Non‐parametric–parametric model for random uncertainties in non‐linear structural dynamics: application to earthquake engineering

This paper deals with the transient response of a non‐linear dynamical system with random uncertainties. The non‐parametric probabilistic model of random uncertainties recently published and extended to non‐linear dynamical system analysis is used in order to model random uncertainties related to the linear part of the finite element model. The non‐linearities are due to restoring forces whose parameters are uncertain and are modeled by the parametric approach. Jayne's maximum entropy principle with the constraints defined by the available information allows the probabilistic model of such random variables to be constructed. Therefore, a non‐parametric–parametric formulation is developed in order to model all the sources of uncertainties in such a non‐linear dynamical system. Finally, a numerical application for earthquake engineering analysis is proposed concerning a reactor cooling system under seismic loads. Copyright © 2003 John Wiley & Sons, Ltd.     

The legacy of initial conditions in landscape evolution

Landscapes subject to constant forcing tend to evolve toward equilibrium states in which individual landforms have similar characteristics. Yet, even in landscapes at or near equilibrium, there can be significant variability among individual landforms. Furthermore, sites subject to similar processes and conditions can have different mean landform characteristics. This variability is often ascribed to on‐going transient evolution, or to heterogeneity in processes, material properties, forcing, or boundary conditions. Three surprising outcomes of landform evolution models suggest, however, that such variability could arise in equilibrium landscapes without any heterogeneity in the physical processes shaping the topography. First, homogeneous systems subjected to constant forcing can generate a heterogeneous distribution of equilibrium landforms. Second, even simple non‐linear systems can have multiple stable equilibrium states. Third, evolving landscapes can exhibit path dependence and hysteresis. We show how these three mechanisms can produce variability in landforms that arises from the characteristics of the initial topographic surface rather than from heterogeneity in geomorphic processes. Numerical experiments on the formation of low‐order fluvial valleys and transportational cyclic steps in erodible streambeds illustrate why it is important to consider the influence of initial conditions when comparing models with natural topography, estimating the uncertainty of model predictions, and studying how landscapes respond to disturbances. Copyright © 2011 John Wiley & Sons, Ltd.     

The application of Karhunen–Loéve,or principal component analysis method,to study the non‐linear seismic response of structures

The Karhunen–Loéve (K–L) method is used to interpret dynamic response data obtained from shaking table and pseudodynamic tests conducted on civil engineering structures subjected to earthquake loading. It is shown how the K–L method can be used to monitor on‐line, or a posteriori, the structural response of non‐linear dynamical systems. Results from these analyses make it possible to quantitatively verify the number and participation factors of non‐linear modes and how they correspond to physical behaviour of the structure. Comments are made regarding the use of this technique in various fields including numerical calculations, experiments and control. Copyright © 2000 John Wiley & Sons, Ltd.     

Use of caesium‐137 data to evaluate SHETRAN simulated long‐term erosion patterns in arable lands

The caesium‐137 method of quantifying soil erosion is used to provide field data for validating the capability of the SHETRAN modelling system for predicting long‐term (30‐year) erosion rates and their spatial variability. Simulations were carried out for two arable farm sites (area 3–5 ha) in central England for which average annual erosion rates of 6·5 and 10·4 t ha^?1 year^?1 had already been determined using caesium‐137 measurements. These rates were compared with a range of simulated values representing the uncertainty in model output derived from uncertainty in the evaluation of model parameters. A successful validation was achieved in that the simulation range contained the measured rate at both sites, whereas the spatial variability was reproduced excellently at one site and partially at the other. The results indicate that, as the caesium‐137 technique measures the erosion caused by all the processes acting at a site, it is relevant to hydrologically based models such as SHETRAN only if erosion by wind, agricultural activities and other processes not represented in the model are insignificant. The results also indicate a need to reduce the uncertainty in model parameter evaluation. More generally, the caesium‐137 technique is shown to provide field data that improve the severity of the validation procedure (accounting for internal as well as outlet conditions) and that add spatial variability to magnitude as a condition for identifying unrealistic parameter sets when seeking to reduce simulation uncertainty. Copyright © 2004 John Wiley & Sons, Ltd.     

Modelling the potential impacts of groundwater hydrology on long‐term drainage basin evolution

Fluvial erosion processes are driven by water discharge on the land surface, which is produced by surface runoff and groundwater discharge. Although groundwater is often neglected in long‐term landscape evolution problems, water table levels control patterns of Dunne runoff production, and groundwater discharge can contribute significantly to storm flows. In this analysis, we investigate the role that groundwater movement plays in long‐term drainage basin evolution by modifying a widely used landscape evolution model to include a more detailed representation of basin hydrology. Precipitation is generated by a stochastic process, and the precipitation is partitioned between surface runoff and groundwater recharge using a specified infiltration capacity. Groundwater flow is simulated by a dynamic two‐dimensional Dupuit equation for an unconfined aquifer with an irregular underlying impervious layer. The model is applied to the WE‐38 basin, an experimental catchment in Pennsylvania, because 60–80 per cent of the discharge is derived from groundwater and substantial hydrologic and geomorphic information is available. The hydrologic model is first calibrated to match the observed streamflows, and then the combined hydrologic/geomorphic model is used to simulate scenarios with different infiltration capacities. The results of this modelling exercise indicate that the basin can be divided into three zones with distinct streamflow‐generating characteristics, and different parts of the basin can have different geomorphic effective events. Over long periods of time, scenarios in which groundwater discharge is large tend to modify the topography in a way that promotes groundwater discharge and inhibits Dunne runoff. Copyright © 2006 John Wiley & Sons, Ltd.     

Possible evidence for underlying non‐linear dynamics in steep‐faced glaciodeltaic progradational ­successions

Steep‐faced glaciodeltaic progradational successions are often studied in order to reconstruct the behaviour of the glacial feeder system, or changes in the sediment sink. This paper analyses the magnitude and frequency of depositional events associated with steep‐faced glacier‐fed progradational successions recorded in Scandinavia and Ireland. The successions exhibit depositional patterns that may be interpreted as a function of underlying non‐linear dynamics. A number of the sequences display fractal scaling in the frequency and thickness of foreset units. Other successions demonstrate chaotic patterns and strong relationships between delta‐front angle and bed thicknesses, suggesting that the progradation of such sequences is self‐organized, and to an extent occurs independently of forcing by the feeder system that provides sediment to the delta front. These patterns of sedimentation appear to be a function of the steepness of the delta front and/or the textural characteristics of the sediment. This paper provides further evidence for the simultaneous presence of order and chaos in Earth surface processes and calls into question the extent to which palaeoenvironmental reconstructions may be made from steep‐faced progradational successions. Copyright © 2000 John Wiley & Sons, Ltd.     

Analysis and performance of a predictor‐multicorrector Time Discontinuous Galerkin method in non‐linear elastodynamics

A predictor‐multicorrector implementation of a Time Discontinuous Galerkin method for non‐linear dynamic analysis is described. This implementation is intended to limit the high computational expense typically required by implicit Time Discontinuous Galerkin methods, without degrading their accuracy and stability properties. The algorithm is analysed with reference to conservative Duffing oscillators for which closed‐form solutions are available. Therefore, insight into the accuracy and stability properties of the predictor‐multicorrector algorithm for different approximations of non‐linear internal forces is gained, showing that the properties of the underlying scheme can be substantially retained. Finally, the results of representative numerical simulations relevant to Duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical scheme and confirm the analytical estimates. Copyright © 2002 John Wiley & Sons, Ltd.     

Multifractal behaviour of long‐term karstic discharge fluctuations

Karstic watersheds are highly complex hydrogeological systems that are characterized by a multiscale behaviour corresponding to the different pathways of water in these systems. The main issue of karstic spring discharge fluctuations consists in the presence and the identification of characteristic time scales in the discharge time series. To identify and characterize these dynamics, we acquired, for many years at the outlet of two karstic watersheds in South of France, discharge data at 3‐mn, 30‐mn and daily sampling rate. These hydrological records constitute to our knowledge the longest uninterrupted discharge time series available at these sampling rates. The analysis of the hydrological records at different levels of detail leads to a natural scale analysis of these time series in a multifractal framework. From a universal class of multifractal models based on cascade multiplicative processes, the time series first highlights two cut‐off scales around 1 and 16 h that correspond to distinct responses of the aquifer drainage system. Then we provide estimates of the multifractal parameters α and C₁ and the moment of divergence q_D corresponding to the behaviour of karstic systems. These results constitute the first estimates of the multifractal characteristics of karstic spingflows based on 10 years of high‐resolution discharge time series and should lead to several improvements in rainfall‐karstic springflow simulation models. Copyright © 2012 John Wiley & Sons, Ltd.     

Simulations of non‐stationary frequency content and its importance to seismic assessment of structures

To verify the importance of the non‐stationary frequency characteristic of seismic ground motion, a joint time–frequency analysis technique of time signals, called chirplet‐based signal approximation, is developed to extract the non‐stationary frequency information from the recorded data. The chirplet‐based signal approximation is clear in concept, similar to Fourier Transform in mathematical expressions but with different base functions. Case studies show that the chirplet‐based signal approximation can represent the joint time–frequency variation of seismic ground motion quite well. Both the random models of uniform modulating process and evolutionary process are employed to generate artificial seismic waves. The joint time–frequency modulating function in the random model of evolutionary process is determined by chirplet‐based signal approximation. Finally, non‐linear response analysis of a SODF system and a frame structure is performed based on the generated artificial seismic waves. The results show that the non‐stationary frequency characteristic of seismic ground motion can significantly change the non‐linear response characteristics of structures, particularly when a structure goes into collapse phase under seismic action. It is concluded that non‐stationary frequency characteristic of seismic ground motion should be considered for the assessment of seismic capacity of structures. Copyright © 2002 John Wiley & Sons, Ltd.     

Asymmetric one‐storey elastic systems with non‐linear viscous and viscoelastic dampers: Earthquake response

Investigated are earthquake responses of one‐way symmetric‐plan, one‐storey systems with non‐linear fluid viscous dampers (FVDs) attached in series to a linear brace (i.e. Chevron or inverted V‐shape braces).Thus, the non‐linear damper is viscous when the brace is considered rigid or viscoelastic (VE) when the brace is flexible. The energy dissipation capacity of a non‐linear FVD is characterized by an amplitude‐dependent damping ratio for an energy‐equivalent linear FVD, which is determined assuming the damper undergoes harmonic motion. Although this formulation is shown to be advantageous for single‐degree‐of‐freedom (SDF) systems, it is difficult to extend its application to multi‐degree‐of‐freedom (MDF) systems for two reasons: (1) the assumption that dampers undergo harmonic motion in parameterizing the non‐linear damper is not valid for its earthquake‐induced motion of an MDF system; and (2) ensuring simultaneous convergence of all unknown amplitudes of dampers is difficult in an iterative solution of the non‐linear system. To date, these limitations have precluded the parametric study of the dynamics of MDF systems with non‐linear viscous or VE dampers. However, they are overcome in this investigation using concepts of modal analysis because the system is weakly non‐linear due to supplemental damping. It is found that structural response is only weakly affected by damper non‐linearity and is increased by a small amount due to bracing flexibility. Thus, the effectiveness of supplemental damping in reducing structural responses and its dependence on the planwise distribution of non‐linear VE dampers were found to be similar to that of linear FVDs documented elsewhere. As expected, non‐linear viscous and VE dampers achieve essentially the same reduction in response but with much smaller damper force compared to linear dampers. Finally, the findings in this investigation indicate that the earthquake response of the asymmetric systems with non‐linear viscous or VE dampers can be estimated with sufficient accuracy for design applications by analysing the same asymmetric systems with all non‐linear dampers replaced by energy‐equivalent linear viscous dampers. Copyright © 2003 John Wiley & Sons, Ltd.     

Modelling the long‐term sediment trap efficiency of small ponds

A numerical model (sediment trap efficiency for small ponds—STEP) is developed to simulate sediment deposition in small ponds (i.e. <1 ha) and to calculate the sediment trap efficiency (STE). The algorithms are kept simple to allow the model to simulate larger time periods (i.e. several years). Eight runs with an experimental pond were executed to test the model. The STEP model produces reasonable predictions of STE as well as the shape and magnitude of the effluent sediment concentration graph. The model efficiency of STEP for the prediction of STE equals 0·38 and the root mean square error equals 4·7%. Similar models, such as DEPOSITS and CSTRS, were inefficient in predicting the experimental results. The STEP model was used to simulate the long‐term (33 years) STE of small retention ponds in central Belgium using 10‐min rainfall data. For a typical pond (1000 m²) with a catchment area of 25 ha, annual STE can vary from 58 to 100%, with a long‐term STE of only 68%. Copyright © 2001 John Wiley & Sons, Ltd.     

Long‐term landscape evolution: linking tectonics and surface processes

Research in landscape evolution over millions to tens of millions of years slowed considerably in the mid‐20th century, when Davisian and other approaches to geomorphology were replaced by functional, morphometric and ultimately process‐based approaches. Hack's scheme of dynamic equilibrium in landscape evolution was perhaps the major theoretical contribution to long‐term landscape evolution between the 1950s and about 1990, but it essentially ‘looked back’ to Davis for its springboard to a viewpoint contrary to that of Davis, as did less widely known schemes, such as Crickmay's hypothesis of unequal activity. Since about 1990, the field of long‐term landscape evolution has blossomed again, stimulated by the plate tectonics revolution and its re‐forging of the link between tectonics and topography, and by the development of numerical models that explore the links between tectonic processes and surface processes. This numerical modelling of landscape evolution has been built around formulation of bedrock river processes and slope processes, and has mostly focused on high‐elevation passive continental margins and convergent zones; these models now routinely include flexural and denudational isostasy. Major breakthroughs in analytical and geochronological techniques have been of profound relevance to all of the above. Low‐temperature thermochronology, and in particular apatite fission track analysis and (U–Th)/He analysis in apatite, have enabled rates of rock uplift and denudational exhumation from relatively shallow crustal depths (up to about 4 km) to be determined directly from, in effect, rock hand specimens. In a few situations, (U–Th)/He analysis has been used to determine the antiquity of major, long‐wavelength topography. Cosmogenic isotope analysis has enabled the determination of the ‘ages’ of bedrock and sedimentary surfaces, and/or the rates of denudation of these surfaces. These latter advances represent in some ways a ‘holy grail’ in geomorphology in that they enable determination of ‘dates and rates’ of geomorphological processes directly from rock surfaces. The increasing availability of analytical techniques such as cosmogenic isotope analysis should mean that much larger data sets become possible and lead to more sophisticated analyses, such as probability density functions (PDFs) of cosmogenic ages and even of cosmogenic isotope concentrations (CICs). PDFs of isotope concentrations must be a function of catchment area geomorphology (including tectonics) and it is at least theoretically possible to infer aspects of source area geomorphology and geomorphological processes from PDFs of CICs in sediments (‘detrital CICs’). Thus it may be possible to use PDFs of detrital CICs in basin sediments as a tool to infer aspects of the sediments' source area geomorphology and tectonics, complementing the standard sedimentological textural and compositional approaches to such issues. One of the most stimulating of recent conceptual advances has followed the considerations of the relationships between tectonics, climate and surface processes and especially the recognition of the importance of denudational isostasy in driving rock uplift (i.e. in driving tectonics and crustal processes). Attention has been focused very directly on surface processes and on the ways in which they may ‘drive’ rock uplift and thus even influence sub‐surface crustal conditions, such as pressure and temperature. Consequently, the broader geoscience communities are looking to geomorphologists to provide more detailed information on rates and processes of bedrock channel incision, as well as on catchment responses to such bedrock channel processes. More sophisticated numerical models of processes in bedrock channels and on their flanking hillslopes are required. In current numerical models of long‐term evolution of hillslopes and interfluves, for example, the simple dependency on slope of both the fluvial and hillslope components of these models means that a Davisian‐type of landscape evolution characterized by slope lowering is inevitably ‘confirmed’ by the models. In numerical modelling, the next advances will require better parameterized algorithms for hillslope processes, and more sophisticated formulations of bedrock channel incision processes, incorporating, for example, the effects of sediment shielding of the bed. Such increasing sophistication must be matched by careful assessment and testing of model outputs using pre‐established criteria and tests. Confirmation by these more sophisticated Davisian‐type numerical models of slope lowering under conditions of tectonic stability (no active rock uplift), and of constant slope angle and steady‐state landscape under conditions of ongoing rock uplift, will indicate that the Davis and Hack models are not mutually exclusive. A Hack‐type model (or a variant of it, incorporating slope adjustment to rock strength rather than to regolith strength) will apply to active settings where there is sufficient stream power and/or sediment flux for channels to incise at the rate of rock uplift. Post‐orogenic settings of decreased (or zero) active rock uplift would be characterized by a Davisian scheme of declining slope angles and non‐steady‐state (or transient) landscapes. Such post‐orogenic landscapes deserve much more attention than they have received of late, not least because the intriguing questions they pose about the preservation of ancient landscapes were hinted at in passing in the 1960s and have recently re‐surfaced. As we begin to ask again some of the grand questions that lay at the heart of geomorphology in its earliest days, large‐scale geomorphology is on the threshold of another ‘golden’ era to match that of the first half of the 20th century, when cyclical approaches underpinned virtually all geomorphological work. Copyright © 2007 John Wiley & Sons, Ltd.     

Application of non‐linear multiple tuned mass dampers to suppress man‐induced vibrations of a pedestrian bridge

This paper presents an application of multiple tuned mass dampers (MTMDs) with non‐linear damping devices to suppress man‐induced vibrations of a 34m long pedestrian bridge. The damping force generated by each of these damping devices is simply a drag force from liquid acting on an immersed section. The quadratic non‐linear property of these devices was directly determined from free vibration tests of a simple laboratory set‐up. Dynamic models of the bridge and pedestrian loads were constructed for numerical investigation based on field measurement data. The control effectiveness of non‐linear MTMDs was examined along with its sensitivity against estimation errors in the bridge's natural frequency and magnitude of pedestrian load. The numerical results indicated that the optimum non‐linear MTMD system was as effective and robust as its linear counterpart. Then, a six‐unit non‐linear MTMD system was designed, constructed, and installed on the bridge. Field measurements after the installation confirmed the effectiveness of non‐linear MTMDs, and the measurement results were in good agreement with numerical predictions. After the installation, the average damping ratio of the bridge was raised from 0.005 to 0.036 and the maximum bridge accelerations measured during walking tests were reduced from about 0.80–1.30 ms^?2 to 0.27–0.40 ms^?2, which were within an acceptable range. Copyright © 2003 John Wiley & Sons, Ltd.     

Gopher bioturbation: field evidence for non‐linear hillslope diffusion

It has generally been assumed that diffusive sediment transport on soil‐mantled hillslopes is linearly dependent on hillslope gradient. Fieldwork was done near Santa Barbara, California, to develop a sediment transport equation for bioturbation by the pocket gopher (Thomomys bottae) and to determine whether it supports linear diffusion. The route taken by the sediment is divided into two parts, a subsurface path followed by a surface path. The first is the transport of soil through the burrow to the burrow opening. The second is the discharge of sediment from the burrow opening onto the hillslope surface. The total volumetric sediment flux, as a function of hillslope gradient, is found to be: q_s (cm³ cm⁻¹ a⁻¹) = 176(dz/dx)³ − 189(dz/dx)² + 68(dz/dx) + 34(dz/dx)^0·4. This result does not support the use of linear diffusion for hillslopes where gopher bioturbation is the dominant mode of sediment transport. A one‐dimensional hillslope evolution program was used to evolve hillslope profiles according to non‐linear and linear diffusion and to compare them to a typical hillslope. The non‐linear case more closely resembles the actual profile with a convex cap at the divide leading into a straight midslope section. Copyright © 2000 John Wiley & Sons, Ltd.     

Asymmetric one‐storey elastic systems with non‐linear viscous and viscoelastic dampers: Simplified analysis and supplemental damping system design

Investigated is the accuracy in estimating the response of asymmetric one‐storey systems with non‐linear viscoelastic (VE) dampers by analysing the corresponding linear viscous system wherein all non‐linear VE dampers are replaced by their energy‐equivalent linear viscous dampers. The response of the corresponding linear viscous system is determined by response history analysis (RHA) and by response spectrum analysis (RSA) extended for non‐classically damped systems. The flexible and stiff edge deformations and plan rotation of the corresponding linear viscous system determined by the extended RSA procedure is shown to be sufficiently accurate for design applications with errors generally between 10 and 20%. Although similar accuracy is also shown for the ‘pseudo‐velocity’ of non‐linear VE dampers, the peak force of the non‐linear VE damper cannot be estimated directly from the peak damper force of the corresponding linear viscous system. A simple correction for damper force is proposed and shown to be accurate (with errors not exceeding 15%). For practical applications, an iterative linear analysis procedure is developed for determining the amplitude‐ and frequency‐dependent supplemental damping properties of the corresponding linear viscous system and for estimating the response of asymmetric one‐storey systems with non‐linear VE dampers from the earthquake design (or response) spectrum. Finally, a procedure is developed for designing non‐linear supplemental damping systems that satisfy given design criteria for a given design spectrum. Copyright © 2003 John Wiley & Sons, Ltd.     

Seismic response of three‐dimensional r/c multi‐storey frame building under uni‐ and bi‐directional input ground motion

This paper deals with seismic analysis of plan‐asymmetric r/c frame multi‐storey buildings. Non‐linear numerical analyses are carried out by using a lumped plasticity model for beams and a multi‐spring model for columns, the latter one introduced to account for axial force–biaxial bending moment interaction. A comparison between numerical analyses and experimental test results is reported in order to calibrate the numerical model, showing that the adopted model is very suitable. In order to study the effects of the earthquake orthogonal component, the seismic response of the modelled structure under uni‐directional excitation is compared to the one under bi‐directional excitation. Such comparison shows that the maximum base shear and the top displacement are not very sensitive to the presence of the orthogonal component, which, conversely, leads to large increase in the column plastic excursions. Copyright © 2007 John Wiley & Sons, Ltd.     

Earthquake response of elastic SDF systems with non‐linear fluid viscous dampers

The steady‐forced and earthquake responses of SDF systems with a non‐linear fluid viscous damper (FVD) are investigated. The energy dissipation capacity of the FVD is characterized by the supplemental damping ratio ζsd and its non‐linearity by a parameter designated α. It is found that the structural response is most effectively investigated in terms of ζ_sd and α because (1) these two parameters are dimensionless and independent, and (2) the structural response varies linearly with the excitation intensity. Damper non‐linearity has essentially no influence on the peak response of systems in the velocity‐sensitive spectral region, but differences up to 14% were observed in the other spectral regions. The structural deformation is reduced by up to 25% when ζ_sd= 5%; and by up to 60% when ζ_sd= 30%. Non‐linear FVDs are advantageous because they achieve essentially the same reduction in system responses but with a significantly reduced damper force. For practical applications, a procedure is presented to estimate the design values of structural deformation and forces for a system with non‐linear FVD directly from the design spectrum. It is demonstrated that the earthquake‐induced force in a non‐linear FVD can be estimated from the damper force in a corresponding system with linear FVD, its peak deformation, and peak relative velocity; however, the relative velocity should not be approximated by the pseudo‐velocity as this approximation introduces a large error in the damper force. Finally, a procedure is presented to determine the non‐linear damper properties necessary to limit the structural deformation to some design value or the structural capacity for a given design spectrum. Copyright © 2002 John Wiley & Sons, Ltd.     

Correlating measured and simulated dynamic responses of a tall building to long‐distance earthquakes

For almost a decade, a 66‐storey, 280m tall building in Singapore has been instrumented to monitor its dynamic responses to wind and seismic excitations. The dynamic characteristics of the tall building have been investigated via both the finite element method and the experimental modal analysis. The properties of the finite element model have been shown to correlate well with those derived from the data recorded during the ambient vibration tests. During the study period, 21 sets of earthquake ground motions have been recorded at the building site. The basement motions may be divided into three categories based on their predominant frequency components with respect to the building's fundamental frequency. The calibrated three‐dimensional finite element model is employed to simulate the seismic response of the tall building. Correlation analysis of the time histories between the recorded data and the simulated results has been carried out. The correlation analysis results show that the simulated dynamic response time histories match well with those of the recorded dynamic responses at the roof level. The results also show that the simulated maximum response at the roof level is close to the peak response recorded during the earthquakes. Copyright © 2004 John Wiley & Sons, Ltd.     

Postglacial topographic evolution of glaciated valleys: a stochastic landscape evolution model

The retreat of valley glaciers has a dramatic effect on the stability of glaciated valleys and exerts a prolonged influence on the subsequent fluvial sediment transport regime. We have studied the evolution of an idealized glaciated valley during the period following retreat of ice using a numerical model. The model incorporates a stochastic process to represent deep‐seated landsliding, non‐linear diffusion to represent shallow landsliding and an approximation of the Bagnold relation to represent fluvial sediment transport. It was calibrated using field data from several recent surveys within British Columbia, Canada. We present ensemble model results and compare them with results from a deterministic linear‐diffusion model to show that explicit representation of large landslides is necessary to reproduce the morphology and channel network structure of a typical postglacial valley. Our model predicts a rapid rate of fluvial sediment transport following deglaciation with a subsequent gradual decline, similar to that inferred for Holocene time. We also describe how changes in the model parameters affect the estimated magnitude and duration of the paraglacial sediment pulse. Copyright © 2005 John Wiley & Sons, Ltd.     

Linearization and first‐order expansion of the rocking motion of rigid blocks stepping on viscoelastic foundation

In structural mechanics there are several occasions where a linearized formulation of the original non‐linear problem reduces considerably the computational effort for the response analysis. In a broader sense, a linearized formulation can be viewed as a first‐order expansion of the dynamic equilibrium of the system about a ‘static’ configuration; yet caution should be exercised when identifying the ‘correct’ static configuration. This paper uses as a case study the rocking response of a rigid block stepping on viscoelastic supports, whose non‐linear dynamics is the subject of the companion paper, and elaborates on the challenge of identifying the most appropriate static configuration around which a first‐order expansion will produce the most dependable results in each regime of motion. For the regime when the heel of the block separates, a revised set of linearized equations is presented, which is an improvement to the unconservative equations published previously in the literature. The associated eigenvalues demonstrate that the characteristics of the foundation do not affect the rocking motion of the block once the heel separates. Copyright © 2008 John Wiley & Sons, Ltd.