Michalis F. Vassiliou

Estimating Time Scales and Length Scales in Pulselike Earthquake Acceleration Records with Wavelet Analysis

This paper is motivated from the need to extract the characteristic time and length scales of strong pulselike ground motions with a mathematically formal, objective, and easily reproducible procedure. The investigation uses wavelet analysis to identify and extract energetic acceleration pulses (not velocity pulses) together with their associated frequency and amplitude. The processing of acceleration records with wavelet analysis is capable of extracting pulses that are not detected visually in the acceleration records, yet they become coherent and distinguishable in the velocity records. Most importantly, the proposed analysis is capable of extracting shorter dura- tion distinguishable pulses (not necessarily of random character) that override the longer near-source pulses that are of significant engineering interest. The study ela- borates on the role of the weighting function in the definition of the wavelet transform and concludes that longer pulses are captured when less suppressive weighting func- tions are implemented in the wavelet transform. We examine the capability of several popular symmetric and antisymmetric wavelets to locally match the energetic accel- eration pulse. We conclude that the exercise to identify the best-matching wavelet shall incorporate, in addition to the standard translation and dilation-contraction of the wavelet transform, a phase modulation together with a manipulation of the oscil- latory character (addition of cycles) of the wavelet. This need leads to the extension of the wavelet transform to a more general wavelet transform during which the mother wavelet is subjected to the four above-mentioned modulations. The mathematical de- finition and effectiveness of this extended wavelet transform is presented in this paper. The objective identification of the pulse period, amplitude, phase, and oscillatory character of pulselike ground motions with the extended wavelet transform introduced in this paper makes possible the immediate use of closed-form expressions published by other investigators to estimate the peak response of elastic and inelastic systems. Online Material: Parameters that maximize the extended wavelet transform of 183 selected records.

Are Some Top-Heavy Structures More Stable?

AbstractThis technical note investigates the dynamic response and stability of a rocking frame that consists of two identical free-standing slender columns capped with a freely supported rigid beam. Part of the motivation for this study is the emerging seismic design concept of allowing framing systems to uplift and rock along their plane in order to limit bending moments and shear forces— together with the need to stress that the rocking frame is more stable the more heavy is its cap-beam, a finding that may have significant implications in the prefabricated bridge technology. In this technical note, a direct approach is followed after taking dynamic force and moment equilibrium of the components of the rocking frame, and the remarkable results obtained in the past with a variational formulation (by the same authors) is confirmed—that the dynamics response of the rocking frame is identical to the rocking response of a solitary, free-standing column with the same slenderness, yet with larger size, which p...

Dynamics of the Rocking Frame with Vertical Restrainers

AbstractThis paper investigates the rocking response and stability analysis of an array of slender columns caped with a rigid beam which are vertically restrained with elastic prestressed tendons that pass through the centerline of the columns while anchored at the foundation and the cap-beam. Following a variational formulation, the nonlinear equation of motion is derived in which the stiffness and the prestressing force of the tendons are treated separately. In this way, the postuplift stiffness of the vertically restrained rocking frame can be anywhere from negative to positive depending on the axial stiffness of the vertical tendons. The paper shows that the tendons are effective in suppressing the response of rocking frames with small columns subjected to long-period excitations. As the size of the columns, the frequency of the excitations, or the weight of the cap-beam increases, the vertical tendons become immaterial given that most of the seismic resistance of tall rocking frames originates primar...

Dynamics of the Vertically Restrained Rocking Column

AbstractThis paper investigates the rocking response of a slender column that is vertically restrained with an elastic tendon that passes through its centerline. Following a variational formulation, the nonlinear equation of motion is derived, in which the stiffness and the prestressing force of the tendon are treated separately. In this way, the post-uplift stiffness of the system can be anywhere from negative to positive depending on the axial stiffness of the vertical tendon. This paper shows that vertical tendons are effective in suppressing the response of smaller columns subjected to long-period excitations. As the size of the column or the frequency of the excitation increases, the effect of the vertical tendon becomes immaterial given that most of the seismic resistance of large rocking columns originates primarily from the mobilization of their rotational inertia.

Dynamics of inelastic base-isolated structures subjected to recorded ground motions

Relations between the strength reduction factor Ry, the displacement ductility μ, and the vibration period Tn of a structure have been presented for fixed-base structures in numerous studies. These relations reflect the ranges of the inelastic response of a fixed-base structure to strong ground motion excitation. The superstructure of a base-isolated structure may also enter the inelastic behavior range when excited by strong ground motions. The goal of this study is to identify similar Ry–μ–Tn relations for base-isolated superstructures. A two-degree-of-freedom model of a base-isolated structure, with the inelastic behavior of the isolators and the isolated superstructure modeled in OpenSees, was used in this study. Recorded ground motions, whose parameters span a wide range of ground motion types, magnitudes and distances, were used to excite this model. Parametric studies were performed to determine the effects of isolator and superstructure design parameters on the response. A Ry–μ–Tn relationship for inelastic base-isolated superstructures is proposed using a format similar to such relations for fixed-base structures.

The Dynamics of the Rocking Frame

This paper investigates the planar rocking response and stability analysis of an array of free-standing columns capped with a freely supported rigid beam. Part of the motivation for this study is the emerging seismic design concept of allowing framing systems to uplift and rock along their plane in order to limit bending moments and shear forces. Following a variational formulation the paper reaches the remarkable result that the dynamic rocking response of an array of free-standing columns capped with a rigid beam is identical to the rocking response of a single free-standing column with the same slenderness; yet with larger size—that is a more stable configuration. Most importantly, the study shows that the heavier the freely supported cap-beam is (epistyles with frieze atop), the more stable is the rocking frame, regardless of the rise of the center of gravity of the cap-beam; concluding that top-heavy rocking frames are more stable than when they are top-light. This “counter intuitive” finding renders rocking isolation a most attractive alternative for the seismic protection of bridges with tall piers.

ROCKING RESPONSE OF SLENDER, FLEXIBLE COLUMNS UNDER PULSE EXCITATION

Early studies of the response of rigid blocks allowed to uplift and rock under seismic motion have shown a scale effect that characterizes the response of rocking blocks subjected to a ground motion. Namely; larger objects need a larger ground acceleration to overturn; and longer dominant period earthquakes have a larger overturning capability than shorter period ones. This is why it has been proposed that rocking can be used as an isolation strategy. However, actual structures are not rigid: structural elements where rocking is expected to occur are often slender and flexible. Modeling rocking of flexible bodies is a challenging task. A finite element model of a rocking elastic body that does not involve explicit modeling of impact is presented in this paper. This model was validated by comparing its response to pulse excitation with an analytical solution. It is concluded that the extensively studied rigid rocking block model provides a good approximation of the seismic response of an elastic rocking column for small-size columns and that it provides a conservative response estimate for larger columns. Guidance for development of rocking column models in ordinary finite element software is provided.

Experimental investigation of the seismic response of a column rocking and rolling on a concave base

Rocking modifies the seismic response of structures, because uplifting works as a mechanical fuse and limits the forces transmitted to the structure. However, the engineering community is in general reluctant to let a structure uplift because it can overturn, and, more important, an unanchored structure has no redundancy against this failure mode. Using a safety factor for the design of a flat rocking foundation (i.e. designing it larger than minimum required to prevent overturning) goes against the essence of the rocking seismic isolation method, as the structure would end up behaving as fixed to the ground. To protect against overturning but preserve the ability to uplift we propose to extend the flat rocking foundation using curved wedges at its ends. This paper presents the results of dynamic tests of small bodies rocking on curved foundations. The results compare relatively well with the analytical solutions, but they are shown to be very sensitive to the coefficient of restitution. Jonas A. Bachmann, Patrick Blöchlinger, Matthias Wellauer, Michalis F. Vassiliou, and Božidar Stojadinović

Comparative Assessment of Two Rocking Isolation Techniques for a Motorway Overpass Bridge

Rocking isolation of structures is evolving as an alternative design concept in earthquake engineering. The present paper investigates the seismic performance of an actual overpass bridge of the Attiki Odos motorway (Athens, Greece), employing two different concepts of rocking isolation: (a) rocking of the piers on the foundation (rocking piers); and (b) rocking of the pier and foundation assembly (rocking footings) on the soil. The examined bridge is an asymmetric 5-span system having a continuous deck and founded on surface foundations on a deep clay layer. The seismic performance of the two rocking isolated bridges is comparatively assessed to the existing bridge, which is conventionally designed according to current seismic design codes. To that end, 3D numerical models of the bridge–foundation–abutment–soil system are developed, and both static pushover and nonlinear dynamic time history analyses are performed. For the latter, an ensemble of 20 records (10 ground motions of 2 perpendicular components each) that exceed the design level are selected. The conventional system collapses in 5/10 of the (intentionally severe) examined seismic excitations. The rocking piers design alternative survives in 8/10 of the cases examined, with negligible residual deformations. The safety margins of the rocking footings design concept are even larger, as it survives in all cases examined. Both rocking isolation concepts are proven to offer increased levels of seismic resilience, reducing the probability of collapse and the degree of structural damage. Nevertheless, in the rocking piers design alternative high stress concentrations at the rotation pole (pier base) are developed, indicating the need for a special design of the pier ends. This is not the case for the rocking footings concept, which however is subject to increased residual settlements but no residual rotations.

Seismic Response and Stability of the Rocking Frame

This paper investigates the planar rocking response of an array of free-standing columns capped with a freely supported rigid beam in an effort to explain the appreciable seismic stability of ancient free-standing columns which support heavy epistyles together with the even heavier frieze atop. Following a variational formulation the paper concludes to the remarkable result that the dynamic rocking response of an array of free-standing columns capped with a rigid beam is identical to the rocking response of a single free-standing column with the same slenderness; yet with larger size – that is a more stable configuration. Most importantly, the study shows that the heavier the freely supported cap-beam is (epistyles with frieze atop), the more stable is the rocking frame regardless the rise of the center of gravity of the cap-beam; concluding that top-heavy rocking frames are more stable than when they are top-light. This “counter intuitive” finding renders rocking isolation a most attractive alternative for the seismic protection of bridges with tall piers; while its potential implementation shall remove several of the concerns associated with the seismic connections of prefabricated bridges.