Analysis of Foundation Damping from Theoretical Models and Forced Vibration Testing

Transcription

1 6 th International Conerence on Earthquae Geotechnical Engineering 1-4 November 2015 Christchurch, New Zealand Analysis o Foundation Damping rom Theoretical Models and Forced Vibration Testing M. J. Givens 1, L.M. Star 2, S. Tileylioglu 3, G. Mylonais 4, J.P. Stewart 5 ABSTRACT Foundation damping incorporates the combined eects o energy loss due to waves propagating away rom the vibrating oundation (radiation damping), as well as hysteretic action in the soil (material damping). Two closed-orm solutions or oundation damping o a leible-based single degree-o-reedom oscillator were derived rom irst principles where the inherent (structural) damping ratio, radiation damping and the soil hysteretic damping ratio appear as variables. Since both ormulations are theoretically deensible, yet provide numerically distinct results, we loo to case histories or validation. In addition to previously established case histories we evaluated data derived rom orced vibration testing o a ield test structure installed at two sites in Caliornia. Parametric system identiication was employed to evaluate the ied-base and leible-base irst mode period and damping ratio during tests perormed across a wide requency range and multiple load amplitudes. Foundation damping derived rom these results is shown to avor one o the theoretical models over the other. Introduction Following the early wor by Parmelee (1967), oundation damping as a distinct component o the damping o a structural system was introduced as part o Biela s (1971) derivation o the replacement (leible-base) single-degree-o-reedom (SDOF) system and was later reined by Veletsos and Nair (1975), Roesset (1980) and others. The wor was predicated on the need to evaluate the eects o soil-structure interaction (SSI) on the seismic response o nuclear power plants. Based on that need, alternative sets o equations were developed to predict oundation damping o a rigid circular oundation resting on a uniorm elastic halspace. Due in part to the convenience o its application in the speciication o seismic demands using response spectrum (orce-based) or pushover (displacement-based) methods o analysis, oundation damping has more recently appeared in several seismic design guidelines or building structures (e.g., ASCE, 2006, 2010). We summarize results o alternative derivations o oundation damping, originally presented by Givens et al. (2015), which match the response o a single-degree-o-reedom (SDOF) equivalent ied-base oscillator to that o the leible-base oscillator. The results show some dierences rom classical solutions. We describe ield-scale orced vibration testing that was perormed in part to measure oundation damping. We briely review system identiication processes and the necessary input-output pairs required to evaluate irst-mode period and 1 Engineer, Geotechnics, Arup, Los Angeles, CA, USA, mjgivens@gmail.com 2 Asst. Pro., Civil Eng. & Construction Eng. Management, Caliornia State Univ., Long Beach, USA, lisa.star@csulb.edu 3 Asst. Pro., Civil Eng., Canaya University, Turey, saliht@canaya.edu.tr 4 Pro., Civil Eng., Univ. Bristol, UK, and Univ. Patras, Greece, and Adjunct Pro., Civil & Environmental Eng., Univ. Caliornia Los Angeles, USA, G.Mylonais@bristol.ac.u 5 Pro. and Chair, Civil & Environmental Eng., Univ. Caliornia Los Angeles, USA, jstewart@seas.ucla.edu

2 damping or the ied-base and leible-base conditions. Foundation damping is derived rom the results and compared to the theoretical derivations. Foundation Damping Solutions The replacement SDOF approach has led to the overall damping o soil-oundation-structure systems ( ο ) being partitioned into components associated with soil-oundation interaction (including hysteretic and radiation damping, reerenced as oundation damping ),, and damping within the structural system, i, (reerenced as structural or inherent damping ) as ollows: o 1 = + ~ n i (1) ( T T ) Equation 1 appears in many seismic design guidelines with n = 3. Givens et al. (2015) show that the eponent n depends on the type o damping (i.e., ideally viscous or general, including hysteretic). Historical precedent and system identiication studies oten dictate values o i in the range o 2-5% in structural design practice or buildings responding in the elastic range (FEMA 440, 2005). The most challenging aspect o deriving ο is the evaluation o since i can be established rom the aorementioned guidelines and period lengthening ( T ~ T ) analysis is relatively straightorward. We consider an SDOF oscillator o height h and stiness supported on a leible-base condition consisting o a horizontal spring ( ) and rotational spring ( ). The total delection ~ ( ) o the oscillator mass (m) rom the application o a horizontal orce (F) results rom horizontal delection o the structural spring and rigid body displacement o the structural mass rom horizontal oundation displacement (u ) and base rotation (θ ). F = + u + θ h F F F h = + + h (2) Considering that the replacement oscillator has delection ~ in response to orce F, the stiness o the replacement oscillator can be taen as the ratio o ~ to F. The ormulation o the damping component o equivalent oscillator requires consideration o the imaginary parts o stiness terms. Damping results rom the phase dierence between the real and imaginary parts o the oscillator response. Givens et al. (2015) provide two derivations or oundation damping. Both begin by dividing the right and let sides o Equation 2 by orce F, recognizing that F is equivalent to an eective leible stiness o the replacement oscillator, and generalizing each term or dynamic loading through the introduction o comple-valued stiness terms (indicated by an overbar), as ollows: h = (3)

3 There are two ways to proceed rom Equation 3. The irst approach is similar in some respects to prior wor by Biela (1971), Aviles and Perez-Rocha (1996) and Maravas et al. (2014). We separate Eq. 3 into its real and comple parts, and upon dropping higher-order ( 2 ) terms, operate eclusively on the imaginary part to evaluate the eective damping o the replacement oscillator. The oundation damping is then readily derived rom the system damping. As shown by Givens et al. (2015), this leads to the ollowing epression or : = 1 ~ s 2 + ~ + 2 ~ 2 (4) ( T T ) ( T T ) ( T T ) where s represents the soil (hysteretic) damping, and represents the radiation damping rom oundation displacement in the horizontal and rotational direction, respectively. In the second approach, both the real and comple parts o Eq. 3 are retained in developing epressions or the dynamic properties o the replacement oscillator (i.e., the replacement oscillator response matches the leible-base oscillator response in both amplitude and phase). The resulting epression or oundation damping is: 1 ( ) ( ) 1 ( ) ( ) ~ + s + 2 ~ s T T T T + = 2 (5) This approach is similar in some respects to that o Veletsos and Nair (1975). Both solutions are presented with ictitious vibration periods in the denominator, calculated as i the only source o the vibration was oundation translation or rotation, with the structure being rigid, as shown in Eq. 6 (Givens et al., 2015). T m m m m = 2 π T = 2π T = 2π T = 2π (6) It should be noted that the absolute values in Eqs. 5 and 6 are to mae the epression or real-valued. Further eplanation o this issue is provided in Givens et al. (2015). In Figure 1, we plot oundation damping derived rom the two solutions against the ratio ( h ( VT s )), which is oten called the wave parameter (Veletsos, 1977). The wave parameter can be thought o as a structure-to-soil stiness ratio because ( h T ) represents the stiness o a structure s lateral orce resisting system in velocity units whereas soil shear wave velocity V s is related to the soil shear stiness. In Figure 1, oundation damping solutions are given or circular oundations and various structure height aspect ratios (h/r). Translation modes o oundation vibration dominate or small height aspect ratios (< 1), and rocing dominates or larger height aspect ratios (> 2). Veletsos and Wei (1971) impedance unctions were used or the two sets o solutions shown in Figure 1. Both the translational and rotational terms are aected by the introduction o imaginary terms in Eq. 5 (relative to the terms in Eq. 4) by amounts ranging rom 0-10% or the translational term and 0-16% or the rotational term. These dierences have negligible inluence at larger height aspect ratios (e.g., h r 2 ), but substantial inluence as the height aspect ratio is

4 reduced. In all cases, Eq. 4 results in more oundation damping than does Eq. 5. Figure 1. Comparison o oundation damping solutions (Equations 4 and 5) results or a rigid, massless, circular disc supported on a homogenous isotropic halspace with ν = 0.33 and hysteretic soil damping a) s = 0% and b) s = 10%. Field Testing The importance o ield testing to measure SSI eects is associated with inherent limits o oundation impedance models, such as those used in Eq. (2)-(5). Such models are typically developed or idealized conditions such as linear soil response, rigid oundations, and depthinvariant soil properties. Thus, testing is needed to evaluate model applicability or realistic ield conditions and to guide the selection o input parameters. NIST (2012) summarizes eisting models or predicting the stiness and damping o oundation-soil interaction and current recommendations or adapting such models to ield conditions. NIST (2012) also summarizes prior testing (laboratory and ield) o soil-structure system responses. Field test results are limited; etending the available dataset thereore represents a substantial research need. A sequence o unique eperiments in which the same structure was subjected to orced vibrations at multiple sites representing varying degrees o base leibility was perormed by Star et al. (2015). The structure consists o a rectangular steel moment rame with removable cross-bracing and a reinorced concrete roo and oundation, as depicted in Figure 2 in its unbraced and braced conigurations. An essentially ied base condition was achieved in testing within a structural laboratory, whereas medium-sti and sot soil conditions representing leible base conditions are present at two ield test sites. The shallow soils at the medium-sti site consist predominantly o sand and silt with some clay with an average shear wave velocity o about 190 m/s. The shallow soils at the sot site consist predominantly o silt and clay with an average shear wave velocity o about 95 m/s. Forced vibrations were applied on the top slab and oundation mat with two shaer systems that impart small- and large-orce demands. Specimen responses were recorded with accelerometers, pressure cells, and displacement transducers (as shown in Figure 2). A data acquisition system was used with precise time stamping, which is important or interpretation o damping eects. The test

5 structures were also instrumented to record earthquaes or several months between tests and ollowing the completion o testing. Details o the test structure coniguration are given in Star et al. (2015). Figure 2. Field test structure depicted in an a) unbraced coniguration and b) braced coniguration. Arrows represent the generalized locations o the recorded accelerations (red), displacements (blue) and applied shaer orce (yellow). Shaer orce was also applied in the transverse direction and on the oundation mat. Pressures cells under mat not shown or clarity. The ull data set rom testing at the laboratory and two ield test sites can be ound online at Each eperiment included multiple trials with varying test conditions. Evaluation o Field Testing Data Foundation damping is not measured directly rom ield perormance data; rather system identiication analyses are perormed to evaluate ied- and leible-base properties o structures, rom which oundation damping is computed using Eq. 1. We apply parametric system identiication procedures in which the properties o a parametric model o the system are evaluated or a given input-output pair o motions (e.g., Stewart and Fenves, 1998). A crucial element o our system identiication is the selection o appropriate input-output motions to evaluate ied-base and leible-base properties o the structure. For the case o orced vibration testing, the input-output pairs or application to SDOF structural systems have been evaluated by Tileylioglu (2008) by solving equations o motion in the Laplace domain, with results in Table 1. Input-output pairs or more general application involving multi-degree o reedom structures have yet to be identiied. Parametric system identiication evaluates a transer unction surace in the Laplace domain by itting an underlying model (Stewart and Fenves, 1998). The locations o peas in this surace can be related to modal periods and damping ratios. Figure 3 shows an eample o transer unctions rom parametric system identiications procedures using data rom the aorementioned orced vibration tests where a large-orce shaer was used. The peas occur at modal requencies, with the irst distinct pea representing irst-mode response. Also shown in Figure 3 are transmissibility unctions rom non-parametric system identiication

6 procedures, which represent the ratio o output/input signals without itting an underlying model. The peas do not always align perectly imperect matches can occur in transer unction space because the parametric model is optimized to match model output to data in the time domain (not requency domain as in Figure 3). The second pea in the short direction is a result o the bracing natural requency and is not considered signiicant or this analysis. Table 2 summarizes irst-mode parameters recovered rom application o these system identiication procedures. Table 2 shows results o analyses o anticipated period lengthening and oundation damping eects using Eqs. (3) to (6) with impedance unction solutions or uniorm halspaces adapted to ield conditions per NIST (2012). Nonlinear eects were approimated in an equivalent-linear sense using an estimate o shear strain as the ratio o PGV to the nearsurace soil shear wave velocity. The two analytical solutions or oundation damping in this test case are generally similar, with the damping rom Eq. (5) being smaller than that rom Eq. (4). When many additional tests are considered, including cases rom the literature, we ind that the smaller damping rom Eq. 5 is generally in better accord with ield perormance data when h/(v s T) > 0.1, which represents the conditions where SSI eects are most signiicant (NIST, 2012). Table 1. Input-output pairs to evaluate leible- and ied-base modal parameters. (Tileylioglu, 2008) Base Fiity Input Output Fleible-Base Fs ms u + h θ + u s Fied-Base Fs ms u h θ u s u Note: F s = shaer orce, m s = structural mass, h = height o the center o m s, = oundation horizontal acceleration, θ = oundation angular acceleration, and u s = top mass acceleration associated with structural stiness/damping. Table 2. Foundation damping results rom orced vibration testing o test structure in unbraced coniguration. Shaing Direction Long direction Short direction Observed Data (1) Results rom Theoretical Models (2) Period Lengthening, T ~ T Foundation Damping, Period Lengthening Equation 4 Result Equation 5 Result % % 9.1% % % 6.1% Note: (1) Bac-calculated using Eq. 1, n = 2, ied-base and leible-base periods and damping rom Figure 3. (2) Solved using Pais and Kausel impedance unctions as presented in NIST 2012and s = 9.0 (long direction) and s = 7.1 (short direction).

7 Figure 3. Transmissibility unctions rom parametric and non-parametric system identiication using response history data rom large-orce orced vibration testing. Conclusions Two epressions or oundation damping (Eq. 4 and Eq. 5) have been presented that were derived rom irst principles elsewhere (Givens et al. 2015). An advantage o the present solution relative to classical solutions in the literature is that oundation damping is epressed as the sum o readily understandable terms representing hysteretic and various orms o radiation damping. On theoretical grounds, there is no clear beneit o one o the aorementioned oundation damping solutions over the other. A practical beneit o Eq. 4 is that it is epressed entirely in terms o real-valued variables, whereas Eq. 5 includes comple variables that are converted to real numbers using absolute values. Forced vibration tests were perormed on a portable steel column structure with concrete top and bottom slabs to measure SSI eects. The test structure was reconigurable to provide alternate structure stinesses and the tests were perormed at three test sites with dierent soil conditions. Acceleration, displacement, and oundation pressure data was recorded and archived at the Networ or Earthquae Engineering Simulations Research (NEESR) website as project NEES Data rom these tests are useul or inerence o oundation stiness and damping in the orm o impedance unctions. In this paper we ocus on oundation damping or a single case. Results are useul to establish general compatibility between model predictions and data. Acnowledgments Funding or the ield testing described here was provided by the National Science Foundation. The ield wor was completed as part o a multi-institutional NSF grand challenge (NSF Program) project, Mitigation o Collapse Ris in Vulnerable Concrete Buildings, under Grant Number CMMI , and was documented in the

8 Networ or Earthquae Engineering Simulations Research (SR) website, NEEShub, as project NEES We than and sta members or maing the ield testing possible. The authors acnowledge the University o Patras, Greece, or providing a ellowship leave to George Mylonais during the academic years. Reerences ASCE, 2006, Seismic Evaluation and Retroit o Eisting Buildings, ASCE/SEI 41-06, American Society o Civil Engineers, Reston, Virginia. ASCE, 2010, Minimum Design Loads or Buildings and Other Structures, ASCE/SEI 7-10, American Society o Civil Engineers, Reston, Virginia. Aviles, J. and Perez-Rocha, L.E., Evaluation o interaction eects on the system period and the system damping due to oundation embedment and layer depth, J. o Soil Dyn. and Eq. Eng 15, Biela, J., Earthquae response o building-oundation systems, Ph.D. Dissertation, Caliornia Institute o Technology, Pasadena. Federal Emergency Management Agency (FEMA), Improvement o Inelastic Seismic Analysis Procedures, FEMA 440, Federal Emergency Management Agency, Washington, D.C. USA. Givens, M.J., Mylonais, G. and Stewart, J.P Modular analytical solutions or damping o structures on shallow oundations, submitted to Earthquae Spectra, in review. Maravas, A., Mylonais, G. and Karabalis, D.L., Simpliied discrete systems or dynamic analysis o structures on ootings and piles, J. o Soil Dyn. and Eq. Eng 61, NIST, Soil-Structure Interaction or Building Structures. Report No. NIST GCR , National Institute o Standards and Technology, U.S. Department o Commerce, Washington D.C,.USA. Parmelee, R.A., Building-Foundation Interaction Eects, J. o Eng. Mechanics Division 93, Roesset, J.M., Stiness and damping coeicients o oundations, Proc. ASCE Geotechnical Engineering Division National Convention, Oct. 30, Star. L.M, Givens, M.J., Nigbor, R.L., and Stewart, J.P., Field testing o structure on shallow oundation to evaluate SSI eects, Earthquae Spectra. DOI: Stewart, J.P. and Fenves, G.L., System identiication or evaluating soil-structure interaction eects in buildings rom strong motion recordings, J. o Eq. Eng. & Struct. Eng. 27, Tileylioglu, S., Evaluation o soil-structure interaction eects rom ield perormance data, Ph.D. Dissertation, University o Caliornia, Los Angeles, Los Angeles, CA, USA. Veletsos, A.S. and Wei, Y.T., Lateral and rocing vibrations o ootings. J. Soil Mech. and Found. Div., 97, Veletsos, A. S., and Nair, V. V., Seismic interaction o structures on hysteretic oundations, J. Struct. Engrg. 101, Veletsos, A. S., Dynamics o Soil-Foundation Systems, in W.J. Hall (ed.) Structural and Geotechnical Mechanics, Prentice-Hall, Inc., Englewood Clis, New Jersey, pp