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    The Generalized Equation of Consolidation Theory and Its Application
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    Based on the small deflection theory of plate and in consideration of the physical nonlinearity of concrete,the dynamic equation of concrete rectangular plate subjected to thermal environment on two-parameters elastic foundation is derived.
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    In conjunction with specific project of DOT running section for Shanghai Rail Transit, based on analyses of on site geologic data and testing data, from PECK equation, we derived a calculation formula in theory of surface deformation happening to DOT tunnel construction, and thence afterwards, a comparative analysis was made to the calculation results against on-site monitoring data.
    本文结合上海轨道交通双圆区间隧道的施工工程,在分析现场地质资料以及试验数据和Peck公式的基础上,推导出双圆盾构隧道施工引起地表变形的理论计算公式,并将计算结果与现场监测数据进行对比分析。
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For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation.
      
The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type.
      
The inverse conductivity problem to the the elliptic equation ${\rm div}((1+(k-1)\chi_D)\nabla u)=0\ {\rm in }\ \Omega$ is considered.
      
As applications, the wave equation on?+ × ?+ and the heat equation in a semi-infinite rod are considered in detail.
      
Pointwise fourier inversion: A wave equation approach
      
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Fineness modulus (F. M.) has served as an index of fineness of aggregates since it was first introduced by Prof. Duff A. Abrams in 1918. In the concrete mix design, the F. M. of sand governs the sand content and hence the proportions of other ingredients. But there are undesirable features in F. M.: it does not represent the grading of sand and manifests no significant physical concept.Prof. suggested an "average diameter" (d_(cp)) in 1943 as a measure of fineness of sand. In 1944, d_(cp) was adopted in 2781-44...

Fineness modulus (F. M.) has served as an index of fineness of aggregates since it was first introduced by Prof. Duff A. Abrams in 1918. In the concrete mix design, the F. M. of sand governs the sand content and hence the proportions of other ingredients. But there are undesirable features in F. M.: it does not represent the grading of sand and manifests no significant physical concept.Prof. suggested an "average diameter" (d_(cp)) in 1943 as a measure of fineness of sand. In 1944, d_(cp) was adopted in 2781-44 as national standard to specify the fine aggregate for concrete in USSR. It was introduced to China in 1952 and soon becomes popular in all technical literatures concerning concrete aggregates and materials of construction.After careful and thorough investigation from ordinary and special gradings of sand, the equation of d_(cp) appears to be not so sound in principle and the value of d_(cp) computed from this equation is not applicable to engineering practice. The assumption that the initial average diameter (ν) of sand grains between consecutive seives is the arithmetical mean of the openings is not in best logic. The value of an average diameter computed from the total number of grains irrespective of their sizes will depend solely on the fines, because the fines are much more in number than the coarses. Grains in the two coarser grades (larger than 1.2 mm or retained on No. 16 seive) comprising about 2/5 of the whole lot are not duly represented and become null and void in d_(cp) equation. This is why the initiator neglected the last two terms of the equation in his own computation. Furthermore, the value of d_(cp) varies irregularly and even inversely while the sands are progressing from fine to coarse (see Fig. 4).As F. M. is still the only practical and yet the simplest index in controlling fineness of sand, this paper attempts to interpret it with a sound physical concept. By analyzing the F. M. equation (2a) in the form of Table 9, it is discovered that the coefficients (1, 2…6) of the separate fractions (the percentages retained between consecutive seives, a1, a2…a6) are not "size factors" as called by Prof. H. T. Gilkey (see p. 93, reference 4), but are "coarseness coefficients" which indicate the number of seives that each separate fraction can retain on them. The more seives the fraction can retain, the coarser is the fraction. So, it is logical to call it a "coarseness coefficient". The product of separate fraction by its corresponding coarseness coefficient will be the "separate coarseness modulus". The sum of all the separate coarseness moduli is the total "coarseness modulus" (M_c).Similarly, if we compute the total modulus from the coefficients based on number of seives that any fraction can pass instead of retain, we shall arrive at the true "fineness modulus" (M_f).By assuming the initial mean diameter (ν') of sand grains between consecutive seives to be the geometrical mean of the openings instead of the arithmetical mean, a "modular diameter" (d_m), measured in mm (or in micron) is derived as a function of M_c (or F. M.) and can be expressed by a rational formula in a very generalized form (see equation 12). This equation is very instructive and can be stated as a definition of mqdular diameter as following:"The modular diameter (d_m) is the product of the geometrical mean ((d_0×d_(-1))~(1/2) next below the finest seive of the series and the seive ratio (R_s) in power of modulus (M_c)." If we convert the exponential equation into a logarithmic equation with inch as unit, we get equation (11) which coincides with the equation for F. M. suggested by Prof. Abrams in 1918.Modular diameter can be solved graphically in the following way: (1) Draw an "equivalent modular curve" of two grades based on M_c (or F. M.) (see Fig. 6). (2) Along the ordinate between the two grades, find its intersecting point with the modular curve. (3) Read the log scale on the ordinate, thus get the value of the required d_m corresponding to M_c (see Fig. 5).As the modular diameter has a linear dimension with a defin

細度模數用為砂的粗細程度的指標,已有三十餘年的歷史;尤其是在混凝土的配合上,砂的細度模數如有變化,含砂率和加水量也要加以相應的調整,才能維持混凝土的稠度(以陷度代表)不變。但是細度模數有兩大缺點,一個是模數的物理意義不明,另一個是模數不能表示出砂的級配來。蘇聯斯克拉姆塔耶夫教授於1943年提出砂的平均粒徑(d_(cp))來,以為砂的細度指標;雖然平均粒徑仍不包含級配的意義,但是有了比較明確的物理意義,並且可以用毫米來度量,這是一種新的發展。不過砂的細度問題還不能由平均粒徑而得到解决,且平均粒徑計算式中的五項,僅首三項有效,1.2和2.5毫米以上的兩級粗砂在計算式中不生作用,以致影響了它的實用效果。本文對於平均粒徑計算式的創立方法加以追尋和推演,發現其基本假設及物理意義,又設例演算,以考察其變化的規律性;認為細度模數還有其一定的實用價值,不能為平均粒徑所代替。至於補救細度模數缺點的方法,本文試由模數本身中去尋找;將模數的計算式加以理論上的補充後,不但能分析出模數的物理意義,並且還發現模數有細度和粗度之別。根據累計篩餘計算出來的F.M.應稱為“粗度模數”,根據通過量計算出來的才是“細度模數”。假定兩隣篩间的顆粒是...

細度模數用為砂的粗細程度的指標,已有三十餘年的歷史;尤其是在混凝土的配合上,砂的細度模數如有變化,含砂率和加水量也要加以相應的調整,才能維持混凝土的稠度(以陷度代表)不變。但是細度模數有兩大缺點,一個是模數的物理意義不明,另一個是模數不能表示出砂的級配來。蘇聯斯克拉姆塔耶夫教授於1943年提出砂的平均粒徑(d_(cp))來,以為砂的細度指標;雖然平均粒徑仍不包含級配的意義,但是有了比較明確的物理意義,並且可以用毫米來度量,這是一種新的發展。不過砂的細度問題還不能由平均粒徑而得到解决,且平均粒徑計算式中的五項,僅首三項有效,1.2和2.5毫米以上的兩級粗砂在計算式中不生作用,以致影響了它的實用效果。本文對於平均粒徑計算式的創立方法加以追尋和推演,發現其基本假設及物理意義,又設例演算,以考察其變化的規律性;認為細度模數還有其一定的實用價值,不能為平均粒徑所代替。至於補救細度模數缺點的方法,本文試由模數本身中去尋找;將模數的計算式加以理論上的補充後,不但能分析出模數的物理意義,並且還發現模數有細度和粗度之別。根據累計篩餘計算出來的F.M.應稱為“粗度模數”,根據通過量計算出來的才是“細度模數”。假定兩隣篩间的顆粒是兩篩篩孔的幾何平均值,以代替數學平均值(即斯氏平均?

The method of complementary I_0/I diagram for simplifying the computations of non-uniform beam constants is presented in this paper. The so-called "complementary I_0/I diagram" is the remaining I_0/I diagram of the haunched or de-haunched (or tapered) parts at the two ends of a beam after the I_0/I diagram of a non-uniform beam has been subtracted from the I_0/I = 1 diagram of a uniform beam.In the method of I_0/I diagram presented previously by the second author, the various momental areas have to be computed...

The method of complementary I_0/I diagram for simplifying the computations of non-uniform beam constants is presented in this paper. The so-called "complementary I_0/I diagram" is the remaining I_0/I diagram of the haunched or de-haunched (or tapered) parts at the two ends of a beam after the I_0/I diagram of a non-uniform beam has been subtracted from the I_0/I = 1 diagram of a uniform beam.In the method of I_0/I diagram presented previously by the second author, the various momental areas have to be computed for the entire length of a beam; in the method of complementary I_0/I diagram, the various momental areas need be computed for the lengths of the non-uniform sections at the two ends of the beam only. Hence the latter method is somewhat simpler than the former and may be considered as its improvement.The angle-change constants are the fundamental constants of a nonuniform beam, and only the coefficients of the angle-change constants need be computed. As any non-uniform beam may be considered as a uniform beam haunched or de-haunched or tapered at its one or both ends, the various anglechange coefficients φ may be computed separately in three distinct parts, viz., of a uniform beam, and φ~a and φ~b of the haunches at its two ends a and b, and then summed up as shown by the following general equation:φ=φ~a-φ~b (A) The values φ~a and φ~b are positive for haunched beams and negative for dehaunched or tapered beams, and either of them is zero for the end which is neither haunched nor de-haunched. To simplify the computations of the values of φ~a and φ~b, the complementary I_0/I diagram at each end of a beam is substituted by a cubic parabola passing through its two ends and the two intermediate points of the abscissas equal to 0.3 and 0.7 of its length. Then the value of φ~a or φ~b is computed with an error of usually less than 1% by the following formula:φ~a or φ~b = K_(0y0)+K_(3y3)+K_(7y7), (B) wherein y0, y3 and y7 are respectively the ordinates at the abscissa equal to 0, 0.3, and 0.7 of the length of the diagram, and the three corresponding values K_0, K_3 and K_7 are to be found from the previously computed tables.A set of the tables of K-values for calculating the values of φ~a and φ~b of the shape angle-changes and the load angle-changes under various loading conditions may be easily computed, which evidently has the following advantages: (1) As indicated by formulas (A) and (B), the computations of φ~a, φ~b and φ with K-values known are very simple; (2) the approximation of the results obtained is very close; (3) A single set of such K-value of the tables is applicable to non-uniform beams of any shape, any make-up, and any crosssection; and (4) as the K-values are by far easier to compute than any other constants, a comprehensive set of the tables of K-values with close intervals and including many loading conditions may be easily computed.Besides, by means of formulas (A), existing tables of constants such as A. Strassner's for beams haunched at one end only may be utilized to compute the shape and load constants for asymmetrical beams with entirely different haunches at both ends.Finally, five simple but typical examples are worked out first by the approximate method and then checked by some precise method in order to show that the approximation is usually extremely close.

本文叙述一种I_0/I余圖法,以簡化变梁常数的計算。所謂I_0/I余圖,即自等截面梁的I_0/I=1圖減去变梁的I_0/I圖后所剩余的兩端梁腋的I_0/I圖。 於本文第二著者前此所建議的I_0/I圖法中,必須計算变梁全長的I_0/I圖的各次矩图面积,於I_0/I余圖法中,則只須計算变梁兩端梁腋的I_0/I余圖的各項积分值。故后法显此前法为簡單,亦可视作系前法的进一步的改善。 角变常数为变梁的基本常数,而所須計算者只是各項角变常数的系数φ,簡称为“角变系数”。任一形式的变梁均可视作一端或兩端的加腋梁或減腋梁。採用I_0/I余圖法,則变梁的各項角变系数φ的計算可分开为等截面梁的φ及其a与b兩端梁腋的φ~a与φ~b三部分而后綜合之,以公式表之,即於加腋梁φ~a与φ~b为正号;於減腋梁φ~a与φ~b为負号,於无梁腋之端則其φ~a或φ~b之值为霉。 計算梁腋的φa或φ~b值时,可用一根三次拋物線以代替I_0/I余圖而計算其各項积分的近似值。由是可得其中y_0,y_3及y_7为a或b端I_0/I余圖的三个豎距。如按变梁的形角变系数及其在各种荷載下的载角变系数將各項K值列成表格,則此項表格显有下列优点:(一)应用步驟簡單,...

本文叙述一种I_0/I余圖法,以簡化变梁常数的計算。所謂I_0/I余圖,即自等截面梁的I_0/I=1圖減去变梁的I_0/I圖后所剩余的兩端梁腋的I_0/I圖。 於本文第二著者前此所建議的I_0/I圖法中,必須計算变梁全長的I_0/I圖的各次矩图面积,於I_0/I余圖法中,則只須計算变梁兩端梁腋的I_0/I余圖的各項积分值。故后法显此前法为簡單,亦可视作系前法的进一步的改善。 角变常数为变梁的基本常数,而所須計算者只是各項角变常数的系数φ,簡称为“角变系数”。任一形式的变梁均可视作一端或兩端的加腋梁或減腋梁。採用I_0/I余圖法,則变梁的各項角变系数φ的計算可分开为等截面梁的φ及其a与b兩端梁腋的φ~a与φ~b三部分而后綜合之,以公式表之,即於加腋梁φ~a与φ~b为正号;於減腋梁φ~a与φ~b为負号,於无梁腋之端則其φ~a或φ~b之值为霉。 計算梁腋的φa或φ~b值时,可用一根三次拋物線以代替I_0/I余圖而計算其各項积分的近似值。由是可得其中y_0,y_3及y_7为a或b端I_0/I余圖的三个豎距。如按变梁的形角变系数及其在各种荷載下的载角变系数將各項K值列成表格,則此項表格显有下列优点:(一)应用步驟簡單,只有几个簡單的乘法与加減法;(二)所得結果的近似程度頗高,差誤一般不超过1%;(三)应用范圍广泛,只一套K值表可用於任何截面及?

Professor Zhmochken chooses cantilever beam as a basic structure for beams on elastic foundations. In his opinion, it is inconvenient to use a simple beam as a basic structure. The calculation of deflection constants is thus involved, and one kind of beam can not be used to represent all kinds of beams on elastic foundations.In this paper the writer tries to use a simple beam with two cantilevers as a basic structure. On the one hand by changing the panel number of cantilevers, one kind of beam may be used to...

Professor Zhmochken chooses cantilever beam as a basic structure for beams on elastic foundations. In his opinion, it is inconvenient to use a simple beam as a basic structure. The calculation of deflection constants is thus involved, and one kind of beam can not be used to represent all kinds of beams on elastic foundations.In this paper the writer tries to use a simple beam with two cantilevers as a basic structure. On the one hand by changing the panel number of cantilevers, one kind of beam may be used to represent all kinds of beams on elastic foundations. On the other hand, the deflection constants computed by the writer are given in the appendix, so the designer may use these constants as easily as he uses Professor Zhmochken's constants. With such a modification the number of normal equations is reduced by two to four.Besides, a method of solving normal equations by utilizing the property of reactive pressures between beams and foundations is suggested. All the normal equations may be solved combinedly by the method of eliminations and successive approximations. Two cycles are usually sufficient for the ordinary purpose.

本文系对苏联学者石氏关於彈性地基梁之理論作了适当的补充。石氏选悬臂梁作为彈性地基梁之輔助結構,本文則选二端附有悬臂之簡支梁。如是,寻求梁与地基之間的反力时,共用於建立法方程的工作,大致与石氏同,而用於求解該項方程的时間,則可大为縮短。

 
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