Rocking Behavior of Irregular Free-Standing Objects Subjected to Earthquake Motion

Free-standing rigid objects and structures are dominantly found to exhibit rocking behavior and can be vulnerable to overturning during an earthquake as demonstrated by numerous past earthquake events. Such objects are typically considered to be displacement sensitive with their rocking response being well presented by the Peak Displacement Demand (PDD) parameter of the supporting floor’s motion. This in turn can be directly related to an object’s width (along the direction of motion) for assessing its vulnerability to overturn. Such findings have been sufficiently justified by refined dynamic analysis supported by experimental evaluations which were based on rigid blocks with uniform geometric format (i.e., regular in their mass distribution). However, vulnerable rocking objects can be asymmetric and accordingly their sensitivity to floor displacement cannot be directly related to their width. The key parameter which defines irregular objects’ response to rocking motion is represented by the degree of eccentricity of their center of mass. In this study, the well-known rocking equation of motion is reconfigured and devised to model the rocking responses for 280 irregular objects undergoing eight earthquake motions which included artificial and recorded earthquakes. Analytical results obtained from solving the adjusted equation of motion were evaluated with sophisticated finite element (FE) models simulating the 280 irregular cases. This experimentally validated FE modeling approach was found to be time- and cost-effective for understating the rocking behavior of asymmetric objects as well as clarifying an interesting relationship between the object’s damping level and the condition of the supporting base (i.e., whether being provided with supports at the points of rotation or not). The rocking response of irregular objects was found to be highly influenced by the level of eccentricity of the object when excited by motions with high displacement amplitudes, while such influence was not found noticeable by wider objects. Based on the developed trends between the maximum top displacement of irregular objects and the PDD, an expression for estimating the rocking amplitudes is proposed which is a function of the object’s eccentricity.     

Contact Dynamics of Masonry Block Structures Using Mathematical Programming

A simple variational formulation for contact dynamics is adopted to investigate the dynamic behavior of planar masonry block structures subjected to seismic events. The numerical model is a two-dimensional assemblage of rigid blocks interacting at potential contact points located at the vertices of the interfaces. A no-tension and associative frictional behavior with infinite compressive strength is considered for joints. The dynamic contact problem is formulated as a quadratic programming problem (QP) and an iterative procedure is implemented for time integration. Applications to analytical and numerical case studies are presented for validation. Comparisons with the experimental results of a masonry wall under free rocking motion and of a small scale panel with opening subjected to in-plane loads are also carried out to evaluate the accuracy and the computational efficiency of the formulation adopted.     

A Simplified Model to Evaluate the Dynamic Rocking Behavior of Irregular Free-Standing Rigid Bodies Calibrated with Experimental Shaking-Table Tests

A simplified model to evaluate the dynamic rocking behavior of free-standing irregular rigid bodies under earthquake-induced forces is proposed. The model analyzes the response of a three-dimensional irregular rigid body using a numerical approach that considers a critical section and two equivalent rectangular rigid blocks. Experimental shaking-table tests were carried out on modular prototypes, which allow the replication of representative mass distributions, sizes, and/or slenderness ratios for typical objects. The tests were used to calibrate the numerical model. It was found that the dynamic behavior under irregular conditions (asymmetrical shape and/or non-uniform mass distribution) can be estimated with the appropriate geometric and density considerations.     

A Simple Model for Estimating Shocks in Unrestrained Building Contents in an Earthquake

Building contents that include cabinets housing electronic equipment are typically not rigidly secured to the floor, nor to the adjacent wall except in regions of high seismic activities. The behavior of unrestrained building contents in an earthquake is a cause of concern because of the consequence of damage to certain equipment or other forms of fragile items. Much of the research reported in the literature has been devoted to studying the rocking and sliding motion behavior of base-excited rigid objects and their risks of overturning. In contrast, this paper is concerned with estimating the impact acceleration that can be generated by the pounding of the rocking object onto the floor. Algebraic expressions for predicting the acceleration level, which can be translated into dynamic force values, are derived and illustrated by case studies. Importantly, the proposed expressions have been verified by comparisons with results from both simulated and physical experiments. In illustrating the use of the proposed analytical procedure, a parametric experimental study has been undertaken on a cushion material to study the sensitivity of its static and dynamic stiffness to changes in the boundary conditions of the cushion. The proposed calculation procedure, while simple to apply, can be used as a means of predicting shock and the dynamic forces that can be generated in an object in the course of the response to an earthquake.     

Rocking of Non-symmetric Rigid Blocks in Buildings Considering Effects Associated with Dynamic Soil-Structure Interaction

A dynamic model for the estimation of the rocking and/or overturning response of a free-standing non-symmetric rigid block considering rotational and horizontal excitation is proposed. The block is situated at different levels of a building with flexible base subjected to earthquakes. Base flexibility introduces the rotational component of the excitation due to dynamic soil-structure interaction (DSSI). The model is used to assess the influence of the dynamic soil-structure interaction on the behavior of the block. An illustrative example of the proposed model for non-symmetric rigid blocks in 5-, 10-, and 15-story buildings located in soft soils considering earthquakes from different seismic sources is presented. Results show that it is important to consider kinematic effects as well as inertial effects of DSSI in the dynamic response of contents. The influence of base flexibility depends on the change of spectral intensities associated to the increase of the building structural period and is larger for higher building levels.     

Impact of Vertical Ground Excitation on a Bridge with Footing Uplift

Ground motions recorded in the epicentral region of an earthquake often have a strong vertical component with dominant high frequencies. Damage to bridges in near-source regions due to strong vertical ground motion has been reported. The beneficial effects of footing uplift on structural performance in form of reduction of seismic response of structural members have been confirmed in previous research. The uplift of bridge piers has been utilised in a very limited number of bridge structures, e.g., the South Rangitikei railway bridge in New Zealand. However, the near-fault seismic behaviour of bridges with footing uplift has been even less addressed. In this study shake table investigations were carried out on the response of a single-span bridge model with footing uplift subjected to simultaneous vertical and horizontal excitations. Near-fault ground motions recorded in the Canterbury earthquake sequences of 2010 and 2011 were used. The experimental results show that inclusion of vertical ground motions produce stronger axial force in the pier and larger bending moment in the deck. Concurrent horizontal and vertical excitations may also cause more frequent footing uplift than the solely horizontal excitations.     

Optimal Control of Structures under Earthquakes Including Soil-Structure Interaction

It is well known that the soil-structure interaction (SSI) changes the dynamic response of a structure supported on flexible soil. The analysis of optimally controlled SSI systems has certain difficulties due to the nature of the SSI and the optimal control problem. In this paper, a two-step iteration-based numerical algorithm is proposed to handle optimally controlled SSI systems under earthquakes. First, the optimal control forces are obtained by using a fixed-base system. Then, the optimal control forces are converted to the frequency domain by the Fourier transform technique to be used in the equations of the SSI system. The lateral displacement and the rocking of the foundation are obtained from the equations of the SSI system containing the optimal control forces in the frequency domain. The lateral displacement and rocking of the foundation are then converted to the time domain by the inverse Fourier transform technique, and the lateral accelerations and the rocking accelerations of the foundation are obtained by the forward finite difference method. During the second step, the optimal control forces are calculated again by using the lateral acceleration and the rocking acceleration of the foundation along with the earthquake ground motion. Using the method explained above, the optimal control forces obtained in the time domain are used in the equations of the soil-structure system from which the behavior of foundation and structure is obtained. In the final section of the paper, a numerical study is conducted for a controlled structure supported on flexible soil.     

Seismic Fragility Evaluation of RC Frame Structures Retrofitted with Controlled Concrete Rocking Column and Damping Technique

Acceleration response of simple yielding structure is proportional to its own weight, but it is limited by yield strength. Thus, using rocking columns that reduces global yield strength, a limited acceleration is achieved. However, the displacement becomes large due to lower strength and higher inelasticity, but it can be controlled by adding damping. Performing fragility analyses, the seismic response of R/C frame structures with rocking columns and viscous dampers is investigated. Near field MCEER ground motions are considered. The analyses show that the story accelerations are reduced by using rocking columns, while the story displacements are controlled by using viscous dampers.     

Rocking Soil-Structure Systems Subjected to Near-Fault Pulses

Seismic performance of rocking soil-structure systems subjected to near-fault pulses is investigated considering foundation uplifting and soil plasticity. An extensive parametric study is conducted including medium-to-high-rise buildings with different aspect ratios based on shallow raft foundation at stiff-to-rock sites. Mathematical directivity and fling pulses are used as input ground motion. The superstructure is assumed to have three different boundary conditions: (a) fixed-base, (b) linear soil-structure interaction (SSI), and (c) nonlinear SSI. Evidently, the prevailing pulse period T_p is a key parameter governing nonlinear SSI effects. The normalized acceleration response spectra reveal that despite beneficial effects of foundation uplifting and soil yielding in most cases, there are some minor regions in which the response accelerations are amplified. In addition, more slender buildings significantly benefit from uplifting and soil yielding when subjected to short- and medium-period directivity pulses compared to squat structures. However, response amplifications with respect to fixed-base structures are considerable in case of slender structures subjected to medium- or long-period directivity pulses. So that neglecting the SSI effects on seismic performance of rocking structures with shallow foundations, as mostly assumed in common practice, may give rise to inaccurate estimations of force demands against near-fault pulselike ground motions. Furthermore, the envelope of residual foundation tilting θ_r is limited to 0.015 rad, in case of directivity pulses.     

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.     

Experimental characterization of the constitutive materials composing an old masonry vaulted tunnel of the Paris subway system

A significant proportion of the Paris metro tunnels comprise a masonry vault built out of stone blocks and mortar joints, and sidewalls and slabs made of unreinforced concrete. In order to provide the necessary data for future structural evaluation, an extensive laboratory testing programme has been conducted to characterize the materials of the tunnel separately, i.e., mortar, stone, and concrete. The tests, carried out on specimens taken from cores extracted from a 1930s tunnel, enabled to determine the mechanical properties, including direct tensile, shear strength, and mode I fracture energy, as well as the properties of the stone-mortar interface. Results show that the masonry mortar joints could reach 10 cm in width, and that blocks of stone varied in composition and porosity, thus producing a wide range of mechanical properties. The concrete was composed of large-sized aggregates and showed low stiffness and strength. Based on these experimental results, ratios between mechanical characteristics are hereby proposed. Perspectives on the use of this experimental data in a finite element model are then discussed.     

Controlled Rocking CLT Walls for Buildings in Regions of Moderate Seismicity: Design Procedure and Numerical Collapse Assessment

Controlled rocking heavy timber walls are designed to rock on their foundations in response to earthquakes. For regions of moderate seismicity, it is proposed that this rocking behaviour can be adequately controlled using only post-tensioning, even with a large force-reduction factor and no supplemental energy dissipation. This article presents a force-based design procedure for controlled rocking cross-laminated timber walls without supplemental energy dissipation, including a method for estimating higher mode effects. Fragility analyses of three prototype walls demonstrate that the procedure can limit the probability of collapse to <10% during a maximum considered earthquake in a region of moderate seismicity.     

REPEATABLE DYNAMIC RELEASE TESTS ON A BASE-ISOLATED BUILDING

The main activities and the relevant results of an experimental program aimed at evaluating the dynamic characteristics of a large seismically isolated building are described. The building is situated in Potenza (Italy) and is the largest of five seismically isolated blocks of the University of Basilicata. Cyclic shear tests on scaled isolators and some small amplitude free vibration tests were preliminarily carried out. The latter ones were aimed at getting a preliminary linear characterisation of the structure. Finally, many in situ release tests on the isolated structure were carried out, using a mechanical device purposely designed to statically move the building and then suddenly release it, thus making it vibrate freely. The variations of the dynamic characteristics of the system while the oscillation amplitude decreases have been evaluated by using the “Short Time Fourier Transform (STFT)”. The results were elaborated in terms of stiffness and equivalent damping and compared to the results of the laboratory tests on scaled isolators, finding an excellent agreement.     

Quarries and transportation routes of Angkor monument sandstone blocks

Sandstone blocks are the main construction materials used in the Angkor monuments in Cambodia. However, a thorough study of the quarries has not yet been carried out. We conducted a field investigation of sandstone quarries from the Angkor period at the southeastern foot of Mt. Kulen, which is approximately 35 km northeast of the Angkor monuments. As a result, we discovered more than 50 sandstone quarries. On the basis of the measurements of magnetic susceptibilities and thicknesses (step heights), we found that they were quarried at different times. These four quarrying areas were identified as the quarries D to G inferred by Uchida et al. (2007). In addition we investigated a canal that was identified on satellite images, connecting quarry sites at the foot of Mt. Kulen to the Angkor monuments. The field investigation suggests a high probability that the canal was used for the transportation of sandstone blocks from Mt. Kulen.     

Simulation of Shake Table Tests on Out-of-Plane Masonry Buildings. Part (IV): Macro and Micro FEM Based Approaches

ABSTRACT

This article presents a study on the out-of-plane response of two masonry structures without box behavior tested in a shaking table. Two numerical approaches were defined for the evaluation, namely macro-modeling and simplified micro-modeling. As a first step of this study, static nonlinear analyses were performed for the macro models in order to assess the out-of-plane response of masonry structures due to incremental loading. For these analyses, mesh size and material model dependency was discussed. Subsequently, dynamic nonlinear analyses with time integration were carried out, aiming at evaluating the collapse mechanism and at comparing it to the experimental response. Finally, nonlinear static and dynamic analyses were also performed for the simplified micro models. It was observed that these numerical techniques correctly simulate the in-plane response. The collapse mechanism of the stone masonry model is in good agreement with the experimental response. However, there are some inconsistencies regarding the out-of-plane behavior of the brick masonry model, which required further validation.     

SOIL DAMPING FORMULATION IN NONLINEAR TIME DOMAIN SITE RESPONSE ANALYSIS

Nonlinear time domain site response analysis is used to capture the soil hysteretic response and nonlinearity due to medium and large ground motions. Soil damping is captured primarily through the hysteretic energy dissipating response. Viscous damp-ing, using the Rayleigh damping formulation, is often added to represent damping at very small strains where many soil models are primarily linear. The Rayleigh damping formulation results in frequency dependent damping, in contrast to experiments that show that the damping of soil is mostly frequency independent. Artificially high damp-ing is introduced outside a limited frequency range that filters high frequency ground motion. The extended Rayleigh damping formulation is introduced to reduce the over-damping at high frequencies. The formulation reduces the filtering of high frequency motion content when examining the motion Fourier spectrum. With appropriate choice of frequency range, both formulations provide a similar response when represented by the 5% damped elastic response spectrum.

The proposed formulations used in non-linear site response analysis show that the equivalent linear frequency domain solution commonly used to approximate non-linear site response underestimates surface ground motion within a period range relevant to engineering applications. A new guideline is provided for the use of the proposed formulations in non-linear site response analysis.     

RESPONSE OF UNREINFORCED MASONRY BUILDINGS

Abstract

A summary of dynamic measurements are presented that illustrate relations between linear seismic demand and true nonlinear response of unreinforced masonry buildings with flexible diaphragms and rocking piers subjected to a series of simulated earthquake motions.     

Cyclic Modeling of Bolted Beam-to-Column Connections: Component Approach

The work is aimed at the prediction of the cyclic response of bolted beam-to-column joints starting from the knowledge of their geometrical and mechanical properties. To this scope a mechanical model is developed within the framework of the component approach already codified by Eurocode 3 for monotonic loadings.

Accuracy of the developed mechanical model is investigated by means of the comparison between numerical and experimental results with reference to an experimental program carried out at Salerno University. The obtained results are encouraging about the possibility of extending the component approach to the prediction of the cyclic response of bolted connections.     

Seismic Displacement Demands on Skewed Bridge Decks Supported on Elastomeric Bearings

The unseating of decks is one of the most prevalent failure modes of bridges after earthquake events, as observed in the 2010 Chile Earthquake. Damaged bridges in Chile often had skew angles and were supported on elastomeric bearings. Similar bridge construction practices with decks supported on elastomeric bearings are also common in the central and eastern U.S. (CEUS). The seismic displacement demands on skewed bridges are more complicated than those on bridges without skew angles due to the coupling of translational modes with the rotational mode of vibration. The study presented in this article seeks to understand the seismic response of skewed bridge decks supported on elastomeric bearings. The scope of the study is limited to one- and two-span bridges, which constitute a large portion of bridge inventory in the CEUS. The vibration modes of skewed bridge decks are derived in closed form and the modes are compared when the gaps between the bridge deck and the abutment are open and when one of the gaps is closed due to seismic excitation. Nonlinear response history analyses are carried out to understand the effects of vertical ground motion, skew angles, aspect ratios, and different ground motion types on the seismic displacement demand in these cases. Amplification factors that approximate the increase in the displacement demand due to the skew angle are proposed.     

Diffraction around a Semi-Parabolic Canyon in an Elastic Half-Space by Out-of-Plane SH-Waves,II: Rocking and Resultant Rotations

This paper is the second of the two papers, discussing the rocking and resultant component of rotations, with the torsional components of motions presented earlier in the first paper for the case of out-of-plane SH waves excitation on a semi-parabolic canyon. The rotational motions are gaining importance in recent years in the field of strong-motion seismology and earthquake engineering. Many previous papers on out-of-plane SH waves excitation presented the associated torsion rotational motions, when in fact there is also a rocking rotational component of motions. Both the rocking and torsion rotational motions, in as much as the translational motions, had shown to play an important role in structural responses, even though rotations are not incorporated into building design codes at present. Studying the rotational motions can thus help to a better understanding of the behavior of surface topographies that are in the vicinity of structures in the event of seismic excitation. This paper included, besides the additional rocking motions, also the resultant rotational motions combining torsion and rocking.