An integrated macroelement formulation for the dynamic response of inelastic deformable rocking bodies

This paper presents a macroelement formulation for the prediction of the planar dynamic response of inelastic deformable rocking bodies. The formulation is based on a previous macroelement developed by the authors able to describe the cyclic response of inelastic rocking bodies, which takes into account the deformability both along the height of the member, as well as near the rocking end. Modifications of this formulation to account for other motion modes of rocking members during their dynamic response, namely, sliding and upthrow, as well as modifications to account for damping in a uniform manner during the whole motion, including impacts, are introduced. The dynamic response predicted by the macroelement for free-standing rigid and deformable rocking bodies is presented and compared with existing theoretical solutions, and the effect of deformability, damping, inelasticity, and friction on the response is discussed.     

Finite Element Analysis of Fault Bend Influence on Stick-Slip Instability along an Intra-Plate Fault

— Earthquakes have been recognized as resulting from stick-slip frictional instabilities along the faults between deformable rocks. A three-dimensional finite-element code for modeling the nonlinear frictional contact behaviors between deformable bodies with the node-to-point contact element strategy has been developed and applied here to investigate the fault geometry influence on the nucleation and development process of the stick-slip instability along an intra-plate fault through a typical fault bend model, which has a pre-cut fault that is artificially bent by an angle of 5.6° at the fault center. The numerical results demonstrate that the geometry of the fault significantly affects nucleation, termination and restart of the stick-slip instability along the intra-plate fault, and all these instability phenomena can be well simulated using the current finite-element algorithm.     

Sliding-uplifting response of rigid blocks to base excitation

In the present paper a two-degree-of-freedom system is considered which allows the simulation of rigid blocks uplifting and sliding on frictional foundations; the monolateral constraint between block and base is schematized by means of a joint model, which allows the contact problem to be discretized. The joint model is governed by normal and shear constitutive laws, which have been derived by the phenomenological behaviour of stone blocks and rock joints, as given by rock mechanics. Furthermore, a numerical procedure has been developed in order to solve the non-linear equations governing the motion of the block-base system, and to analyse the dynamic response of this system under seismic excitation; particular attention has been paid to the influence of the vertical displacement on the slip response.     

Experimental evaluation of piecewise exact solution for predicting seismic responses of spherical sliding type isolated structures

The use of base isolation for enhancing seismic resistibility has been proven as an efficient method in experimental and theoretical studies. It is usual to insert a flexible device in the horizontal direction to permit the most of relative deformation of a structure at this level. Because the rigidity of the superstructure is much higher than that of the base isolator underneath the structure, the behavior of the superstructure can be idealized as a rigid body during earthquakes. In general, hundreds of degrees of freedom and a step‐by‐step time history analysis are the basic requirements for calculating the seismic response of a base‐isolated structure under earthquakes. In order to develop a simple tool which can be easily adopted for calculating the seismic responses of the spherical sliding type isolated structures, a piecewise exact solution for predicting the seismic responses of base‐isolated structures has been derived in this study. The comparison between the experimental results conducted at the National Center for Research on Earthquake Engineering (NCREE) in Taiwan and the analytical results obtained from the piecewise exact solution show that the formulation derived in this study can predict the seismic responses of the base‐isolated structure with a very high accuracy. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite element model for seismic response analysis of deformable rocking frames

A new finite element model to analyze the seismic response of deformable rocking bodies and rocking structures is presented. The model comprises a set of beam elements to represent the rocking body and zero‐length fiber cross‐section elements at the ends of the rocking body to represent the rocking surfaces. The energy dissipation during rocking motion is modeled using a Hilber–Hughes–Taylor numerically dissipative time step integration scheme. The model is verified through correct prediction of the horizontal and vertical displacements of a rigid rocking block and validated against the analytical Housner model solution for the rocking response of rigid bodies subjected to ground motion excitation. The proposed model is augmented by a dissipative model of the ground under the rocking surface to facilitate modeling of the rocking response of deformable bodies and structures. The augmented model is used to compute the overturning and uplift rocking response spectra for a deformable rocking frame structure to symmetric and anti‐symmetric Ricker pulse ground motion excitation. It is found that the deformability of the columns of a rocking frame does not jeopardize its stability under Ricker pulse ground motion excitation. In fact, there are cases where a deformable rocking frame is more stable than its rigid counterpart. Copyright © 2016 John Wiley & Sons, Ltd.     

Plastic velocity vector diagrams applied to indentation and transpression in the Alps

Alpine collision tectonics has affected the European Alps for more than 110 Ma and is probably still active today. The cumulative large-scale convergence in the Alps adds up to 1100 km. About 500 km of this convergence can be budgeted by escape tectonics, 300 km by continental and oceanic subduction, and an equal amount by crustal thickening. Considerable uncertainty still exists as to the kinematics of the collision; a transpressive indentation of a more or less rigid “Italy/Adria” block into a deformable European plate appears likely. In recent surveys of horizontal mass transfer in the Alps, as inferred from structural analyses, a dominant westwards directed flow regime has been recognized. Theoretical indentation models have been applied to explain the palaeo-flow field. The flow can be explained by: Case I normal indentation without transpression but a space restriction to the east. A palinspastic space restriction did not exist, on the contrary, remnant oceanic terrain survived longest in the east; Case II dextral transpression with no significant indentation. This would require an age progression of westward flow from the east (oldest) to the west (youngest) which is not observed; and Case III transpressive indentation with a sinistral shear component. This seems to be the most satisfactory solution as it explains large-scale sinistral lineaments, and leaves palinspastic space for the rapid rotation of Corsica and Sardinia.     

Experimental assessment of contact forces on a rigid base following footing uplift

A number of investigations in the recent decades have shown that footing uplift can reduce the seismic loading on a structure. Guidelines to design a structure with seismic uplift capability have been proposed. However, these studies mainly focus on the structural response and neglect the impact forces on the footing from re‐contact between the footing and the supporting medium. A small number of computational studies of the induced forces on the footing have been performed. This paper presents the results of free vibrations and shake table tests on a single degree‐of‐freedom model of a bridge pier with footing uplift on a rigid base. Two support conditions are considered, that is, footing fixed to the base and footing free to uplift on a rigid base. Load cells were placed at the interface of the footing and rigid base to measure the contact forces during structural vibration. The footing responses of both flexible and rigid structures due to free vibration are compared. The results show that the flexibility of the structure has significant effects on footing uplift duration and amplitude and reduces the contact force, in some cases very significantly. The flexible structure was also subjected to harmonic base excitations. It is found that varying the characteristics of the excitation changes the uplift amplitude but does not affect the contact force significantly. Copyright © 2017 John Wiley & Sons, Ltd.     

Pounding of structures modelled as non‐linear impacts of two oscillators

A new formulation is proposed to model pounding between two adjacent structures, with natural periods T₁ and T₂ and damping ratios ζ₁ and ζ₂ under harmonic earthquake excitation, as non‐linear Hertzian impact between two single‐degree‐of‐freedom oscillators. For the case of rigid impacts, a special case of our analytical solution has been given by Davis (‘Pounding of buildings modelled by an impact oscillator’ Earthquake Engineering and Structural Dynamics, 1992; 21 :253–274) for an oscillator pounding on a stationary barrier. Our analytical predictions for rigid impacts agree qualitatively with our numerical simulations for non‐rigid impacts. When the difference in natural periods between the two oscillators increases, the impact velocity also increases drastically. The impact velocity spectrum is, however, relatively insensitive to the standoff distance. The maximum relative impact velocity of the coupled system can occur at an excitation period T_n^* which is either between those of the two oscillators or less than both of them, depending on the ratios T₁/T₂ and ζ₁/ζ₂. Although the pounding force between two oscillators has been primarily modelled by the Hertz contact law, parametric studies show that the maximum relative impact velocity is not very sensitive to changes in the contact parameters. Copyright © 2001 John Wiley & Sons, Ltd.     

Rockfall rebound: comparison of detailed field experiments and alternative modelling approaches

The accuracy of rockfall trajectory simulations mainly rests on the calculation of the rebound of fragments following their impact on the slope. This paper is dedicated to the comparative analysis of two rebound modelling approaches currently used in rockfall simulation using field experiments of single rebounds. The two approaches consist in either modelling the rock as a single material point (lumped mass approach) or in explicitly accounting for the fragment shape (rigid body approach). A lumped mass model accounting for the coupling between translational and rotational velocities and introducing a slope perturbation angle was used. A rigid body approach modelling the rocks as rigid locally deformable (in the vicinity of the contact surface) assemblies of spheres was chosen. The comparative analysis of the rebound models shows that both of them are efficient with only a few parameters. The main limitation of each approach are the calibration of the value of the slope perturbation (‘roughness’) angle, for the lumped mass approach, and the estimation of the rock length and height from field geological and historical analyses, for the rigid body approach. Finally, both rebound models require being improved in a pragmatic manner to better predict the rotational velocities distribution. Copyright © 2012 John Wiley & Sons, Ltd.     

Transmitting boundary for water-saturated transversely isotropic strata based on the u–U formulation

A new transmitting boundary in a cylindrical coordinate system has been developed for modeling the elastic waves radiating out to an infinite boundary in water-saturated transversely isotropic soil strata over a rigid bedrock. The saturated soil strata are assumed to consist of a porous material and modeled as a transversely isotropic two-phase medium, based on the u−U formulation. The newly developed transmitting boundary is combined with the finite-elements model of the near-field region, using the same u−U formulation, and applied to the study of the dynamics of a rigid circular foundation in porous isotropic or transversely isotropic layered strata, either fully or partly saturated with water. The verification and application examples give valuable insights into new and interesting aspects of the dynamic behavior of rigid circular foundations in fully or partly saturated two-phase ground in terms of permeability, transverse anisotropy, and ground-water table level.     

Arbitrary Lagrangian Eulerian formulation for soil structure interaction problems

Simulation of soil structure interaction problems becomes more and more the focus of computational engineering in the last decades in the civil engineering community. In many applications in civil engineering, the Lagrangian formulation is the main approach for soil structure interaction problems. For the last decades, various approaches have been investigated in fluid and structure mechanics, to solve fluid structure interaction problems, including Arbitrary Eulerian Lagrangian (ALE) formulation and meshless methods as Smoothing Particle Hydrodynamics (SPH). These two approaches are more common in fluid mechanics than structure and civil engineering, where the classical Lagrangian formulation is still dominant for numerical simulations. The Lagrangian formulation, where the mesh moves with material is mainly used to solve problems in solid mechanics, and civil engineering for soil structure interaction problems. For small deformation, Lagrangian formulation can solve soil structure interface and material boundary accurately; the main limitation of the formulation is high mesh distortion for large deformation and moving structure. One of the commonly used approaches to solve these problems is the ALE formulation which has been used with success in the simulation of fluid structure interaction with large structure motion such as sloshing fuel tank in automotive industry and bird impact in aeronautic industry. In this paper, devoted to ALE formulation for civil engineering problems, the mathematical and numerical implementation of the ALE formulation and the contact algorithm are described. In order to validate the ALE algorithms, and prove its ability for civil engineering applications, we consider two different applications. First we perform the simulation of a shock wave propagation generated by a detonation of an explosive. In the second application, we consider a soil structure coupling problem, using an ALE formulation for the soil which undergoes large mesh deformation, and a Lagrangian formulation for the structure. A contact algorithm is performed at the soil structure interface.     

Finite Element Analysis of a Sandwich Friction Experiment Model of Rocks

-- Sandwich friction experiments are one of the most widely used standard methods for measuring the frictional behavior between rocks. A finite element code for modeling the nonlinear friction contact between elastoplastic bodies has been developed and extended to analyze the sandwich friction experiment model with a rate- and state-dependent friction law. The influences of prescribed slip velocity and variation of movement direction and state on the friction coefficient, the relative slip velocity, the normal contact force, the frictional force, the critical frictional force and the transition of stick-slip state between the deformable rocks are thoroughly investigated, respectively. The calculated results demonstrate the usefulness of this code for simulating the friction behavior between rocks.     

An analytical model of a deformable cantilever structure rocking on a rigid surface: experimental validation

This paper describes an experimental program to examine the dynamic response of deformable cantilevers rocking on a rigid surface. The primary goal of the tests is to verify and validate a dynamic rocking model that describes the behavior of these structures. The benchmark response data was obtained from shaking‐table tests on deformable rocking specimens with different natural vibration frequencies and different aspect ratios excited by analytical pulses and recorded ground motions. The responses computed using the model are found to be in good agreement with the benchmark test results. Widely used impact, restitution and damping assumptions are revisited based on the experiment results and the analytical model findings. Copyright © 2015 John Wiley & Sons, Ltd.     

Stresses produced by a normal load moving over a transversely isotropic layer of ice lying on a rigid foundation

Summary In this paper, the problem of a normal load moving over the surface of a layer of ice, lying on a rigid foundation, has been considered. FollowingHearmon, the ice is taken to be transversely isotropic. Two cases have been considered-in the first case, the rigid foundation is taken to be smooth, and in the second case, it is taken to be in welded contact with the layer of ice. Expressions have been obtained in both cases for the stresses at any point of the layer. From detailed numerical calculations, it has been found that the value of the normal stress at a point directly below the moving load falls of approximately as that of 1/z with increase in the depthz below the surface. On calculating the normal stress at the boundary between the layer of ice and the foundation, it is found that the normal stress falls off rapidly as we move away along the boundary from the point directly below the load.     

Out-of-plane (SH) soil-structure interaction: Semi-circular rigid foundation revisited

The model studied in this paper presents an extension of previous work for a shear wall on a semi-circular rigid foundation in an isotropic homogeneous and elastic half-space. The objective is to develop a soil-structure interaction model that can later be applied to the case of a flexible foundation. As shown in the Introduction below, Luco considered the case of a rigid foundation subjected to vertical incident plane SH waves, and Trifunac extended the solution for the same rigid foundation subjected to SH waves but for arbitrary angles of the incidence. In this paper, a new approach and model are presented for the same semi-circular rigid foundation with a tapered-shape (instead of rectangular) superstructure. The analytical expression for the deformation of the semi-circular rigid foundation below this tapered shear wall with soil-structure interaction in an isotropic homogeneous and elastic half-space is thus derived. Results are then compared with those of Trifunac discussed in the section below. This problem formulation can and will later be extended in the case of a flexible foundation that is semi-circular or arbitrarily shaped.     

Development of deformable connection for earthquake‐resistant buildings to reduce floor accelerations and force responses

This paper presents the development of a deformable connection that is used to connect each floor system of the flexible gravity load resisting system (GLRS) with the stiff lateral force resisting system (LFRS) of an earthquake‐resistant building. It is shown that the deformable connection acts as a seismic response modification device, which limits the lateral forces transferred from each floor to the LFRS and allows relative motion between the GLRS and LFRS. In addition, the floor accelerations and the LFRS story shears related to the higher‐mode responses are reduced. The dispersion of peak responses is also significantly reduced. Numerical simulations of the earthquake response of a 12‐story reinforced concrete shear wall example building with deformable connections are used to define an approximate feasible design space for the deformable connection. The responses of the example building model with deformable connections and the example building model with rigid‐elastic connections are compared. Two configurations of the deformable connection are studied. In one configuration, a buckling restrained brace is used as the limited‐strength load‐carrying hysteretic component of the deformable connection, and in the other configuration, a friction device is used. Low damping laminated rubber bearings are used in both configurations to ensure the out‐of‐plane stability of the LFRS and to provide post‐elastic stiffness to the deformable connection. Important experimental results from full‐scale tests of the deformable connections are presented and used to calibrate numerical models of the connections. Copyright © 2016 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.     

Seismic response of rocking frames with top support eccentricity

The seismic response of rocking frames that consist of a rigid beam freely supported on rigid freestanding rectangular piers has received recent attention in the literature. Past studies have investigated the special case where, upon planar rocking motion, the beam maintains contact with the piers at their extreme edges. However, in many real scenarios, the beam‐to‐pier contact lies closer to the center of the pier, affecting the overall stability of the system. This paper investigates the seismic response of rocking frames under the more general case which allows the contact edge to reside anywhere in‐between the center of the pier and its extreme edge. The study introduces a rocking block model that is dynamically equivalent to a rocking frame with vertically symmetric piers of any geometry. The impact of top eccentricity (ie, the distance of the contact edge from the pier's vertical axis of symmetry) on the seismic response of rocking frames is investigated under pulse excitations and earthquake records. It is concluded that the stability of a top‐heavy rocking frame is highly influenced by the top eccentricity. For instance, a rocking frame with contacts at the extreme edges of the piers can be more seismically stable than a solitary block that is identical to one of the frame's piers, while a rocking frame with contacts closer to the centers of the piers can be less stable. The concept of critical eccentricity is introduced, beyond which the coefficient of restitution contributes to a greater reduction in the response of a frame than of a solitary pier.     

Thermal stresses in a finite cylinder having temperature varying linearly along the length prescribed on the curved surface

Summary The problem of a finite cylinder of isotropic material in which the temperature on the curved surface varies linearly along its length and has its ends in contact with smooth rigid insulating plates has been investigated in this paper.     

Investigation of the sliding behavior between steel and mortar for seismic applications in structures

The friction developed between a steel base plate and a mortar base contributes shear resistance to the building system during a seismic event. In order to investigate the possible sliding behavior between the base plate and the mortar, a shake table study is undertaken using a large rigid mass supported by steel contact elements which rest on mortar surfaces connected to the shake table. Horizontal input accelerations are considered at various magnitudes and frequencies. The results provide a constant friction coefficient during sliding with an average value of approximately 0.78. A theoretical formulation of the friction behavior is also undertaken. The theoretical equations show that the sliding behavior is dependent on the ratio of the friction force to the input force. The addition of vertical accelerations to the system further complicates the sliding behavior as a result of the varying normal force. This results in a variable friction resistance which is a function of the amplitude, phase, and frequency of the horizontal and vertical input motions. In general, this study showed a consistent and reliable sliding behavior between steel and mortar. Copyright © 2009 John Wiley & Sons, Ltd.