Equation of planar vibrations of rigid structure on kinematic supports after A.M. Kurzanov
The author derives the equation of planar vibrations of rigid structure resting on kinematical pendulum supports with planar bottom (after A.M. Kurzanov). Both support and the surface below are assumed rigid; no sliding assumed. One of the coefficients in the equation (i.e. coordinate of the rotation center) proves to be piece-wise constant. The equation is of the hyperbolic type with parametric terms. Even linearization of this equation does not bring it to the conventional equation of the SDOF oscillator. Principal difference is that the free vibration period depends on the amplitude. The equation is checked for free and forced vibrations. Similar problem is for the seismic response of the unanchored items. For the further research the experimental data about damping are of great importance: both for rotation and the gap closing.
Combination of modal responses in linear spectral method: comparison of different formulae for correlation coefficients
Linear-spectral method (LSM) is still the common method for the seismic design analysis. "One-component one-mode" responses, obtained by static analysis in the conventional variant of LSM, are combined twice: first for different modes but for each single excitation component separately, then for the different excitation components. In the alternative LSM variant presented in the Russian code SP 14.13330, first one chooses the "most dangerous" direction of the one-component excitation for each mode; then calculates the "one-mode" response for this excitation, and finally these responses are combined. In both cases the combination is performed using the complete quadratic combination (CQC) rule. Different documents suggest different formulae for the correlation coefficients. In the paper different formulae are compared to each other. The goal is to limit the number of calculated coefficients and decrease the amount of calculations.
Planar vibrations of rigid structure on kinematic supports after A.M. Kurzanov
Issue: #6 2020
The author carries out
parametric studies for the equation of planar vibrations of rigid structure
resting on kinematical rolling supports with planar bottom (after A.M. Kurzanov).
Both support and the surface
below are assumed rigid; no sliding assumed. Varied parameter is the width of
the bottom. Horizontal structural acceleration is studied. Three variants of
the possible behavior are shown: (i) minor rocking with little decrease in
response accelerations as compared to the initial excitation; considerable
rocking with considerable decrease in the response accelerations; intensive
rocking leading to the overturn of the supports. In vertical direction there
appear shocks (infinite accelerations) during gap closings of the supports. The
importance of the problem for the seismic response analysis of the unanchored
items is noted. The author gives recommendations for the experimental program,
aimed to obtain data about damping both for rotation and for the gap closing,
and also about the impact of the flexibility of the supports and underlying
PLANAR VIBRATIONS OF RIGID STRUCTURE ON KINEMATIC SUPPORTS: GENERAL GEOMETRY
Issue: №4 2020
The author derives
the equation of planar vibrations for kinematical rolling supports of varying
curvature radius. Both support and the surface below are assumed rigid; no
sliding assumed. Generalization means arbitrary geometry of the support and
rolling surface (the building is attached to the supports by hinges, as
previously). The equation of motion developed is checked with free and forced
vibrations of the support with two curvature radii (great radius in the central
part and small radius in the side parts). Such a support in the limit case
(when great radius goes to infinity, and small radius goes to zero) models the
support after A.М.
Kurzanov with planar bottom and finite size in plan.
Planar Vibrations of Rigid Structure Resting on Kinematical Supports of Yu. D. Cherepinsky
Issue: №3 2020
The author derives the equation of motion for a structure resting on kinematical pendulum supports of Yu.D.Cherepinsky. Both structure and supports are assumed to be rigid; no
sliding is assumed during rolling. Two components of seismic excitation are considered (horizontal one and vertical one). Equation of motion for free vibrations looks like that of the free vibrations for massive pendulum support standing alone (it was studied earlier). It is fact the equation of motion for pendulum, but center of rotation, inertia moment and stiffness are varying with time. This equation may be simplified to the linear one by skipping the second
order terms. The equation of motion for seismic response after linearization is the extension of the Mathieu-Hill’s equation, where horizontal component is responsible for the right-hand part (in the conventional Mathieu-Hill’s equation it is zero), and vertical component creates parametric excitation in the left-hand part. In fact, vertical seismic acceleration modifies gravity acceleration g, which controls the effective natural frequency for pendulum.
Thus, there might appear dynamic instability (though without infinite response due to the finite duration of excitation). The author presents numerical example.
Free Vibrations of Rigid Kinematic Support of Yu.D. Cherepinsky
Issue: №2 2020
The author derives the equation of free vibrations for kinematical rolling support of Yu.D.Cherepinsky. Both support and the surface below are assumed rigid. It is shown that Lagrange equation is similar to the equation of motion for rotational oscillator where the rotation centre, rotational inertia and stiffness are changing every moment, depending on displacements. This equation can be further simplified to the linear form with the error proportional to the second degree of displacements. This equation looks somewhat like the equation for classical pendulum, but effective length in our case is controlled by curvature radii of support and of the rolling surface, as well as by the position of vertical load relative to the centre of the support. Non-linear characteristic is soft. The main nonlinearity is in the inertial term, and not in the stiffness term.
Site Response Analysis for “Side” Soil Profiles
Issue: №1 2020
The authors suggest a new procedure of Site Response Analysis (SRA) for the so-called “side” (or additional) soil profiles – Low Boundary (LB) and Upper Boundary (UB). Standards require the analyses of these profiles in addition to the Best Estimate profile (BE) to account for the uncertainty in the input data about soil properties. The authors suggest stopping using the same input time history for all three profiles as a control motion at the surface, because it corresponds to the different physical seismic excitations coming form the depth. This is not in line with the ideology of Standards. Instead the authors suggest using the same time history as a control motion at the outcropped surface of the underlying half-space. This is also not completely correct, because for these three profiles (BE, UB and LB) the underlying half-spaces are also different. However, due to the physical considerations if all half-spaces are stiff enough, the error should not be so important. The effect of the
proposed change is demonstrated on a particular site. The changes in the velocity and damping profiles have proved to be negligible, but the difference in the resulting response spectra at the outcropped surface of the foundation mat has proved to be significant. Generally, the response spectra for the “side” profiles came closer to spectrum for the BE profile. This result reflects the real world logic.
Issue: №5 2018
Issue: №4 2018
Issue: №3 2018
“Old” and “New” Nodes in Modeling of Complicated Soil Environment
Issue: №6 2019
Complicated soil environment” means that under the basement there is a certain finite soil volume with properties breaking horizontally-layered initial soil composition. The author discusses two alternatives in modeling such volume. The “old nodes” approach means that one and the same set of internal nodes is used both for the outcropped soil and for the substituting soil. The “new nodes” approach means that a new set of the internal nodes (with the same coordinates) is introduced for the substituting soil. The author shows that the difference in the results is caused by special feature of the SASSI program: after the outcrop the old internal nodes stay with certain “residual” non-zero stiffness. If the substituting soil
is hard enough, this difference is negligible: “new nodes” approach will not give new results. But in case of soft substituting soil this “residual” stiffness can spoil the results of the “old nodes” approach. In such cases the “new nodes” approach is justified in spite of the increase of the problem size. Combined Asymptotic Method (CAM) is extended for the case of the “new nodes” approach; the capability of CAM is further extended. The topic is important because of the implementation of the upgraded soil pads under the base mats of the heavy structures in seismic regions.
Some Comments on the New Generation of Standards in Earthquake Engineering. Part I: General Requirements and Seismic Input. Part II: Seismic Forces in Linear-spectral Method.
Issue: №5 2019
The author gives his comments on the already started process of the revision of Russian Standards in earthquake engineering. He submits certain recommendations on the process and some recommendations of the general format of the new generation of Standards. Then the author in details discusses the seismic input in the Standards. The goal of the present text is not to give answers, but rather to stress main questions and to list the points where the authors of the future Standard will have to make a certain choice between alternatives. The author describes the alternative solutions for these choices. Some of such issues follow. The multi-level seismic input; the format of the seismic input required from seismologists, the non-exceedance level for seismic input; the location of the control point where seismic input is given. Besides, the issues of the intensity degrees and peak accelerations, the shape of the normalized response spectra, the composition of the wave fild (i.e. non-vertical seismic waves) are also discussed. Finally, the authors comments on the «dangerous directions of the seismic excitation» concept.
The author submits certain recommendations on the description of the linear-spectral analysis. Principal text is accompanied by the Comments. The author suggests the following changes. Matrix equations are introduced, as matrix calculations correspond to the current level of analysis. Mass matrix is populated (and not diagonal, as previously used). Residual mode is introduced as an alternative to the achievement of 95% total mass by the accumulation of modal masses. Seismic excitation is three-component without non-physical “dangerous directions” concept. Normalized spectra, responsibility coeffient, modifid damping coeffient are explained explicitly. Coeffient of non-linear response is applied to certain parts, and not to the whole structure. In the combination of modal responses the inter-correlation of the low-frequency modal responses is explicitly accounted for. If the suggested terms are fulfiled, the linear-spectral analysis should give the results like the time-domain analysis (provided response spectra are calculated from the actual components used in the time domain analysis). At the same time there is a certain smooth transition from the previous codes.
“Dangerous Directions of Seismic Excitation” and Combination of the Close Modal Responses in Linear Spectral Analysis
Issue: №4 2019
This paper continues the discussion on the linear spectral analysis using “dangerous directions of seismic excitation” concept. It is shown that in case all components of seismic excitation have similar response spectra, this “directional” approach, though non-physical, gives similar results to the conventional more physical approach, accounting for the statistical independence of diffrent components. This statement, previously proved for the modal responses with separated frequencies, here is extended for the case of close modal frequencies. If the excitation is diffrent, the “directional” approach leads to the systematic errors. The author once again considers the inertial load vectors in the nodes and studies which of them rotate along with the rotation of the “seismic excitation direction” set up by the analyst, and which of them keep their direction and change only module. Besides, the author notes certain limitation of the implementation of the D’Alambert principle to the quasi-static analysis of the damped systems. It turns out, that this approach, treating inertial loads as quasi-static and neglecting damping in the left-hand parts of the equation, gives reasonable results only for the internal forces, but not displacements, and only for systems either with homogeneous damping, or statically-determined.
«Dangerous Directions of Seismic Excitation» in Linear Spectral Analysis
Issue: №3 2019
This paper opens the discussion on the «dangerous directions of seismic excitation» concept. This concept is popular for linear spectral analysis in our country. The author compares this approach with conventional «multi-component» approach used in the international practice for nuclear structures. It is shown that in case all components of seismic excitation have similar response spectra, both approaches give similar results, in spite of the principal physical difference in excitations (no correlation between components in the «multi-component» approach versus full correlation in the «directional» approach). The reason of this similarity is a math analogy between SRSS (Square Root of Sum of Squares) rule used for (i) the combination of maximums for the statistically independent functions; (ii) combination of vector’s components along orthogonal axes to get absolute value of vector. Similarity of the component response spectra is commonly adopted for two horizontal axes. However, vertical component usually has different spectrum; therefore the «directional» approach in the initial form cannot be applied. On the other hand, if spectral shapes are not very different (e.g. for the widespread case when vertical spectrum is taken as 2/3 of horizontal spectrum), the «directional» approach can be easily modified. One has to envelop normalized spectral shape curves and also scale participation factors for all natural modes in vertical direction.
Operational Basis Earthquake and Design Basis Earthquake: Normal and Abnormal Relations
Issue: №2 2019
The authors study the unexpected effect occurred in the practical design: the in-structure response spectrum for OBE proved to exceed the response spectrum for DBE at certain frequencies. It turns out that the reason is in the «abnormal» relation between Fourier spectra for the OBE and DBE time-histories. Such behavior was the result of the independent syntheses of the OBE and DBE time-histories matching target response spectra. «Normal» behavior of the response spectra does not guarantee «normal» behavior of the amplitude Fourier spectra throughout the whole frequency range, if phase Fourier spectra are independent. The «abnormal» effect under consideration is nonphysical. The authors give some recommendations to avoid it in the future analyses.
Linear Spectral Seismic Analysis of High-rise Building
Issue: №1 2019
Summary: Linear Spectral Method (LSM) by origin is accurate only for single-degree- of-freedom systems. For more complicated systems (and real systems are always more complicated) LSM uses certain double summation rules to combine single-mode responses to one-component excitations. These rules are not precise, they are rather statistical; in different codes they are different. In fact, these rules are based on experience with structures of certain type. That is why the author thinks special check is necessary when one deals with structures having certain peculiarities (like high-rise buildings). This is essentially a check for the applicability of the summation rules. Algorithm for such a check – to compare the results obtained in the time domain for excitation timehistories with LSM results provided input spectra for LSM are obtained from the same time-histories and damping in oscillators is equal to the structural damping. In the present paper seismic responses are compared in the format of the integral response forces under the base mat. The main result is that conventional summation rules in LSM lead to the considerable deviation in the results including underestimation of the response forces.
Seismic Isolation Under the Base Mat of Structure Interacting With Soil. Part I. One‑Dimensional Linear Model
Seismic isolation under the base mat has a special feature: the lower plate of the seismic isolation system (SIS) interacts with soil foundation. These SSI effects are of special importance for heavy structures. In terms of “platform model” there is a “soil support” with certain stiffness and damping installed under the lower base mat. SIS provides another support sequentially linked to the “soil support” via the lower base mat. The combined effect of these two supports depends on the combination of their parameters. In the paper the author uses the simplest one-dimensional model to study the impact of SIS on the seismic structural response considering SSI effects.
Non-Classical Damping in the Soil-Structure System and Applicability of the Linear-Spectral Method to the Calculation of Forces
Previously the author has published several papers with analyses of three-mass system for the kinematical excitation applied at the platform. This is a rough model of a soil-structure system: lower spring and dashpot model soil, the rest of springs dashpots and masses model structure. The author compared direct and modal approaches with the initial and the cut-off damping values. The results were compared in the format of transfer functions and accelerations. In this paper the author compares the results in the format of internal forces, adding linear-spectral method to those mentioned above. It turns out that for narrow-banded excitation in certain frequency range the abnormal damping effect (increasing of response along with increasing of damping) is valid for internal forces, and not only for accelerations. Spectral approach gives the results close to the time-domain approach and does not change the main conclusions. For broad-banded excitations (like seismic one) the effect of damping is “normal”: cut-off damping leads to the increase in resulting forces. Results both for cut-off modal damping and for cut-off dashpot viscosity parameter are conservative as compared to the initial values. Spectral approach gives the results close to the time-domain approach, keeping the overall effect of damping. The main conclusion is that the frequency content of the excitation plays crucial role in the applicability of modal and spectral approaches to the dynamic analysis.
“Normal” and “Abnormal” Effect of Damping on the Dynamic Response of Linear System
Specialists working in earthquake engineering generally are sure that increase in damping causes the decrease in response; e. g. response spectra for increasing damping decrease in the whole frequency range. Hence, underestimation of damping in analysis leads to the conservative results. The author shows that even elementary SDOF oscillators sometimes in certain frequency ranges demonstrate “abnormal” effect of damping on the dynamic response (i. e. response increases along with the increase of damping). For the modes with frequencies in such frequency ranges, the underestimation of modal damping leads to the underestimation of response. This effect is more pronounced for the narrow-banded excitations. Broad-banded excitation mitigates this effect, shifting the frequency range of abnormal damping to the low natural frequencies of oscillators. The abnormal damping effects may be of special importance in two special cases: for seismo-isolated structures with very low natural frequencies, and for high-frequency dynamic excitations (like aircraft impact).
Modal Damping Considering Soil Foundation
The simplest soil-structure platform model is considered in the paper. Soil foundation is modeled by six pairs of springs and viscous dampers. Impact of these soil dampers on the modal damping values is calculated by various methods and compared. First modal damping is calculated by direct approach, adding the contribution of the soil dampers to the modal damping value calculated for stand-alone structure model. Then alternative calculation is carried out using the substitution of the viscous damping by certain material damping — this approach makes it easy to obtain total modal composite damping for the further dynamic analysis. However it is demonstrated that for the first natural modes the alternative approach means the underestimation of modal damping, leading to the non-conservative results.
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