Rocking Components Research Articles

Out-of-plane load–displacement model for two-way spanning masonry walls

This paper describes a methodology for modelling the nonlinear, inelastic load–displacement behaviour of two-way spanning unreinforced masonry walls subjected to out-of-plane loading. The model utilises a simplified macroblock approach that starts with the assumption of a collapse mechanism based on the wall’s boundary conditions. It then treats the wall as having zero tensile strength and assumes that the resistance comes entirety from two gravity-based resistance components: elastic rigid block rocking, and inelastic friction, with the total load resistance of the wall taken as the sum of these individual components. Analytical expressions for calculating the load and displacement capacities of the elastic rocking component of response are derived from the principles of statics using an integration approach well suited for the treatment of two-way mechanisms. Expressions for the associated frictional capacity component are obtained using the virtual work method. Comparison of the theoretical load–displacement response with experimentally measured data is favourable as demonstrated using data obtained via quasistatic cyclic tests on two-way spanning walls; the model is shown to provide an acceptable lower bound estimate of actual behaviour. The developed approach could be used to construct pushover curves for a range of different collapse mechanisms and therefore has the potential to be assimilated into a simplified displacement-based seismic design/assessment technique for two-way spanning walls against out-of-plane collapse.

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.

Surface Rotations on and Around a Semi-Parabolic Canyon in an Elastic Half-Space Induced by Out-of-Plane SH-Waves, I: Torsion

The surface displacement amplitudes excited by out-of-plane SH waves on and around a semi-parabolic canyon had solution that was introduced over twenty years ago. In 2015, such solution was updated to much (5 times) higher frequencies, using a transformation involving the Wronskian. The same excitations also introduce rotations on and around the parabolic canyon. This article is the first of two that discuss the torsional component of rotations, with the rocking components discussed in the subsequent article. The rotational amplitudes of motions, as much as the displacement amplitudes, play an important role in studying the surface responses of structures, and can lead to a better understanding of the behavior of surface topographies in the event of seismic excitation.

Cold orbital forging of gear rack

At present, gear rack is manufactured mainly by cutting method while its precision plastic forming method has not been reported. This paper proposes an innovative cold orbital forging method for manufacturing gear rack, in which the complicated teeth of gear rack are forged by the highly dynamic rocking tool under straight line path. Firstly, a universal method for designing the rocking tool under straight line path is proposed according to the forming mode and contacting mode between the rocking tool and component, i.e., one side of the component is ultimately overall formed by the overall contact (it is achieved by the incremental deformation) between the corresponding side of the rocking tool and component. Based on this universal method, the rocking tool in cold orbital forging of gear rack is accurately designed. Secondly, the mathematical model and method for interference examinations between the rocking tool and gear rack are developed and the interference amount and interference zone are calculated. Thirdly, to eliminate the interference between the rocking tool and gear rack so as to guarantee the forming accuracy of gear rack, a dynamic envelope forming principle is proposed and an innovative forming mode of gear rack is correspondingly developed, i.e., all the time-varying interference zones in the rocking tool are cut one by one and the ultimate tooth surface (planar) of gear rack is incrementally formed by the local contact between a series of modified surfaces (nonplanar) of the rocking tool and tooth surface of gear rack. Based on this proposed dynamic envelope forming principle, the mathematical model and method for modifying the rocking tool are developed and the dynamic envelope forming of gear rack is achieved successfully. The FE simulations and experiments of cold orbital forging of gear rack are conducted. The results demonstrate that the proposed cold orbital forging method for manufacturing gear rack in this paper is effective and cold orbital forging shows great potential in manufacturing gear rack.

Revised Seismic Intensity Parameters for Middle-Field Horizontal and Rocking Strong Ground Motions

AbstractThis paper presents a new method for determining rocking ground motion considering the effects of both time delay and loss of coherency in spatially variable seismic ground motions. Acceleration, velocity, and displacement response spectra resulting from the rocking component are also derived. Next, new seismic intensity parameters are proposed to estimate the combined action of the middle-field horizontal and rocking motions on the earthquake excitation of structures. These seismic intensity parameters are revised forms of the widely used ones for characterizing translational components, namely, peak ground velocity, peak ground displacement, cumulative absolute displacement, acceleration response spectrum, acceleration spectrum intensity, Fajfar intensity, and Kappos spectrum intensity. In addition, an expression for the estimation of the effect of the base tilt due to the rocking ground motion on the structural response is presented. The main advantage of the proposed relations is that they can...

Modeling of ground motion rotational components for near-fault and far-fault earthquake according to soil type

The motion of a point is specified completely through six components, three translational and three rotational. Three rotational components of ground motion are classified to two rocking components and one torsional component. The transitional components of ground motion are easily measurable by standard techniques, whereas the rotational components are not directly accessible. In this study, the rotational component of production technique was described, and the influences of soil type and the distance from the fault on the rotational component were investigated. The results showed that the effect of the rotational components is more significant in low periods. Also, the value of the normalized response spectra for soft soil is more than stiff soil, but for far-fault earthquake, it is less than near-fault earthquake.

Dynamic Analysis of Elevated Water Storage Tanks due to Ground Motions’ Rotational and Translational Components

Rotational components of ground motion including rocking and torsional components have rarely been accounted for in designing and analyzing water storage tanks. In this paper, the effects of these components on the dynamic response of water storage tanks are studied. The rotational components of ground motion acceleration have been obtained using an improved approach from the corresponding available translational components based on the transversely isotropic elastic wave propagation and classical elasticity theories. Based on this approach, it becomes possible to consider the frequency-dependent wave velocities and incident wave angle to generate the rotational components. For this purpose, the translational components of four earthquakes have been selected to generate their relative rotational components based on SV and SH wave incidence. The translational and computed rotational motions were then applied to the concrete elevated water storage tank with different water elevations considering fluid–structure interaction using the finite element method. The linear responses of these structures considering the full six components of the ground motion show that the rotational components of ground motion can affect the displacement and reaction force of the structure, depending on the frequency of structure and predominant frequencies of translational and rotational components of ground motion.

Simplified relations for the application of rotational components to seismic design codes

In this paper, after reviewing the characteristics of earthquake rotational components, the acceleration response spectra of rocking and torsional components are presented. In addition, the proposed relations in the author’s previously published paper on the rotational loading of structures are extended, and application of the new relations is examined. The numerical results of this study show that the contribution of earthquake rocking components to the rotational loading of multistory buildings is strongly sensitive to structural irregularity, structural height and seismic excitation. The rocking component contribution to story shear can be one-third of the horizontal component contribution, and should be included in seismic design codes. Further, the value of the eccentricity of 0.05 plan dimension, which is prescribed in the most current seismic design codes for accidental torsional effects, is not a conservative approximation for the accidental eccentricity caused just by the influences of the torsional component, particularly in symmetric, torsionally stiff, multistory buildings.

Seismic response of aboveground steel storage tanks: comparative study of analyses by six and three correlated earthquake components

Ground motions at a point on the ground surface can be decomposed to six components, namely three translational components and three rotational components; translational components include two components in the horizontal plane, and one in the vertical direction. Rotation about horizontal axes leads to rising of rocking, while the rotational component about a vertical axis generates torsional effects even in symmetrical buildings. Due to evident and significant contribution of ground shakings to the overall response of structures, rocking and torsional components of these motions resulted by strong earthquakes are recently subjected to widespread researches by engineering and research communities. In this study, first rotational components of ground motion are determined using a method developed by Hong-Nan Li and et al (2004). This method is based on frequency dependence on the angle of incidence and the wave velocity. In consequence, aboveground steel storage tanks (ASSTs) with different water elevations have been analyzed with the effects of these six components of earthquake. Three translational components of six important earthquakes have been adopted to generate relevant rotational components based on SV and SH wave incidence by the Fast Fourier Transform (FFT) with the discrete frequencies of time histories of translational motion. Using finite element method, linear properties of tank material including steel for cylindrical tanks have been taken into with considering fluid-structure interaction. Numerical linear dynamic analysis of these structures considering six components of earthquake motions is presented; results are compared with cases in which three translational components are considered.

Assessment of soil–pile–structure interaction influencing seismic response of mid-rise buildings sitting on floating pile foundations

The role of the seismic soil–pile–structure interaction (SSPSI) is usually considered beneficial to the structural system under seismic loading since it lengthens the lateral fundamental period and leads to higher damping of the system in comparison with the fixed-base assumption. Lessons learned from recent earthquakes show that fixed-base assumption could be misleading, and neglecting the influence of SSPSI could lead to unsafe design particularly for structures founded on soft soils. In this study, in order to better understand the SSPSI phenomena, a series of shaking table tests have been conducted for three different cases, namely: (i) fixed-base structure representing the situation excluding the soil–structure interaction; (ii) structure supported by shallow foundation on soft soil; and (iii) structure supported by floating (frictional) pile foundation in soft soil. A laminar soil container has been designed and constructed to simulate the free field soil response by minimising boundary effects during shaking table tests. In addition, a fully nonlinear three dimensional numerical model employing FLAC3D has been adopted to perform time-history analysis on the mentioned three cases. The numerical model adopts hysteretic damping algorithm representing the variation of the shear modulus and damping ratio of the soil with the cyclic shear strain capturing the energy absorbing characteristics of the soil. Results are presented in terms of the structural response parameters most significant for the damage such as foundation rocking, base shear, floor deformation, and inter-storey drifts. Comparison of the numerical predictions and the experimental data shows a good agreement confirming the reliability of the numerical model. Both experimental and numerical results indicate that soil–structure interaction amplifies the lateral deflections and inter-storey drifts of the structures supported by floating pile foundations in comparison to the fixed base structures. However, the floating pile foundations contribute to the reduction in the lateral displacements in comparison to the shallow foundation case, due to the reduced rocking components.

Approximate formulas for rotational effects in earthquake engineering

The paper addresses the issue of researching into the engineering characteristics of rotational strong ground motion components and rotational effects in structural response. In this regard, at first, the acceleration response spectra of rotational components are estimated in terms of translational ones. Next, new methods in order to consider the effects of rotational components in seismic design codes are presented by determining the effective structural parameters in the rotational loading of structures due only to the earthquake rotational components. Numerical results show that according to the frequency content of rotational components, the contribution of the rocking components to the seismic excitation of short period structures can never be ignored. During strong earthquakes, these rotational motions may lead to the unexpected overturning or local structural damages for the low-rise multi-story buildings located on soft soil. The arrangement of lateral-load resisting system in the plan, period, and aspect ratio of the system can severely change the seismic loading of wide symmetric buildings under the earthquake torsional component.

Rocking Component of Earthquake Induced by Horizontal Motion in Irregular Form Ground

Two horizontal components and a vertical component are usually employed in seismic observations. Therefore, only those components are used as the input ground motions in seismic design code. However, there must be rotational or rocking component together with basic three components in ground motion, because of the existence of Rayleigh wave, phase difference between vertical components at grounds with different stiffness, and no mechanism preventing ground motion from being horizontal for excitations from various directions. Seismographs are generally installed more than several hundred meters apart from each other even in arrayed observations. This distance is too long to identify the existence of rotational or rocking component. And there is no seismograph which can measure this component. Since there has been no information on this component, such component cannot be taken into account in seismic design code. However such component may have an impact on some rigid structures as well as flexible structures. It is indispensable to examine the effects of this component on structural response. In this study, experiments using the model ground excited by a shaking table and simulations of ground motion using FEM are carried out, in which existence of rocking components are examined in earthquake motion generated at various grounds such as the ground with irregular form foundation. Consequently, the experiment and analysis verified existence of rocking motion component. It is shown that the vibration of horizontal direction is the generating factor of rocking motion components in the experiment and that the vibration of vertical direction is the amplifying factor of rocking motion components. Moreover, rocking motion component is large near the border between two different grounds, and depends on the vertical amplitude of vibration and on excitation frequency.

THREE-DIMENSIONAL SHAKING TABLE TESTS ON SEISMIC RESPONSE OF REDUCED-SCALE STEEL FRAMES WITH YIELDING BASE PLATES ALLOWED TO UPLIFT

The seismic response of three-story one-third-scale steel frames with columns allowed to uplift is evaluated and compared with that of fixed-base frames by three-dimensional shaking table tests. The test frames were retested using four different frame conditions: uplift-base and fixed-base frames; plan symmetric and asymmetric frames. The results are: 1) While the column uplift displacements of the uplift frames is increased by the vertical component of the input motion, the horizontal response values such as the roof drifts and base shears are not affected by that; 2) The difference of the roof floor rotations between the symmetric and asymmetric uplift frames are not increased in spite of the increase of the input motion intensity; 3) The base shears of the uplift frames are effectively reduced from those of the fixed-base frames; and, 4) The deformation of the uplift frames excluding the rocking component is nearly equal to or smaller than the elastic response value of the fixed-base frames.

Rotational Seismic Load Definition in Eurocode 8, Part 6, for Slender Tower-Shaped Structures

This note describes the rotational seismic load definition as included in Part 6 of Eurocode 8 (EC8.6, 2005). The Eurocode 8, Part 6 (EC8.6, 2005), definition of the rotational ground-motion component depends upon the structural subsoil compliance, which is controlled by the shear-wave velocity in the top 30 m of ground. A comparison of the effects of the rocking ground motion and the horizontal ground motion on the response of a 160 m reinforced concrete chimney shows that for the Eurocode 8, Part 6 (EC8.6, 2005), definition of the rotational seismic ground motion, the rocking excitations contribute significantly to the overall response of the structure. The engineering code formulas for the rocking component of ground motion, however, should be calibrated and reconciled with the results of the latest empirical research.

Empirical Scaling of Rotational Spectra of Strong Earthquake Ground Motion

Abstract Many empirical scaling equations have been developed for scaling Fourier spectrum amplitudes of strong earthquake accelerations and for generation of artificial strong-motion accelerograms for the translational, torsional, and rocking components of strong motion. It has also been shown that rotational components of strong motion significantly contribute to the overall response of structures; however, little progress has been made in the development and deployment of strong-motion instruments to measure rotations. This article presents a simple approximate algorithm for generating torsional and rocking Fourier spectral amplitudes from the corresponding translational motions. The method can be used to generate torsion and rocking spectra from the translational Fourier spectra of actual records. Inverse Fourier transform can be used to generate torsional and rocking time histories.

Response Spectra for Near-Source, Differential, and Rotational Strong Ground Motion

It is shown that the pseudorelative spectral velocity (PSV) of an equiva- lent oscillator, which can be represented by a single degree-of-freedom system for excitation by synchronous horizontal excitation, can be extended to describe the PSV spectra of the same oscillator when excited by simultaneous action of horizontal, vertical, and rocking components of strong ground motion. At short periods, the new spectra are governed by differential ground motion and peak ground velocity, and they depend on the transit time of the waves along the length of the structure and on the oscillator frequency. At long periods, PSV spectral amplitudes tend toward an asymp- tote with amplitude proportional to the maximum rocking angle of ground motion.

Response of pendulums to complex input ground motion

Dynamic response of most seismological instruments and many engineering structures to ground shaking can be represented via response of a pendulum (single-degree-of-freedom oscillator). In most studies, pendulum response is simplified by considering the input from uni-axial translational motion alone. Complete ground motion however, includes not only translational components but also rotations (tilt and torsion). In this paper, complete equations of motion for three following types of pendulum are described: (i) conventional (mass-on-rod), (ii) mass-on-spring type, and (iii) inverted (astatic), then their response sensitivities to each component of complex ground motion are examined. The results of this study show that a horizontal pendulum similar to an accelerometer used in strong motion measurements is practically sensitive to translational motion and tilt only, while inverted pendulum commonly utilized to idealize multi-degree-of-freedom systems is sensitive not only to translational components, but also to angular accelerations and tilt. For better understanding of the inverted pendulum's dynamic behavior under complex ground excitation, relative contribution of each component of motion on response variants is carefully isolated. The systematically applied loading protocols indicate that vertical component of motion may create time-dependent variations on pendulum's oscillation period; yet most dramatic impact on response is produced by the tilting (rocking) component.

FREQUENCY DISPERSION CHARACTERISTICS OF PHASE VELOCITIES IN SURFACE WAVE FOR ROTATIONAL COMPONENTS OF SEISMIC MOTION

When rotational components of ground motion produced by seismic surface waves are computed, the phase velocities must always be dealt with in earthquake engineering. In this paper, appropriate methods are presented to obtain the calculation formulas for the phase velocities of surface waves by applying the theory of elastic wave propagation. Frequency dispersion characteristics of phase velocities are discussed. The rocking component around a horizontal axis and the torsional component around a vertical axis, which are generated, respectively, by the Rayleigh and Love waves, are reasonably given. A procedure is developed to calculate the time histories of these rotational components.

Seismic Soil-Structure Interaction in Buildings. I: Analytical Methods; Seismic Soil-Structure Interaction in Buildings. II: Empirical Findings

The two papers describe trends of the apparent system periods, , and damping factors, , in recorded response of 56 ˜˜ T z0 buildings (A1 to A45, and B1 to B14 in Table 1 of part II) to 15 California earthquakes. The building, foundation, and soil are viewed as an equivalent linear system, excited by horizontal ground motion. Only inertial interaction is considered as ‘‘the more important effect for foundations without large, rigid base slabs or deep embedment’’ (by allowing one translation and rocking of the foundation). The kinematic interaction (modification of the free-field motion by the foundation in absence of inertial forces) is neglected as ‘‘second order.’’ Rocking and torsional excitation components of the incident motion, differential motions of extended foundations, warping of flexible foundations, and nonlinear soil response due to soilstructure interaction (SSI) are not mentioned. The papers aim to verify the ‘‘simplified analytical procedures similar to those in the BSSC and ATC codes.’’ This discussion (1) requests some clarifications; (2) suggests that time-dependent changes in (caused by nonlinear response of soil during SSI) should ˜ T have been included in the analysis; and (3) comments on the conditions when kinematic SSI should not be neglected. In view of the bold simplifications adopted in parts I and II, one would expect at least fair to good agreement of the predicted trends with the data, as a proof (albeit qualitative and rough) that the assumptions are justified. However, as the authors admit, there is ‘‘significant scatter in the data.’’ Table 1 in part II lists the confidence (A-acceptable; L-low) characterizing the quality of the available geotechnical data and the accuracy of identification. An L was assigned 15% of the 44 cases with base acceleration 0.25g. Does the proposed simplified analysis break down for progressively larger strains in the soil, e.g., base accelerations >0.15g?

A dynamic analysis procedure for concrete-faced rockfill dams subjected to strong seismic excitation

There is presently little analytical and essentially no observational evidence on the behavior of the concrete-face of the very popular concrete-faced rockfill (CFR) dams during strong seismic shaking. The choice of slab thickness and steel reinforcement is based solely on precedent, with performance under static loads being the only design consideration. To fill this gap, a dynamic finite element (FE) procedure is presented to design the concrete-face slab which involves a realistic modeling for the embankment material, the face slab, and the slab–rockfill interface. A set of historic accelerograms, with peak ground accelerations (pga) of about 0.60 g, is used as excitation. Numerical results highlight key aspects of the seismic response of CFR dams with emphasis on the internal forces developing in the slab. It is shown that slab distress may be produced only from axial tensile forces, developing mainly due to the rocking component of dam deformation. For the 0.60 g shaking tensile stresses much higher than the likely tensile strength of concrete development in the slab. All analyses in this FE study are performed with the ADINA special interface elements to model the slip and Newmark’s time-integration algorithm for a direct step-by-step solution.