Non-linear Analysis Method of Ground Response Using Equivalent Single-degree-of-freedom Model

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1 Proceedings of te Tent Pacific Conference on Eartquake Engineering Building an Eartquake-Resilient Pacific 6-8 November 25, Sydney, Australia Non-linear Analysis Metod of Ground Response Using Equivalent Single-degree-of-freedom Model K. Sakai & Y. Murono Center for Railway Eartquake Engineering, Railway Tecnical Researc Institute, Tokyo, Japan. ABSTRACT: Tis paper proposes a simple metod to evaluate surface ground waveforms using bot te natural period of te ground and waveforms in te engineering bedrock. In tis researc, te "Equivalent Single Degree of Freedom metod" is proposed and is based on te results of static pus-over analyses of many grounds wit various properties. A comparison was made of waveforms of multi-layered ground and tat obtained from te proposed metod using dynamic analysis. It was ten confirmed tat tese waves are almost identical, and tat te proposed metod is applicable to te evaluation of te surface motion. Even if only a few data wit regard to soil properties property is available, particularly in case of an existing railway facility, te proposed metod enables economical dense calculation of trackside surface motion. Tese sets of waveforms will be utilized to identify locations were devastating damage is will be expected in te case of a strong eartquake. INTRODUCTION Social infrastructure facilities consist of not only tose constructed according to eartquake resistance standards enacted following te Soutern Hyogo Prefecture Eartquake, but also a large number of older structures. Te eartquake resistance and te eartquake related risks of eac structure differ greatly from one structure to anoter. An efficient metod to reduce eartquake related risk of wide spread infrastructures is to locate weak structures were appropriate countermeasures are needed. Acieving tis would require te compreensive evaluation of te eartquake resistance of a very large number of structures, and te effective evaluation of eartquake motion of eac of tese points, followed by precise individual calculation of eartquake related risk for eac structure. Te multi-dimensional or multiple-degree-of-freedom response analysis metods migt used in order to evaluate te beaviour of te structure and te amount of damage to te structure under given eartquakes. However, it is not realistic to analyze suc a large group of structures at once. For tis reason, it is necessary to carry out a response evaluation using more simple metod. For example, it as been confirmed tat te accurate simulation is carried out for a relatively simple structure suc as a viaduct, if parameters are given suc as natural period, yield strengt, and so on. In addition, tese parameters can be estimated by only a few information available suc as eigt of structure or ground condition (Murono and Sato 27). If detailed data concerning te structure is available, it will be possible to calculate te natural period or te yield strengt wit iger accuracy by implementing simple static non-linear analysis (pus-over analysis (Priestley et al. 996)). In addition, a metod for easily calculating te natural period and te yield strengt from only a small number of parameters, suc as te eigt of te viaduct and te ground conditions, as also been proposed. Likewise, simple modelling and response evaluation are being studied for embankments (Furukawa et al. 23). Consequently, by using tese metods, it is possible to evaluate te beaviour and te potential degree of damage of te large number of upper structures by simple and minimal calculations. On te oter and, wen evaluating te eartquake motion for tese structures, it is important to consider ground beaviour. Te most commonly used metod for analyzing te ground response is sequential non-linear analysis were te ground is regarded as one-dimensional tin layers (Konder 963) or equivalent linearization analysis (Scnabel et al. 972). To apply tis metod, owever, it is necessary to obtain te detail information suc as sear wave velocity Vs, te unit weigt, non-linear caracteristics (te G - relationsip, te - relationsip, te adesion c, and te internal friction angle ), and oter parameters for te location were eartquake motion is being evaluated. In addition, suc a dynamic analysis requires a multi-degree-of-freedom model. However, wen Paper Number 73

2 considering te evaluation of eartquake motion applied to an existing facility, very often adequate ground data is unavailable. Also, if te evaluation target consists of a group of many structures, it is assumed tat tis will require a vast number of calculations. Many researcers ave been obliged terefore to evaluate ground beaviour using simple metods, most of wic only allow evaluation of a single eartquake motion index or amplification factor of te response spectrum wile oter simple metods ave been introduced on te assumption tat te amplification factor is te same regardless of te input (Midorikawa et al. 992, Nogami et al. 22). Consequently, tis situation called for te development of a simple metod to evaluate te time istory waveform of te ground surface taking into consideration te non-linearity of te soil. In an attempt to meet tis need, tis study proposes a simple metod to evaluate te surface waveform considering te complicated nonlinear beaviour of te strata of te ground. Tis paper proposes a new metod for analyzing te static non-linearity of te overall ground system using a simple equivalent single-degree-of-freedom model. Despite te mass distribution of te ground is rougly uniform,te response caracteristics cange abruptly due to te localized non-linearization of te strata, wic is clearly a contrast to te viaduct. Tat is to say, a static non-linear analysis metod for bridges tat applies a set of static load equivalent to te dynamic inertial force under eartquake, is not appropriate for estimating ground beaviour. Accordingly, te new static non-linear analysis metod was proposes tat is able to consider canges in te mode sape according to an increase in ground deformation. Furtermore, by using a dynamic analysis metod wic employs an equivalent singledegree-of-freedom model, it is possible to confirm tat te eartquake motion of te ground surface is accurately evaluated. Te proposed static non-linear analysis can ten be employed to evaluate te beaviour of grounds aving different natural periods and strata compositions. In addition, te proposed metod is employed to estimate te necessary parameters for te equivalent single-degree-of-freedom models using only te natural period of te ground, wic is available in most situations. Finally te effectiveness of te response obtained using te proposed metod was confirmed. It is found tat tis approac will permit te evaluation of te eartquake motion of te ground surface wit just a simple investigation and response analysis. 2 METHOD FOR STATIC NON-LINEAR ANALYSES OF GROUND RESPONSE Tis section first proposes a metod for evaluating te cange of stiffness and damping of te overall ground depending on te displacement of te surface soil. Here, in te case of a general bridge or viaduct, it is possible to rougly evaluate te load displacement relation by a static nonlinear analysis, in wic te acceleration acting on eac node is equal. On te oter and, in case of evaluating ground motion, it is conceivable tat te weigt distribution is rougly uniform, but tat te distribution and te response acceleration and response displacement vary in a complicated way according to te degree of non-linearity resulting from te canges in te strata composition and input level of eartquake motion. Consequently, it is not appropriate to carry out static non-linear analysis using te same procedure as tat used for bridges or viaducts. For tis reason, te evaluation of canges in stiffness and damping of te overall ground is necessary to consider canges in te mode sape along wit deformation progression as sown in Fig.. Te actual procedure is sown below. () Assuming te ground as k degrees of freedom and te values of te pysical properties of eac stratum (stratum tickness, weigt, initial sear stiffness, damping, and non-linear caracteristics) are given. At te initial step, te ground is assumed to be in an elastic region, and sear stiffness G i () is assumed to be equal to G (i). Here, G i () is te initial (st step) sear stiffness of te i t stratum, and G (i) is te initial sear stiffness of te i t stratum. (2), (3) Eigenvalue analysis is performed using te stratum tickness, weigt and sear stiffness G i (N) of eac stratum, and te primary natural frequency (N) and corresponding eigenvector { u (N) } are calculated. Here, te eigenvector { u (N) } is normalized so tat te response at te ground surface position is.. Also, superscript (N) indicates tat te result is at te N t step. (4) Te displacement distribution is assumed so tat te increment of deformation at te ground 2

3 6 Perform an eigenvalueanalysis once again using te calculated tangential stiffness. Ground surface () G =G () u N () G 2 =G (2). G G G (N+) G 2 (N+) (N+) 2 (N+) G k () =G (k) Bedrock Construction of a ground model 2 Evaluation of mode sape by means of eigenvalue analysis 3 Setting te displacement distribution sape to be utilized 4 Static analysis according to displacement increase (loading a microdisplacement) G G k (N+) 5 Calculation of stiffness and damping corresponding to strain at eac position k (N+) G/G. Evaluation of reduction of natural period (secant stiffness) accompanying te increase in displacement G/G relation Evaluation of te increase of damping accompanying te increase in displacement (average using mode sape) relation Displacement Construction of an equivalent single degree of freedom model Displacement Fig. Procedure for performing a static non-linear analysis of te ground surface is Here, te top to bottom displacement distribution is assigned according to te primary mode sape obtained from (3) above, as sown in Eq.(). N N N u () (5) Te tangential stiffness G (N+) i at eac stratum and also te damping (N+) i are updated. (6) Te procedures (2) to (5) are repeated until te predetermined displacement is obtained at te ground surface position. Also, as sown in Figure, te relationsips G/G - and - for te overall ground system were obtained by using te aforementioned static non-linear analysis. G/G - relationsip: Te reduction of te stiffness of te overall ground system is evaluated at eac step from te cange in te st mode natural frequency (N) of te soil deposit, wic is related to te increase in te ground surface displacement, as sown in Eq.(2). 2 / G N N G (2) / - relationsip: Te average damping of te overall ground for eac displacement condition is calculated from te weigted average of te damping i (N) of eac stratum corresponding to te strain distribution { (N) } in te primary mode. According to (5) of te above, it is given by following. N N N eq i N N N,, k, k N N u u2 / D / Dk N N uk uk N uk / Dk Were is te surface displacement, D i is te tickness of te i t layer. In addition, it is necessary to take into consideration te participation function (PF) in order to convert te response of te primary mode into response at te ground surface position. For tis reason, wen performing te static non- (3) 3

4 G/G δ(cm).. (a) G/G - relationsip (b) - relationsip Fig. 2 Results of static non-linear analysis of multiple ground locations wit different caracteristics linear analysis of te ground, te primary mode participation function for te displacement at eac step is to be also calculated δ(cm) 3 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM MODEL 3. Standard parameters as a function of natural period of ground Te proposed static analysis metod requires te detail information wit regard to te target soil deposit as mentioned in Section. Suc information, owever, is not necessarily available particularly wen evaluating te existing structures. In order to overcome te problem, te simple equivalent single-degree-of-freedom model is proposed to evaluate eac ground location using only te natural period T g of te ground. Tis T g can be easily evaluated from microtremor measurements. In te study, 6 ground locations aving a large variety of period caracteristics and strata compositions were used, and a static non-linear analysis for eac ground location was performed. Troug te analysis, te G/G -, te -, and PF - relationsips were evaluated and an equivalent single-degree-of freedom model of te ground was constructed from te natural period T g alone. Note tat tese ground locations are also used to evaluate te ground surface design eartquake motion of railway structures, wose natural periods are between. and.7 seconds. Te non-linear caracteristics of eac ground location are expressed by GHE-S model, and te parameters assigned to eac stratum of eac ground location are canged according to te soil quality classification suc as sandy soil or coesive soil. Static non-linear analysis was carried out at all of tese 6 ground locations. Te conditions of te analysis were te same as tose of te case described in te previous section, were analyses were performed in two ground locations. Figure 2 sows te obtained G/G - and - relationsip. From tis data, it can be seen tat tere is an overall tendency tat stiffness was falling and damping was increases according to te increase of te displacement. However, te degree of displacement differs greatly depending on te ground conditions. Consequently, it is difficult to create a simple model from tese results only. On te oter ands, It is generally understood tat if te results of element tests on te ground material are normalized and arranged using a strain value (reference strain r ) suc tat G/G =.5, te results can be expressed using te same parameters irrespective of te ground material, and tere is a possibility tat tis metod can be applied to te overall ground system. Accordingly, te relationsips G/G -, and - were normalized by a reference displacement r suc tat G/G =.5, as sown in Fig. 3 (a) and (b). Likewise, Fig. 3 (c) sows normalized participation function (PF), also using a reference displacement r. First, from Fig. 3 (a) and (b) it can be seen tat by normalizing te results of te static non-linear analysis of multiple ground locations by means of reference displacement r, it is possible to express te cange in stiffness and damping using rougly te same relationsips, irrespective of te natural period, te stratum tickness, and te stratum composition of te ground. Also, from Fig. 3 (c), it can be seen te overall trend of participation function PF is rougly te same regardless of te ground condition. From te above results, it can be said tat if te reference displacement r is known, it is possible to predict to a certain extent te non-linear caracteristics of 4

5 G/G Eac ground μ μ±σ Fitting result Standard displacement δ/δ r (a) G/G - relationsip Standard displacement δ/δ r (b) - relationsip PF Eq.(4).5.. Standard displacement δ/δ r δ r (cm) Tg(s) Eac ground Eq.(5) (c) PF - / r relationsip (d) T g - r relationsip Fig. 3 Results of te static non-linear analysis after normalization using reference displacement r Table Results after setting standard parameters C () C 2 () C ( ) C 2 ( ) C () C 2 () max te overall ground system, regardless of te period and stratum composition of te ground. Accordingly, te standard soil parameters tat can be utilized in common regardless of te ground conditions were evaluated. First, te G/G - and - relationsips, were expressed by means of a GHE-S model, as in te previous section, and model parameters tat satisfied te average values of all of te results (blue lines in te figures) were determined by trial and error. Te identified parameters are indicated in Table Based on tese, te G/G - and - relationsips, expressed using te proposed metod are indicated by red lines in Fig. 3 (a) and (b). It can be seen tat tese eigt parameters ave made it possible to reproduce te average value of eac result wit good accuracy. Next, participation function PF was also calculated as sown in Fig. 3 (c) wic is expressed by Eq.(4). Note tat eac coefficient in Eq.(4) as been determined by te non-linear least-squares metod..4.7 PF (4) r Results obtained using Eq.(4) are also indicated by te red line in Fig. 3 (c). It is found tat Eq.(4) expresses te overall trend of te participation factor in a good accuracy. Fig. 3 (d) sows te relationsip between te reference displacement r and te natural period T g of te ground. It can be seen tat te reference displacement r increases along wit te increase in te natural period T g. Despite te reference displacement r is affected by many factors suc as te natural period, te stratum tickness and stratum composition, r was likely to be evaluated only by using te natural period T g. Tat is to say, te reference displacement r was estimated by a regression expression wit respect to te natural period T g of Fig. 3 (d) wic is determined by a non-linear least-squares metod log.45 (5) r Tg 5

6 Te relationsip between te natural period T g and te reference displacement r expressed by Eq.(5) is also indicated by te red line in Fig. 3 (d). It is found tat Eq.(5) traces te general tendency for te reference displacement to increase along wit te increase in te period. By integrating tese results, it is possible to construct an equivalent single-degree-of-freedom model using only te natural period T g as a parameter. 3.2 Simulation of te ground response using an equivalent singledegree-of-freedom model In order to verify te accuracy of tis metod, dynamic response using te equivalent single-degree-of-freedom model was compared wit te reference responses obtained by employing a multi degree of freedom model. Te two different ground caracters were selected for te simulation, as sown in Fig. 4. Te proposed equivalent single-degreeof-freedom model was constructed according to te steps described below, and te ground surface eartquake motion was calculated. Dept (m) Ground A (Tg=.36s ) Ground B (Tg=.86s ) Vs (m/s) Fig. 4 Velocity construction of te ground used in te calculation Step Set te initial stiffness K from te natural period T g of te ground. Step 2 Set te GHE-S parameters and te natural period T g from Table, and also set te reference displacement r (cm) from Eq. (5). Step 3 Construct an equivalent single-degree-of-freedom model from Steps and 2, and conduct a non-linear dynamic analysis. Step 4 Calculate te participation function PF from te maximum displacement max obtained from Step 3 of te above and Eq.(4). Step 5 Multiply te response waveforms of te relative acceleration, te relative velocity and te relative displacement by te participation function PF, and convert te results into te response at te ground surface position. Te comparison of te ground surface eartquake motion obtained using te proposed procedure and tat from detailed non-linear ground response analysis are sown in Fig. 5 (ground A) and Fig. 6 (ground B). As seen in tese figures, te proposed simple metod was able to estimate te time istories and frequency caracters of te surface waveform from te reference model in a good accuracy.. In addition, te same comparison was carried out at all te 6 ground locations used in tis study. Figure 7 sows te comparison of te maximum displacements of te surface. As seen in tis figure, bot results sow good agreement It consequently follows tat tis metod is effective wen carrying out a simple prediction of te ground surface in a region were te results of te ground investigation are incomplete. 4 CONCLUSION Tis paper proposes a metod for performing non-linear dynamic analysis using an equivalent singledegree-of-freedom model tat can be used even for ground tat as a complex stratum composition. Te results obtained are sown below: () A metod was proposed for performing static non-linear analysis of te ground, wic makes it possible to successively take into consideration te accumulation of localized strain accompanying non-linear beaviour of te ground. As a result, it was possible to evaluate te decrease of te stiffness and te increase of te damping according to te displacement. (2) Based on te results of te static non-linear analysis of te ground, a simple single-degree-offreedom model of te ground was proposed. It was confirmed troug numerical simulations tat te proposed metod gave almost identical waveforms to tose obtained from detail dynamic analysis wit a multi-degree-of-freedom model. (3) As a result of te application of te static non-linear analysis metod to te ground, it became clear 6

7 Acc (gal) Vel (kine) Dis (cm) Response acceleration Response velocity Response displacement Detailed model Equivalent SDOF model Time (s) Resp. acc. (gal) =.5 Detailed model Equivalent SDOF model..5 5 Period (s) (a) Time istory waveform (b) Response spectrum Fig. 5 Results of te prediction of te ground surface waveform by using only te natural period T g (Ground A (T g =.36 sec)) Acc (gal) Vel (kine) Dis (cm) Response acceleration Response velocity Response displacement Detailed model Equivalent SDOF model Time (s) Resp. acc. (gal) =.5 Detailed model Equivalent SDOF model..5 5 Period (s) (a) Time istory waveform (b) Response spectrum Fig. 6 Results of te prediction of te ground surface waveform by using only te natural period T g (Ground B (T g =.86 sec)) tat by normalizing ground displacement wit a reference displacement r, it was possible to uniformly express te tendency for stiffness to decrease and for damping to increase, irrespective of te softness or ardness of te ground, or te stratum composition. In addition, te standard parameters wic express te G/G - / r and - / r relationsips were proposed, and te reference displacement r needed for tose relations was expressed by a function of te natural period of te ground Tg. As a result, it was possible to construct an equivalent singledegree-of-freedom model from te natural period T g of te ground alone. Te efficacy of te proposed metod was confirmed by te numerical analysis using 6 ground conditions. Even if only a few data wit regard to soil properties property is available, particularly in case of an existing railway facility, te proposed metod enables economical dense calculation of trackside surface motion. Tese sets of waveforms will be utilized to identify locations were PGD (cm) (Equivalent SDOF model) PGD (cm) (Detailed model) Fig. 7 Results of te evaluation te ground surface displacement using te proposed metod 7

8 devastating damage is will be expected in te case of a strong eartquake. Consequently, by using tis metod, it is possible to rationally extract locations wic require priority attention in an eartquake, wen te target of te measurement is an existing group of structures for wic ground data is inadequate, and to predict ground surface eartquake motion immediately after a large scale eartquake. REFERENCES: Furukawa, A., Mizukami, A. and Kiyono, J., Fundamental study on a simple evaluation metod of seismic safety of road embankment structures, Journal of JSCE, Vol.69, No.2, I_457-I_468, 23 (in Japanese). Konder, R. L., Hyperbolic stress-strain response; Coesive soils, Proc.ASCE, EM, Vol.89, No., pp.5-43,963. Midorikawa, S., Matsuoka, M. and Sakugawa, K., Evaluation of site effects on peak ground acceleration and velocity observed during te 987 Ciba-ken-Too-Oki eartquake, Journal of Struct. Engng. AIJ, No.442, pp.7-78, 992 (in Japanese). Murono, Y. and Sato, T., Applicability of non-linear response spectrum for verification of seismic performance of structure, JSCE Journal of Eartquake Engineering, Vol.29, pp , 27 (in Japanese). Nogami, Y., Sakai, K., Murono, Y. and Morikawa, H., evaluation metod of seismic site amplification considering predominant periods of subsurface soil and bedrock motion, Journal JSCE, Ser.A, Vol.68, No., pp.9-2, 22 (in Japanese). Priestley, M. J. N., F. Seible, G. M. Calvi, Seismic Design and Retrofit of Bridges. Jon Wiley &Sons, New York, USA, 996. Scnabel, P.B., Lysmer, J. and Seed, H.B., SHAKE a computer program for eartquake response analysis of olizontally layered sites, EERC, pp.72-2,