10-10-2014, 02:03 PM
Earthquakes constitute an important source of dynamic action on engineering
structures. Considerable progress has been achieved over the last few decades
in the aseismic design of engineering structures using time history, response
spectra and stochastic process models for earthquake loads. More recently,
stochastic field models which allow for spatial variability in earthquake loads
on extended structures have also been developed. Seismic risk analysis
procedures to assess the safety of important, structures are also available.
Notwithstanding this progress, it is important to note that, in specifying the
earthqu4ke,.lq4!k for the design of important structures, one has to reckon with
three conflicting requirements:
• Scarcity of recorded earthquake accelerograms,
• A high level of uncertainty associated with mechanisms producing
ground motions at a given location and
• High level of confidence with which the engineering structures have to be
designed to withstand earthquake loads.
This would mean that, aseismic design of engineering structures is an ill-posed
problem. The difficulties are further accentuated when one has to deal with
multi-support and multi-component support motions which demands the
description of loads at a much finer level for which the available data is all the
more scarce. Thus, the robustness and accuracy of the available earthquake
load models are open to question. In such a situation, it is natural to ask what
is the worst which n-Light happen to a given structure under the action of an
incompletely specified earthquake excitation. The method of critical excitation
has been developed to answer this type of questions. This method is based on a
realistic premise that earthquake inputs can only be partially specified with
confidence and the emissing information in the input, which is essential for a
complete response analysis, is that damage to a given structure is maximized.
The term damage here denotes any unfavorable behavior of the structure.
Critical excitations are, thus, by definition, dependent on the system, nature of
partial information available and the damage variable chosen for maximization.
The present thesis contributes to the state of the art in the development. of
method of critical excitations. The attention is limited to the linear system
behavior. The thesis is divided into five chapters and an appendix. Firstly, the
study of uniform support excitation is considered and subsequently, the analysis
is extended to cover multi-support / multi-component seismic excitation
problems. The layout of the thesis is as follows:
A review of literature on the method of critical excitations has been presented in
Chapter 1. The analysis procedures and their applications reported in the
literature have been surveyed and the need for investigation to develop the
method further is discussed. A systematic I study of some of the open questions
in this area of research is brought out in the subsequent chapters.
In Chapter 2, the study of spatially uniform critical excitation is carried out. The
earthquake ground motion is modeled as a non-stationary random process
obtained as a product of a prescribed deterministic envelope and a stationary
Gaussian random process having an unknown power spectral density. The
excitation is taken to satisfy constraints on the total average power and zerocrossing
rate. In this case, the critical excitation is defined as the excitation which
maximizes the response variance of a given linear system under the constraints
of total average power and zero-crossing rate. The unknown optimal power
spectral density of the stationary part of the excitation is obtained using linear
programming methods. The resulting solutions are shown to display a highly
deterministic structure and, therefore, fail to capture the stochastic nature of the
input. A modification to the definition of critical excitation is proposed which
takes into account the entropy rate as a measure of uncertainty in the
earthquake loads. The resulting problem is solved using calculus of variations
and also within a multiple objective linear programming framework. Illustrative
examples on specifying seismic inputs for a nuclear power plant and a tall earth
dam are considered and the resulting solutions are shown to be realistic.
The highest response of multi-supported structures subjected to partially
specified multi-component earthquake support motions is considered in Chapter
3. The seismic inputs are modeled as incompletely specified vector Gaussian
random processes with known auto spectral density functions but unknown
cross spectral densities. This type of situations are easily conceivable when the
seismic inputs are specified through a set of response spectra which, by
definition, do not consider the effect of cross-spectral densities between different
excitations. These unknown cross-spectral density functions are determined
such that the steady state response variance of a given linear system is
maximized. The resulting coherence functions are shown to be dependent on the
system properties, auto spectra of excitation and the response variable chosen
for maximization. It emerges that the highest system response is associated
neither with fully correlated support motions, nor with independent motions,
but, instead, specific forms of coherence functions are shown to exist which
produce bounds on the response of a given structure. Application of the
proposed results is demonstrated by examples on a ground based extended
structure, namely, a 1578 m long, three span, suspension cable bridge and a
secondary system, namely, an idealized piping structure of a nuclear power
plant.
The problem of multi-support critical seismic excitations when the prior
knowledge on the inputs is further restricted to only the first two spectral
moments is considered in Chapter 4. A method for determining critical power
spectral density matrix models for earthquake excitations which maximize
steady state response variance of multiply supported extended structures and
which also satisfy constraints on input variance, zero crossing rates, frequency
content and transmission time lag has been developed. The optimization
problem for this case is shown to be nonlinear in nature and solutions are
obtained using an iterative technique, which is based on linear programming
method. A constraint, on entropy rate as a measure of uncertainty which can be
expected in realistic earthquake ground motions is proposed which makes the
critical excitations more realistic. Illustrative examples on critical inputs and
responses of single degree of freedom systems and a long span suspended cable
which demonstrate the various features of the approach developed are
presented.
Chapter 5 presents the conclusions emerging from the above studies and makes a
few suggestions for further research.
In the Appendix, a newly developed computational scheme for determining the
dynamic stiffness coefficients of a linear, inclined, translating and
viscously/hysteretically damped cable element is outlined which takes into
account. the coupling between in plane transverse and longitudinal forms of
cable vibration. The numerical examples on cable systems considered in Chapter
3 and 4 are based on the algorithm outlined in this appendix. This
computational scheme is based on conversion of the governing set of quasistatic
boundary value problems into a larger equivalent set of initial value problems
which are subsequently integrated numerically in spatial domain using
marching algorithms. Numerical results which bring out the nature of the
dynamic stiffness coefficients are also presented.
Based on the work described above the following papers (with Dr C. S. Manohar
as co-author) have been submitted for publication:
• Critical earthquake input power spectral density function models for
engineering structures (accepted for publication, scheduled to appear in
October 1995 in Earthquake Engineering and Structural Dynamics, a copy
of the paper is appended to the thesis).
• Critical coherence functions and the highest response of multi-supported
structures subjected to multi-component earthquake excitations (under
review, Earthquake Engineering and Structural Dynamics)
• Critical seismic vector random excitations for multiply supported
structures (under review, Probabilistic Engineering Mechanics)
Dynamic stiffness matrix of a general cable element. by numerical integrations in
spatial domain (under review, Archive of Applied Mechanics)