In this paper, in order to establish the dynamic failure criteria of multi-degree structures with structural instability, the true dynamic ultimate strength of two-degree flexural systems with structural instability as well as the response parameters which determine these ultimate strength are investigated numerically. In particular, effects of the collapse mode shape on the dynamic ultimate strength and the amount of energy are examined.
As a result, it is revealed that the effective input energy governs the dynamic ultimate state of two-degree systems with structural instability and that the input energy depends on the collapse mode shape.
Furthermore, the method of estimating the effective input energy of two-degree systems is proposed from the results of the single-degree systems.

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In is known that for more precisely calculating the deflection of continuous reinforced concrete beams under service load, the redistribution of internal forces at this stage should be considered. For this purpose, the author suggests two new formulas to calculate the adjusting coefficients of support moments. As establishing the formulas, the translation of inflection points due to moment redistribution has been taken into account. It is reasonable theoretically and can give calculated results in good accordance with tests. In this paper, three diagrams of deflection coefficient are given so as to simplify the calculation greatly. Besides, the consideration of long-term influence on deflection is also proposed.

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A new and powerful material nonlinear analysis method for rigid plane frames is proposed on the basis of the principle of minimum complementary energy and mathematical programming algorithms. The analysis problem is formulated as the total complementary energy minimization problem subjected to the equilibrium equations at the free nodes in terms of the unknown forces acting at the ends of member elements. The unknown member end forces are determined by solving the energy minimization problem using a modified sequential quadratic programming algorithm.
The problem formulation and analysis algorithm of the proposed method are applicable for rigid frame structures with any types of nonlinear materials. The power, reliability, efficiency and practical usefulness of the method presented are demonstrated by comparing the results obtained with ones by the displacement method of analysis for several statically indeterminate rigid frame structures with three types of nonlinear materials

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This paper presents an expert system which uses “ranking method” to solve bridge superstructure selection problem. The “ranking method” is usefully used in bridge selection problem and also can deal with fuzzy set which is used to handle fuzziness in decision knowledge which are collected from various sources.
In the selection, several appropriate alternatives are chosen in consideration of the appropriate combination of superstructures and construction method. The final selected results are shown in the form of a list of ranked alternatives after they are ranked based on judgement of decision knowledge and designer's opinion.

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The curved member is assumed to be an assemblage of straight members connected to each other at nodal points whose coordinates are introduced in the initial configuration. Although it has been recognized by many researchers that the straight beam element cannot always model the curved member properly, the present work proves that this conclusion is not true provided that the usual geometric stiffness of the thin-walled straight beam element is adjusted by taking into consideration the out-of-balance of the internal forces at the joint of two adjacent elements meeting at an angle

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The behavior of axially loaded cylindrical shells is studied using a general purpose non-linear shell element. The well-known nine-node degenerated shell element was selected among the variety of shell finite elements proposed and developed in the past by several researchers, because it satisfies all the compatibility conditions between adjacent elements directly and gives accurate results. The behavior of axially loaded cylindrical shells are observed through computations of some fabricated pipes which were tested in Nagoya University. The measured initial imperfections are included in the calculations. The effects of initial imperfections on the cylinder strength are also discussed.

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A new formulation of membrane elements with drilling degrees of freedom is presented. The Lagrange multiplier method is employed to obtain the modified functional. Since the independent rotational field is used, the present formulation is applicable to any shape of elements. The resulting stiffness matrix has negative diagonals associated with drilling rotations. This deficiency is remedied by introducing a positive parameter. It is shown that the numerical results are not affected by the value of the parameter. The present element has no zero-energy mode, and no locking phenomena are observed. Numerical results demonstrate an excellent performance of the membrane elements with drilling rotations.

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Elastodynamic behavior of a two-dimensional composite rectangular beam, which has regularly-spaced fibers, is formulated by a couple of finite difference equation based on FEM formulation of rectangular element. Making use of modal analysis and Duhamel integral equation, the impulse response of the composite rectangular beam is obtained. The transient response from the first incident wave through the whole oscillation of the beam is found by the series of simulation analysis. The effects of coupling oscillation of the transducer and the beam on the impulse response of incident stress wave are obtained in experiments of the ultrasonic pulse method. In addition, the effect of internal friction of the beam and the frequency dependence of damping coefficient are discussed.

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Experiments for total of 11 specimens are carried out to observe the behavior of elasto-plastic out-of-plane buckling of the arch structures with open cross section which are subjected to uniform vertical load. The effects of several factors on ultimate strength are investigated, that is, slenderness ratios of arch rib, load directions, types of bracing system and braced length ratios. The experimental results are compared with the theoretical ones. In general, the theoretical predictions show good correspondence with the experimental results in ultimate strength and buckling modes and so on. Validity and efficiency of the theoretical procedure are confirmed.

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The classical impact damper (CID) and the pendulum impact damper (PID) are studied for the suppression of horizontal vibrations of civil engineering structures. Experiments on the PID to verify the validity of the mathematical model are done and comparisons with simulation results are provided. Parametric study under harmonic excitation shows that the effectiveness of the CID is sensitive to changes in the amplitude of the excitation. On the other hand, the PID is insensitive to changes in the amplitude of the excitation when its frequency is close to the structural frequency, and it is less sensitive to deviations in tuning than the tuned mass damper (TMD). Comparison of the CID, PID and TMD shows optimum efficiency in terms of mass ratio, under steadystate motion and ideal conditions, to be highest for the TMD and lowest for the PID. Included is a discussion of the practical constraints that may govern the actual choice of damper.

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This paper presents the fatigue strength of fillet weld joints at the web and flange plates in a horizontally curved plate girder. Firstly, an approximate formula to estimate the maximum out-of-plane bending stress in the web plate is proposed for fatigue design. Secondly, five curved plate girder models were fabricated according to the design criteria of the Japanese Specification for Highway Bridges (hereafter referred to as JSHB). Then, the model girders were tested up to the fatigue failure after having checked the stress distribution and displacements in the web plate through static loading tests. Finally, the fatigue strength of these fillet weld joints is examined based on the test results.

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For the purpose of defining the initial crack size used in estimating the fatigue life of fillet welded joints, fatigue tests of fillet welded joints with a longitudinal attachment were carried out under constant and computer-simulated highway variable amplitude loadings, and the initiation and early-stage propagation behavior of fatigue cracks at the fillet weld toe was investigated. Facet-like fracture surfaces were observed where fatigue cracks initiated and the depth of them was independent of stress range and welding material. Fatigue life analyses applying the fracture mechanics concept were performed on the assumption that such a fracture surface is the initial crack. Good agreement was obtained between the estimated fatigue life and the experimental one.

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Usual Distinct Element Method (DEM), in which soil is represented as a system of numerous discrete particles, does not account for two factors; the continuity of the medium and wave propagation. The modified method, which is proposed in this paper, has another physical structure which presents the effect of the internal material between particles as pore water or clay. We simulated the two dimensional dynamic fracture of a cliff using this method. The fracture process is as follows; many small cracks occur widely then they form fracture lines. Results confirmed that this method can simulate the continuous and discontinuous medium.

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Proposed is an inspection method which is capable to assess seismic vulnerability of a number of existing highway bridges without complex calculation. Total of 124 highway bridges which suffered damages during four earthquakes including the Miyagi-ken-oki Earthquake of 1978 is statistically analyzed, and factors affecting seismic damages are studied. Because fatal damages such as falling-off of superstructures are most likely developed from an excessive relative movement between the superstructure and the substructure, and from a failure of the substructures due to inadequate strength, an inspection method for the seismic vulnerability is developed based on vulnerability to develop excessive deformation and failure of substructures. Accuracy of the method is also studied.

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Resolving power of a strong-motion array for inferring the history of slip on an unidentified earthquake fault is defined as the accuracy of a source inversion.
The accuracy of the source inversion is efficiently calculated using a scheme constructed on the basis of Wolberg's prediction analysis. The resolving power of arrays for identified faults is also estimated to gain an appreciation for unidentified events. We find that the resolving power of the array is gradually enhanced as the station density increases on a square-grid array surrounding an inland fault. Effects of ocean bottom stations deployed for an offshore fault are recognized.

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It is an old saying that vibration and wave propagation are the same in principle. The saying is applied to vibration of an elastic layer as well as to dispersive Rayleigh wave in the layer. Based on vibration mode analysis for the layer of finite length, the Rayleigh wave characteristics such as dispersion and rotation of particle motion are discussed. Formulation of the impulsive response of the layer leads to familiar relationships between applied force and resulting deflection. The amplitude of the dispersive Rayleigh wave generated by a horizontal and/or vertical impulse is also discussed in reference to Poisson's ratio.

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Damping characteristics are significant factors for seismic design of a cable stayed bridge. It is general in seismic design to assume a damping ratio of approximately 2-5% of critical for superstructure. However, damping ratio which has so far been estimated from field observation such as forced vibration test is generally much smaller than those values. This study presents an analysis on the damping characteristics of a cable stayed bridge where strong motion records have been accumulated. Damping characteristics are analyzed through analytical simulation for the measured response by varying the damping ratio assumed in the analysis. It is concluded from the analysis that the damping ratio which gives the best correlation in response analysis simulation is 2% and 0-1% of critical in longitudinal and transverse direction, respectively, for the tower, and 5% in both directions for the deck.

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A new closed-open-loop optimal control algorithm is proposed that has been derived by minimizing the sum of the quadratic time-dependent performance index and the seismic energy input to the structural system. This new control law provides feasible control algorithms that can easily be implemented for applications to seismic-excited structures. We developed optimal control algorithms, taking into account the nonlinearity of the structural system and the effect of the time delay, for applying a control force to a structural system subjected to general dynamic loads. These optimal algorithms are simple and reliable for on-line control operations and effective for a structural system with a base isolation mechanism. The control efficiency affected by two weighting matrices included in the performance index is investigated in detail. Numerical examples are worked out to demonstrate the control efficiency of the proposed algorithms.

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We have proposed the most general higher-order equations of three-dimensional static and dynamic theories for a circular cylindrical shell by expanding the displacement into infinite Power series with respect to the radial coordinate of the shell.
In this paper, the shell theories by our reduction are applied to the free ring analysis, and the natural frequencies of vibration of free rings are computed. The theoretical results are compared with the results of various theories and with experimental data.

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Interpolation functions using finite Fourier series are formulated and their applications to two dimensional boundary elements are proposed.
Interpolation functions proposed here are derived by calculating the finite Fourier series of relative displacement and traction vectors between adjacent two nodes on the boundary. These interpolation functions may vary in higher order, including linear variation.
Numerical examples are examined to evaluate the efficiency of the proposed method.

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By using of the formulation of the authors' general higher-order theory of plates which satisfy exactly the boundary (loading) conditions of upper and lower surfaces of the plate, we solve the particular problems of a semi-infinite, isotropically elastic plate of uniform thickness with a sinusoidally varying in-plane (stretching) and out-of-plane (bending) edge loads. Two types of approximate three-dimensional corrections to the elementary two-dimensional plane stress solution are obtained as closed form solutions. Numerical results for these solutions of appropriate truncated terms are compared with the plane stress and plane strain solutions which depend on Poisson's ratio.

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This study is for the clarification of the cableaerodynamics of a cable-stayed bridge induced by wind and rain, that is called “Rain-Wind Induced Vibration”, which is one of recent majoring subject in bridge aerodynamics. In conclusion, an inclined cable shows aerodynamic instability because of intensive axial flow in its near wake. This axial flow plays similar role of splitter plate submerging in the wake which can make a circular cylinder aerodynamically unstable.

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In this study, the effects of turbulence on “Torsional Flutter” of 2-D H-section cylinders with some slenderness ratios (3≤B/D<10, B: chord length, D: height of section) were experimentaly investigated from the frequency effect of turbulence point of view in smooth flow, grid-turbulent flow and a sinusoidal fluctuating flow. In conclusion it was clarified that the higher frequency fluctuating components of turbulence amplifies the entrainment effect of separated shear layer around comparatively bluff body such as B/D<5.5, on the other hand, comparatively low frequency fluctuating components of turbulence which is the specified frequency region subject to “shear layer instability enhancement frequency region”, can significantly affect torsional flutter of comparatively slender body such as B/D≥5.5.

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This paper presents both experimental and analytical approaches for the dynamic response of steel circular fixed arch under impact for the purpose of developing the impact resistant design of the steel arched Sabo dam. First of all, the static and impact experiments have been performed in order to measure the relationship between load and deformation, and the absorbed energy has been obtained by integrating the area of the load-deformation curve. Secondly, the impact response analysis has been performed by modelling the impact phenomenon into the two-degrees of freedom system accounting for the local deformation. As the results, it is confirmed that the load-deformation relation and the absorbed energy calculated by the impact response analysis are relatively in good agreements with the ones measured by the impact experiment. Therefore, it is found that these results will provide the useful data for the energy design from the viewpoint of the impact resistant design.

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Because of their excellent characteristics against wind and fluid loading and because of their appearance, steel tubular members are frequently used in a variety of structures including offshore oil drilling platforms, bridge piers and so on. There exist large discrepancies among currently available design recommendations for the design of such members against local buckling. This is mainly due to the wide scatter of experimental data. This paper is to present a series of carefully conducted failure tests on fabricated short cylindrical tubes in compression and in bending. Based on the test results, a design formula is proposed for local buckling strength of cylindrical pipes. Furthermore, a study on the interaction strength between local and overall buckling is also presented

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Several investigations on practical use of the time domain boundary element method for in-plane elastodynamics are carried out. Treatment of tractions at corner point are examined for both continuous and discontinuous element in view of coupling boundary element method and finite element method. Numerical results for wave diffraction in half-infinite plane are discussed relating with several essential factors, that is, incident wave length, element size, time increment and differences in the form of the truncation of matrix coefficients. Some useful informations are presented to development of efficient strategy on accuracy, CPU time and memory requirements.

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In this paper, a stiffness method using a large-size plate elements covering one panel is applied for the three dimensional analysis of composite structures of plate and gril-lage. A method proposed here starts with the presentation of large rectangular plate element with all edges which are able to afford arbitrary displacements and forces. The stiffness matrices of each element are derived by the combination of analytical method and numerical method (point-matching method), and afterwards a block element transversely connected is made using a direct stiffness procedure. In addition, a relaxation technique is also used for jointing the block elements in longitudinal direction. The employment of such a relaxation technique permits to analysis of multi-panel plate structures by a small computer irrespective of the number of block elements. Numerical examples prove high precision of elements and usefulness of the presented method.

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A finite element solution procedure is presented for solving the interaction problems of viscous incompressible fluid and a rigid body supported by elastic springs. The Arbitrary Lagrangian-Eulerian formulation is employed in order to treat the boundary conditions on the surface of the moving body strictly. A step-by-step solution procedure is derived by using the mixed finite element method as well as explicit time integration schemes. The solution procedure includes rediscretization of solution domain according to the displacements of the moving body. The pattern of the rediscretization is prescribed so as to avoid excessive element deformation. The characteristics of the present procedure are verified through solving the problems of an oscillating circular cylinder in a closed circular domain and an oscillating square cylinder in a uniform flow.

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Characteristic equations, such as stability equation and frequency equation in structural mechanics, are often reduced to transcendental equations, and the practically usable solver of the equations is anticipated.
Although the DKA (Durand-Kerner-Aberth)-scheme is already established as a solution method of high order algebraic equations and appraised high in the field of numerical analyses, the scheme has been considered not to be applicable to transcendental equations. However, further improving a solver which one of the authors discussed and expanded the DKA-scheme earlier, the authors succeeded to obtain good solutions of transcendental equations in specified domain with sufficient accuracy.
In this paper, the authors wish to describe the algorithm and the numerical examples of the proposed transcendental equation solver.

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In this paper, the basic beam strength curves are investigated numerically by the finite element technique. At first, an imperfection sensitivity function for beam strength is introduced and various coefficients in the function are numerically identified by a finite displacement elasto-plastic analysis with both geometrical and material imperfections. And using a number of imperfection sets generated by Monte Carlo simulation from measured imperfection data, the strength distributions of steel beams with rolled and welded I-section are evaluated from a probabilistic view. These numerical results are compared with the experimental buckling strength database. Secondly the effects of bending moment distribution on beam strength are examined, and an equivalent slenderness formula applicable to inelastic range is developed. Finally, the mean strength curves and the resistance factors for rolled and welded beams are proposed.

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A refined control scheme with an optimal observer is presented for active optimal control of civil engineering structures. The purpose is to increase the accuracy and robustness of the control system. The structural stiffness and mass distribution cannot be evaluated with 100% accuracy, because there is gradual (aging, shift of live and dead load, etc.) or abrupt (plasticity due to extraodinary external force) change in structural behavior. In the proposed control scheme, stable control is achieved in an asymptotic manner, if there is large mass variation. However, the proposed scheme is not applicable for decrease of stiffness; and it is concluded that the use of servo-control scheme is recommendable in that occasion.

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Vertical oscillation of a circular disk on an elastic half space is solved. For this purpose, a comprehensive method has been developed on the basis of the direct integral equation which relates explicitly the displacement of the disk to the stress of contact. This integral equation and the differential equation of motion of the elastic disk are discretized separately. As a consequence, a method of wide applicability is obtained; the stiffness and the mass density of the disk as well as the external force can assume arbitrary distribution. The obtained linear algebraic equation, whose coefficient matrix has originally been virtually singular, is modified theoretically and is made numerically robust. As a demonstration, the new algorithm is applied to axisymmetrical and rocking oscillations of a uniform elastic disk and its result discussed.

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Due to the inelastic property of beam-to-column connections, the stability of semirigid frames is affected not only by the loading and the unloading characteristics of connections at the critical points, but also by their load history up to these points, thus resulting in a complicated buckling behavior. Herein, utilizing the accurate method of analysis developed by authors, we precisely examined how the stability of semi-rigid frames is influenced by the modeling of connections, the load conditions and the load history caused by the cyclic wind load. With the information, so obtained, we further discussed the way of stability check to be used in design.

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With the increasing capacity of electronic computers, elastic-plastic, large displacement analysis of steel structures has now become reality in research level. However, there remain some problems that must be overcome. One of those is to reduce the computation time. It is a usual practice to divide the cross-section of a member into many elements so that the spread of yielding in the cross-section may be taken into account. This is considered to be the main reason why a huge computation time is required. In this paper a new constitutive law is introduced which is expressed as direct functions of stress resultants. With this constitutive law the cross-section need not be divided, yet the penetration of yielding into the cross-section as well as along the member may be taken into consideration in an approximate manner. It has been demonstrated that this newly developed scheme saves computation time considerably.

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Identification problems on coefficient matrices of an autoregressive model for multivariate and one-dimensional nonstationary Gaussian random processes are investigated, by appling the Kalman filter incorporated with a weighted global iteration.
The major contributions of the paper are the use of Kalman filter for estimating time varying model parameters and the development of an effective method in terms of computer time.
The results indicate that the coefficients of this recursive equation are identified extremely well at the stage of their stable convergency to optimal ones.

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This paper presents an appropriate design formula for steel beam-columns based on the ultimate strength. The strength of beam-columns is evaluated by an elasto-plastic finite displacement analysis considering characteristic values of measured initial imperfections. The strength analysis is carried out for in-plane and out-of-plane failure having welded box section and I-section, respectively. For the design formula of in-plane failure the conventional linear interaction equation is acceptable providing that the column strength used in the equation should be given appropriately. On the other hand, in the case of out-of-plane failure a non-linear interaction equation is proposed. It is made clear that the resistance factor for columns and beams can also be adopted effectively to those for beam-columns according to the SGST-format. Finally, the proposed formula is verified by the comparison between the prescribed results and the experimental data.

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Many studies on tension-type connections have been carried out. However, there are still many points which must be clarified. Fundamentally, this type connection transmits the external loads by “metal-touch”. Therefore, it is very important to investigate effects of contact surface conditions on mechanical behaviours. In this paper, the effects of contact surface flatness on the split-tee connection behaviours are investigated as a contact-problem by finite element method. It is revealed that the bolt axial force, axial stiffness and stress in T-flange are influenced considerably by the degree of contact surface flatness.

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This study aimed at developing an effective approach to evaluating the durability of existing water supply pipes. The influential factors were clarified and classified into two groups, i. e. general factors and corrosion environment factors. Then, an approach using fuzzy sets theory was proposed to combine the evaluations on each group of factors. In the approach, the general factors are evaluated using an existing statistical model and the corrosion environment ones using experts' knowledge. A small number of field surveys of water supply pipes are required to be carried out for the combination of evaluations. The approach was used to establish an evaluation model for the cast iron pipes in Nagoya City and the model showed higher accuracy than the statistical model.

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The strength of beam-columns depending on local buckling of plate elements is investigated by the FEM analysis including both geometrical and material nonlinearities. The local buckling strength of thin-walled sections is evaluated by considering interactive effects between flanges and webs. By introducing the local buckling effects to European interaction formula, a simple and exact method to evaluate the load carrying capacity of beam-columns with box section is developed. And the proposed method is compared with several existing specifications.

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The mixed boundary value problem of the thin plate is analyzed for a strip under concentrated bending and torsion. The strip has a part of the boundary where the displacement is constrained, and a crack initiating from an end of the constrained part. A rational mapping function which maps the strip with a crack into a unit circle and the complex stress functions for the deflection are used. The first derivative of the complex stress function is obtained in a closed form without integral term. The stress distributions before and after crack initiation, and the stress intensity factors of the thin plate for the bending and torsional modes are obtained from short to long cracks. The stress intensity factors are compared with those of other strips which are different in the constrained degree of the strip edge, and the effect of Poisson's ratio is studied.

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A calculation method of additional bolt axial force in long bolt tension-type connections by a spring model is investigated. The spring model is very useful in design use because of its simplicity. The estimation method of the spring stiffness of the model is established by the parametric analysis using finite element method. Fig. 12 or Eq. (7) gives the key value C_R being a multiplier of K_RIB.

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This paper is concerned with the three-dimensional dynamics of group piles in layered soils with emphasis on the pile-soil-pile interaction. The analysis, based on the substructure technique, is made by the finite element method for piles and the Green function method for soils which makes use of the thin layer formulation. Points for investigation is placed on the pile head impedance function and their group effect. The engineering aspect is also addressed to compare the present solution with the current design formula with respect to the pile head spring constant.

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The dynamic earth pressure acting on the rigid caisson foundation was studied by the model vibration tests. The model caisson used in the experiment enabled to measure the dynamic earth pressure. In order to account for the characteristics of the dynamic earth pressure influenced by the backfill ground vibration, frequency of an input sinusoidal wave was chosen very wide. The fundamental information of the dynamic earth pressure with frequency, the effect of the caisson mass on the dynamic earth pressure, its phase difference from the inertia force of the caisson and its distribution, were given in this experiment. And authors also regarded the dynamic earth pressure as a variation from the static one during vibration and compared experimental results with the Mononobe-Okabe formula.

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Response of an elastic dam-foundation system to a horizontal impulse is analyzed by a FE-BE method in time domain. The method has proved to be effective not only in calculation of the response itself but also in getting many vibration modes of the system from the response. As eigen values pertaining to the system are complex, the present procedure based on the time domain analysis seems to be superior to the determinant search procedure applied to a frequency domain analysis. From a practical point of a view, the present analysis places a main focus on effects of an impedance ratio on vibration characteristics such as radiation damping, natural frequency and vibration mode shape. The vibration shape which is obtained in a complex form is plotted in terms of its real and imaginary parts, showing interaction between the dam and foundation.

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The problem of system identification has become increasingly important in the area of structural engineering, particularly in connection with the assessment of damage and deterioration of existing structures.
In this paper, a method is presented to identify natural frequencies and critical damping ratios of existing structures on the basis of a multivariate ARMA model by appling the Kalman Filter incorporated with a global iteration.
The method is verified by appling it to microtremor records which are measured on a four story reinforced concrete building.

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Several stress approximation formulas of truss structures are investigated analytically. Several approxinflation formulas of the stress of i-th member are shown analytically and compared with each other in the following four special cases, (1) statically determinate structures, (2) only the sectional area of i-th member is variable, (3) the sectional areas of the members except for the i-th member are variables, and (4) the sectional area of i-th member approached to zero. It is concluded that the approximation formulas with respect to reciprocal variables, especially the force approximation with respect to reciprocal variables, are superior to the others for the stress approximations of the truss optimization.

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The method of estimating tension of bridge cables by measuring their natural frequency, so-called vibration method, is convenient on construction site of structures with cables, Concerning the method some papers have been reported hitherto. In these reports cables are assumed to be clamped on both ends. However, for the estimated tension of cable supported by open sockets, vibration of the sockets has a notable effect, which has not been investigated. In order to get reliable and accurate estimation for tension of cables with open socket, some experimental works were performed and reported in this paper.

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A method of seismic hazard estimation using the polygonal source-area models is developed. In the Cornell's type seismic hazard analysis, the seismic region surrounding the analytical point is divided into small source-areas. In each source-area, parameters characterizing its seismic activities are assumed to be uniform and earthquakes are assumed to occur independently and randomly. Although the circular, fan-shaped, or rectangular models are commonly used as the shape of source-area models in Japan, they are too restrictive to represent the complicated shape of faults and plate boundaries. Hence, polygonal models are expected to be practical to represent them. In this paper, the formulation of the probability of epicentral distance by the use of a polygonal model is mainly described.

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