Reliability-based limit state design is one of strong candidates for performance-based design; the performance level of a structure can be explicitly controlled in terms of limit state probability. Although estimating limit state probability generally requires complicated calculations, a load and resistance factor design (LRFD) is developed as a simple method and adopted in many countries. However, the performance level of only structural members can generally be considered in LRFD. The authors have been conducting research on a framework of a practical reliability-based limit state design that can control the performance level of a whole structure with limit state probability considering the effects of seismic hazard, characteristic of ground motions, and structural capacity.

 Several methods have been proposed to predict the maximum inter-story drifts of multi-story frames such as the Calculation of Response and Limit Strength method introduced in the Japanese Building Code in 2000 as a seismic design rule for ordinary building structures, Inelastic Modal Predictor (IMP), and Modal Pushover Analysis (MPA). The latter two estimation methods are based on the square-root-of-sum-of-squares (SRSS) rule of modal combination. These predictors utilize the post-elastic first-mode shape, φ^l_1,i, that is approximated by the distribution of the story drifts in the nonlinear static pushover analysis. The story drifts are determined so that the roof drift corresponds to the maximum displacement of the equivalent inelastic oscillator, S^l_D,1.

 In order to develop the practical seismic performance-design format on the basis of the above predictors, the following problems need to be considered.

 1. Development of a simple design format like LRFD

 2. Estimation for a probability distribution of the maximum S^l_D,1

 3. Estimation for a probability distribution of a first-mode response using φ^l_1,i (noted as P_1st hereafter)

 In this paper, the first and third problems are discussed.

 Mori and Nakano proposed a LRFD format for a serviceability limit state considering the sum of the square of each elastic mode response estimated by SRSS as if it is a load combination. A similar approach can be taken to develop a practical ultimate limit state design format adopting the predictors of maximum inelastic response of structures such as IMP and MPA. However, it is not an easy task to estimate P_1st on the basis on the probability distribution of S^l_D,1 because unlike an elastic response, φ^l_1,i is a complicated function of S^l_D,1.

 This paper first presents the design format based on IMP, which is expressed in the form of LRFD. Then, it proposes to approximate P_1st with a shifted lognormal probability distribution function. Although the approximation requires to solve simultaneous nonlinear equations, it is shown that they can be solved analytically under certain conditions. By adjusting three parameters rather than two, the tail behavior of the exceedance probability of an inelastic mode response can be captured well. Despite of the three parameters, the load factor can still be expressed in a closed form. The accuracy and applicability of the proposed method are investigated by using the Monte Carlo simulation on a series of 3-story, 6-story and 12-story steel moment resisting frames assuming that the probability distribution of the maximum S^l_D,1 during 50 years is available.

 This study deals with an evaluation method of performance to transmit horizontal seismic loads to substructures of double layer truss domes built in heavy snowfall region considering the vertical load resistant capacity. The purpose of this study is to propose the evaluation method based on the dynamic analysis of the dome due to horizontal and vertical earthquake motions. Both the static and dynamic characteristics of shell-like structures vary depending on their configuration. However, membrane actions are dominant in those structures. Among efficient spatial structures, which have membrane actions similar to shells, typical examples are single layer reticular domes and double layer domes composed of many straight members, which are subjected to member buckling. After undertaking the static behavior, dynamic buckling behavior is analyzed in order to determine the collapse mechanism of these reticular domes. The present analysis adopts an assumption that members buckle due to compression and yield under tension. The present analysis considers the effects of material nonlinearity of members and the geometrical nonlinearity of the domes.

 The purpose of this study is also to investigate the earthquake response such as the acceleration, the velocity, the displacement and the axial force of the domes subjected to vertical and horizontal earthquake motions. Based on the dynamic response, the practical calculation method is shown to predict the equivalent static load for the earthquake-proof design of the dome. As long as the accuracy verification of the equivalent static load is concerned, the collapse mechanism and the axial stress of the dome subjected to the static loads show a good agreement with the earthquake response analyses subjected to the earthquake motions.

 In the practical calculation method, the distribution and the value of the equivalent static load are calculated by means of the participation vector and the earthquake acceleration response spectrum. The static earthquake-proof design is able to carry out using the equivalent seismic loads practically. In general, it is not easy to predict the distribution of the acceleration response because of the complicated dynamic behavior combined with the vertical and horizontal response of the dome built in heavy snow region. This is a reason why the earthquake static load is required for the earthquake-proof design using the static analysis.

 The characteristic response of the dome occurs subjected to vertical and horizontal earthquake motions. The design seismic coefficient of the applied load distribution considering the snow weight effect is also actually necessary for the safety study in the earthquake proof design. The study proposes the estimation method of calculating the design seismic coefficient for the applied load for the earthquake-proof design of the dome. The coefficient is easily obtained by means of the eigenvalue analysis. The accuracy is also verified by a good agreement with the earthquake response analysis of the dome subjected to the artificial vertical and horizontal earthquake motions varying the PGA. The proposed method can be used to predict the performance to transmit horizontal seismic loads by means of the elastic static analysis of the truss dome subjected to the equivalent static load.

 The topology optimization method using voxel finite element method is an effective method to create various morphologies from rectangular parallelepiped design domain. Fujii et al. [1, 2] created morphologies of building structures using such method. Also, Fujii et al. [3] applied such method to create link mechanisms that amplify input displacement such as toggle damping device. However, these link mechanisms increase manufacturing cost and maintenance cost as the number of links increases. Therefore, it is desirable to develop compliant mechanisms that amplify input displacement by elastic deformation. However, almost topology optimization methods to create these compliant mechanisms [4, 5] are based on the infinitesimal deformation theory. Therefore, in this paper, we develop a topology optimization method considering finite deformation in order to develop a vibration control device using compliant mechanism.

 In this paper, we use a particle method instead of finite element method, because when using finite element method, the computation time becomes enormous, and the convergence solution often cannot be obtained because the calculation becomes unstable by the large distortion of elements. Manabe and Fujii [6] have proposed a method that using CA-ESO method for topology optimization and MPS method [11] for particle method. Also, Manabe et al. [7] have developed amethod using Level Set method for topology optimization. However, these methods target two-dimensional problems, and methods for three-dimensional problems have not been developed yet. Therefore, the purpose of this study is to extend this approach to three-dimensional problem.

 The proposed method in this paper is an extension of the proposed method in ref. [6]. That is, HMPS (Hamiltonian Moving Particle Semi-implicit) method [12-14] is used instead of MPS method [11], and IESO (Improved Evolutionary Structural Optimization) method with finishing algorithm is used instead of CA-ESO method.

 In Section 2, the formulation of HMPS method is shown so that our created program can be understood. In Section 3, the outline of IESO method with finishing algorithm using HMPS method is explained. In Section 4, we verify the effectiveness of the proposed method by numerical examples in which jumping buckling occurs. In Section 5, the above results are summarized.

 The conclusions are as follows.

 (1) In the analysis of infinitesimal deformation range, the solutions of the proposed method almost agree with the solutions of the method using voxel finite element method.

 (2) In the analysis of finite deformation, the solutions corresponding to buckling cannot be obtained by IESO method alone. However, the solutions obtained by IESO evolve into solutions corresponding to the buckling by the finishing algorithm (CA+IESO method). Also, the two-dimensional solutions have the same topology as the solutions in ref.[7], [9]. And, the shapes of two and three dimensional solutions change according to the magnitude of the load. Furthermore, the solutions of two-dimensional problem and three-dimensional problem show similarities in topology.

 In addition, the proposed method is very robust and the computation time does not become enormous. Therefore, the proposed method can be practically implemented as a three-dimensional topology optimization method that can consider finite deformation.

 In the next step, we plan to apply the proposed method to the topology optimization problem of compliant mechanisms.

 1. Introduction

 In recent years, mechanical splices (hereafter, called as couplers) are widely used in RC buildings; thus, RC beams without stirrups on couplers have many advantages in design and construction. This paper focuses on the shear strength of the above RC beams. The experiments preceded by the authors were outlined and simulated by three-dimensional nonlinear FEM analyses to discuss the shear-resisting mechanisms. Furthermore, based on the analytical findings, shear design equations proposed in the authors’ previous study were modified and verified.

 2. Outline of the preceded experiments for analytical study

 Four specimens with different details on couplers which were tested in the authors’ preceded study were outlined: a control specimen without couplers named NN1, and other three specimens having couplers with different lengths of 180 mm, 280 mm, and 380 mm named MI1, MI1A, and MI1B, respectively (Fig. 1 and Tables 1-3). These specimens were designed assuming a specific shear failure mode with compressive failure of concrete. As a result, the maximum strengths of NN1 and MI1 were similar to each other, while those with longer couplers were lower compared to NN1 and MI1 (Fig. 4).

 3. Three-dimensional nonlinear FEM analysis

 Three-dimensional nonlinear FEM analyses were conducted using FINAL Ver. 11 for four experimental specimens and four additional ones with longer couplers (Fig. 5). Core and cover concrete was modeled using eight-node isoparametric solid elements, and longitudinal bar and stirrup were modeled using two-node truss elements. Coupler was modeled using eight-node isoparametric solid elements to reproduce bearing pressure at coupler ends (Fig. 6).

 4. Analytical results

 The shear force-drift angle relationships of four experimental specimens were compared with the test results (Fig. 8). The analytical results agreed well with the experimental ones; thus, differences in the maximum strengths were limited at -6% to +4%. Major findings are summarized as follows:

 [Effects of the presence or absence of couplers] The averaged stress of stirrups was lower in MI1 (Fig. 9), while the maximum strengths and minimum principal stress distributions were similar to each other. It indicated that the shear-resisting mechanisms might be different between NN1 and MI1.

 [Effects of coupler length] In the cases with longer couplers, the truss mechanism deteriorated but the arch mechanism remained in the ultimate states (Fig. 10).

 5. Modification of design equations for the shear strength of RC beams of interest

 Based on the above analytical results, the shear design equations of RC beams without stirrups on couplers were modified considering 1) existence of arch mechanism remaining after deterioration of truss mechanism by Eq. (7), and 2) confinement to core concrete along couplers by Eq. (9).

 6. Verification of the proposed modifications

 The modified shear design equations were verified comparing with the experimental results of seventeen RC beam specimens from the authors’ previous study. Consequently, a better agreement was obtained between the test results and the estimations considering the modifications (Fig. 20). The average and coefficient of variation on the experimental value/calculated value were 1.29 and 18.4%.

 A concrete-filled tubular (CFT) column is a synthetic structure of a steel tube filled with concrete, which not only possess superior hardness and load-bearing capabilities but is also considered to be highly feasible for use. Therefore, CFT columns are used in steel structures for a wide range of application. CFT columns without fire proofing (unprotected CFT columns), which take advantage of the high heat capacity of concrete, are in high demand for the architectural planning reasons, such as improved aesthetics and reduction in cross sections, and also for the construction reasons, such as their ability to omit the fireproofing process.

 In contrast, to a large extent, the performance of unprotected CFT columns is unknown with respect to its fire-resistant ability. In experiments of fire resistance performance of unprotected CFT columns, a fire resistance test subjected to a combined load is conducted, taking into consideration the horizontal displacement in the capital caused by the thermal expansion of the beam, in addition to the long-term axial force inflected during the heating process. In the present study, the authors performed a more adequate fire resistance test subjected to a combined load and a double curvature bending than the existing combined load test, by evaluating the behavior of a full size CFT column (600 mm × 600 mm) in a framework during a fire. The result showed that the local buckling of the steel tube occurred in the capital portion or the pedestal portion with a high degree of curvature, and most of the breaks in the concrete occurred in these parts. However, the mechanism of these behaviors remained unknown.

 In the present study, we verify the temporal behavior in a state of combined load with double curvature bending, which was confirmed in the experiment reported in the previous study with 3D finite element analysis; and subsequently, we discuss the mechanism of CFT columns breaking in a fire. To consider the elongation and local buckling of the steel tubes during the increase in temperature, the steel tube and concrete were separated in the model, and contact conditions were applied between the two elements. Subsequently, we used the load induced thermal strain (LITS) model for estimating the shrinkage strain of concrete during the process which involved increase in the temperature.

 The results obtained from the analysis showed that, the model proposed in this paper accurately recreates the deformation behavior observed in the combined load experiment with double curvature bending. Based on analyses, we were able to confirm the elongation and local buckling of the steel tube, and the change in the state of concrete from shrinkage to breaking, based on the existing experiment. Our study also showed that in the state of combined load and double curvature bending, the uneven distribution of axial stress due to bending stress occurring on the cross-section of the concrete at the capital with a large curvature, was concentrated within a small area of the cross section due to the impact of the transient strain on the concrete, causes the concrete to break.