(文档含英文原文和中文翻译)中英文对照外文翻译基于拉格朗日乘数法的框架结构合理线刚度比的研究【摘要】框架结构是一种常见的多层高层建筑结构；列的合理线刚度比研究是框架结构优化设计中的一个重要方面。
本论文研究合理线刚度比时，框架梁、柱的侧移刚度根据拉格朗日乘数法结构优化的理论和在框架梁、柱的总物质的量一定的前提下，取得最高值。
与传统的估计方法和试算梁柱截面尺寸不同，梁、柱的合理的截面尺寸可以在初步设计阶段由派生的公式计算。
这种方法不仅作为计算框架梁、柱的截面尺寸基础，确认初步设计阶，而且也被用做类似的结构梁柱合理线刚度比研究的参考。
此外，在调整帧梁、柱的截面尺寸的方法的基础上，降低柱的轴向的压缩比，从而达到剪切压缩比和提高结构的延展性。
【关键词】拉格朗日数乘法框架结构刚度比截面尺寸1 引言在混凝土框架结构初步设计的期间，通常，框架梁截面高度通过跨度来估算，和截面宽度根据高宽比估算; 框架柱的截面尺寸是根据柱轴压缩的支持柱的面积的比率估算[1]。
然而，在估计过程中，初步设计阶段中的一个重要的链，未考虑到柱侧移刚度的影响[2]。
列侧移刚度越大，结构层间位的刚度越大，剪切型框架结构的层间位移将越较小。
所以，总结构越小的侧向位移将减少地震灾害[3]所造成的损失。
论文的核心是如何得到列侧移刚度的最大值。
同时，列侧移刚度的值与框架梁-柱线刚度直接相关。
本论文的目的是为了得到一个合理的框架梁 - 柱的线刚度比，在某个控制范围内获得列侧移刚度的最大值。
计算列横向位移的方法有两种方法：刚度拐点点法和修改拐点法。
拐点的方法假定关节的旋转角度为0（当梁柱线性刚度比是大于或等于3时，柱的上端和下端的关节的旋转角度可以取为0，因为它实际上是相当小），即梁的弯曲刚性被视为无穷大。
拐点的方法主要是应用于具有比较少层的框架结构。
但对于多层、高层框架结构，增加柱截面会导致梁柱线刚度比小于3，在水平荷载作用下，框架结构的所有关节的旋转角度的横向位移会发生不可忽视。
因此，一位日本教授武藤提出修改拐点法[4]，即D-值方法。
本文采用D-值列侧移刚度的计算法，因为它着重于多层、高层框架结构。
少数在国内外对框架梁柱合理线刚度比的研究，只有梁七黹，源于列侧移刚度的计算方法，比D-值法更加应用广泛；申得氏指出在多层、高层框架结构的柱侧向刚度计算中存在的问题，补充和修改底部和顶部层的列侧向刚度计算公式;应用于史密斯和库尔博士法，焱鑫田源于梁 - 柱线刚度比的合理值，由计算的最大等效刚度框架柱。
本文计算列侧移刚度的最大值,第一次通过采用在约束条件下结构优化理论，那就是，约束Lagrange乘子法优化理论对框架梁-柱的材料是有一定价值[5]。
因此，混凝土框架梁-柱的合理线刚度比在一定范围内可以得到。
在初步设计和参考阶段，得出的结论可以作为梁-柱的截面尺寸的一个决定性因素，类似在梁柱框架结构设计的研究2 用约束拉格朗日乘子法计算框架梁柱的合理线性刚度比2.1 侧向刚度的框架梁-柱的D值法以标准层框架结构的中间关节作为一个例子，由D值法计算出梁-柱的侧向位移刚度：:关节旋转的影响系数；:框架柱的线性刚度;:层高；:梁-柱的平均线性刚度;:梁柱的线性刚度;假设所有的梁-柱线性刚度都是，所以，考虑到框架梁铸在原址层的钢筋混凝土框架结构受限的影响，中间框架梁的转动惯量[6]；是梁截面的转动惯量，所以，因此，标准的框架柱侧移刚度得出以下公式：因为,侧移刚度列进一步推导出下述公式：2.2 基于拉格朗日乘数法得到合理线刚度比为了获得框架柱的侧移刚度最大D值，我们需要找到目标函数：（1）假设截面框架梁是和截面框架柱是，在材料总量是A的前提下，在材料总量是A的前提下，公式满足这个约束条件，得：（2）通过拉格朗日数乘法来获得目标函数：因为所以（3）E : 混凝土的弹性模量；同理，柱线性刚度可以用下面的公式推导：因此我们得到：（4）把公式（3）和公式（4）代入公式（2），得，可以进一步推导：（5）(T是定值)在一定约束条件下，根据拉格朗日数乘法，目标函数可以由公式（5）得到：（6）分别对求各自的偏导，并且令其偏导数为0，得：整理上述方程，得，（7）从等式（7）对开根号，得：（8）上面的公式是在框架结构标准层中间接头的柱的侧移刚度是最大时的梁-柱线性刚度比，即合理梁柱线刚度比。
同理，我们可以在框架结构柱的侧移刚度是最大时，得出标准层侧接头的合理线刚度比。
（9）3 工程梁柱合理线刚度的应用3.1多层和高层框架结构的合理线刚度应用在消耗材料量是一个定值前提下，框架结构的标准框架层的中间关节作为梁-柱线性刚度比满足式（8），且框架结构标准层框架侧接头作为线性刚度比满足公式（9），那么框架结构侧向位移刚度将一直保持最大值。
显然，那么总的侧向位移是最小的结构就在这时[7]; 其工程应用价值不言而喻。
在一般的框架结构中，柱高和梁跨度满足下面的公式：梁-柱的截面高度比满足下式[8]：对于框架结构的标准层的中间节点，我们可以由等式（8）推出：（10）上文中的计算公式在合理线刚度比的梁-柱框架结构的标准层中间接头的应用范围。
结果表明，梁-柱的截面尺寸可被相应的计算出来，如果梁-柱线性刚度比满足等式（10）时，框架柱的侧向位移则是最大值。
3.2 例子在一般负载下，一个10层的、4跨度的钢筋混凝土浇筑在原址框架结构，每层的高度是3.6米，梁跨度为7.2m，梁-柱的混凝土强度等级是一样的。
在材料量是定值的前提下，中间框的标准层梁 - 柱中间关节的截面尺寸通用估计方法估计，然后将估计值与由式8和式9计算的截面尺寸的结果相比较。
根据一般方法估算梁-柱的截面尺寸：梁:柱:则梁-柱材料的量一共是：那么，柱侧向位移的刚度比是：然而，线性刚度比是在等式10的应用范围之内。
基于等式10，在A是定值的条件下，计算梁柱的截面尺寸，且调整梁柱的截面尺寸，如下：然后，现在侧向位移刚度是：然后，显然，此时，梁柱的线性刚度比在等式10的范围之内，所以它就是合理线性刚度比并且在工程应用范围内。
4 结论（1）在上述的例子可以得出：在梁柱的材料消耗总量A是定值前提下，在标准框架结构的最初设计阶段，如果梁柱的截面尺寸可以调整，梁宽保持不变。
在这个例子中，梁宽保持不变，把柱高从650毫米调整到600毫米，然后把柱高从500毫米调整到560毫米，如果柱截面宽度保持不变，则会得到柱侧向位移刚度的最大值。
证明梁柱的线性刚度比此时满足等式（10），所以在标准框架结构的最初设计阶段，得到的梁柱截面尺寸在应用范围内的合理线性刚度比内。
(2) 该研究方法通过拉格朗日数乘法来获得柱侧向位移刚度最大值被广泛应用于研究类似的框架结构。
例如，可以用于研究中间框架底部、侧向框架、类似工程结构的合理线性刚度比(3) 这个研究的结论可以为框架结构其他方面的研究提供一定的参考。
例如，通过调整截面尺寸获得柱侧向位移刚度的最大值在框架结构的抗震设计中变得越来越重要。
增加柱的截面尺寸可以有效地控制轴压比，剪压缩比，从而提高结构延性和减少地震灾害造成的损失。
参考文献[1]Tao Ji, Zhixiong Huang, Multi-story and High-rise Reinforced ConcreteStructure Design, Michanical Industry Publishing House, Beijing, 2007.[2] Shihua Bao, High-rise Building Structure of New Edition.WaterResource and Hydropower Publishing House of China, Beijing, 2005.[3]Ahmed Ghobarah and A. Said, “Shear strengthening of Beam-columnJoints”, Journal of Engineering Structures,2002, 24（7）,pp.881-888.[4] Ahmed Ghobarah, Seism Resistance Design & Seism Resistance Methods[M], Maruzen Company, Limited. 1963.[5] Aichuan. Jiang, Structural Optimization Design, Qinghua PublishingHouse, Beijing, 1986.[6]Xi’an Zhao, High-rise Reinforced Concrete Structure Design,Architecture＆Building Press Beijing, China, 2003.[7]P.G. Bakir and H.M. Boduro.lu, “A New Design Equation for Predictingthe Joint Shear Strength of Monotonically Loaded ExteriorBeam-column Joints”, Journal of Engineering Structures, 2002, 24（8）, pp.1105-1117.[8]Huanling Meng and Pusheng Shen, “Research on Behaviors of Frame-shearWall Structures Based on Stiffness Degradation”, Journal of Railway Science and Engineering,2006, 3(1), pp.12–17.Study on Reasonable Linear Stiffness Ratio in FrameStructure Based on Lagrange Multiplier Method AbstractFrame structure is a common structure of multistory and high-rise buildings; research on column’s reasonable linear stiffness ratio is an important aspect on frame structure optimization design. The thesis researches on reasonable linear stiffness ratio when the frame beam-column’s lateral displacement stiffness reaches its maximum value based on Lagrange Multiplier Method structure optimization theory and on the premise that total material quantity of framework beam-column is definite. Different from traditional estimation methods and trial calculation on section dimension of beam-column, the reasonable section dimension of beam-column can be calculated by the derived formulas on preliminary design stage. This method is not only used as basis for the frame beam-column’s section dimension confirmation on preliminary design stage, but also taken as reference for research on beam-column’s reasonable linear stiffness ratio for similar structure. In addition, adjusted frame beam-column’s section dimension based on the method reduces the column’s axial compression ratio, shear compression ratio and improve the structural ductility.1 IntroductionDuring the preliminary design of concrete frame structures, generally, the section height of the frame beam is estimated by its span, and section width is estimated according to the height-width ratio; the section dimension of the frame column is estimated by the column axial compression ratio according to the column-supported floor area [1]. Therefore, effects from the column lateral displacement stiffness [2] are not taken into consideration in the process of section estimation, an important chain in the preliminary design stage. The bigger the column lateral displacement stiffness is,the bigger stiffness of structure story displacement will be, but the smaller story displacement in shear-type frame structure will be. As a result, smaller total structure lateral displacement will reduce the loss caused by earthquake disaster [3].The core of the thesis is how to get the maximum value of column lateral displacement stiffness. Meanwhile, column lateral displacement stiffness value is directly related with linear stiffness of frame beam-column.The purpose of the thesis is to get a reasonable linear stiffness ratio of frame beam column within a certain control range after deriving the maximum value of column lateral displacement stiffness.There are two methods of calculating the column lateral displacement stiffness-inflexion point method and modified inflexion point method. Inflexion point method assumes joint rotation angle as 0 ( when linear stiffness ratio of beam-column is more than or equal to 3, joint rotation angle of upper and lower ends of the column can be taken as 0 since it is actually quite small), namely flexural rigidity of beam is regarded as infinity.Inflexion method is mainly applied to the frame structures with fewer stories.But for multi-story and high-rise frame structures, since increasing column section makes beam-column linear stiffness ratio be less than 3, lateral displacement will occur on frame structures and rotation angle of all joints can not be neglected under horizontal load. Accordingly, Muto, a Japanese professor puts forward the modified inflexion point method [4], namely D-value method.The thesis adopts D-value method of calculating the column lateral displacement stiffness because it focuses onmulti-story and high-rise frame structures.Research on reasonable linear stiffness ratio of frame beam-column is few at home and abroad, only Liang Qizhi derives the calculation method of column lateral displacement stiffness which is applied more widely than D-value;Shen Dezhi points out the existing problems in column lateral stiffness calculation for multi-story and high-rise frame structure, supplements and modifies the column lateral stiffness calculation formula on bottom and top layer;applying Smith & Coull method, Yanxin Tian derives the reasonable value of beam-column linear stiffness ratio by calculating the maximal equivalent stiffness of frame column.The thesis calculates the maximal value of column lateral displacement stiffness for the first time by adopting the structure optimization theory under constraint conditions, that is, the constraint Lagrange Multiplier Optimization Theory when the material of frame beam-column is a definite value [5]. Thus, a reasonable lines stiffness ratio of concrete frame beam-column within a certain scope can be obtained.The conclusion can be taken as a decisive factor for section dimension of frame beam-column during preliminary design and the reference on the research of beam-column in similar structure design.2 Reasonable Linear Stiffness Ratio of Frame Beam-column Calculated by Constraint Lagrange Multiplier Method2.1 Lateral Displacement Stiffness of Frame Beam-Column by D Value MethodTaking the middle joint of standard floor frame structure as an example, lateral displacement stiffness of beam-column calculated by D value method:: Influence coefficient of joint rotation；: Linear stiffness of frame column;: Story height;: Average linear stiffness of floor beam-column;: Linear stiffness of beam-column；Suppose all linear stiffness of beam-column is ,Then,Considering the restriction effect on frame beam from cast-in-situ floor of reinforced concrete frame structures, the inertia moment of the middle frame beam [6];is the inertia moment of the beam section, then,Thus, lateral displacement stiffness of standard frame column is derived by the following formula:For , lateral displacement stiffness of columnis further derived from the following formula:2.2 Deriving the Reasonable Linear Stiffness Ratio Based On Lagrange Multiplier MethodIn order to get the maximal lateral displacement stiffness of frame column D value, we need to find the objective function:(1)Suppose the section of frame beam is and section of frame column is , on the premise of total amount of material is definite value A, formula meets the constraint condition, then,(2)To find objective function by Lagrange Multiplier Method:For,Then,(3)E : Elastic modulus of concrete;In the same way, the linear stiffness of column is derived by the following formula:Therefore we get：(4)It can be further derived:(5)(T is a definite value) According to the Lagrange Multiplier Methodunder certain constraints, the objective function can be derived by formula (5):(6)Calculation the partial derivatives of ,respectively and make the results equal to 0, then，Further derive the above formula, then，(7)Derive from Eq.7,(8)The above formula is linear stiffness ratio of beam-column standard floor middle joint in frame structure when the lateral displacement stiffness of column is maximal, namely reasonable linear stiffness ratio of beam-column. In the same way, we can derive the reasonable linear stiffness ratio of standard floor side joint in the frame structure when the lateral displacement stiffness of column is maximal.(9)3 Application of Reasonable Linear Stiffness of Beam-column in Engineering3.1 Application of Reasonable Linear Stiffness in Multi-layer and High-rise Frame StructuresOn the premise that consumed material amount is a definite value, to the middle joints of standard frame floor of frame structures as linear stiffness ratio ofbeam-column satisfies the formula (8) and to side joint of frame standard floor of frame structures as linear stiffness ratio satisfies the formula (9), the lateral displacement stiffness of frame structures will remain the maximum value all the time. Obviously, then the total lateral displacement of the structure is minimal at this moment [7]; its application value in engineering goes without saying. In common frame structures, column height and beam span satisfy the followingformula:h/l=0.3~0.6.The section height ratio of beam-column satisfies the formula [8]:To standard floor middle joint of frame structure, it can be derived from Eq.8,(10)The formula hereinabove is the applied scope of reasonable linear stiffness ratio of beam-column standard floor middle joint in frame structures. The result shows that the section dimension of beam-column can be calculated accordingly if only the linear stiffness ratio of beam-column satisfies Eq.10 and hence the lateral displacement stiffness of the frame column is maximal.3.2 ExampleA 10-story 4-span reinforced concrete cast-in-situ frame structure under general load, the height of each story is 3.6m, beam span is 7.2m, and concrete strength grade of beam-column is the same. On the premise that material amount is a definite value, the section dimension of beam-column middle joint of standard floors in middle frame is estimated by general estimation method and then compare it with calculated section dimension by the results of Eq.8 and Eq.9. Estimate the section dimension ofbeam-column according to general methods:Beam: Column:Then the material amount of beam-column isTherefore, the lateral displacement stiffness of column isHowever, the linear stiffness ratio is beyond the application scope of Eq.10.On the basis of Eq.10, calculate the section dimension of beam-column under the condition that A is a definite value, and adjust the section dimension of beam-column, as:,Then, Now the lateral displacement stiffness isThen, obviously,Now,The linear stiffness of beam-column is within the scope of Eq.10, so it is reasonable linear stiffness ratio and within the engineering application scope.4 Conclusion(1) It can be seen from the example hereinabove: on the premise that the total consumed material quantity A of frame beam-column is a definite value during the preliminary design in the standard frame structure, the beam width remains unchangeable if the section dimension of the beam-column is adjusted slightly. In this example, beam width remains unchangeable, adjust the column height from 650mm to 600mm and then adjust column height from 500mm to 560mm if the column dimension width remains unchangeable, the maximum value of the column lateral displacement stiffness will be obtained. proves that the linear stiffness ratio ofbeam-column at this moment satisfies the formula (10), so the derived beam-column section dimension satisfies the applied scope of the reasonable linear stiffness ratio during the preliminary design in the standard frame structure.(2) The research method obtaining the maximum value of column lateral displacement stiffness by Lagrange Multiplier Method can be widely used to research on the similar frame structures. For example it can be used to research the reasonable linear stiffness ratio of middle frame bottom, side frame, and similar engineering structures.(3) The conclusion of the research will provide certain reference to research on other aspects of frame structure. For example, the method obtaining the maximum value of column lateral displacement stiffness by adjusting its section dimension has great importance in anti-seismic design of frame structures. Increasing section dimension of columns can effectively control the axial compression ratio and shear compression ratio, and hence improve structural ductility and reduce loss due to earthquake disasters.References[1] Tao Ji, Zhixiong Huang, Multi-story and High-rise Reinforced Concrete StructureDesign, Michanical Industry Publishing House, Beijing, 2007.[2] Shihua Bao, High-rise Building Structure of New Edition.Water Resource andHydropower Publishing House of China, Beijing, 2005.[3] Ahmed Ghobarah and A. Said, “She ar strengthening of Beam-column Joints”,Journal of Engineering Structures,2002, 24（7）, pp.881-888.[4] Ahmed Ghobarah, Seism Resistance Design & Seism Resistance Methods [M],Maruzen Company, Limited. 1963.[5] Aichuan. Jiang, Structural Optimization Design, Qinghua Publishing House,Beijing, 1986.[6] Xi’an Zhao, High-rise Reinforced Concrete Structure Design, Architecture＆Building Press Beijing, China, 2003.[7] P.G. Bakir and H.M. Boduro.lu, “A New Design Equation for Predicting the JointShear Strength of Monotonically Loaded Exterior Beam-column Joints”, Journal of Engineering Structures, 2002, 24（8）, pp.1105-1117.[8] Huanling Meng and Pusheng Shen, “Research on Behaviors of Frame-shear WallStructures Based on Stiffness Degradation”, Journal of Railway Science andEngineering,2006, 3(1), pp.12–17.。