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After giving a suitable model for the cutting strips problem, we present a branchandprice algorithm for it by combining the column generation technique and the branchandbound method with LP relaxations.


Some theoretical issues and implementation details about the algorithm are discussed, including the solution of the pricing subproblem, the quality of LP relaxations, the branching scheme as well as the column management.


Let U(R,S) denote the class of all (0,1)m × n matrices having row sum vector R and column sum vector S.


Enumeration of (0,1)matrices with constant row and column sums


Let fs,t(m,n) be the number of (0,1)  matrices of size m × n such that each row has exactly s ones and each column has exactly t ones (sm = nt).

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 Single storied industrial buildings composed of steel trusses and reinforcedconcrete columns are very common. As the upper joints are hinged, the stresses in the columns are not influenced by the elastic properties of the trusses; while the upper joints are rigid, methods of analysis are usually based on the assumption that the moment of inertia of a steel truss may be taken as equivalent to that of a beam. In this paper, the author making use of the principle of least work reviews the equations... Single storied industrial buildings composed of steel trusses and reinforcedconcrete columns are very common. As the upper joints are hinged, the stresses in the columns are not influenced by the elastic properties of the trusses; while the upper joints are rigid, methods of analysis are usually based on the assumption that the moment of inertia of a steel truss may be taken as equivalent to that of a beam. In this paper, the author making use of the principle of least work reviews the equations for calculating the anglechanges at the ends of a truss, and then illustrates their applications with two practical examples: one with flat roof and the other with gabledroof. They are solved respectively by the method of slopedeflection for the cases of nosidesway, sideswaycorrection and sidesway included by solving the elastic equations of unit deformation. The results are compared with those obtained with usual assumptions.  單層廠房採用鋼架砼柱之混合結構,極為普遍,上端鉸結時柱之應力並不受桁架彈性常數之影響,上端剛結時之分析方法,對於鋼架之截面慣矩每予假定,並代之以假想變梁,本文作者用最小功能定律以求計算桁架兩端角變常數之公式;再取二實例:一平頂屋架,一脊式屋架,用角變位移法分就無側欹、侧欹校正及單位變位彈性方程解三步驟解出共結果,與一般假定方法所得相互比較。  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 socalled "truss rigid frames" are those rigid frames with trusses as their horizontal beams, of which the two ends are rigidly connected to columns. Within the author's knowledge, all the methods available at present for analyzing such rigid frames are based on Certain special assumptions such as (1) that the positions of the points of contraflexure in all the columns are previously known; (2) that the end rotations of a truss may be reprensented by that of its assumed line of axis as in the... The socalled "truss rigid frames" are those rigid frames with trusses as their horizontal beams, of which the two ends are rigidly connected to columns. Within the author's knowledge, all the methods available at present for analyzing such rigid frames are based on Certain special assumptions such as (1) that the positions of the points of contraflexure in all the columns are previously known; (2) that the end rotations of a truss may be reprensented by that of its assumed line of axis as in the case of an ordinary beam; or (3) that the end verticals of trusses may be given certain prescribed deformations. Of course, the adoption of any of such assumptions leads to only approximate results inconsistent with the actual deformations of such rigid frames under any loading. Heretofore, the author did not know any correct method for analyzing such rigid frames. In this paper, the author presents two principles of the correct analysis of truss rigid frames. The first principle is that of "moment action on column" for computing the angle change constants of columns, and the second principle is that of "effect of spanchange in truss" for computing the angle and span change constants of trusses.As, for computing the angle change constants of a truss, the dummy unit moment is a couple applied to its end verticals, so, for computing the angle change constants of a column, the dummy unit moment must also be a couple applied to the section of column rigidly connected to the end of a truss, in order to effect a consistent deformation at the joint of the two. This is the first principle.A truss just like a curved or gabled beam of which the effect of spanchange can not be neglected, so truss rigid frames belong to the same category of what may be called "spanchange" rigid frames such as rigid frames with curved or gabled beams. Therefore the spanchange constants of trusses should be included besides their anglechange constants for analyzing truss rigid frames. This is the second principle.With the constants of columns and trusses are all computed in accordance with respectively the first and second principles mentioned above, truss rigid frames may be analyzed by any method including the effect of spanchange as in the case of rigid frames with curved or gabled beams, and the results thus obtained will be exactly the same as by the method of least work or deflections without any special assumptions.In this paper, after the two principles are described and the formulas for computing the constants of columns and trusses are derived, the correctness of the two principles are then proved by the methods of least work, deflections and slopedeflection. A twospan truss rigid frame is analyzed under the following three conditions:Ⅰ. Applying both of the two principles to obtain the correct results.Ⅱ. Applying only the first principle to show the discrepancies of neglecting the effect of spanchange in trusses as born out by comparing the results of Ⅱ with Ⅰ.Ⅲ. Applying neither of the two principles, and the truss rigid frames being analyzed by the special assumption (2) mentioned above with the line of axis at the bottom chord of truss, in order to show the discrepancies of neglecting the moment action on column as born out by comparing the results of Ⅲ with Ⅱ. For the sake of brevity, only the results are given in Tables 1 to 5 without computations in details.Although the discrepancies of neglecting the moment acticn on column are only slight as shown by comparing the results of Ⅲ with Ⅱ in Tables 2, 4 and 5, there is no reason why special assumptions should not be replaced by the correct principle of moment action on column to obtain correct results. As shown by comparing the results of Ⅱ with Ⅰ in Tables 2, 4 and 5, the discrepancies by neglecting the span change in trusses are generally considerable and, in certain particular part, as large as 3000%. Therefore, for the safe and economical design of truss rigid frames, the effect of spanchange in trusses should not be neglected in their analysis.Finally, for analyzing co  所謂“桁架剛構”即以桁架為横梁与柱相剛接之剛構。現下採用分析剛構之任一方法,以分析此項剛構时,均須採用種種特殊之假定而得近似之結果。據著者所知,中外書刊中似尚无此項剛構之正確分析法。於本文中,著者發表关於桁架剛構正確分析之兩項原理,即柱頂力矩作用与桁架跨变影響之兩项原理。前項原理使柱頂段之角夔与桁架端豎桿相同,以符合柱与桁架剛接处之連续性。後項原理指出桁架与曲梁(即拱)及折梁(即山墙式梁)相同係一種“跨变横梁”,故桁架刚構亦与拱式及山墙式剛構相同,係一種“跨变剛構”。若根據此兩项原理,分别计算柱与桁架兩端的撓曲常数,再用分析跨变刚構之任一分析法以分析此項刚構,則所得之枯果,与不作任何特殊假定用最少功法或变位法所得者完全相同。本文先說明此兩项原理及根據此兩項原理计算柱与桁架撓曲常數之方法。次取一最簡單之桁架刚構为例,證明此丙項原理之正確性。桁架刚構既与拱式及山墙式刚構同属於跨变刚構一類型,分析後者之任何方法均可用以分析前者,本文无須贅述。但取一兩跨之桁架刚構為例,列举所得之正確結果,与用近似法所得者相比较,藉以顯出近似法有相當巨大之差誤。關於階形之複式桁架刚構之分析,本文用“代替桁架”之辦法,但只說明其原則,不... 所謂“桁架剛構”即以桁架為横梁与柱相剛接之剛構。現下採用分析剛構之任一方法,以分析此項剛構时,均須採用種種特殊之假定而得近似之結果。據著者所知,中外書刊中似尚无此項剛構之正確分析法。於本文中,著者發表关於桁架剛構正確分析之兩項原理,即柱頂力矩作用与桁架跨变影響之兩项原理。前項原理使柱頂段之角夔与桁架端豎桿相同,以符合柱与桁架剛接处之連续性。後項原理指出桁架与曲梁(即拱)及折梁(即山墙式梁)相同係一種“跨变横梁”,故桁架刚構亦与拱式及山墙式剛構相同,係一種“跨变剛構”。若根據此兩项原理,分别计算柱与桁架兩端的撓曲常数,再用分析跨变刚構之任一分析法以分析此項刚構,則所得之枯果,与不作任何特殊假定用最少功法或变位法所得者完全相同。本文先說明此兩项原理及根據此兩項原理计算柱与桁架撓曲常數之方法。次取一最簡單之桁架刚構为例,證明此丙項原理之正確性。桁架刚構既与拱式及山墙式刚構同属於跨变刚構一類型,分析後者之任何方法均可用以分析前者,本文无須贅述。但取一兩跨之桁架刚構為例,列举所得之正確結果,与用近似法所得者相比较,藉以顯出近似法有相當巨大之差誤。關於階形之複式桁架刚構之分析,本文用“代替桁架”之辦法,但只說明其原則,不列出公式及算例。   << 更多相关文摘 
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