Tsuneyoshi Nakamura

Optimum viscous dampers for stiffness design of shear buildings

The problem in this paper is to find the optimum sets of story stiffnesses and of damping coefficients of the dampers of an elastic planar shear building with viscous dampers (SBVD model) subject to constraints on maximum interstory drifts due to a set of spectrum-compatible earthquake motions, on upper bounds for each damping coefficient of dampers, and on the sum of damping coefficients of dampers. Two basic characteristics of an ordered set of optimum SBVD designs have been disclosed: (1) greater damping coefficients are distributed within the specified upper bounds among the dampers placed on stories with greater interstory drifts; (2) the effect on stiffness reduction of the optimization of the damping coefficients of dampers from an initial design of an SBVD model with uniform dampers is greater when non-uniform maximum interstory drifts with larger differences between the minimum and maximum values are specified. A design guideline for the effective configuration of viscous dampers for stiffness design of an SBVD model is proposed.

The secondary buckling and post-secondary-buckling behaviours of rectangular plates

Abstract Secondary buckling and post-secondary-buckling behaviours are theoretically studied for simply-supported rectangular plates, whose primary buckling modes of deflection contain more than half-waves in the load acting direction. Modal coupling effects more complex than one two-term-coupling effects are incorporated into the secondary buckling and post-secondary-buckling analyses. Then, unstable or stable symmetric secondary branching points are found on the post-primary-buckling paths and “snap through” motions involving an abrupt change in wave-form are shown to be possible. Wave-form variations along post-secondary-buckling paths are also disclosed by means of a numerical analysis of equilibrium paths.

Sequential optimal truss generator for frequency ranges

Abstract The concept of an ordered set of optimum designs is introduced here and all the design variables and behavioral variables belonging to the optimum designs are represented by successive piecewise Taylor series. A most natural, direct, and efficient way of generating or sweeping out all the optimum designs sequentially in the set is devised. The procedure is started with the eigenvalue analysis on the truss defined by the set of all the minimum cross-sectional areas. Neither any further eigenvalue analysis nor application of any optimization technique is required in the proposed procedure. It is demonstrated that the proposed method is efficient not only for an ordered set of optimum designs of a large truss associated with the single lowest eigenvector but also for that associated with multiple lowest eigenvectors.

Symmetry limit theory for cantilever beam-columns subjected to cyclic reversed bending

Abstract The behavior of a linear strain-hardening cantilever beam-column subjected to completely reversed plastic bending of a new idealized program under constant axial compression consists of three stages: a sequence of symmetric steady states, a subsequent sequence of asymmetric steady states and a divergent behavior involving unbounded growth of an anti-symmetric deflection mode. A new concept “symmetry limit” is introduced here as the smallest critical value of the tip-deflection amplitude at which transition from a symmetric steady state to an asymmetric steady state can occur in the response of a beam-column. A new theory is presented for predicting the symmetry limits. Although this transition phenomenon is phenomenologically and conceptually different from the branching phenomenon on an equilibrium path, it is shown that a symmetry limit may theoretically be regarded as a branching point on a “steady-state path” defined anew. The symmetry limit theory and the fundamental hypotheses are verified through numerical analysis of hysteretic responses of discretized beam-column models.

SEQUENTIAL STIFFNESS DESIGN FOR SEISMIC DRIFT RANGES OF A SHEAR BUILDING–PILE–SOIL SYSTEM

A shear building supported by a prescribed pile-soil system is subjected to bedrock earthquake input. A new design procedure is presented for generating a sequence of stiffness designs satisfying the constraints on interstorey drifts. The mean peak interstorey drifts of the shear building subjected to a set of spectrum-compatible ground motions at the bedrock are evaluated by a modal combination rule. Tuning of the fundamental natural period of a shear building with a fixed base with that of a shear beam ground results in a non-monotonic sequence of stiffness designs with respect to a ground stiffness parameter and previous approaches cannot be applied to such a problem. This difficulty in finding such a non-monotonic sequence is overcome by utilizing the ground stiffness parameter and the superstructure stiffness parameter alternately in multiple design phases and by developing a new multi-phase perturbation technique. Fundamental characteristics of this sequence of stiffness designs and the effect of ground stiffnesses on the design of the shear building are disclosed. It is further shown that the stiffness contour method is also useful for the design procedure such that a scattering effect in the estimates of ground stiffnesses is taken into account. The usefulness of the proposed procedure of sequential stiffness design and contour line method is demonstrated through several sequential design examples.

Sequential generator of earthquake-response constrained trusses for design strain ranges

Abstract A new concept called “an ordered set of earthquake-response constrained designs” is introduced for trusses with respect to a parameter representing design strain levels. An efficient computational method is developed of sequentially generating such an ordered set with the use of successive piecewise Taylor series expansion of all the design variables and of formulas for estimating mean maximum responses. The numerical procedure is started with eigenvalue analysis of the truss defined by the set of all the minimum cross-sectional areas. No further eigenvalue analysis or any successive improvement is needed. The efficiency of the proposed method is demonstrated through the ordered sets of designs for a 480-bar truss and an 800-bar truss.

Shape optimization of a double-layer space truss described by a parametric surface

A method is presented for finding the optimal configuration of a double-layer space truss described by a parametric surface. The number of design variables is drastically reduced, without sacrificing the smoothness of the upper layer surface, by using the control net parameters as design variables. The coordinates of the lower nodes are defined by using a vertical or normal offset vectors from the upper surface. Curvatures of the curve associated with the lower polygon are calculated for evaluating feasibility and regularity of the a double-layer lattice space truss. In the examples, a feasible optimum design, which minimizes compliance under constraints on structural volume, are successfully found by using the proposed method. It is shown that the optimal solutions strongly depend on the specified ratios of the cross-sectional areas of the members.

A Hybrid Inverse Mode Problem for Fixed-Fixed Mass-Spring Models

A hybrid inverse mode problem is formulated for a fixed-fixed mass-spring model. A problem of eigenvalue analysis and its inverse problem are combined in this hybrid inverse mode formulation. It is shown if all the masses and the mid-span stiffnesses of the model are prescribed, then the stiffnesses of the left and right spans (side-spans) can be found for a specified lowest eigenvalue and a specified set of lowest-mode drips in the side-spans. Sufficient conditions are introduced and proved for a specified eigenvalue and a specified set of drifts in the side-spans to provide positive stiffnesses of the side-spans and to be those in the lowest eigenvibration. A set of solution stiffnesses in the side-spans is derived uniquely in closed form.

Seismic Frame Design via Inverse Mode Design of Frame-Ground Systems

A two-step stiffness design procedure is developed for a moment-resisting planar frame supported by a prescribed two-dimensional finite-element ground-pile system. In the first step, a hybrid inverse eigenmode problem is formulated and its solution is derived in an analytical form. A difficulty resulting from the existence of multiple interface nodes is overcome by incorporating a deformation constraint into a set of linear equations for finding the lowest-mode displacements at the interface nodes and in the ground. In the second step, the fundamental natural frequency of the combined system and the lowest mode-strain ratios in the frame specified in the first step are regarded as the parameters for adjusting the mean peak seismic member-end strains to their specified values. If the fundamental natural frequency of the frame with a fixed-base happens to be close to that of the ground, a difficulty arises in the two-step stiffness design procedure because of an irregular response amplification and of the non-predominance of the lowest-mode components. A new practical design procedure of rapid convergence is proposed such that an initial design is found for a stiff ground and that a sequence of stiffness designs is generated with respect to a ground stiffness parameter without any differential coefficient of series expansion. The accuracy of the model utilized in this paper and the validity of the present stiffness design procedure are verified through time-history response analysis.

Optimum Design of Multistory Multispan Frames for Prescribed Elastic Compliance

ABSTRACT This paper presents an analytical method for minimum cost design of regular rectangular building frames for constrained elastic compliance. The method, consisting of a semi-inverse method and a design region extension procedure, is illustrated by two classes of exact solutions. It is shown that the relative story displacements are almost uniformly distributed in an optimally designed frame with almost uniform story heights and that the solutions enable one to calculate all maximum member-end stresses from member-end curvatures. It is suggested that the proposed design formulas may be utilized for design problems, subject to relative story displacement and stress constraints. Since these solutions are shown to be dependent upon the nature of the prescribed minimum stiffnesses, a method of finding a practically reasonable set of minimum stiffnesses is also presented.