09-10-2014, 04:15 PM
The problem of specifying ground motions for which engineering structures are
to be designed to resist the action of earthquakes has remained a challenging
problem in earthquake engineering research. Important engineering structures,
such as large dams and nuclear power plants, are to be built so as to ensure
probabilities of failure that could be as low as 10-5 to 10-\'. In order that the
engineer is able to predict such low failure probabilities at the design stage,
elaborate and reliable models for the anticipated earthquake loads are needed.
The construction of such models, however, is beset with significant difficulties
that stem because of the following facts:
Lack of adequate engineering models that capture the complex physics involved
in seismic energy built-up in the earth\'s crust and production of ground motions
during earthquakes.
Paucity of adequate database of recorded strong motions for many parts of the
world that constitutes a stumbling block in arriving at acceptable engineering
models for strong ground motions.
Consequently, specifying design ground motions for important structures
becomes essentially an il1-posed problem. In this context, it is of interest to note
that the method of critical excitations has been developed in the existing
literature as a notable counter-point to the more widely employed earthquake
load specifications in terms of smooth design response spectra and power
spectral density function models. The starting point in the method of critical
excitations is the assumption that adequate description of earthquake inputs, that
enable a desired response or reliability analysis to be satisfactory carried out, is
not available, method advocates that the missing information in the earthquake
inputs be determined such that a specified response Variable of a given structure
is maximized. This leads to tailor-made earthquake load models that are least
favorable to a given structure and at the same time they possess all the reliably
known features of a future earthquake load. The development of critical
earthquake excitation models, thus, depends upon the following influencing
factors:
Framework adopted for response analysis: deterministic or probabilistic.
Dynamic characteristics of engineering structure including the effects that might
arise due to the possibility of nonlinear mechanical behavior of the structure.
Nature of partial information available on the earthquake inputs.
The response variable chosen for maximization. In a deterministic analysis this
could be the highest value of displacement, stress or stress resultant at a specified
location in the structure over a given time period. In stochastic analysis, the
response could be described either in terms of second order statistics or in terms
of probability of violation of specified limit states.
Inclusion of spatial variability and multi-component nature of earthquake loads.
Possibility of earthquake loads appearing as parametric excitations in the
governing equations of motion.
Research into development of critical earthquake excitations is now more than
three decades old. A majority of these studies have focused on the analysis of
single point seismic excitations for linear structures. Both deterministic and
probabilistic frameworks have been employed in these studies. Notwithstanding
these developments, currently there exists a need to develop critical excitation
models that allow for spatial variability and multi- component nature of
earthquake loads, possibility of earthquake loads driving the structure
parametrically, nonlinear behavior of the structure and response characterization
in terms of reliability measures. The present thesis, constitutes an attempt
towards addressing some of these research needs. The thesis is organized into
seven chapters and two annexure.
A review of literature covering existing random process models and critical
earthquake excitation models is provided in chapter 1. The review begins by
summarizing the salient features of random process models for earthquake
ground motions encompassing issues pertaining to transient nature, spatial
variability and multi-component nature of earthquake ground motions. The
review of papers on critical earthquake load modeling covers, deterministic /
stochastic models linear / nonlinear structural behavior and spatial variability of
ground motions. It is observed from this review that literature on deterministic.
The nonlinear optimization problem in both these approaches is solved by using
the sequential quadratic programming (SQP) method. The procedures
developed are illustrated by considering seismic response of a tall chimney and
an earth dam. It is concluded that the have lower and upper bounds on Fourier
coefficients in the first approach and constraints on amount of disorder in the
second approach are crucial in arriving at realistic critical excitations.
The methods developed in chapter 2 are extended to determine vector critical
earthquake excitation models for multi-supported linear structures in chapter 3.
The formulation of structural equations of motion is carried out using the finite
element method. The pseudo-static and dynamic components of the response are
analyzed separately. Accordingly, response variables for maximization are
chosen to reflect the decomposition of the total response into pseudo-static and
dynamic components. The extension of deterministic procedures outlined in
chapter 2 to multi-supported structures is fairly straightforward. Here, each
component of support motion is expressed in terms of a modulated Fourier
series with undetermined coefficients. Constraints on the input are ex w pressed
in terms of these coefficients and are taken to reflect, as before, peak values of
ground accelerations, velocities and displacements, total energy and upper and
lower bound on Fourier amplitude spectra determining stochastic critical
excitation models, the input is taken to be a vector of partially specified nonstationary
Gaussian random processes. The critical psd matrix models that
produce highest response variance are developed taking into account constraints
on total energy, zero crossing rate and entropy rate. The numerical solutions to
the nonlinear optimization problem is again obtained using the SQP method as
has been done in chapter 2
The probabilistic critical earthquake load models in chapters 2 and 3 essentially
the linear input-output relations for second order response statistics that are
readily available in the standard random vibration literature. In the aseismic
analysis of engineering structures, however, it is also of interest to characterize
structural responses in terms of reliability measures. For linear structures under
Gaussian inputs, theories based on extreme value statistics of Gaussian random
processes can be employed for this purpose. Similar theories, however, are not
available for nonlinear structures and for structures that are parametrically
excited by earthquake loads. Studies reported in chapters 4 to 6 propose the use
of time-variant reliability methods in arriving at critical earthquake excitation
models. Accordingly, chapter 4 reports on deriving random critical excitation
models for linear structures under single point Gaussian inputs. Here, the
critical excitation is taken in the critical psd matrix models as well as in the
excitation at the check point and the sensitivity vector. Chapter 6 describes the
extension of reliability index based critical excitation models developed in the
previous chapter to problems involving nonlinear structural behavior. This
study covers both singlv supported and multi-supported nonlinear structures,
but, however, attention is limited to single-degree-of-freedom (sdof) systems
with cubic nonlinear force-displacement relations. As before, critical excitations
are defined as those that minimize probability of violation of limit state defined
on a specified internal force in the structure. This probability is again
characterized indirectly in terms of the Hasofer-Lind reliability index that is
derived from a quadratic response surface that fits the implicitly defined failure
surface at the design point. The results on critical psd functions show a
dominant peak at the linear structural frequency and also an additional peak at
three times the linear structure natural frequency. This latter peak being
attributed to the presence of structural non-linearity. For the case of the doubly
nonlinear sdof system that is subjected to differential support motions, the
governing equation of motion is shown to contain non-Gaussian parametric and
external excitations. The results on critical excitations for this case are shown to
reflect the influence of both parametric and nonlinear resonances. Issues related
to treating structural capacities as being random are also examined in this
chapter. Chapter 7 is the concluding chapter of the thesis in which a brief resume
of new improvements to existing critical load modeling as well as novel elements
of the thesis are given along with a set of suggestions for future research.
A few computational details of reliability indices and response surface modeling
are provided in annexure A and B. It is hoped that these annexure improve the
readability of the thesis.
Based on the work carried out in this thesis, the following papers, with Prof. C. S.
Manohar as a co-author, have been prepared.
1. A comparative study on deterministic and stochastic critical earthquake load
models, Proceedings of the International Conference the Civil Engineering, ICCE
2001, Volume II, Interline publishing, Bangalore, India, 173-180.
2. Investigation into critical earthquake load models within deterministic and
probabilistic frameworks. Earthquake Engineering and Structural Dynamics.
2002. 31. 813-832.
3. Critical spatially varying earthquake load models for extended structures.
Provisionally accepted, Journal of Structural Engineering SERC Madras.
4. Reliability-based critical earthquake load models for linear structures, to be
submitted, journal of Sound and Vibration.
5. Reliability-based critical earthquake load models for nonlinear structures, to be
submitted, ASCE, Journal of Engineering Mechanics.
6. Deterministic/Probabilistic critical earthquake load models for parametrically
excited structures, to be submitted, Probabilistic Engineering Mechanics