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Journal ArticleDOI

A critical look into Rayleigh damping forces for seismic performance assessment of inelastic structures

01 Nov 2014-Engineering Structures (Elsevier)-Vol. 78, pp 28-40

AbstractRayleigh damping forces are commonly introduced in the numerical simulations of nonlinear structures run to assess structural performance in case of an earthquake. Their purpose is to account for energy dissipative mechanisms not otherwise explicitly represented in the model. When caused by interactions between the structure and its surrounding environment, energy dissipation is external to the structure, whereas it is internal when resulting from energy absorption mechanisms activated in the structure. In this paper, the concept of discrepancy forces is introduced in the framework of computational dynamics. Then, damping forces are presented as a model of these so-called discrepancy forces to represent internal energy dissipation. On the other hand, the discrepancy forces are identified from a set of experimental data recorded during shaking-table test of a ductile moment-resisting frame, which provides the rationale for a critical look into Rayleigh damping forces. It is in particular observed that, for the structure tested, the Rayleigh damping model used is inaccurate as a representation of the discrepancy forces. Besides, while the knowledge of the discrepancy forces allows for rationally discussing the capabilities of the inelastic structural model to represent the actual behavior of the structure, this is only possible to a limited extent with the Rayleigh damping model used.

Topics: Dissipation (50%)

Summary (2 min read)

1. Introduction

  • Whether they are pertaining to the ground motion signal or to the structural response, uncertainties are numerous and can dramatically impact the conclusions of seismic risk analyses.
  • Then, two inelastic structural models of the tested momentresisting reinforced concrete frame are presented.
  • The discrepancy forces are calculated for both structural models in sections 5 and 6.

2. Damping forces revisited – Discrepancy forces

  • The authors also assume that displacement , velocity , and accelerationproportional forces contribute to the structural response (left-hand side of the equation): (2) Mü(tn) +C(tn)u̇(tn) + F hys(u; tn) = F ext(tn) where M and C are the mass and damping matrices, Fhys is the structural hysteretic restoring force vector, and Fext is the external loading vector.
  • The external forces along with the displacements y(t) and accelerations ÿ(t) recorded during shaking table test are imposed to the system, which yields R̃ = 0 because y(t) and ÿ(t) are the “true” displacements and accelerations; then Fdis can be directly computed with equation (10), also known as In other words.
  • Whereas, on the other hand, the direct problem R̃(u, ü) = 0 is solved dynamically.
  • G being a valid model does not imply the plastic deformations in steel rebars are accurately simulated.

3. Shaking table tests

  • Experimental data recorded during the shaking-table test of a ductile moment-resisting reinforced concrete frame is used [7].
  • These latter additional masses induced service cracks.
  • Mode 1 is preponderant in the sense that it accounts for more than 90% of the total mass.
  • The structure is subjected to two types of loading: vertical static loading due to the dead load of the frame along with the additional masses, and horizontal dynamic forces induced by the seismic acceleration time history imposed on its base.

4. Two inelastic structural models: H1 and H2

  • All the numerical simulations are performed with the finite element computer program FEAP [22] where the various elements and material behavior laws have been implemented.
  • For model H2, the additional nodes are considered as massless.
  • Uniaxial material behavior law is then assigned to every concrete layer and steel fiber.
  • All the material models considered in this work have been developed using some of the ingredients of the more general model presented in [12].
  • As illustrated in figure 6, beam-column joint elements momentrotation response is modeled by upper and lower bars.

5. Model G1 = H1⋆D1

  • Only an approximation of the discrepancy forces can be provided because experimental data are missing to calculate the complete discrepancy forces vector.
  • The authors use this information to check whether model H1 accurately represents structural mass and stiffness distribution in the initial state before seismic loading.
  • To account for the contribution of static loading to the discrepancy forces, these displacements are approximated performing quasi-static numerical analysis and storing the displacements pertaining to the N e DOFs monitored during shaking table test, hereafter referred to as usta,e.
  • Discrepancy and hysteretic forces looks like they are symmetric with respect to the x-axis.
  • Considering the accelerations time histories, the numerical response is in good accordance with the experimental data for the 2nd level, especially between 8 and 20 s, as for the displacements.

6. Model G2 = H2⋆D2

  • The joint elements are parameterized by the elastic modulus Ej and the limit stress σj (post-yielding slope is set to less than 0.1Ej).
  • Their purpose here is only to develop a structural modelH2 that is different from previous structural model H1, and not to rigorously identify the beam-tocolumn joint parameters that would lead to the best representation of the frame.
  • This would require much more advanced identification procedures that are out of the scope of this work.
  • Displacements at both levels are fairly good simulated, while larger errors are observed for the accelerations.
  • It only means that the discrepancy forces for model G2 are not as accurately computed as for model G1.

8. Conclusions

  • The general concept of discrepancy forces has been introduced in the framework of computational dynamics.
  • Better knowledge of how additional damping forces should be computed thus is highly desirable.
  • The purpose of this proposed shift of perspective is to provide a critical look into Rayleigh damping forces.
  • While discrepancy forces provide insight into the capability of the structural model for representing the actual structural behavior, the modeled Rayleigh damping forces only shed light on the structural model to a more limited extent; .
  • As a final remark, this work illustrates how experimental data can be effectively used to identify the forces that are usually expected to be represented by Rayleigh damping in seismic structural analyses.

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A CRITICAL LOOK INTO RAYLEIGH DAMPING FORCES FOR
SEISMIC PERFORMANCE ASSESSMENT OF INELASTIC
STRUCTURE S
Pierre Jehel
12
pierre.jehel[at]ecp.fr
August 2
nd
, 2014: Paper accepted for publication in Engineering Structures
Abstract. Rayleigh damping forces are commonly introduced in the numerical
simulations of nonlinear structures run to a ssess structural performance in case
of an earthquake. Their purpose is to account for ene rgy dissipative mechanisms
not otherwise explicitly represented in the model. When caused by interactions
betwee n the structure and its surrounding environment, energy dissipation is
external to the structure, whereas it is internal when resulting from energy ab-
sorption mechanisms activated in the structure. In this paper, the concept of
discrepancy forces is introduced in the framework of computational dynamics.
Then, damping forces are presented as a mode l of these so-called discrepancy
forces to represent internal ener gy dissipation. On the other hand, the dis -
crepancy forces are identified from a set of experimenta l data recorded during
shaking-table test of a ductile moment-resisting frame, which provides the ratio-
nale for a critical look into Rayleigh damping forces. It is in particular observed
that, for the structure tested, the Rayleigh damping model used is ina ccurate as
a representation of the discrepancy forces. Besides, while the knowledge of the
discrepancy forc es allows for rationally discussing the capabilities of the inela stic
structural model to represent the actual behavior of the structure, this is only
possible to a limited extent with the Rayleigh damping model used.
keywords: seismic performance, damping fo r ces, no nlinear structure, compu-
tational mechanics, experimental test design, earthquake engineering.
1. Introduction
1.1. Seismic structural performance assessment. Seismic p erf ormance as-
sessment of inelastic structures is a key step in a seismic risk management process
that aims at mitig ating t he risks for the populations and the infrastructures in
seismically active regions. Seismic risk is a combination of seismic hazard and
1
Laboratoire MSSMat / CNRS-UMR 8579,
´
Ecole Centrale Paris, Grande voie des Vignes,
92295 Chˆatenay-Malabry Cedex, France
2
Department of Civil Engineering and Engineering Mechanics, Columbia University, 630 SW
Mudd, 500 Wes t 120th Stree t, New York, NY, 10027, USA
1

2 A CRITICAL LOOK INTO RAYLEIGH DAMPING FORCES
structural vulnerability and can be effectively formalized and communicated in a
probabilistic setting (see e.g. [6]):
(1) P
PL
=
Z
P [EDP EDP
PL
| IM = x] · P [IM = x] dx
In t his equation, PL is the performance level associated to a certain value of a n
engineering demand parameter (EDP) of interest. For instance, in the case of
moment-resisting frame structures, maximum interstory drift is often used as the
EDP of interest (see [6, 16, 5] among others) because it is possible to map it to
meaningful PL such as “immediate occupancy”, “structural damage” or “collapse
prevention” [
6]. In equation (1), IM refers t o the intensity measure of the seismic
ground motion, and the probability to observe an earthquake with an IM equal to
x in the region of interest, that is P [IM = x], is given by seismic hazard maps.
This paper focusses on the role of Rayleigh damping forces in the vulnerability
assessment of nonlinear structures, that is on t he computation of the conditional
probability that a structural performance criteria is exceeded given a certain IM of
the ground motion, when the EDPs are computed through time-history analyses.
1.2. Uncertainties in the performance assessment. Whether t hey are per-
taining to the ground motion signal or to the structural response, uncertain-
ties are numerous and can dramatically impact the conclusions of seismic risk
analyses. To identify which of the potential uncertainty sources have to b e ac-
counted for in the communication of risk analyses results, a series of sensitivity
analyses has been conducted in the structural earthquake engineering community
(e.g. [
16, 5, 18, 15, 24, 2, 23] among others). In particular, the studies in [18, 15, 2]
explicitly account for Rayleigh damping as a potential contributor to t he uncer-
tainty in the EDP of interest.
The sources of uncertainty that most strongly affect the repair cost in an earth-
quake have been sought in [
18] using sensitivity analyses. A high-rise reinforced
concrete nonductile moment-resisting frame is studied. Structural Rayleigh damp-
ing ratio is found to be a minor source of uncertainty in the adopted structural
performance measure with respect to the capacity of the structural elements to
damage and the seismic ground motion intensity.
In [
15], the authors use FOSM method to investigate the sensitivity of a series of
EDPs to uncert ain parameters among which Rayleigh damping ratio. The building
studied is a seven-story reinforced concrete shear-wall structure. It is concluded
that, for t he local EDPs considered (curvature in critical sections), viscous damping
is the second most significant source of uncertainty after the intensity o f the ground
motion.
In [
2], a sensitivity analysis of the maximum interstory drift to inelastic frame
element properties, beam-column joint properties as well a s structural viscous
damping ratio is performed for a reinforced concrete frame structure at various
seismic ha zar d levels. Although the uncertainty in the ground mot ion dominates

A CRITICAL LOOK INTO RAYLEIGH DAMPING FORCES 3
the overall uncertainty in the interstory drift, Rayleigh damping ratio is found to
be one of the most significant other contributors to the ED P of interest.
1.3. Objective and scope of the paper. On the one hand, it has been ob-
served that Rayleigh damping can be a significant contributor to t he overall un-
certainty in the EDPs of interest for seismic performance assessment of inelastic
structures [15, 2]. On the other hand, it has been shown that using Rayleigh
damping forces a long with an inelastic structural model can be problematic and
lead to unintended consequences tha t can compromise the validity of the analyses
outputs [10, 3]. Therefo r e, the objective of this paper is to provide a rational
discussion on the validity of Rayleigh damping forces in the time history analyses
of inelastic structures and to shed light on a potential strategy to model realis-
tic damping forces in inelastic simulations along with improving the predictive
capabilities of the structural models.
Rayleigh damping can be used in seismic simulations either to account for energy
dissipation mechanisms that are external to the structure or for energy absorption
mechanisms that are internal to the structure. This work focusses on internal
energy absorption only. Besides, in case internal energy absorption has t o be mod-
eled, we ado pt the viewpoint of Rayleigh damping forces being added to complete
the seismic energy absorption capacity of the inelastic structural model. In other
words, Rayleigh damping forces are not considered in this pa per as intrinsic to
the structural respo nse but as some ad hoc correction of deficiencies of the inelas-
tic structural model to accurately represent the actual structural response to the
seismic action.
The approach adopted in this work is fundamentally different from what is de-
veloped in studies focused on structural system identification (see e.g. [
9, 20, 21]).
In t hese la t ter studies, the structure is considered as a system that modifies the
seismic ground mot ion (input signal) into the dat a (e.g. displacements) recorded
at monitored points (output signals). In such analyses, t here is no explicit inelastic
structural model used to simulate the structural response: the structural system is
represented by linear differential equations characterized by modal damping ratios
and frequencies that can be identified in the process. Hereafter however, an inelas-
tic structural model is constructed to approximate the response of t he structure,
and the damping forces time history is identified.
1.4. Outline of the paper . The outline of the paper is a s follows. In t he next
section, basic equations of nonlinear dynamics are first recalled a s a baseline for
introducing the concept of discrepancy forces in the framework of computational
nonlinear dynamics. In particular, the need for experimental data to calculate
these discrepancy fo rces is pointed out. Section 3 is devoted to a short description
of the shaking table tests during which the experimental data that are used there-
after were recorded. Then, two inelastic structural models of t he tested moment-
resisting r einforced concrete frame are presented. They are developed using fiber

4 A CRITICAL LOOK INTO RAYLEIGH DAMPING FORCES
frame elements and simple inelastic beam-to-column connections. The discrepancy
forces are calculated for both structural models in sections 5 and 6. How discrep-
ancy forces can be used to improve structural models is in particular discussed
and used to parameterize the improved structural model used in section 6. Before
closing the paper with some conclusions, section 7 presents a critical discussion on
Rayleigh damping forces based on the rationale provided by the knowledge of the
discrepancy forces.
2. Damping forces revisited Discrepancy forces
2.1. Classical computational nonlinear dynamics. We assume that the dy-
namic nonlinear structural pro blem is cast in a standar d finite element form. We
also a ssume that displacement (hysteresis), velocity (viscosity), and acceleration-
proportional (inertia) forces contribute to the structural response (left-hand side
of the equation):
(2) M
¨
u(t
n
) + C(t
n
)
˙
u(t
n
) + F
hys
(u; t
n
) = F
ext
(t
n
)
where M and C are the mass and damping matrices, F
hys
is the structural hys-
teretic restoring force vector, and F
ext
is the external loading vector . t
n
T with
T = {n × t | n [0, 1, .., N], t = T /N > 0} is a discrete process. F
ext
typ-
ically consists of the static loadings (dead and service loads), the forces induced
by the seismic ground motion, and the reactions at t he connections between t he
structure and its environment. Also, if some energy dissipation sources that are
external to the structure are present in the system (structure equipped with energy
dissipation device that has known physical properties), they ar e considered here
to act as external loading, so that the viscous and hysteretic fo r ces only account
for mechanisms that are internal to the structure.
Because the structural resp onse is possibly nonlinear, we rewrite equation (
2)
as a residual vector R that has to be iteratively set to zero:
(3) R (u,
˙
u,
¨
u; t
n
) = 0
At iteration k, with the subscript n referring to t
n
, the Newton-Raphson updating
residual reads
(4) R
(k+1)
n
= R
(k)
n
+
dR
du
(k)
n
du
(k)
n
= 0
where:
(5) R
(k)
n
= F
ext
n
M
¨
u
(k)
n
C
n
˙
u
(k)
n
F
hys
(u
(k)
n
)
and the total tangent matrix
S
(k)
n
=
dR
du
(k)
n
=
R
u
(k)
n
R
˙
u
(k)
n
d
˙
u
du
R
¨
u
(k)
n
d
¨
u
du
= K
(k)
n
+ c
C
C
(k)
n
+ c
M
M(6)

A CRITICAL LOOK INTO RAYLEIGH DAMPING FORCES 5
where K = dF
hys
/du is the structural tangent stiffness matrix, c
C
= d
˙
u/du
and c
M
= d
¨
u/du are coefficients dependent on both the time step t and the
parameters of any one-step time integration algor it hm.
2.2. Damping forces revisited. The presence of velocity-proportional forces in
equation (2) does not result fro m continuum mechanics principles but from the
introduction of viscous forces at a structural level. These latter forces are re-
ferred to as damping forces because they are designed for representing damping
features resulting from some viscous mechanisms activated during the dynamic
response of the structure. In earthquake engineering, there is few evidence that
actual structural damping results from viscous phenomena and several researchers
have claimed that the energy dissipative phenomena that contribute to the over-
all structural damping should rat her be explicitly accounted for in the hysteretic
response F
hys
(t) (e.g. [
3]).
A shift of perspective is proposed here: Instead of adding viscous forces in the
equations of equilibrium, discrepancy forces F
dis
(t) are added. Discrepancy f orces
not necessarily are proportional to the velocity. They have to be understood as
completing the hysteresis and inertia forces in the balance equation so that the
model accurately predicts the displacements and accelerations that would be exper-
imentally observed. Then, denoting y and
¨
y the displacements and accelerations
that would actually b e observed during experimental tests, the residual vector and
total tangent stiffness matrix, compare with equations (
5) and (6), now read:
0 :=
˜
R
n
= F
ext
n
M
¨
y
n
F
dis
n
F
hys
(y
n
)(7)
˜
S
n
= K
n
+ c
M
M(8)
2.3. Discrepancy forces. The comparison between experiment a nd simulation
observations can generically be formulated as:
(9) q(t) + η(t) = G(t) + ǫ(t)
On the left-hand side is the experimental observation of the “truth” q (to reuse the
term employed in [
19]), polluted by some necessary observation errors η. On the
right-ha nd side is the output of the numerical model G along with the simulation
errors gathered in ǫ. Hence, ǫ accounts for the effects of a possible lack of physics in
G as well as for any other reason why the output of G does not fit the experimental
observations, such as numerical errors due to digitized space and time integration
procedures for instance.
Here, we focus on acceleration and displacement time histories as quantities
of interest. Accelerations provide insight in the resisting forces developed in the
structure; displacements are, as mentioned in the introduction, often used as in-
dicators of the structural respo nse t hat can be mapped to p erformance levels.
Besides, we do not explicitly account for any measurement error. Consequently,
the right-hand side in equation (
9) takes t he form q + η = {y
¨
y}
T
. On a nother
hand, we set G = H D, where denotes the dual actio ns of model H (inertia

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Abstract: This paper is intended to evaluate the seismic performance of a twelve-story reinforced concrete moment-resisting frame structure with shear walls using 3D finite element models according to such seismic design regulations as Federal Emergency Management Agency (FEMA) guideline and seismic building codes including Los Angeles Tall Building Structural Design Council (LATBSDC) code. The structure is located in Seismic Zone 4, considered the highest-seismic-risk classification established by the U.S. Geological Survey. 3D finite element model was created in commercially available finite element software. As part of the seismic performance evaluation, two standard approaches for the structure seismic analysis were used; response spectrum analysis and nonlinear time-history analysis. Both approaches were used to compute inter-story drift ratios of the structure. Seismic fragility curves for each floor of the structure were generated using the ratios from the time history analysis with the FEMA guideline so as to evaluate their seismic vulnerability. The ratios from both approaches were compared to FEMA and LATBSDC limits. The findings revealed that the floor-level fragility mostly decreased for all the FEMA performance levels with an increase in height and the ratios from both approaches mostly satisfied the codified limits.

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Abstract: When performing a nonlinear time-history analysis of a reinforced concrete structure, it is necessary for the used structural model to dissipate the correct amount of energy. For the sake of computational efficiency, viscous damping models are still commonly used to account, partially or not, for non-viscous dissipations (e.g. friction between the crack surfaces, bond slip at the steel-concrete interface). In order to improve the physical relevance of such a substitution, an evolving equivalent viscous damping ratio estimated for a simply supported reinforced concrete beam is proposed in this paper. This work takes place in the scope of a moderate seismicity context for which steel yielding is not expected. The results are not directly identified from experimental results but rather from numerical simulations carried out thanks to an equivalent single-degree-of-freedom model, itself calibrated by means of quasi-static experiments. To begin with, the experimental setup used to calibrate the single-degree-of-freedom model and the equivalent viscous damping ratio assessment method are presented. Then, the single-degree-of-freedom model and the identification procedure are exposed. The resulting outputs are presented and commented. Finally, numerical experiments are performed in order to obtain equivalent viscous damping ratio values corresponding to a given maximum time-history curvature and a curvature demand.

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Abstract: This paper presents an Enhanced Rayleigh damping model for dynamic analysis of inelastic structures. The conventional Rayleigh damping model has been extensively used to represent inherent ...

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  • ...Terming from the age when the seismic analyses were first developed for elastic structures, considering its computational efficiency and its convenient implementation, adopting a viscous damping model is, most of time, reasonable.(13) However, the nonlinear time-history analysis, which is based on the nonlinear constitutive model, is the most appropriate way to account for the hysteretic behaviors of structures....

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TL;DR: The IDEFIX experimental campaign (French acronym for Identification of damping/dissipations in RC structural elements ) has been carried out on RC beams set up on the Azalee shaking table of the TAMARIS experimental facility operated by the French Alternative Energies and Atomic Energy Commission (CEA).
Abstract: Despite their now well documented drawbacks, viscous damping based models to describe the dissipations occurring in reinforced concrete (RC) structures during seismic events are popular among structural engineers. Their computational efficiency and their convenient implementation and identification are indeed attractive. Of course, the choice of a viscous damping model is, most of the time, reasonable, but some questions still arise when it comes to calibrate its parameters: how do these parameters evolve through the nonlinear time-history analysis? How do they interact when several eigenmodes are involved? To address these questions, the IDEFIX experimental campaign (French acronym for Identification of damping/dissipations in RC structural elements) has been carried out on RC beams set up on the Azalee shaking table of the TAMARIS experimental facility operated by the French Alternative Energies and Atomic Energy Commission (CEA). First, this experimental campaign is positioned within an overview of related experimental campaigns in the literature. Second, the IDEFIX experimental campaign is presented. In particular, noticeable results are given by examples of first post-treatments, including an improved so-called “areas method”, which lead to very different identified damping ratio depending on the method used.

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Abstract: A new family of unconditionally stable one-step methods for the direct integration of the equations of structural dynamics is introduced and is shown to possess improved algorithmic damping properties which can be continuously controlled. The new methods are compared with members of the Newmark family, and the Houbolt and Wilson methods.

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"A critical look into Rayleigh dampi..." refers methods in this paper

  • ...Two inelastic structural models: H1 and H2 All the numerical simulations are performed with the finite element computer program FEAP [22] where the various elements and material behavior laws have been implemented....

    [...]

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Abstract: The primary goal of seismic provisions in building codes is to protect life safety through the prevention of structural collapse. To evaluate the extent to which current and past building code provisions meet this objective, the authors have conducted detailed assessments of collapse risk of reinforced-concrete moment frame buildings, including both ‘ductile’ frames that conform to current building code requirements, and ‘non-ductile’ frames that are designed according to out-dated (pre-1975) building codes. Many aspects of the assessment process can have a significant impact on the evaluated collapse performance; this study focuses on methods of representing modeling parameter uncertainties in the collapse assessment process. Uncertainties in structural component strength, stiffness, deformation capacity, and cyclic deterioration are considered for non-ductile and ductile frame structures of varying heights. To practically incorporate these uncertainties in the face of the computationally intensive nonlinear response analyses needed to simulate collapse, the modeling uncertainties are assessed through a response surface, which describes the median collapse capacity as a function of the model random variables. The response surface is then used in conjunction with Monte Carlo methods to quantify the effect of these modeling uncertainties on the calculated collapse fragilities. Comparisons of the response surface based approach and a simpler approach, namely the first-order second-moment (FOSM) method, indicate that FOSM can lead to inaccurate results in some cases, particularly when the modeling uncertainties cause a shift in the prediction of the median collapse point. An alternate simplified procedure is proposed that combines aspects of the response surface and FOSM methods, providing an efficient yet accurate technique to characterize model uncertainties, accounting for the shift in median response. The methodology for incorporating uncertainties is presented here with emphasis on the collapse limit state, but is also appropriate for examining the effects of modeling uncertainties on other structural limit states.

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"A critical look into Rayleigh dampi..." refers methods in this paper

  • ...For instance, in the case of moment-resisting frame structures, maximum interstory drift is often used as the EDP of interest (see [6, 16, 5] among others) because it is possible to map it to meaningful PL such as “immediate occupancy”, “structural damage” or “collapse prevention” [6]....

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Abstract: The investigation reported in this paper looks into the application of a number of system-identification techniques to problems of earthquake engineering. A number of techniques for structural-system identification have been developed over the past few years. Many of these techniques have been successful at identifying properties of linearized and time-invariant equivalent structural systems. Most of these techniques were verified using mathematical models simulated on the computer. In this paper, a number of structural-identification algorithms are reviewed and applied to the identification of structural systems subjected to earthquake excitations. The algorithms are applied to experimental data obtained in controlled laboratory conditions. The data pertain to the acceleration records from two building models subjected to various loading conditions. The performance of the various identification algorithms is critically assessed, and guidelines are obtained regarding their suitability to various engineeri...

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Journal ArticleDOI
Abstract: Rayleigh damping is commonly used to provide a source of energy dissipation in analyses of structures responding to dynamic loads such as earthquake ground motions In a finite element model, the Rayleigh damping matrix consists of a mass-proportional part and a stiffness-proportional part; the latter typically uses the initial linear stiffness matrix of the structure Under certain conditions, for example, a non-linear analysis with softening non-linearity, the damping forces generated by such a matrix can become unrealistically large compared to the restoring forces, resulting in an analysis being unconservative Potential problems are demonstrated in this paper through a series of examples A remedy to these problems is proposed in which bounds are imposed on the damping forces Copyright © 2005 John Wiley & Sons, Ltd

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"A critical look into Rayleigh dampi..." refers background in this paper

  • ...Although there presently is a consensus in the earthquake engineering community that using Rayleigh damping forces is far from reaching this ultimate goal [3, 10], common implementation of inelastic time history seismic analyses adds Rayleigh damping forces to the hysteretic structural forces....

    [...]

  • ...as a requirement for a good modeling of damping in other works [10, 3]....

    [...]

  • ...On the other hand, it has been shown that using Rayleigh damping forces along with an inelastic structural model can be problematic and lead to unintended consequences that can compromise the validity of the analyses outputs [10, 3]....

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