本稿の主題は、最短木の辺の約90%は第一近で構成されていることを示すことである。より具体的には69%が第一近、22％が第二近、残り９％は第三近以遠の辺から構成されていることが示された。この事実から、一辺１の正方形内に N個の点がランダムに分布しているときの最短木の長さの下限値は、従来よりも改善されて、0.64√N以上と推定されることが示された。

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本稿の主題は、地域構造を表現するネットワークの一つである近隣木を取り上げ、その長さの上限値と下限値を理論的に考察し、さらに実用的な長さの推定式を導出することである。本稿で得られた結果は次の 3つに要約できる。結果１「頂点数Ｎが 20個未満のとき、近隣木は第一近で構成され、第三近が使用される可能性はきわめて低い」結果２「一辺１の正方形内にＮ個の点がランダムに一様分布しているとき、近隣木の長さの上限値は 0.695√Ｎ以下と推定できる」結果３「一辺１の正方形内にＮ個の点がランダムに一様分布しているとき、近隣木の長さの実用的推定値は 0.666√Nである」

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本稿の主題は、最短木の辺の約90%は第一近で構成されていることを示すことである。より具体的には69%が第一近、22%が第二近、残り9%は第三近以遠の辺から構成されていることが示された。この事実から、一辺１の正方形内にN個の点がランダムに分布しているときの最短木の長さの下限値は、従来よりも改善されて、0.64√N以上と推定されることが示された。

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Timoshenko(1959) set up the two supported conditions by the 1^st ; the simple support along edges, and the 2^nd ; the fixed support along edges, and suggested the calculation formulas through the concentrated load on a deflection in the center of rectangular flat slabs. This method, however, was only a solution of the specific slenderness ratio, and the support conditions were not restraint along edge of the actual slabs, but using his calculation formulas, it was insufficient. The objective of this study is to frame calculation formulas of a deflection in the center of rectangular flat slabs of arbitrary slenderness ratios when the concentrated load is made up in the center of rectangular flat slabs. The author calculated such support conditions as the following, the 1^st ; the simply support along edges, the 2^nd ; the fixed support along edges, the 3^rd ; the simple support along two adjoining and the other edges built-in edges, the 4^th ; the semi-fixation.

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直交異方性平板を対象とした既往の研究では, Huber の仮定に代表されるような, 振り剛性に関する何らかの仮定を導入しているものが殆どである. しかし, Huber の仮定に拘束されない任意の曲げ・振り剛性を有する一般的なスラブについての解析的な研究は殆どなされていないのが現状のようである. このような研究の現況に鑑み, 本研究は任意の曲げ・涙り剛性を有する周辺固定平板, 三辺固定一辺自由平板, 二固定二辺自由平板の3種の直交異方性平板に対する解析について論じ, Huber の仮定の妥当性と適用範囲についての考察を行ったものである.

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The basic measure against a vibration proof design is to take effective steps to vibration proofs and to keep vibrating generators and precision machines apart as far as possible. As one of the effective measures against the vibration proof, a vibrating generator is installed on the underground stories. But it's impossible to install all the vibrating generators there. The author can't help installing the medium vibrating generators on the flat slabs near the precision instruments. The purpose of this study is limited to solve the effective mass for the slabs with arbitrary thin length ratio. As support conditions the author calculated the 1^st; the simply support along edges, the 2^nd; the fixed support along edges, the 3^rd; two adjacent edges simply supported and the other edges built-in, the 4^th; the semi-fixation.

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Two types of the single fourier series solution on the bending of the plate with adjacent edges clamped and the other two edges free when the lather corner is either perfectly free or simplay supported are obtained by means of the fourier transformations. Numerical values are given for uniformly loaded square plates and the two types of solutions-the sine series solution and the modified cosine series solution-are compared with each other as to their convergencies.

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Based on Mindlin's plate theory, free vibration of rectangular plates with combinations of free and clamped edges is investigated by using the method of superimposition. For the Mindlin plate the boundary conditions on an edge x=constant are as follows : for clamped edge, w=θ_x=θ_y=0 : for free edge, M_x=M_<Xy>=Q_x=0. The rectangular plates under consideration are as follows ; plates with clamped three edges and other free ; plates with two adjacent clamped edges and others free. Eigenvalues of four digit accuracy are provided for a wide range of plate ratios, thickness to width ratios and modal shapes. Numerical results are compared with those of classical solutions.

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This is a study of calculation formulas of first natural frequency of rectangular flat slabs of arbitrary width with various edge conditions. Formerly, only the calculations formulas on both the first natural frequencies the rectangular flat slabs with simple supported edges and all edges built-in have been proposed as the edge conditions. The arbitrary edge conditions except both of them have not been proposed so far. So an author derived the generalized calculation formulas of the first natural frequencies from the various edge conditions. The first natural frequencies of the ordinary flat slabs are intermediate between the simple supported edges and all edges built-in. So the author has defined the restraint along edges of the rectangular flat slabs as the former is assigned 1 and the latter is assigned 2. From the above-mentioned, the author has estimated that the first natural frequencies of the ordinary flat slabs would be one and a half times of the simple supported edges.

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In the light of the writers' measurements on load-bearing behavior of variedly dimensioned reinforced concrete slab models with free edges, the modified Johansen method with the effect of concrete tensile strength taken into account and Park's method are examined herein respecting their validity in such cases of two-way slabs. In consequence, that of the former method, including its relative accuracy and feasible extent of application, is shown to be substantially limited because in the present cases the general mode of failure is more dominated by torsion and/or shear types than in the case of many otherwise supported slab systems subject to failures of generally bending types.

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本稿の主題は、地域構造を表現するネットワークの一つである近隣木を取り上げ、その長さの上限値と下限値を理論的に考察し、さらに実用的な長さの推定式を導出することである。本稿で得られた結果は次の3つに要約できる。結果１「頂点数Ｎが20個未満のとき、近隣木は第一近で構成され、第三近が使用される可能性はきわめて低い」結果２「一辺１の正方形内にＮ個の点がランダムに一様分布しているとき、近隣木の長さの上限値は0.695√Ｎ以下と推定できる」結果３「一辺１の正方形内にＮ個の点がランダムに一様分布しているとき、近隣木の長さの実用的推定値は0.666√Nである」　

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In this paper, it is presented in-plane vibration analysis of rectangular plates. By using the method of superimposition, it is shown that solutions which satisfy identically the differential eqaution and which satisfy the boundary conditions with any desired degree of accuracy are obtained. The rectangular plates under consideration are as follows : a) Free- Clamped-Free-Claraped Plate, b) Clamped-Clamped-Clamped-Clamped Plate, c) Clamped-Free-Clamped-Clamped Plate and d) Clamped-Free-Free-Clamped Plate. Eigenvalues of four digit accuracy are provided for a wide range of plate aspect ratios. Accurate modal shapes are provided for the response of each rectangular plates.

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This paper presents a numerical method, the BF-spline Ritz method, for free vibration analysis of skew Mindlin plates with arbitrary boundary conditions. The proposed method utilizes admissible functions comprising the B-spline functions multiplied by a boundary function to define the trial function for the transverse displacement and rotations of the Mindlin plates. The geometric boundary conditions of the plate at edges are automatically satisfied using boundary functions. To demonstrate the validity and accuracy of the BF-spline Ritz method, several examples are solved, and results are compared with those obtained by other numerical methods. Excellent convergence and high accuracy are obtained by the present method regardless of boundary conditions. The effects of skew angles, thickness to width ratio, and aspect ratio on frequency parameters of skew Mindlin plates with clamped and free edges are shown in tabular forms to serve as benchmark data for future development of new numerical methods.

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 Formerly, the calculation formulas of the primary natural frequency of the both (A) all edges simply supported and (B) all edges clamped on the rectangular slabs have only been proposed. So Kojima(2012) had derived the calculation formulas of the primary natural frequency on the generalized support conditions of the rectangular flat slabs. The subjects of studies for support conditions on the rectangular flat slabs are (A) all edges simply supported, (B) all edges clamped, (C) two opposite edges simply supported and the other two edges clamped, (D) one edges simply supported and three edges clamped, (E) three edges simply supported and one edge clamped, (F) two adjacent edges simply supported and the other edges clamped, (G) all edges semi-clamped by Kojima(2012) between (A) all edges simply supported and (B) all edges clamped. Since the primary natural frequency of the ordinary flat slabs is intermediate between the simple support and the fix along edges, Kojima(2012) had defined restraints along edges of the flat slabs which are set to 1 by simple support along edges of the flat slabs and set to about 2 by fix along edges. Through the definition of the new restraints along edges, Kojima(2012) had estimated the primary natural frequency of the ordinary flat slabs is intermediate between all edges simply supported and clamped on the rectangular flat slabs, which are the numerical value is 1.5 times of all edges simply supported slabs. Kojima(2013a) had proposed the calculation formulas of the effective mass coefficient that the concentrated load affected the rectangular flat slabs. As the support conditions, Kojima(2013a) had calculated (A) all edges simply supported, (B) all edges clamped, (F) two adjacent edges simply supported and the other edges clamped and (G) all edges semi-clamped by Kojima(2012). Through a series of these processes, Kojima(2013b) had derived the calculation formulas of the maximum deflections from the concentrated load on the rectangular flat slabs. Timoshenko(1959) proposed the accurate solutions of the deflection on the rectangular flat slabs through both the all edges simply supported and clamped with the both the concentrated load and the uniformly distributed load. Kojima(2106) has proposed the following calculation formulas of the effective mass coefficients and the maximal deflection amount with the uniform load on the rectangular flat slabs. The support conditions on the rectangular flat slabs are the support conditions of (A) to (G).
 It's said to a support conditions on flat slabs of real buildings is the (G) all edges semi-clamped by Kojima(2012) between (A) all edges simply supported and (B) all edges clamped. The calculation formulas of the maximum deflection with the uniform load under the support conditions of (A) to (F) was the solutions of the Fourier polynomial proposed by Timoshenko (1959).
 This paper is the following calculation formulas of the maxim deflection with the uniform load on the rectangular flat slabs. The subjects of studies for support conditions on the rectangular flat slabs are (A) to (F) and (G) all edges semi-clamped by Kojima(2012). The support conditions on the flat slabs of the real buildings have been said to be (G) all edges semi-clamped between (A) all edges supported and (B) all edges clamped. The author has proposed the solutions incorporating the effective mass coefficient and the polynomial solutions of the function on the slenderness ratio λ by using the deflection amount at the center of the rectangular flat slabs with uniformed load under the various support conditions by Timoshenko(1959) and FEM analysis.

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This paper is the second report in a series of studies on finite element analysis of square column footings under vertical load. In this paper, many square column footings with and without footing girders were analyzed by three dimensional finite element method within the limits of elasticity. In addition, two of those column footing specimens experimented by F. E. Richart were analyzed by three dimensional finite element method within the limits of elasticplasticity and the results of the analysis were compared with the experimental data. After examining the results of the analysis described above, the following items were confirmed : a) The magnitude of stresses and the stress distributions obtained by elastic finite element analysis change according to the condition of soil and footing slab. b) The ratios of the moment obtained by elastic analysis to the moment calculated on the assumption of uniform distribution of soil reaction have a tendancy to become small with decreasing the relative rigidity of footing slab to soil. Under certain circumstances, the stress change due to the variation of this relative rigidity is more than that produced by cracking. c) In the case of the footing slab with the footing girders of which the stiffnesses are large enough, it is considered that the footing slab can be designed as a thin plate with two adjacent free sides and two adjacent fixed sides on the theory of thin plates, but in that case the footing girders should be designed considering the soil reaction under the girders. d) As the results of elastic-plastic analysis of column footings on the assumption that the elastic coefficient of soil is 10 kg/(cm)^3, the relationships of load-average strain in bar close to the experimental data at small strain levels.

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