Vibration of viscoelastic foundations

Simple approximate solutions are presented for the steady-state response of a massless, rigid disk, which is supported at the surface of a linear viscoelastic halfspace and is excited by a harmonically varying horizontal or vertical force, or by a harmonically varying moment about an axis in the plane of the disk. The halfspace is idealized either as a standard Voigt solid or as a constant hysteretic solid. The solution for the massless disk is then applied to the analysis of foundations with mass, and for each mode of excitation, comprehensive numerical data are presented to illustrate the effects of viscoelastic action on the response of both the massless disk and of foundations with mass It is shown that the principal effects of material damping are to increase the damping capacity of the foundation and to reduce its ‘natural’ frequency. In the mathematically equivalent representation of the elastically supported massless disk as a spring-dashpot combination without mass, it is shown that the effective stiffness of the spring may become negative over significant ranges of the exciting frequency, and that it is preferable to model the disk-halfspace system as an oscillator with mass, taking its spring constant equal to the static stiffness of the original system.     

Response of fixed offshore structures in random sea

An integrated model is developed for the short-term and long-term dynamic response of an offshore structure subjected to random wave excitation. A discrete linear, elastic model of the upper structure is coupled with an iterative linear quasi-three-dimensional finite element model for the pile-soil medium, and the system is subjected to stochastic storms described by mean rate of arrival, joint probability distribution of storm duration and average intensity, and a random process that describes the variations of a statistical wave height measure during each storm. Sarpkaya's 1977 experimental information is used to specify values for C_M and C_D, in Morison's equation, consistent with the flow and response characteristics. Soil (clay) degradation, due to cyclic excitation, is followed during the passage of a storm. The indicator of soil degradation is taken to be the value of the horizontal foundation stiffness and the stiffness degradation is modelled by a homogeneous Markov process.     

Equivalent mechanical models of sloshing fluid in arbitrary‐section aqueducts

This paper reports on a semi‐analytical/numerical method to model sloshing water in an arbitrarily shaped aqueduct. The water motion is assumed to be inviscid, compressible, and linear (small displacement). The transverse sloshing fluid in an aqueduct is equivalently simplified as a fixed rigid mass M₀ and a mass–spring system (M₁, K₁). According to a rule that the actual fluid (computed with finite element model) and its equivalent mechanical model have the same first sloshing frequency and acting effects on the aqueduct, the analytical solutions of the fixed (impulsive) mass M₀, sloshing (convective) massM₁, spring stiffness K₁, and their locations in the aqueduct body are acquired by the least squares (curve fitting) algorithm. Applying this equivalent principle, the equivalent mechanical models are respectively obtained for the sloshing water in rectangular, semicircular, U‐shaped, and trapezoid aqueducts. The equivalent principle and fluid models are validated through comparison investigations involving rectangular and U‐shaped aqueducts. The dynamic properties and seismic responses of the original and equivalent systems are simulated, compared, and discussed for a U‐shaped aqueduct bridge. The main purpose of this paper is to provide a simplified model of sloshing fluid for the seismic/wind‐resistant computation of the support structures of the aqueduct bridge. Copyright © 2011 John Wiley & Sons, Ltd.     

Malleable disc base for direct infiltration measurements using the tension infiltrometry technique

The correct use of the tension disc infiltrometer requires the membrane of the disc base to be completely in contact with the soil surface. To achieve this contact, a thick layer of sand is commonly placed between the soil surface and the disc base. This paper presents an alternative disc (M_DB), which, by incorporating a malleable membrane, allows direct infiltration measurements without using a contact sand layer. Infiltration curves obtained with this new design in a soil under three different tillage management treatments were compared with the corresponding curves obtained with a conventional disc (C_DB) that uses a contact sand layer. The cumulative infiltration curves measured with C_DB were analysed by the differentiated linearization (DL) method, and the corresponding curves obtained with M_DB were analysed using both the DL and the cumulative linearization (CL) models. The values of hydraulic conductivity (K₀) and sorptivity (S₀) estimated with C_DB were also compared with those obtained with M_DB. Finally, the cumulative infiltration curves measured with C_DB and M_DB were compared with the corresponding modelled function for the respective K₀ and S₀ values calculated with the CL and DL models. The results show that, compared with C_DB without a contact sand layer, M_DB allows complete soil surface wetting even when non‐smoothed soil surfaces are used. The C_DB, which yielded average K₀ values 18% lower than those estimated with M_DB, gave the highest values of standard error for the hydraulic parameters calculated. Furthermore, the subjective method employed in the C_DB‐DL technique, which requires the first points of the differential infiltration line corresponding to the sand layer to be manually removed, introduces additional uncertainties in estimating S₀ and K₀. Comparison between the modelled and measured infiltration curves demonstrates that the DL or CL methods applied to M_DB gave excellent estimates of S₀ and K₀. Copyright © 2012 John Wiley & Sons, Ltd.     

Dynamic behaviour of rigid foundations of arbitrary shape on a halfspace

An approximate method for computation of the compliance functions of rigid plates resting on an elastic or visco-elastic halfspace excited by forces and moments in all degrees of freedon is presented. The method is based on a Green's function approach. These functions are given for all degrees of freedom in form of well-behaved integrals. The numerical procedure is described and is used to evaluate the vertical, horizontal, rocking and torsion compliance functions of rectangular plates with side ratios 1 ≤ b/a ≤ 10 and non-dimensional frequency 0≤a₀≤10. It is shown how this method can be extended to problems concerning a linear visco-elastic halfspace and a halfspace with variable stiffness.     

Vertical vibration of ring foundations

A comprehensive analysis is made of the harmonic response of vertically excited, massless, rigid ring foundations supported at the surface of an homogeneous elastic halfspace. The parameters considered include the thickness to radius ratio for the ring, the frequency of the exciting force and of the resulting steady-state response, and Poisson's ratio for the supporting medium. The response quantities examined include the stiffness and damping coefficients of the system in an equivalent spring–dashpot representation of the supporting medium, the displacements of the foundation and of points on the ground surface away from the foundation, and the normal pressure at the foundation–medium interface. The results in each case are compared with those obtained for a solid disk having the same radius as the outer radius of the ring, and a simple approximate model is used to interpret the results and to provide insight into the action of the system. The reported data are computed by a method of analysis that takes due account of the mixed boundary conditions at the surface of the halfspace, and are believed to be of high accuracy.     

Dynamic stiffness of foundation embedded in layered halfspace based on wave propagation in cones

An Erratum has been published for this article in Earthquake Engineering & Structural Dynamics 33(6) 2004, 793. The dynamic stiffness of a foundation embedded in a multiple‐layered halfspace is calculated postulating one‐dimensional wave propagation in cone segments. In this strength‐of‐materials approach the sectional property of the cone segment increases in the direction of wave propagation. Reflections and refractions with waves propagating in corresponding cone segments occur at layer interfaces. Compared to rigorous procedures the novel method based on cone segments is easy to apply, provides conceptual clarity and physical insight in the wave propagation mechanisms. This method postulating one‐dimensional wave propagation in cone segments with reflections and refractions at layer interfaces is evaluated, calculating the dynamic stiffness of a foundation embedded in a multiple‐layered halfspace. For sites resting on a flexible halfspace and fixed at the base, engineering accuracy (deviation of ±20%) is achieved for all degrees of freedom with a vast parameter variation. The behaviour below the cut‐off frequency in an undamped site fixed at its base is also reliably predicted. The accuracy is, in general, better than for the method based on cone frustums, which can lead to negative damping. Copyright © 2003 John Wiley & Sons, Ltd.     

An experimental investigation of vertical vibration of model footings on sand

Many free and forced vertical vibrations tests were conducted on surface and embedded models for footings on dry and moist poorly graded sand that has the following properties: D₁₀ = 0·21 mm, D₃₀ = 0·305 mm, D₆₀ = 0·423 mm, C_u = 2, C_c = 1·05, G_s = 2·66, γ_max = 1·74 cm⁻³ and γ_min = 1·4 g cm⁻³. The tests were conducted at relative density Dr = 82%. The effect of mass, area, geometry, embedment, saturation, load amplitude and frequency were studied. For this purpose square, rectangular and circular models of concrete footings were chosen. Swieleh sand was chosen as the foundation soil. Results have been obtained for models having different mass, same base shape and area; models of different base area and equal base shape and mass; and models of different base shape geometry and about equal masses and base areas.Forced vertical vibration tests results showed an increase in natural frequency and a reduction in amplitude with the increase in embedment depth, degree of saturation and footing base area. Increasing the mass of model footing resulted in a decrease in the natural frequency while the dynamic response increased. Also, results showed that the circular model footing gives low values of dynamic response in comparison to other models.Free vertical vibration test results showed an increase in damping ratio with increase in the base area of the model footing, depth of embedment and saturation of sand. On the other hand, the results showed a decrease in damping ratio with increase in the footing mass. Circular footing gives the highest value of damping ratio among other footings.Results showed that the best method for evaluating the dynamic stiffness for vertical vibration is using Dobry and Gazetas¹ (J. Geotech. Engng, ASCE, 1986, 112, 109–133) equations together with the formula proposed by Ronald and Bojan² (J. Geotech. Engng, ASCE, 1995, 121, 274–286) for evaluating the equivalent shear modulus.     

M TSU : Recovering Seismic Moments from Tsunameter Records

We define a new magnitude scale, M_TSU, allowing the quantification of the seismic moment M₀ of an earthquake based on recordings of its tsunami in the far field by ocean-bottom pressure sensors (``tsunameters') deployed in ocean basins, far from continental or island shores which are known to affect profoundly and in a nonlinear fashion the amplitude of the tsunami wave. The formula for M_TSU, M_TSU = log₁₀ M₀ − 20 = log₁₀ X (ω) + C_D^TSU + C_S^TSU + C₀, where X (ω) is the spectral amplitude of the tsunami, C_D^TSU a distance correction and C_S^TSU a source correction, is directly adapted from the mantle magnitude M_m introduced for seismic surface waves by Okal and Talandier. Like M_m, its corrections are fully justified theoretically based on the representation of a tsunami wave as a branch of the Earth's normal modes. Even the locking constant C₀, which may depend on the nature of the recording (surface amplitude of the tsunami or overpressure at the ocean floor) and its units, is predicted theoretically. M_TSU combines the power of a theoretically developed algorithm, with the robustness of a magnitude measurement that does not take into account such parameters as focal geometry and exact depth, which may not be available under operational conditions in the framework of tsunami warning. We verify the performance of the concept on simulations of the great 1946 Aleutian tsunami at two virtual gauges, and then apply the algorithm to 24 records of 7 tsunamis at DART tsunameters during the years 1994–2003. We find that M_TSU generally recovers the seismic moment M₀ within 0.2 logarithmic units, even under unfavorable conditions such as excessive focal depth and refraction of the tsunami wave around continental masses. Finally, we apply the algorithm to the JASON satellite trace obtained over the Bay of Bengal during the 2004 Sumatra tsunami, after transforming the trace into a time series through a simple ad hoc procedure. Results are surprisingly good, with most estimates of the moment being over 10²⁹ dyn-cm, and thus identifying the source as an exceptionally large earthquake.     

Stress Changes due to Recent Seismic Events in the Central Apennines (Italy)

The Central Apennines, Italy, are characterized by moderate seismic activity on normal faults, oriented in directions parallel to the Apenninic chain. The subject of this study is the Umbria-Marche Apennines, a segment approximately 200-km long, where three main seismic events occurred in the last three decades. The 1979 Norcia earthquake was a M_w = 5.8 event, taking place at the south end of the considered segment. The 1984 Gubbio earthquake was a M_w = 5.6 event which took place at the north end. The 1997-1998 Colfiorito sequence constituted 8 main shocks with magnitudes M_w between 5 and 6 and epicenters located between the Gubbio and the Norcia earthquake areas. A model made of an elastic half-space is considered, in which the seismic sources are represented by rectangular dislocations which have the appropriate values of source parameters, and in which the static stress field produced by each event is calculated. The analysis of the Coulomb stress change (ΔC) as a function of time shows that the coseismic stress transfer and fault interaction played an important role in the region during the past three decades: 7 earthquakes of the 9 considered took place where ΔC>0. Such an interaction has been confirmed by the analysis of the aftershocks in the Colfiorito zone post September 26, 1997: about the 61% of the aftershocks considered took place where ΔC>0. The comparison between the ΔC^s due to the coseismic stress transfer and the rate ΔĊ^t due to the tectonic stress allows us to quantify the time advance of the earthquakes. The ΔC^s pattern shows positive values in two areas that can be regarded as historical seismic gaps.     

A study of rupture length vs. moment for synthetic earthquakes

From the events synthesized from the one-dimensional dynamical mass-spring model proposed byBurridge andKnopoff (1967), the relation between rupture length and earthquake momentM is studied for various model parameters. The earthquake moment is defined to be the total displacement of a connected set of mass elements which slide during an event. A parameter stiffness ratios is defined as the ratio of the spring constant between the two mass elements to that between one mass element and the moving plate. The velocity-dependent friction law (including weakening and hardening processes) is taken to control the sliding of a mass element. The distribution of the breaking strengths over the system is considered to be a fractal function. The cases for severals values and different velocity-dependent friction laws with different decreasing ratesr _w of the frictional force with sliding velocity are studied numerically. The weakening process of the frictional force from the static one to the dynamic one obviously affects theM– relation. Meanwhile, a rapid weakening process rather than a slow weakening process can result in aM– relation, which is comparable to the observed one. Although an increase in thes value can yield an increase in the upper bound of the value and the number of events with largeM and values, the scaling of theM– relation is not affected by the change of thes value. For the cases in this study, the theoretical –M relations for small events withM<1 are almost in the form: M ^1/2, while those for large events withM>1 have a scaling exponent less than but close to 1. In addition, the fractal dimension, the friction drop ratio and the roughness of the distribution of the breaking strengths over the fault surface are the minor parameters influencing the –M relation. A comparison between the theoreticalM– relation and the observed one for strike-slip earthquakes shows that for large events the theoreticalM– relation is quite consistent with the observed one, while for small events there is a one-order difference in the two relations. For the one-dimensional model, the decreasing rate of the dynamic frictional force with velocity is the main factor in affecting the characteristic value of the earthquake moment, at which the scaling of theM– relation changes.     

Kinematic response functions and dynamic stiffnesses of bridge embankments

Recognizing that soil–structure interaction affects appreciably the earthquake response of highway overcrossings, this paper compares approximate analytical solutions and finite element results to conclude on a simple procedure that allows for the estimation of the kinematic response functions and dynamic stiffnesses of approach embankments. It is shown that the shear‐wedge model yields realistic estimates for the amplification functions of typical embankments and reveals the appropriate levels of dynamic strains which are subsequently used to estimate the stiffness and damping coefficients of embankments. The shear‐wedge model is extended to a two‐dimensional model in order to calculate the transverse static stiffness of an approach embankment loaded at one end. The formulation leads to a sound closed‐form expression for the critical length, L_c, that is the ratio of the transverse static stiffness of an approach embankment and the transverse static stiffness of a unit‐width wedge. It is shown through two case studies that the transverse dynamic stiffness (‘spring’ and ‘dashpot’) of the approach embankment can be estimated with confidence by multiplying the dynamic stiffness of the unit‐width wedge with the critical length, L_c. The paper concludes that the values obtained for the transverse kinematic response function and dynamic stiffness can also be used with confidence to represent the longitudinal kinematic response function and dynamic stiffness, respectively. Copyright © 2002 John Wiley & Sons, Ltd.     

Case 25 application of the concentration parameter of seismoactive faults to Southern California

In this paper we evaluate the present state of the seismic regime in Southern California using the concentration parameter of seismogenic faults (K _sf ,Sobolev andZavyalov, 1981). The purpose of this work is to identify potential sites for large earthquakes during the next five or ten years. The data for this study derived from the California Institute of Technology's catalog of southern California earthquakes, and spanned the period between 1932 to June 1982. We examined events as small asM _L 1.8 but used a magnitude cutoff atM _L =3.3 for a detailed analysis. The size of the target earthquakes (M _M ) was chosen as 5.3 and 5.8.The algorithm for calculatingK _sf used here was improved over the algorithm described bySobolev andZavyalov (1981) in that it considered the seismic history of each elementary seismoactive volume. The dimensions of the elementary seismoactive volumes were 50 km×50 km and 20 km deep. We found that the mean value ofK _sf within 6 months prior to the target events was 6.1±2.0 for target events withM _L 5.3 and 5.41.8 for targets withM _L 5.8. Seventy-three percent of the targets withM _L 5.8 occurred in areas whereK _sf was less than 6.1. The variance of the time between the appearance of areas with lowK _sf values and the following main shocks was quite large (from a few months to ten years) so this parameter cannot be used here for accurate predictions of occurrence time.Regions where the value ofK _sf was below 6.1 at the end of our data set (June, 1982) are proposed as the sites of target earthquakes during the next five to ten years. The most dangerous area is the area east of San Bernardino whereK _sf values are presently between 2.9 and 3.7 and where there has been no earthquake withM _L 5.3 since 1948.     

城市轨道桥梁桩基抗震简化模型的比较研究

以郑州城郊铁路工程中独柱高架车站为例,将土体化为一系列弹簧,描述土体的变形性质。通过比较《城市轨道交通结构抗震设计规范》中的非线性土弹簧、《铁路桥涵地基和基础设计规范》中的m系数法弹簧及Mindlin解弹簧进行建模计算得到的车站结构的地震响应,由结果可知结构地震响应对承台处弹簧刚度最为敏感。另外,将分布弹簧模型等代为六弹簧模型进行地震反应计算,结果表明桩体质量的影响与承台质量相比很小。     

二维地基波阻板隔振分析

本文采用了以薄层法层状半空间基本解答作为格林函数的边界元法,对均质弹性半空间和层状半空间中二维波阻板隔振设计进行了详细的参数分析。分析结果表明,均质弹性半空间和层状半空间内采用波阻板隔振均可取得较好的隔振效果;应保证波阻板具有合理的宽度和较小的埋深,才能得到理想的隔振效果;同时增加波阻板的厚度和模量是提高隔振效果的最有效的2种措施;此外,分层性对波阻板隔振效果也具有一定的影响。     

Discrete modelling of vertical track–soil coupling for vehicle–track dynamics

This paper presents a coupled lumped mass model (CLM model) for the vertical dynamic coupling of railway track through the soil. The well-known Winkler model and its extensions are analysed and fitted on the result obtained numerically with a finite–infinite element model in order to validate the approach in a preliminary step. A mass–spring–damper system with frequency independent parameters is then proposed for the interaction between the foundations, representing the contact area of the track with the soil. The frequency range of track–soil coupling is typically under 100 Hz. Analytical expressions are derived for calibrating the system model with homogeneous and layered half-spaces. Numerical examples are derived, with emphasis on soil stiffness and layering. The dynamic analysis of a track on various foundation models is compared with a complete track–soil model, showing that the proposed CLM model captures the dynamic interaction of the track with the soil and is reliable to predict the vertical track deflection and the reaction forces acting on the soil surface.     

Experimental evidence for flexibility of a building foundation supported by concrete friction piles

This article explores the possibility to measure deformations of building foundations from measurements of ambient noise and strong motion recordings. The case under study is a seven-storey hotel building in Van Nuys, California. It has been instrumented by strong motion accelerographs, and has recorded several earthquakes, including the 1971 San Fernando (M_L=6.6, R=22 km), 1987 Whittier–Narrows (M_L=5.9, R=41 km), 1992 Landers (M_L=7.5, R=186 km), 1992 Big Bear (M_L=6.5, R=149 km), and 1994 Northridge (M_L=6.4, R=1.5 km) earthquake and its aftershocks (20 March: M_L=5.2, R=1.2 km; 6 December, 1994: M_L=4.3, R=11 km). It suffered minor structural damage in 1971 earthquake and extensive damage in 1994. Two detailed ambient vibration tests were performed following the Northridge earthquake, one before and the other one after the 20 March aftershock. These included measurements at a grid of points on the ground floor and in the parking lot surrounding the building, presented and analyzed in this article. The analysis shows that the foundation system, consisting of grade beams on friction piles, does not act as a “rigid body” but deforms during the passage of microtremor and therefore earthquake waves. For this geometrically and by design essentially symmetric building, the center of stiffness of the foundation system appears to have large eccentricity (this is seen both from the microtremor measurements and from the earthquake recordings). This eccentricity may have contributed to strong coupling of transverse and torsional responses, and to larger than expected torsional response, contributing to damage during the 1994 Northridge, earthquake.     

Continental influence on the spatial distribution of particulate loading over the Indian Ocean during winter season

Advection of continental aerosols over to the Arabian Sea and Indian Ocean regions during winter (when the prevailing synoptic wind is north-easterly) has been studied by examining the spatial distribution of the near-surface aerosol mass concentration (M_C). In the northern part of Arabian Sea and north of the inter-tropical convergence zone (ITCZ) M_C is significantly larger than that in the southern side of ITCZ. A prominent high in M_C is observed near the mid-Arabian Sea region ∼12° to 15°N. Associated with local convergences introduced by mesoscale weather systems, pockets of high M_C are observed at different locations over the oceanic regions. Significantly low values of M_C are observed in the ITCZ. Surface streamlines and wind patterns indicate advection of continental aerosols from the adjoining landmass over to the oceanic environment. Aerosol mass concentration in the far oceanic region is also influenced by in situ production of sea-spray aerosols, which depends on wind speed. The nature of advection of continental aerosols has been studied by separating the wind-induced component of M_C and chemical nature of the aerosols. The spatial variation of the wind-independent component matches well with that of the non-sea-salt aerosols estimated from chemical composition.     

The elastic properties of single-crystal fayalite as determined by dynamical measurement techniques

We present new elasticity measurements on single-crystal fayalite and combine our results with other data from resonance, pulse superposition interferometry, and Brillouin scattering to provide a set of recommended values for the adiabatic elastic moduliC _ij and their temperature variations. We use a resonance method (RPR) with specimens that were previously investigated by pulse superposition experiments. The nineC _ij of fayalite are determined from three new sets of measurements. One set of our newC _ij data is over the range 300–500 K. We believe that the relatively large discrep ncies found in someC _ij are due in large part to specimen inhomogeneities (chemical and microstructural) coupled with differences in the way various techniques sample, rather than only systematic errors associated with experimental procedures or in the preparations of the specimens.Our recommendeaC _ij's (GPa) and (C _ij/T) _p (GPa/K) are: The resulting values for the isotropic bulk and shear moduli,K _s and , and their temperature derivatives are:K _s=134(4) GPa; =50.7(0.3) GPa; (K _s/T) _p =–0.024(0.005) GPa/K; and (/T) _p =–0.013(0.001) GPa/K. An important conclusion is thatK _s increases as the Fe/(Fe+Mg) ratio in olivine is increased.     

Feasibility of grain-size analysis methods for determination of vertical hydraulic conductivity of streambeds

Accurate estimation of streambed vertical hydraulic conductivity (K_v) is of great importance in the analysis of water quantity exchange and solute transfer between a stream and its sediments. The paper analyzed the inaccuracy of hydraulic conductivity values of sediments derived from grain-size distribution (K_g), which were determined from eight empirical grain-size methods to represent streambed K_v. In this study, the values of K_v for a streambed were derived using falling-head standpipe permeameter tests conducted at eight study sites in the Elkhorn River, Nebraska, and the tested streambed columns were then collected for grain-size analysis by sieving. These empirical methods were used to calculate the K_g values of the streambed from grain-size distribution data of sediments. Unlike many other studies, this study verifies K_g from grain-size distribution with K_v from permeameter tests on the basis of the same samples of streambed sediments. The K_g values derived from the eight empirical methods were larger than the K_v from permeameter tests; there are five methods that give K_g values of about 3–6 times larger than these K_v. The K_g values from the Kozeny formula followed by the Hazen formula give the largest overestimation error if they are used to represent the K_v of the streambed. The USBR and Shepherd formulas generated K_g values close to K_v, but these K_g values are still larger in general than the K_v values. Moreover, the new values of coefficient C for the empirical formulas were revised so that they can be used to calculate the approximate K_v of a streambed. Among the eight methods, the ratios of the original C values to the average new C range from 1.3 to 5.9. It can be hypothesized that smaller C values must be used in the estimation of K_v for general soil samples if these empirical formulas are used to calculate K_v.