A note on the statistics of ordered peaks in stationary stochastic processes

Numerical simulation experiments have been carried out to evaluate the relative performance of three approximate formulations owing to Amini and Trifunac,[1] Gupta and Trifunac[2] and Basu et al.[3] for probability distributions of ordered peaks in stationary stochastic processes. The first two formulations are based on the assumption that the unordered peaks are statistically independent; whereas the formulation of Basu et al.[3] considers the dependence via Markov transition probability. In the formulation of Gupta and Trifunac,[2] the probability distribution of nth order peak is inherently conditioned by the fact that at most (n−1) peaks can occur with higher amplitudes and, thus, the ordered peaks are necessarily dependent. In this paper, ensembles of stationary time-histories are generated for three PSDFs representing a narrow-band, a broad-band and a band-limited white noise type of processes. Comparison of the results for the expected value and standard deviation for various orders of peaks, obtained by averaging over these ensembles, with the corresponding results obtained from the above mentioned three approximations, defined in terms of the moments of the PSDFs, has shown that the formulation of Gupta and Trifunac[2] describes the data well.     

Errors caused by peak factor assumptions in response‐spectrum‐based analyses

The modal combination rules commonly used in response spectrum analyses implicitly assume that the peak factor associated with the response quantity of interest is equal to the peak factors of the contributing modal responses. In this paper, we examine the validity of this assumption and demonstrate that it causes the modal combination rules to over‐represent the contribution of the higher modes of vibration to the total response and under‐represent the contribution of the lower modes. Consequently, a response‐spectrum‐based analysis can yield a biased estimate for the peak value of a response quantity when two or more well‐separated modal frequencies make significant contributions to the total response. To correct this potential bias in response‐spectrum‐based estimates, we develop a procedure for estimating the peak factors that is suitable to the response spectrum analysis calculations commonly used in the current design practice. Examples are presented to demonstrate the proper use and potential impact of the proposed procedure. Copyright © 2015 John Wiley & Sons, Ltd.     

Non-stationary seismic response of MDOF systems by wavelet transform

A wavelet-based formulation has been presented in this paper for the stochastic analysis of a linear multi-degree-of-freedom (MDOF) classically damped system subjected to earthquake ground motion. The ground motion has been modelled as a non-stationary process (both in amplitude and frequency) using wavelets. Closed-form expressions of the moments of the instantaneous Power Spectral Density Function (PSDF) of the response have been derived and used to predict the statistics of the response peak of any desired order. For illustration of the formulation, an example torsionally coupled multistoried building has been considered along with the twenty synthetically generated time-histories corresponding to an example ground motion process. © 1997 John Wiley & Sons, Ltd.     

Response spectrum analysis for floor acceleration

The peak floor acceleration (PFA) is a critical parameter influencing the performance of non‐structural elements in buildings. This paper develops a response spectrum analysis method based on the complete quadratic combination (CQC) rule to estimate the PFA. The method accounts for the rigid contribution of truncated higher modes and the cross‐correlations between all pairs of modes. The approximation is introduced in the time domain and then formulated in the frequency domain by CQC. Application of the method to a continuous cantilever beam idealizing a building with shear walls is presented and compared with alternative formulations. The proposed method is able to provide a consistent estimation of the PFA along the entire structure, not only where the PFA is principally influenced by the first few flexible modes but also where the PFA is mainly related to the rigid response of the structure, for example, near its base. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic soil–structure interaction effects on the seismic response of suspension bridges

A stochastic approach has been formulated for the linear analysis of suspension bridges subjected to earthquake excitations. The transfer functions of various responses have been formulated while including the effects of dynamic Soil–Structure Interaction (SSI) via the use of the fixed-base modes of the structure. The excitation has been characterized by the ‘equivalent stationary’ processes corresponding to the free-field motions at each support and by an assumed coherency function between these motions. The proposed formulation considers the non-stationarity in the structural response due to sudden application of excitation by considering (i) the time-dependent frequency response functions, and (ii) the order statistics formulation for the peak factors in evolutionary response processes. The formulation has been illustrated by analysing the seismic response of the Golden Gate Bridge at San Francisco for two example excitations conforming to USNRC-specified design spectra. The significance of various governing parameters on the dynamic soil–structure interaction effects on the seismic response of suspension bridges has also been studied. It has been found that the contribution of the vertical component of ground motion to the bridge response increases with increasing soil compliance. Also, the extent to which the spatial variation of ground motion affects the bridge response depends on how significant the SSI effects are. Copyright © 1999 John Wiley & Sons Ltd.     

Probabilistic analysis of peak response of MDOF systems with uncertain PSD function

The use of uniform hazard spectra which have the same probability of exceedance at different frequencies has been proposed for the future version of the National Building Code of Canada. Commonly used combination rules to estimate the peak responses of multi‐degree‐of‐freedom (MDOF) systems are the square root of sum of squares rule and the complete quadratic combination rule. However, the probability that the peak response of a MDOF system exceeds the one estimated by using these rules with the peak modal responses from the uniform hazard spectra cannot be inferred directly. The assessment of the probability of exceedance of the peak response of MDOF systems is presented by considering that the uncertainty in seismic excitation due to all potential earthquakes can be lumped in the power spectral density function of the ground acceleration with uncertain model parameters. This probability is evaluated based on the random vibration of linear systems and the first‐order reliability method. It is found that the under‐ or over‐estimations are less than about 5 or 10% if the modal contributions are not within 10–90% of, or not within 20–80% of, the absolute sum of the effective modal peak responses, respectively. Otherwise, severe under‐ or over‐estimation could result. Copyright © 2002 John Wiley & Sons, Ltd.     

A probabilistic assessment of seismic damage in ductile structures

The current practices in ductility-based earthquake design ignore the damage caused by the repeated random inelastic excursions. A ductile structure may however suffer different degrees of damage depending on the number and amplitudes of these inelastic excursions. The information about the largest response peak as available from the response spectra may not thus be enough as the higher-order peaks are likely to play an important role in the progressing damage. A probabilistic model is proposed here to estimate the damage in a structure with a given amount of ductility by using the order statistics of the higher-order peaks. The proposed formulation relates damage to the entire response process, not just to the largest response. It thus accounts for the different roles played by the total number of peaks in the process, the relative amplitudes and number of excursions, and frequency content of the response process.     

Stochastic seismic response of multiply-supported secondary systems in flexible-base structures

A formulation has been proposed for the transfer function of a secondary system response while the primary system is supported on a compliant soil and the excitation comprises of translational ground motion at its base. For this purpose, the earlier formulation of the authors for the fixed-base case, which exactly considers the interaction between the two sub-systems and is based on the use of their individual modal properties, has been extended. Also, the concept of modifying the input excitation for the interaction accelerations (associated with the soil–structure interaction) has been used. An example P–S system and three example earthquake excitations have been considered to illustrate the proposed formulation and to estimate the expected response peak amplitudes in the secondary system. This study shows that ‘detuning’ of the tuned systems may occur in case of significant soil–structure interaction. Further, for the reasons of both safety and economy, ignoring the interaction effects in designing the secondary systems may not always be justified. Copyright © 1999 John Wiley & Sons, Ltd.     

Damping formulation for nonlinear 1D site response analyses

Measurements and observations of ground shaking during large earthquakes have demonstrated the predominant role of site effects in the response of infrastructure during a seismic event. Despite significant efforts to model the hysteretic response and nonlinearity of soils due to medium and large ground motions, the most widely accepted nonlinear site response methods are not able to represent simultaneously the changes of stiffness and energy dissipation (damping) observed in both laboratory tests and during earthquake events. This paper presents two new soil damping formulations implemented in nonlinear one-dimensional site response analysis for small and large strains. The first formulation introduces an approach to construct a frequency-independent viscous damping matrix which reduces the over-damping at high frequencies, and therefore, the filtering at those frequencies. The second formulation introduces a reduction factor that modifies the extended Masing loading/unloading strain–stress relationship to match measured modulus reduction and damping curves simultaneously over a wide range of shear strains. A set of examples are introduced to illustrate the effect of using the two proposed formulations, separately and simultaneously, in nonlinear site response analyses.     

Wavelet‐based non‐stationary response analysis of a friction base‐isolated structure

A wavelet‐based stochastic formulation has been presented in this paper for the seismic analysis of a base‐isolated structural system which is modelled as a two‐degree‐of‐freedom (2‐DOF) system. The ground motion has been modelled as a non‐stationary process (both in amplitude and frequency) by using modified Littlewood–Paley basis wavelets. The proposed formulation is based on replacing the non‐linear system by an equivalent linear system with time‐dependent damping properties. The expressions of the instantaneous damping and the power spectral density function (PSDF) of the superstructure response have been obtained in terms of the functionals of input wavelet coefficients. The proposed formulation has been validated by simulating a ground motion process. The effect of the frequency non‐stationarity on the non‐linear response has also been studied in detail, and it has been clearly shown how ignoring the frequency non‐stationarity in the ground motion leads to inaccurate non‐linear response calculations. Copyright © 2000 John Wiley & Sons, Ltd.     

Variations in the dynamic response of structures founded on piles induced by obliquely incident SV waves

Although the seismic actions generally consist of a combination of waves, which propagates with an angle of incidence not necessarily vertical, the common practice when analyzing the dynamic behavior of pile groups is based on the assumption of vertically incident wave fields. The aim of this paper is to analyze how the angle of incidence of SV waves affects the dynamic response of pile foundations and piled structures. A three-dimensional boundary element-finite element coupling formulation is used to compute impedances and kinematic interaction factors corresponding to several configurations of vertical pile groups embedded in an isotropic homogeneous linear viscoelastic half-space. These results, which are provided in ready-to-use dimensionless graphs, are used to determine the effective dynamic properties of an equivalent single-degree-of-freedom oscillator that reproduces, within the range where the peak response occurs, the response of slender and nonslender superstructures through a procedure based on a substructuring model. Results are expressed in terms of effective flexible-base period and damping as well as maximum shear force at the base of the structure. The relevance and main trends observed in the influence of the wavefront angle of incidence on the dynamic behavior of the superstructure are inferred from the presented results. It is found that effective damping is significantly affected by the variations of the wave angle of incidence. Furthermore, it comes out that the vertical incidence is not always the worst-case scenario.     

A new model for spectral velocity ordinates at long periods

The estimation of peak linear response via elastic design (response) spectra continues to form the basis of earthquake‐resistant design of structural systems in various codes of practice all over the world. Many response spectrum‐based formulations of peak linear response require an additional input of the spectral velocity (SV) ordinates consistent with the specified seismic hazard. SV ordinates have been conventionally approximated by pseudo spectral velocity (PSV) ordinates, which are close to the SV ordinates only over the intermediate frequency range coinciding with the velocity‐sensitive region. At long periods, PSV ordinates underestimate the SV ordinates, and this study proposes a formulation of a correction factor (>1) that needs to be multiplied by the PSV ordinates in order to close the gap between the two sets of ordinates. A simple model is proposed in the form of a power function in oscillator period to estimate this factor in terms of two governing parameters which are in turn estimated from two single‐parameter scaling equations. The parameters considered for the scaling equations are (1) the period at which the PSV spectrum is maximized and (2) the rate of decay of the pseudo spectral acceleration (PSA) amplitudes at long periods. For a given damping ratio, four regression coefficients are determined for the scaling equations with the help of 205 ground motions recorded in western USA. A numerical study undertaken with the help of several design PSA spectra and ensembles of spectrum‐compatible ground motions illustrates the effectiveness of the proposed correction factor, together with the proposed scaling models, in comparison with the PSV approximation in a variety of design situations. Both the input parameters mentioned above can be easily obtained from the specified design spectrum, and thus the proposed model is convenient to use.     

Dynamics of three-block assemblies with unilateral deformable contacts. Part 1: contact modelling

As far as the dynamics of multibody systems is concerned, a brief review has been performed in order to frame the dynamic response of a trilith (the simplest scheme of a colonnade belonging to a temple) into a wide theoretical background. Under the assumption of rigid bodies, two different approaches can be found in the literature: rigid or deformable contacts formulation. In this paper, an effort is made at outlining the principle of rigid contact formulation and of deformable contact formulation. The latter approach can be assumed within the framework of the distinct element method; for this purpose a model of deformable contact has been proposed in order to simulate the real behaviour of stone joints. The sample application referred to the trilith will be presented in Part 2. Copyright © 1999 John Wiley & Sons, Ltd.     

Principal axes of M-DOF structures Part Ⅱ: Dynamic loading

This paper is the second in a two-part series that discusses the principal axes of M-DOF structures subjected to static and dynamic loads. The primary purpose of this series is to understand the magnitude of the dynamic response of structures to enable better design of structures and response modification devices/systems. Under idealized design conditions, the structural responses are obtained by using single direction input ground motions in the direction of the intended response modification devices/systems, and by assuming that the responses of the structure is decoupleable in three mutually perpendicular directions. This standard practice has been applied to both new and retrofitted structures using various seismic protective systems. Very limited information is available on the effects of neglecting the impact of directional couplings (cross effects of which torsion is a component) of the dynamic response of structures. In order to quantify such effects, it is necessary to examine the principal axes of structures under both static and dynamic loading. In this two-part series, the first paper is concerned with static loading, which provides definitions and fundamental formulations, with the conclusion that cross effects of a statically loaded M-DOF structure resulting from the lack of principal axes are of insignificant magnitude. However, under dynamic or earthquake loading, a relatively small amount of energy transferred across perpendicular directions is accumulated, which may result in significant enlargement of the structural response. This paper deals with a formulation to define the principal axes of M-DOF structures under dynamic loading and develops quantitative measures to identify cross effects resulting from the non-existence of principal axes.     

Principal axes of M-DOF structures Part II: Dynamic loading

This paper is the second in a two-part series that discusses the principal axes of M-DOF structures subjected to static and dynamic loads. The primary purpose of this series is to understand the magnitude of the dynamic response of structures to enable better design of structures and response modification devices/systems. Under idealized design conditions, the structural responses are obtained by using single direction input ground motions in the direction of the intended response modification devices/systems, and by assuming that the responses of the structure is decoupleable in three mutually perpendicular directions. This standard practice has been applied to both new and retrofitted structures using various seismic protective systems. Very limited information is available on the effects of neglecting the impact of directional couplings (cross effects of which torsion is a component) of the dynamic response of structures. In order to quantify such effects, it is necessary to examine the principal axes of structures under both static and dynamic loading. In this two-part series, the first paper is concerned with static loading, which provides definitions and fundamental formulations, with the conclusion that cross effects of a statically loaded M-DOF structure resulting from the lack of principal axes are of insignificant magnitude. However, under dynamic or earthquake loading, a relatively small amount of energy transferred across perpendicular directions is accumulated, which may result in significant enlargement of the structural response. This paper deals with a formulation to define the principal axes of M-DOF structures under dynamic loading and develops quantitative measures to identify cross effects resulting from the non-existence of principal axes.     

Biot理论的唯象修正对S波特性的影响

将复模量引入Biot方程后，在一维条件下通过S波的波动方程研究了S波的传播特性，S波的数值分析显示在频率域或温度域上都能获得热弛豫衰减峰和Biot衰减峰. 在频率域上由于温度的变化引起两峰相向位移,在温度域上,因频率的变化也发生相对移动. 随着温度和频率的不断提高，两峰发生叠加，叠加后两峰互换位置. 低频或低温段的热弛豫峰移到了高频或高温段，高频或高温段的Biot峰移到了低频或低温段.由于两峰的衰减机制不同，导致S波波速随频率或温度变化规律的复杂性. 这些规律已部分被共振实验所证实，证实该理论模型具有实验基础.     

Shaking‐table tests of flat‐bottom circular silos containing grain‐like material

According to Eurocode 8, the seismic design of flat‐bottom circular silos containing grain‐like material is based on a rough estimate of the inertial force imposed on the structure by the ensiled content during an earthquake: 80% of the mass of the content multiplied by the peak ground acceleration. A recent analytical consideration of the horizontal shear force mobilised within the ensiled material during an earthquake proposed by some of the authors has resulted in a radically reduced estimate of this load suggesting that, in practice, the effective mass of the content is significantly less than that specified. This paper describes a series of laboratory tests that featured shaking table and a silo model, which were conducted in order to obtain some experimental data to verify the proposed theoretical formulations and to compare with the established code provisions. Several tests have been performed with different heights of ensiled material – about 0.5 mm diameter Ballotini glass – and different magnitudes of grain–wall friction. The results indicate that in all cases, the effective mass is indeed lower than the Eurocode specification, suggesting that the specification is overly conservative, and that the wall–grain friction coefficient strongly affects the overturning moment at the silo base. At peak ground accelerations up to around 0.35 g, the proposed analytical formulation provides an improved estimate of the inertial force imposed on such structures by their contents. Copyright © 2015 John Wiley & Sons, Ltd.     

Dependence between flood peaks and volumes: a case study on climate and hydrological controls

Abstract

The aim of this paper is to understand the causal factors controlling the relationship between flood peaks and volumes in a regional context. A case study is performed based on 330 catchments in Austria ranging from 6 to 500 km² in size. Maximum annual flood discharges are compared with the associated flood volumes, and the consistency of the peak–volume relationship is quantified by the Spearman rank correlation coefficient. The results indicate that climate-related factors are more important than catchment-related factors in controlling the consistency. Spearman rank correlation coefficients typically range from about 0.2 in the high alpine catchments to about 0.8 in the lowlands. The weak dependence in the high alpine catchments is due to the mix of flood types, including long-duration snowmelt, synoptic floods and flash floods. In the lowlands, the flood durations vary less in a given catchment which is related to the filtering of the distribution of all storms by the catchment response time to produce the distribution of flood producing storms.

Editor Z.W. Kundzewicz     

A simple model of hillslope response for overland flow generation

This paper deals with the derivation of the hydrological response of a hillslope on the assumption of quick runoff by surface runoff generation. By using the simple non‐linear storage based model, first proposed by Horton, an analytical solution of the overland flow equations over a plane hillslope was derived. This solution establishes a generalization for different flow regimes of Horton's original solution, which is valid for the transitional flow regime only. The solution proposed was compared successfully with that of Horton and, for the turbulent flow regime, to the one derived from kinematic wave theory. This solution can be applied easily to both stationary and non‐stationary rainfall excess events. An analytical solution for the instantaneous response function (IRF) was also derived. Finally, simple expressions to compute peak and time to peak of IRF are proposed. Copyright © 2001 John Wiley & Sons, Ltd.