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A frequency-domain method is developed for evaluating the earthquake input energy to two building structures connected by viscous dampers. It is shown that the earthquake input energies to respective building structures and added viscous connecting dampers can be defined as works done by the boundary forces between the subsystems on their corresponding displacements. Fourier and inverse Fourier transformations enable one to transform the earthquake input energy in time domain to that in frequency domain. The earthquake input energy in frequency domain has an advantage that, while the input energy cannot be decomposed into the term for the structural parameters and that for the parameters of input ground motions in the conventional time-domain approach, it is possible in the proposed frequency-domain approach. It is demonstrated that the proposed energy transfer function is very useful for clearer understanding of dependence of energy consumption ratios in respective buildings and connecting viscous dampers on their properties. It is concluded that the total input energy to the overall system including both buildings and connecting viscous dampers is approximately constant regardless of the quantity and location of connecting viscous dampers. This property leads to an advantageous feature that, if the energy consumption in the connecting viscous dampers increases, the input energies to the buildings can be reduced drastically. DOI: 10.1061/(ASCE)0733-9445(2007)133:5(620) CE Database subject headings: Dynamic response; Structural control; Damping; Earthquake resistant structures; Energy methods; Frequency analysis; Structural design.

The input energy to a soil-structure interaction (SSI) system during earthquake shaking is taken as a structural performance measure and is formulated in the frequency domain. The purpose of this paper is to derive the closed-form expression of the sensitivity of the input energy to the SSI system with respect to uncertain parameters representing soil stiffness and damping. It is demonstrated first that the input energy expression can be of a compact form consisting of the product between the input motion component (Fourier amplitude spectrum of acceleration) and the structural model component (so-called energy transfer function). With the help of this compact form, it is shown that the formulation of earthquake input energy in the frequency domain is essential for deriving the closed-form expressions of the sensitivity of the input energy to the SSI system with respect to uncertain parameters in contrast to the time-domain formulation including inevitable numerical error and instability. This formulation is then extended to a multidegree-of-freedom superstructure model. Numerical examples support the fact that the closed-form expressions enable one to find in a reliable and efficient way the most critical combination of the uncertain parameters that leads to the maximum energy input. DOI: 10.1061/(ASCE)0733-9399(2007)133:4(389) CE Database subject headings: Closed-form solutions; Earthquake resistant structures; Frequency analysis; Sensitivity analysis; Soil-structure interaction; Structural design; Energy methods.

Since earthquake ground motions and their input effects on structures are very uncertain even with the present state of knowledge, it is desirable to develop a 'robust' structural design method taking into account these uncertainties. Approaches based on critical excitation methods have been proven to be promising for such robust structural design. A new critical excitation method is developed here in which the mean earthquake energy input rate is chosen as a measure of criticality. The earthquake energy input rate is closely correlated with the story deformation and this supports the suitability of the energy input rate as a criticality measure in the case where the deformation is crucial in the design. The ground motion is described as a uniformly modulated nonstationary random process. The power [area of power spectral density (PSD) function] and the intensity (magnitude of PSD function) are fixed and the critical excitation is found under these restrictions. The key for finding the new random critical excitation is the interchange of the order of the double maximization procedures with respect to time and to the PSD function. Examples for a specific envelope function of the ground motion are presented for demonstrating the validity of the proposed method. Extension of the proposed method will be discussed for a more general ground motion model, i.e., nonuniformly modulated nonstationary models, and for a more general problem for variable envelope functions and variable frequency contents. DOI: 10.1061/(ASCE)0733-9399(2006)132:9(990) CE Database subject headings: Dynamic response; Excitation; Ground motion; Vibration; Frequency analysis; Earthquakes; Probabilistic methods.

A more reliable and stable method, compared to the previous one developed by the present writers, of stiffness-damping simultaneous identification of shear-type building structures is proposed using stationary random records under limited observation. It is shown that when stationary random horizontal accelerations are recorded at the floors just above and below a specific story in a shear building model, the story stiffness and the damping ratio can be identified uniquely. The viscous damping coefficient and the material damping ratio can also be identified simultaneously in a numerical model structure. It is also shown that unfavorable effects by extraneous noises, including measurement noises, due to wind excitation, building facilities, and the motion of the occupants can be eliminated partially by using the present method. The accuracy of the present identification method is investigated through the actual records under limited observation in a base-isolated building. DOI: 10.1061/(ASCE)0733-9399(2005)131:10(1027) CE Database subject headings: Stiffness; Damping; Noise; Base isolation; Buildings.

A new general critical excitation method is developed for a damped linear elastic single-degree-of-freedom structure. In contrast to previous studies considering amplitude nonstationarity only, no special constraint of input motions is needed on nonstationarity. The input energy to the structure during an earthquake is introduced as a new measure of criticality. It is shown that the formulation of earthquake input energy in the frequency domain is essential for solving the critical excitation problem and deriving a bound on the earthquake input energy for a class of ground motions. It is remarkable that no mathematical programming technique is required in the solution procedure. This enables structural engineers to use the method in their structural design practice without difficulty. The constancy of earthquake input energy for various natural periods and damping ratios is discussed based on an original sophisticated mathematical treatment. Through numerical examinations for four classes of recorded ground motions, the bounds under acceleration and velocity constraints (time integral of the squared base acceleration and time integral of the squared base velocity) are clarified to be meaningful in the short and intermediate/long natural period ranges, respectively. CE Database subject headings: Excitation; Earthquakes; Velocity; Damping.

Since earthquake ground motions involve various intrinsic and epistemic uncertainties, it is difficult even with the present knowledge to predict forthcoming events at a specific site in a reasonably accurate way. It is therefore desirable to develop a robust structural design method taking into account these uncertainties even partially. Critical excitation or worst-case analysis approaches are making remarkable progress recently and seem to be promising as a candidate to overcome such difficulties. In this paper, the power (area of power spectral density function) and the intensity (magnitude of power spectral density function) are fixed and the critical excitation is found under these restrictions. A design problem for restricted variable design earthquakes is formulated as a minimum-maximum problem which is expected to lead to the maximum global performance design for variable critical excitations. The elastic-plastic response characteristics of the building models designed by the present method are revealed for a broader class of excitations and code-specified design earthquakes. DOI: 10.1061/(ASCE)0733-9445(2002)128:12(1565) CE Database keywords: Earthquake excitation; Stiffness; Sensitivity analysis; Elastoplasticity; Building design.

A critical review of methods for critical excitation (worst-case input) is presented to enhance the robustness of structural design of aerospace, mechanical, and civil engineering structures. These structures are often subjected to disturbances including inherent uncertainties due mainly to their occurrence scarcity and worst-case analysis is expected to play an important role in avoiding difficulties induced by such uncertainties. During the last three decades, various critical excitation methods have been proposed extensively and applied to some structural design problems, mainly for important structures of which the failure or collapse must be absolutely avoided. Comparison of the critical responses with the corresponding structural responses to recorded ground motions, deterministic or stochastic treatment, elastic or elastic-plastic responses, system-dependent critical excitations are major subjects of discussion. DOI: 10.1061/(ASCE)0733-9445(2002)128:5(665) CE Database keywords: Robust design; Excitation; Random vibration; Ground motion; Uncertainty; Structural design.

For a simplified model of a structure-pile-soil system, semiexplicit forms of the responses to random earthquake inputs are derived together with sensitivity expressions of the random responses with respect to a certain model parameter. Originally this model has a non-tridiagonal stiffness and damping matrices with single earthquake input. It is shown for the first time that independent treatment of the free-field ground and multiple imposition of the free-field ground response into the structure-pile system allow for a transformation of the governing equations so that the final form has tridiagonal stiffness and damping matrices. Closed-form expressions of the inverse of the tridiagonal matrix lead to semiexplicit expressions of the random responses and their sensitivities with respect to a certain model parameter.

An analytical procedure is developed for deriving the solution to a new hybrid inverse eigenmode problem such that multiple subassemblages to be designed (design subassemblages) are supported on a foundation with given stiffnesses and the number of equations [[N.sup.i].sub.E] of equilibrium of the ith design subassemblage as a whole may not coincide with the number [[N.sup.i].sub.I] of degrees of freedom of interface displacements. Due to the incompatibility of these two numbers, that is, [[N.sup.i].sub.E] and [[N.sup.i].sub.I], previous approaches determining the interface displacements only from the equations of equilibrium of the subassemblage as a whole are not applicable to this problem. It is shown that division of interface displacements and specified lowest-mode representative deformations into two groups is essential for developing the analytical procedure for the hybrid inverse eigenmode problem. It is also shown that decomposition of nodal displacements in the design subassemblage and subsidiary interface displacements into 'rigid-body displacements' and 'actual deformation components' plays an important role in this procedure. A systematic procedure for finding stiffness parameters is then proposed. Application of this analytical procedure to two example models is presented to demonstrate the validity and usefulness of this procedure.

A class of hybrid inverse eigenmode problems is formulated such that a set of stiffnesses of a structure supported by a prescribed shear-type foundation system at an interface is determined for a specified lowest eigenvalue and a specified set of lowest-mode deformation-quantity ratios in the structure. The key point of the hybrid inverse eigenmode formulation is to take full advantage of the dynamic equilibrium of the whole superstructure, including the interface, and to express the lowest-mode interstory-drift components in the foundation system in terms of the specified lowest eigenvalue and the specified lowest-mode deformation quantities in the superstructure. A property on signs of the lowest-mode components is clarified for a general finite-element model supported by a shear-type foundation system. A systematic procedure for determining the stiffnesses of the superstructure is devised. The condition for uniqueness of the stiffnesses is then disclosed. Finally an example of application of this formulation is presented for a finite-element beam model supported by a shear-type foundation system.

The critical excitation method is promising as a robust method for accounting for inherent uncertainties in predicting forthcoming earthquake events and for constructing design earthquake ground motions in a reasonable way. Most of the proposed theories are based on deterministic approaches and deal with displacement responses. A stochastic acceleration response index is treated here as the objective function to be maximized. The power (area of power spectral density function) and the intensity (magnitude of power spectral density function) are fixed and the critical excitation is found under these restrictions. It is shown that the original idea for stationary random inputs can be utilized effectively in the procedure for finding a critical excitation for nonstationary acceleration responses of nonproportionally damped structural systems. Several numerical examples are presented to demonstrate the characteristics of generalized time-varying frequency response functions for models with various stiffness and damping distributions.

Cites a study regarding the closure to bound of earthquake input energy. Analysis of the possibility of occurrence of the maximum value of the earthquake input energy at an arbitrary time before the end of the input; Assessment of the modified approximation of the earthquake input energy; Factors affecting earthquake input energy.

Presents a correction to the article, 'Robust Building Stiffness Design for Variable Critical Excitations,' by Izuru Takewaki, published in the December 2002 issue of the 'Journal of Structural Engineering.'

Presents a corrected version of the article 'Seismic Critical Excitation Method for Robust Design: A Review,' by Izuru Takewaki published in the May 2002 issue of the 'Journal of Structural Engineering.'