh towers under extreme seismi loading with the damage severity is laking, and the damage state of the tower remains unlear. Therefore, one of the objetives in this paper is to define the damage states of suh strutures under earthquake loading based on the performane analysis. Strutural seismi performane is the strutural apaity to resist seismi loading, inluding load-bearing apaity, deformation apaity and energy dissipation apaity and so on. In this paper, the seismi apaity is desribed in terms of strutural deformation. The key issue is to obtain the harateristi performane indies of the transmission towers under various seismi exitations in order to define different damage levels. Considering the unertainties of the member manufaturing and environmental ondition, the performane index is disretely distributed rather than deterministi. Nonlinear bukling analysis and dynami analysis are arried out to aquire the performane of the transmission tower whih is subjeted to horizontal seismi loading, and Monte Carlo (MC) method is utilized for simulation of the hanging environment and property variation of the strutural material. Based on a suffiient number of simulations, the probabilisti relationship between seismi intensity and strutural response of the tower (in terms of maximum deformation of tower) an be determined Tower desription and finite element model The nonlinear finite element analysis program ANSYS is utilized in this paper for evaluating the performane of the spae frame, onsidering the material nonlinearity and geometri nonlinearity of towers. For the numerial analysis, a lattie steel tower is onsidered as shown in Fig. 1. The tower has a total height of m with an m m square base area. The leg and diagonal members in the tower are steel pipes and the braing members are steel bars with L-shape. Modelling the tower members using beam elements provides better numerial auray of nonlinear responses than those using truss elements. Eah member of the transmission tower is modelled by beam elements (BEAM188 in ANSYS), whih is based on Timoshenko beam theory onsidering shear deformation effets. The elasto-plasti property of the steel material is represented by an bilinear kinemati model, with the elasti modulus of MPa up to yield and 790MPa after yielding. The yield strength for leg and diagonal members and the braing members is 345MPa and 235MPa, respetively. The finite element (FE) model onsists of 505 nodes and 1378 elements, whih is shown in Fig.1. The analysis of idealized onfiguration models is performed to obtain the pre-ultimate behaviour and the limit loads of the transmission tower without onsidering the oupling effet of the ondut lines. Figure 1. The finite element model of transmission tower

3 2.2.The nonlinear bukling analysis and the definition of damage states The transmission towers are spae steel strutures and in many ases steel strutures fail due to instability. Prasad Rao et al. (2010) present different types of premature failures observed in full-sale testing. Different types of failures are modelled using finite element software and the analytial results and the test results are ompared with various ode provisions. It is onluded that many failures of towers are aused by bukling of ompression leg or braing members and it is possible to predit the probable strutural apaity of the tower by finite element non-linear analysis. Albermani et al. (1992) presented a non-linear analytial method aounting for both material and geometri non-linearity to predit transmission tower failure. In this study, nonlinear bukling analysis onsidering material and geometrial nonlinearity is adopted to determine the limit loads of the tower. With a geometrially nonlinear analysis, the stiffness matrix of the struture is automatially updated to inorporate deformations whih affet the strutural behaviour. In ANSYS, the proedure for nonlinear bukling analysis is simple: it gradually inreases the applied load until the struture beomes unstable (ie. a very small inrease of the load will ause very large defletion of the struture). The nonlinear analysis inorporates the modelling of geometri imperfetions, load perturbations, material nonlinearities and so on. Imperfetion suh as eentri loads or initially deformed shape is introdued to perform the nonlinear bukling analysis. Firstly, eigenvalue bukling analysis (linear) is performed to predit the theoretial bukling strength of an ideal elasti struture and aquire the bukling mode shape. Then nonlinear bukling analysis is onduted after updating the geometri information of the finite element model based on the previous results of linear analysis. The suggested earthquake load is alibrated based on the assumption that the load pattern is unhanged during an earthquake event. Aording to the Code for seismi design of buildings in China, an inverted triangle pattern is used to ompute the lateral seismi load: GH i i Fi FEK (1 n )( i1,2,... n) n GH j1 j j (1) where F i is lateral load of eah segment, F EK is harateristi value of seismi load, G is representative value of gravity load, H is the height, is seismi oeffiient at the top of the tower. Considering the strutural harateristis, the tower is divided into five segments and the seismi load ating on eah segment is alulated using Eq. (1). The inverted triangle seismi load is imposed on eah segment of the tower, whih is illustrated in Fig. 2. The modal information from an eigenvalue analysis of the tower is listed in Table 1 and mode shapes in X diretion are illustrated in Fig. 3. Table 1. Natural frequenies and mode shapes of transmission tower in X diretion Mode Frequeny (Hz) Mode shape First bending mode in lateral diretion First bending mode in longitudinal diretion First torsion mode n An eigenvalue bukling analysis is first used to determine the theoretial limit load and the buking mode shape of the struture. By gradually inreasing the lateral seismi load, a nonlinear bukling analysis is performed to predit the failure mode of the struture. Fig. 4 illustrates the bukling mode shape whih is most likely to our. It shows that the tower failure ours due to the out-of-plane instability of the ompressed leg and braing members, whih is similar to the failure modes in literature (Prasad Rao et al., 2010).

4 1 DISPLACEMENT STEP=1 SUB =1 FREQ=1.184 DMX = DISPLACEMENT STEP=1 SUB =2 FREQ=1.207 DMX = MAR :16:32 MAR :18:41 1 Tower Modal STEP=1 SUB =3 FREQ=1.745 DMX = DISPLACEMENT Y Z X Tower Modal Tower Modal Y Y Z X X Z (1) (2) (3) MAR :19:27 Figure 2. The horizontal load pattern Figure 3. Mode shapes Figure 4. The first bukling mode shape Fig.5 gives apaity urve of the tower represented by the loading-deformation urve: the total base shear fore versus the top rotation angle (horizontal displaement/height, RDA) of the tower. The rotation angle of the top node inreases steadily with the inreasing seismi load before the base shear fore reahes N. After that, the tower experienes rapid and large deformation with the small inrease of the loading. The simulation program finally halts due to exessive deformation of the tower, whih indiates the instability happens ausing the strutural failure. Apparently, the strutural instability happens to the tower at the base shear fore of N. After the apaity urve is determined, the elasti displaement limit, yielding displaement limit and the ultimate displaement limit of the transmission tower an be determined from the results. Then in the following step, it is possible to define the damage state of the transmission tower aording to the apaity of the tower. In this paper, three damage states are defined: minor damage, major damage, and ollapse state. The ultimate rotation angle (RDA) of the tower top is defined as RDAo of the ollapse state, when the deformation response of the tower is greater than this value, RDA RDA o, the tower is in the state of ollapse. When the tower begins to yield, the RDA of the tower top is defined as RDA ma, the displaement limit of the major damage state. When RDAma RDA RDAo, the tower is in the major damage state. For the minor damage state is less ritial and more ambiguous than the other two states, the authors define 0.5RDA ma as the displaement limit of minor damage state ( RDA mi ).

5 2.3. Probabilisti analysis Figure 5. The base shear fore versus the top rotation angle In strutural engineering, there exist all kinds of unertainties, whih are the inherent harateristis of the nature. The seismi performane of the struture is not deterministi but stohasti due to the unertainties of the struture and environment. In this paper, the unertainties of the struture are represented in terms of stohasti variables of material properties and geometri parameters, suh as elasti modulus, yielding strength, passion ratio, density and the dimension of member setion. In ANSYS,Probabilisti Design System (PDS) an produe probability distribution funtion of the aforementioned stohasti variables by Monte Carlo sampling methods. These stohasti variables are assumed to obey Gaussian distribution by setting the design value as its mean and 5% of its mean as its standard deviation. For example, the sampling of the elasti modulus of steel is illustrated in Fig R.225 e l.2 a t.175 i v e.15 MEAN E+12 STDEV E+11 SKEW E-02 KURT E-01 MIN E+12 MAX E+12 F.125 r e.1 q u.075 e n.05 y E E E E E+12 Figure 6. The sampling distribution of elasti modulus of steel Taking the advantage of Monte Carlo (MC) methods, nonlinear bukling analysis is arried out for eah sampling of the stohasti variables, and for eah simulation the apaity urve is obtained. After aquiring suffiient data of simulation, displaement limits of minor damage state, major damage state and ollapse state an be determined. The results are analyzed by using statistial tools, and statistial histogram of the displaement limits of three damage states are plotted in Fig.7.

6 (a) Minor damage state (b) Major damage state () Collapse damage state Figure 7. Statistial histogram of the displaement limits (rad) Fig. 7 learly shows that the displaement limits of minor damage state and major damage state is well distributed, while that of ollapse state is disretely distributed with double peaks. This demonstrates that the unertainty inreases greatly when struture is approahing the ollapse limit, and the apaity of the transmission tower is unstable. Two-parameter lognormal distribution funtions are used to represent the displaement limit of eah damage state. From regression analysis of the data, the results are listed in Table 2. Table 2. Lognormal distribution parameter of displaement limits Level Mean value of Logarithm standard deviation of Expetation (rad) (rad) Logarithm (rad) Minor damage state β mi = Major damage state β ma = Collapse damage state β o = SEISMIC RESPONSE ANALYSIS In order to determine the fragility urve of the transmission tower, it s neessary to determine seismi responses of the tower indued by different magnitude of earthquakes. Due to the randomness of the ground motion, the seismi performane of the building will respond with unertainty as well. The randomness of the ground motion is realized by building a pakage of various ground motions overing a wide range of peak intensity, time-varying amplitude, strong-motion duration and frequeny ontent. In this study, the maximum rotation angle of the top is taken as the seismi response parameter and

7 peak ground aeleration (PGA) taken as seismi intensity parameter. A pakage of seismi reords overing different onditions is hosen for the ground inputs of nonlinear dynami analysis. By using regression analysis of the results, the relationship between the seismi intensity parameter and seismi response parameter of the struture is established. The ground motion reords are downloaded from the website of PEER (Paifi Earthquake Engineering Researh), eah of them has different site onditions, PGA, spetrum harateristis and duration to represent the variation of nature as muh as possible. The maximum of PGA in all reords is 1.779G and the minimum one is 0.109G. One reord in whih PGA= 0.753G, is taken for example, as illustrated in Fig. 8. The modal damping ratio is assumed to be 1%, and the seismi input is applied only in X diretion. Corresponding to the seismi input shown in Fig. 8, the displaement response of the tower at the top is plotted in Fig. 9. Figure 8. Time history of ground aeleration Figure 9. Displaement time history of top node The maximum rotation angle of the top orresponding to the PGA of eah seismi wave an be obtained. And the maximum RDA of the transmission tower is plotted against its orresponding PGA, as shown in Fig. 10. Transforming the oordinates into logarithm sale, the relation between RDA and PGA is demonstrated in Fig.11. Figure 10. Seismi intensity vs. Seismi performane Figure 11. Logarithmi desription The linear orrelation between PGA and RDA an be determined by regression analysis, as shown in Eq. (2). ln( RDA) 0.993ln( PGA) (2) The statistial histogram of RDA in Fig.12 shows that the seismi response parameter has a lognormal distribution, whih is oinident with the original assumption. Two-parameter lognormal distribution funtions are used to represent the probability model of response parameters. By estimation method we ould obtain the statistial parameters of the seismi performane, the mean value of its logarithm is and the standard deviation of its logarithm

8 Figure 12. Histogram of seismi performane (rad) 4. FRAGILITY ANALYSIS The fragility desribes the probability of strutural failure or damage states under a ertain seismi intensity; it depends on the strutural integrity, damage onditions and other fators. Commonly, the fragility urve with respet to seismi intensity is assumed to have a lognormal distribution haraterized by two parameters. The probability that strutural seismi response Sd exeeds the strutural apaity R ( R is the displaement limit of eah damage state whih has been determined in the previous performane analysis) an be alulated by the Eq. (3). R Pf Pf( 1) (3) S d Both R and S d obey a logarithm normal distribution. Therefore, probability of ollapse or damage states an be determined by the Eq. (4). P f R -ln( ) S d 2 2 d (4) where R is the mean value of R, S is strutural response (seismi demand), is logarithm d standard deviation of seismi apaity, is logarithm standard deviation of strutural response, and d is a funtion of standard normal distribution. Aording to the previous results whih are shown in Fig. 10 and 11, the probability of minor damage, major damage and ollapse state an be derived by substituting the orresponding data, the seismi fragility urve of the tower an be depited in Fig.13. Even under the extremely intensive earthquake whose PGA is equal to 1.0G, the probability of tower ollapse is less than 5%, the probability of major damage happened to the transmission tower is less than 15% and that of minor damage is less than 30%. It is obvious that the seismi apaity of transmission tower is very robust and the tower is not easy to ollapse under seismi load.

9 Figure 13. The seismi fragility urve of the transmission tower 5. CONCLUSIONS AND DISCUSSIONS The paper presents a numerial method to obtain a seismi fragility urve of the transmission tower, whih is very important to evaluation of the integrity and reliability of transmission towers. Considering the internal unertainty of the tower, the randomness of ground motion and the variation of its seismi performane, seismi performane is analysed by using nonlinear bukling analysis method and nonlinear dynami analysis. And the performane limits of different damage states are determined. Finally, the seismi fragility urve of the transmission tower is aquired by numerial Monte Carlo simulation. By the seismi fragility urve, the failure probability of the transmission tower under different magnitudes of earthquake an be visually predited. However, there are several issues should be studied further in the future: 1. The harateristis of earthquake motion have three elements: time-varying amplitude, strong-motion duration, and frequeny ontent. In this paper, the amplitude of ground motion (PGA) is the only fator desribing the seismi intensity, while the relation between the other two harateristis and strutural seismi response is left untouhed. 2. The transmission tower itself is studied in this paper, negleting the oupling effet of the ondutor lines with the tower. In the long-span transmission-line system, the integrity and fragility of tower-line system needs to be studied further. 3. The damage state of the transmission tower is defined by the stati nonlinear bukling analysis; however, the seismi load is dynami having more ompliate stability and safety onditions, for example, the dynami instability. The damage state should be examined by dynami analysis. AKCNOWLEDGEMENT This study was supported by Ph.D. Programs Foundation of Ministry of Eduation of China (Grant No ) and Guanghua Tongji Civil Funding in China. REFERENCES Applied Tehnology Counil. (1985). Earthquake damage evaluation data for California. Report ATC-13, Redwood City, CA, USA. Li, H. (2009). Seismi analysis and design of overhead transmission tower (in Chinese). China eletri power publisher, Beijing, China. Prasad Rao,N., Samuel Knight, G.M., Lakshmanan, N. and Iyer, Nagesh R. (2010). Investigation of transmission

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