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An example shows that the differentiable structures can indeed be different, even though they are isomorphic.


the volume computations involve several different methods according to the parity of dimension, subgroup relations and arithmeticity properties.


Our proof is based on a different approach and is much more rigorous.


In the process we study the properties of different homogeneous models for ${\mathbb H}H(n).$


The constants obtained are independent of the dimension n and depend only on k,p, and the number of different eigenvalues of the matrix B.

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 It is pointed out in this paper that the following apparent discrepancies exist in Coulomb's Theory: (1) In any problem in mechanics, a force to be definite must have all the three factors involved under consideration. In Coulomb's Theory, however, the point of application of the soil reaction on the plane of sliding is somehow neglected, thus enabling the arbitrary designation of the obliquity of the earth pressure on the wall to be equal to the friction angle between the wall surface and soil. As a matter... It is pointed out in this paper that the following apparent discrepancies exist in Coulomb's Theory: (1) In any problem in mechanics, a force to be definite must have all the three factors involved under consideration. In Coulomb's Theory, however, the point of application of the soil reaction on the plane of sliding is somehow neglected, thus enabling the arbitrary designation of the obliquity of the earth pressure on the wall to be equal to the friction angle between the wall surface and soil. As a matter of principle, the point of application should never be slighted while the obliquity of the earth pressure could only have a value that is compatible with the conditions for equilibrium. (2) If the point of application of the soil reaction is taken into account in the problem, the sliding wedge would only tend to slide either on the plane of sliding or on the surface of wall but not on both at the same time, thus frustrating the very conceptidn of sliding wedge upon which Coulomb's Theory is founded. (3) The above discrepancies arise from the fact that the shape of the surface of sliding should be curvilinear in order to make the wedge tend to slide as desired, while Coulomb, however, adopted a plane surface instead. (4) Coulomb, in finding the plane of sliding, made use of the maximum earth pressure on the wall (for active pressure), which refers to the different magnitudes of pressure corresponding to different assumed inclinations of the plane of sliding. But from the relation between the yield of wall and amount of pressure, this maximum value is really the minimum pressure on the wall, which it is the purpose of the theory to find. In engineering books, however, this terminology of maximum pressure has caused considerable confusion, with the result that what is really the minimum pressure is carelessly taken as the maximum design load for the wall. How can a minimum load be used in a design?This paper also attempts to clarify some contended points in Rankine's Theory: (1) It is claimed by Prof. Terzaghi that Rankine's Theory is only a fallacy because of the yield of wall and that of the soil mass on its bed. This charge is unjust as it can be compared with Coulomb's Theory in the same respect. (2) Some books declare that Rankine's Theory is good only for walls with vertical back, but it is proved in this paper that this is not so. (3) It is also generally believed that Rankine's Theory is applicable only to wall surfaces with no friction. This is likewise taken by this paper as unfounded and illustration is given whereby, in this regard, Rankine's Theory is even better than Coulomb's, because it contains no contradiction, as does Coulomb's.  本文從力學觀點對庫隆理論提出下列問題:(1)在解算力學問題時,每個力有三個因素都該同時考慮,但庫隆對土楔滑動面上土反力的施力點竟置之不理,因而才能對擋土墙上土壓力的傾斜角作一硬性假定,使它等於墙和土間的摩阻角,然而施力點是不能不管的,因而土壓力的傾斜角是不能離開平衡條件而被隨意指定的。(2)如果考慮了土反力的施力點,則土楔祇能在滑動面上,或在墙面上,有滑動的趨勢,而不能同時在兩個面上都有滑動的趨勢,因而庫隆的基本概念“滑動土楔”就站不住了。(3)問題關鍵在滑動面的形狀;如要使土楔在滑動面和墙面上同時有滑動趨勢,則滑動面必須是曲形面,然而庫隆採用了平直形的滑動面。(4)庫隆的土楔滑動面是從墙上最大的土壓力求出的(指主動壓力),這裏所謂“最大”是指適應各個滑動面的各個土壓力而言,但對適應墙在側傾時土壓力應有的變化來說,這個最大土壓力却正是墙上極限壓力的最小值。一般工程書籍,以為這土壓力既名為最大,就拿它來用作設計擋土墙的荷載,荷載如何能用最小的極限值呢?本文對朗金理論中的下列問題作了一些解釋:(1)朗金理論在擋土墙的位移問題上所受的限制,是和庫隆理論一樣的,竇薩基教授曾就此問題認為朗金理論是幻想,似乎是無根據的。... 本文從力學觀點對庫隆理論提出下列問題:(1)在解算力學問題時,每個力有三個因素都該同時考慮,但庫隆對土楔滑動面上土反力的施力點竟置之不理,因而才能對擋土墙上土壓力的傾斜角作一硬性假定,使它等於墙和土間的摩阻角,然而施力點是不能不管的,因而土壓力的傾斜角是不能離開平衡條件而被隨意指定的。(2)如果考慮了土反力的施力點,則土楔祇能在滑動面上,或在墙面上,有滑動的趨勢,而不能同時在兩個面上都有滑動的趨勢,因而庫隆的基本概念“滑動土楔”就站不住了。(3)問題關鍵在滑動面的形狀;如要使土楔在滑動面和墙面上同時有滑動趨勢,則滑動面必須是曲形面,然而庫隆採用了平直形的滑動面。(4)庫隆的土楔滑動面是從墙上最大的土壓力求出的(指主動壓力),這裏所謂“最大”是指適應各個滑動面的各個土壓力而言,但對適應墙在側傾時土壓力應有的變化來說,這個最大土壓力却正是墙上極限壓力的最小值。一般工程書籍,以為這土壓力既名為最大,就拿它來用作設計擋土墙的荷載,荷載如何能用最小的極限值呢?本文對朗金理論中的下列問題作了一些解釋:(1)朗金理論在擋土墙的位移問題上所受的限制,是和庫隆理論一樣的,竇薩基教授曾就此問題認為朗金理論是幻想,似乎是無根據的。(2)有些工程書中認為朗金理論是專為垂直的墙?  This paper is a supplement to the author's previous paper "The Constants and Analysis of Rigid Frames", published in the first issue of the Journal. Its purpose is to amplify as well as to improve the method of propagating joint rotations developed, separately and independently, by Dr. Klouěek and Prof. Meng, so that the formulas are applicable to rigid frames with nonprismatic bars and of closed type. The method employs joint propagation factor between two adjacent joints as the basic frame constant and the... This paper is a supplement to the author's previous paper "The Constants and Analysis of Rigid Frames", published in the first issue of the Journal. Its purpose is to amplify as well as to improve the method of propagating joint rotations developed, separately and independently, by Dr. Klouěek and Prof. Meng, so that the formulas are applicable to rigid frames with nonprismatic bars and of closed type. The method employs joint propagation factor between two adjacent joints as the basic frame constant and the sum of modified stiffness of all the barends at a joint as the auxiliary frame constant. The basic frame constants at the left of right ends of all the bars are computed by the consecutive applications of a single formula in a chain manner. The auxiliary frame constant at any joint where it is needed is computed from the basic frame constants at the two ends of any bar connected to the joint, so that its value may be easily checked by computing it from two or more bars connected to the same joint.Although the principle of this method was developed by Dr. Klouěek and Prof. Meng, the formulas presented in this paper for computing the basic and auxiliary frame constants, besides being believed to be original and by no means the mere amplification of those presented by the two predecessors, are of much improved form and more convenient to apply.By the author's formula, the basic frame constants in closed frames of comparatively simple form may be computed in a straightforward manner without much difficulties, and this is not the case with any other similar methods except Dr. Klouěek's.The case of sidesway is treated as usual by balancing the shears at the tops of all the columns, but special formulas are deduced for comput ing those column shears directly from joint rotations and sidesway angle without precomputing the moments at the two ends of all the columns.In the method of propagating unbalanced moments proposed by Mr. Koo IYing and improved by the author, the unbalanced moments at all the barends of each joint are first propagated to the barends of all the other joints to obtain the total unbalanced moments at all the barends, and then are distributed at each joint only once to arrive at the balanced moments at all the barends of that joint. Thus the principle of propagating joint rotations with indirect computation of the barend moments is ingeneously applied to propagate unbalanced moments with direct computation of the barend moments, and, at the same time, without the inconvenient use of two different moment distribution factors as necessary in all the onecycle methods of moment distribution. The basic frame constant employed in this method is the same as that in the method of propagating joint rotations, so that its nearest approximate value at any bar end may be computed at once by the formula deduced by the author. Evidently, this method combines all the main advantages of the methods proposed by Profs.T. Y. Lin and Meng ChaoLi and Dr. Klouěek, and is undoubtedly the most superior onecycle method of moment distribution yet proposed as far as the author knows.Typical numerical examples are worked out in details to illustrate the applications of the two methods.  本文為著者前文“剛構常數與剛構分析”之補充,其目的在將角變傳播法及不均衡力矩傳播法加以改善,以便實用。此二法均只需一個公式以計算剛構中所有各桿端之基本剛構常數(即任何二相鄰結點间之角變傳播係數),將此項公式與柯勞塞克之公式相比較,藉以指出前者較後者為便於應用,並亦可用之以直接分析較簡單之閉合式剛構,此外補充說明此法中之剛構常數與定點法之關係,剛構有側移時計算各結點角變所需之各項公式亦行求出。不均衡力矩傳播法係顧翼鹰同志最近研究所得者,既係直接以桿端力矩為計算之對象,而且只須採用不均衡力矩分配比將各結點作用於各桿端不均衡力矩之總和,一次分配,即得所求各桿端分配力矩之總值,實係力矩一次分配法之一大改進,著者將顧氏之法加以推廣与改善,使其原則簡明而計算便捷,著者認為此法係將林、柯、孟三氏法之所有優點熔冶於一爐,實可稱為现下最優之力矩一次分配法。最後列舉算例,以說明此二法在實際工作中之應用。  The method of complementary I_0/I diagram for simplifying the computations of nonuniform beam constants is presented in this paper. The socalled "complementary I_0/I diagram" is the remaining I_0/I diagram of the haunched or dehaunched (or tapered) parts at the two ends of a beam after the I_0/I diagram of a nonuniform 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 nonuniform beam constants is presented in this paper. The socalled "complementary I_0/I diagram" is the remaining I_0/I diagram of the haunched or dehaunched (or tapered) parts at the two ends of a beam after the I_0/I diagram of a nonuniform 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 nonuniform 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 anglechange constants are the fundamental constants of a nonuniform beam, and only the coefficients of the anglechange constants need be computed. As any nonuniform beam may be considered as a uniform beam haunched or dehaunched 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~aφ~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 dehaunched. 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 Kvalues for calculating the values of φ~a and φ~b of the shape anglechanges and the load anglechanges 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 Kvalues known are very simple; (2) the approximation of the results obtained is very close; (3) A single set of such Kvalue of the tables is applicable to nonuniform beams of any shape, any makeup, and any crosssection; and (4) as the Kvalues are by far easier to compute than any other constants, a comprehensive set of the tables of Kvalues 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?   << 更多相关文摘 
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