Matthew J. DeJong

Revisiting the rocking block: closed-form solutions and similarity laws

In this paper, the dynamic response of the rocking block subjected to base excitation is revisited. The goal is to offer new closed-form solutions and original similarity laws that shed light on the fundamental aspects of the rocking block. The focus is on the transient dynamics of the rocking block under finite-duration excitations. An alternative way to describe the response of the rocking block, informative of the behaviour of rocking structures under excitations of different intensity, is offered. In the process, limitations of standard dimensional analysis, related to the orientations of the involved physical quantities, are revealed. The proposed dimensionless and orientationless groups condense the response and offer a lucid depiction of the rocking phenomenon. When expressed in the appropriate dimensionless–orientationless groups, the rocking response becomes perfectly self-similar for slender blocks (within the small rotations range) and practically self-similar for non-slender blocks (larger rotations). Using this formulation, the nonlinear and non-smooth rocking response to pulse-type ground motion can be directly determined, and need only be scaled by the intensity and frequency of the excitation.

Rocking Stability of Masonry Arches in Seismic Regions

This study evaluates the susceptibility of masonry arches to earthquake loading through experimental testing and progresses toward a specific criterion by which arches can be quickly assessed. Five different earthquake time histories, as well as harmonic base excitations of increasing amplitude, were applied to model arches, and the magnitude of the base motion resulting in collapse was determined repeatedly. Results are compared with failure predictions of an analytical model which describes the rocking motion of masonry arches under base excitation. The primary impulse of the base excitation is found to be of critical importance in causing collapse of the masonry arch. Accordingly, a suite of failure curves are presented which can be used to determine the rocking stability of masonry arches under a primary base acceleration impulse which has been extracted from an expected earthquake motion.

Overturning of Retrofitted Rocking Structures under Pulse-Type Excitations

AbstractNumerous existing structures exhibit rocking behavior during earthquakes, and there is a continuing need to retrofit these structures to prevent collapse. In addition, while rocking behavior is typically prevented instead of utilized, an increasing number of structures are being designed or retrofitted to allow rocking motion as a means of seismic isolation. This paper investigates the use of viscous damping to limit the rocking motion by characterizing the fundamental behavior of damped rocking structures through analytical modeling. A single rocking block analytical model is used to determine the viscous damping characteristics, which exploit the beneficial aspects of the rocking motion, while dissipating energy and preventing overturning collapse. To clarify the benefits of damping, overturning envelopes for pulse-type ground accelerations are presented and compared with the pertinent envelopes of the free rocking block. A semianalytical solution to the linearized equations of motion enables ra...

Amplification of Rocking Due to Horizontal Ground Motion

This paper addresses the fundamental behavior of rocking structures by investigating the rate of energy input due to horizontal ground motion. This perspective indicates that the assumption of initial at-rest conditions may be inappropriate, and identifies ground motions which cause ‘rocking resonance’ for various constraints. Furthermore, the ability of multiple impulses to either reduce or amplify the rocking response is demonstrated, and motivates the use of statistical assessment procedures to predict rocking stability during earthquakes. In this context, past earthquake data is used to generate an array of synthetic earthquakes of a given intensity which are applied to determine the overturning probability of the rocking block. The results emphasize the time dependence of the rocking response. Finally, simulation of the rocking response to real earthquake records confirms that multiple impulse rocking amplification may be significant.

The rocking response of large flexible structures to earthquakes

The rocking response of structures subjected to strong ground motions is a problem of ‘several scales’. While small structures are sensitive to acceleration pulses acting successively, large structures are more significantly affected by coherent low frequency components of ground motion. As a result, the rocking response of large structures is more stable and orderly, allowing effective isolation from the ground without imminent danger of overturning. This paper aims to characterize and predict the maximum rocking response of large and flexible structures to earthquakes using an idealized structural model. To achieve this, the maximum rocking demand caused by different earthquake records was evaluated using several ground motion intensity measures. Pulse-type records which typically have high peak ground velocity and lower frequency content caused large rocking amplitudes, whereas non-pulse type records caused random rocking motion confined to small rocking amplitudes. Coherent velocity pulses were therefore identified as the primary cause of significant rocking motion. Using a suite of pulse-type ground motions, it was observed that idealized wavelets fitted to velocity pulses can adequately describe the rocking response of large structures. Further, a parametric analysis demonstrates that pulse shape parameters affect the maximum rocking response significantly. Based on these two findings, a probabilistic analysis method is proposed for estimating the maximum rocking demand to pulse-type earthquakes. The dimensionless demand maps, produced using these methods, have predictive power in the near-field provided that pulse period and amplitude can be estimated a priori. Use of this method within a probabilistic seismic demand analysis framework is briefly discussed.

Multi-Temporal InSAR Structural Damage Assessment: The London Crossrail Case Study

Spaceborne multi-temporal interferometric synthetic aperture radar (MT-InSAR) is a monitoring technique capable of extracting line of sight (LOS) cumulative surface displacement measurements with millimeter accuracy. Several improvements in the techniques and datasets quality led to more effective, near real time assessment and response, and a greater ability of constraining dynamically changing physical processes. Using examples of the COSMO-SkyMed (CSK) system, we present a methodology that bridges the gaps between MT-InSAR and the relative stiffness method for tunnel-induced subsidence damage assessment. The results allow quantification of the effect of the building on the settlement profile. As expected the greenfield deformation assessment tends to provide a conservative estimate in the majority of cases (~71% of the analyzed buildings), overestimating tensile strains up to 50%. With this work we show how these two techniques in the field of remote sensing and structural engineering can be synergistically used to complement and replace the traditional ground based analysis by providing an extended coverage and a temporally dense set of data.

Experimental Identification of the Dynamic Characteristics of a Flexible Rocking Structure

This article presents the results of free vibration and earthquake excitation tests to investigate the dynamic behavior of freely rocking flexible structures with different geometric and vibration characteristics. The primary objective of these tests was to identify the complex interaction of elasticity and rocking and discuss its salient effects on the rocking and vibration mode frequencies, shapes and excitation mechanisms. The variability of response is discussed, including critical investigation of the repeatability of the tests. It was found that the variability in energy dissipation and energy transfer to vibrations at impact may lead to significantly different responses to almost identical excitations.

Site amplification in the Kathmandu Valley during the 2015 M7.6 Gorkha, Nepal earthquake

The 25th April 2015 M7.6 Gorkha earthquake caused significant damage to buildings and infrastructure in both Kathmandu and surrounding areas as well as triggering numerous, large landslides. This resulted in the loss of approximately 8600 lives. In order to learn how the impact of such events can be reduced on communities both in Nepal and elsewhere, the Earthquake Engineering Field Investigation Team (EEFIT) reconnaissance mission was undertaken, aiming to look at damage patterns within the country. Passive, microtremor recordings in severely damaged areas of the Kathmandu Valley, as well as at the main seismic recording station in Kathmandu (USGS station KATNP) are used to determined preliminary shear wave velocity (Vs) profiles for each site. These profiles are converted into spectral acceleration using the input motion of the Gorkha earthquake. The results are limited, but show clear site amplification within the Siddhitol Region. The resulting ground motions exceed the design levels from the Nepalese Building Codes, indicating the need for site-specific hazard analysis and for revision of the building code to address the effect of site amplification.

Dynamic response of masonry arch with geometrical irregularities subjected to a pulse-type ground motion

Ancient masonry structures often rely on the masonry arch as a load bearing element. The understanding of its response under seismic actions is a first fundamental step towards the comprehension of the behaviour of more complex structures. It is well known that the stability of masonry arches is primarily related to the geometry. The safety assessment under seismic actions is usually carried out by considering known deterministic geometrical parameters, such as thickness, rise and span, and the voussoirs are assumed with equal dimensions. However, many factors, like defects or irregularities in the shape of the voussoirs and imprecise construction, produce variations of the geometry with respect to the nominal one and, as a consequence, may effect the ability of the arch to resist seismic actions. In this paper, the effect of geometrical irregularities on the dynamic response of circular masonry arches is considered. Irregular geometries are obtained through a random generation of the key geometrical parameters, and the effect of these irregularities is quantified by analysing the dynamic response to ground motion. The masonry arch is modelled as a four-link mechanism, i.e. a system made of three rigid blocks hinged at their ends. The position of the hinges at the instant of activation of the motion is determined through limit analysis. Lagrange’s equations of motion have been written for the generated irregular geometries and solved through numerical integration. The results are summarised by a fragility surface that quantify the extent to which geometrical uncertainties can alter the dynamic response of the masonry arch and increase its seismic vulnerability.

Vibration modes and equivalent models for flexible rocking structures

Predicting the displacement and force response of flexible rocking structures to ground motion is important for their assessment and design. Insofar as practical, it is desirable to use simple mechanical models to make these predictions. However, the complex coupling between rocking and vibration makes accurate predictions with simple models difficult. In this paper, the use of semi-coupled equivalent models to approximate the dynamic response of multi-mass structures rocking on rigid ground is evaluated. These equivalent models feature a two-degree of freedom coupled rocking oscillator to describe the interaction of rocking and the first mode of vibration, and uncoupled linear elastic oscillators to describe higher mode vibration response. To evaluate these equivalent models, the modal components of the dynamic response of multi-mass structures are first determined. These components highlight the critical influence of the excitation of vibration modes at impact. Then, further investigations are carried out by comparing equivalent model simulations to recent shake table tests and multi-mass analytical model simulations. These comparisons reveal that the equivalent models can capture the rocking response accurately for a realistic range of displacements, if a new ground acceleration scaling term is adopted. However, the uncoupled linear elastic oscillators do not consider excitation at impact and consequently, the equivalent models do not capture the acceleration response adequately. Therefore, on the basis of the analytically identified modal components, a further modification that improves the equivalent model acceleration predictions is proposed and validated.