基于拉格朗日乘數法的框架結構合理線剛度比的研究外文翻譯

1、<p>　　畢業(yè)設計（論文）外文文獻翻譯</p><p>　　基于拉格朗日乘數法的框架結構合理線剛度比的研究</p><p>　　本文譯自：Zhiqin LIU,Guoliang BAI. Study on Reasonable Linear Stiffness Ratio in Frame Structure Based on Lagrange Multiplier Metho

2、d[J].Information Management,Innovation Management and Industrial Engineering,2008.ICIII’08.International Conference on,2008,9(6):368-371.</p><p>　　【摘要】框架結構是一種常見的多層高層建筑結構；列的合理線剛度比研究是框架結構優(yōu)化設計中的一個重要方面。本論文研究合理線剛

3、度比時，框架梁、柱的側移剛度根據拉格朗日乘數法結構優(yōu)化的理論和在框架梁、柱的總物質的量一定的前提下，取得最高值。與傳統(tǒng)的估計方法和試算梁柱截面尺寸不同，梁、柱的合理的截面尺寸可以在初步設計階段由派生的公式計算。這種方法不僅作為計算框架梁、柱的截面尺寸基礎，確認初步設計階，而且也被用做類似的結構梁柱合理線剛度比研究的參考。此外，在調整幀梁、柱的截面尺寸的方法的基礎上，降低柱的軸向的壓縮比，從而達到剪切壓縮比和提高結構的延展性。</p

4、><p>　　【關鍵詞】 拉格朗日數乘法 框架結構 剛度比 截面尺寸</p><p><b>　　1 引言</b></p><p>　　在混凝土框架結構初步設計的期間，通常，框架梁截面高度通過跨度來估算，和截面寬度根據高寬比估算; 框架柱的截面尺寸是根據柱軸壓縮的支持柱的面積的比率估算[1]。然而，在估計過程中，初步設計

5、階段中的一個重要的鏈，未考慮到柱側移剛度的影響[2]。列側移剛度越大，結構層間位的剛度越大，剪切型框架結構的層間位移將越較小。所以，總結構越小的側向位移將減少地震災害[3]所造成的損失。論文的核心是如何得到列側移剛度的最大值。</p><p>　　同時，列側移剛度的值與框架梁-柱線剛度直接相關。本論文的目的是為了得到一個合理的框架梁 - 柱的線剛度比，在某個控制范圍內獲得列側移剛度的最大值。</p>

6、<p>　　計算列橫向位移的方法有兩種方法：剛度拐點點法和修改拐點法。拐點的方法假定關節(jié)的旋轉角度為0（當梁柱線性剛度比是大于或等于3時，柱的上端和下端的關節(jié)的旋轉角度可以取為0，因為它實際上是相當?。?，即梁的彎曲剛性被視為無窮大。拐點的方法主要是應用于具有比較少層的框架結構。但對于多層、高層框架結構，增加柱截面會導致梁柱線剛度比小于3，在水平荷載作用下，框架結構的所有關節(jié)的旋轉角度的橫向位移會發(fā)生不可忽視。因此，一位日本教

7、授武藤提出修改拐點法[4]，即D-值方法。本文采用D-值列側移剛度的計算法，因為它著重于多層、高層框架結構。</p><p>　　少數在國內外對框架梁柱合理線剛度比的研究，只有梁七黹，源于列側移剛度的計算方法，比D-值法更加應用廣泛；申得氏指出在多層、高層框架結構的柱側向剛度計算中存在的問題，補充和修改底部和頂部層的列側向剛度計算公式;應用于史密斯和庫爾博士法，焱鑫田源于梁 - 柱線剛度比的合理值，由計算的最大等

8、效剛度框架柱。</p><p>　　本文計算列側移剛度的最大值,第一次通過采用在約束條件下結構優(yōu)化理論，那就是，約束Lagrange乘子法優(yōu)化理論對框架梁-柱的材料是有一定價值[5]。因此，混凝土框架梁-柱的合理線剛度比在一定范圍內可以得到。在初步設計和參考階段，得出的結論可以作為梁-柱的截面尺寸的一個決定性因素，類似在梁柱框架結構設計的研究</p><p>　　2 用約束拉格朗日乘子法計

9、算框架梁柱的合理線性剛度比</p><p>　　2.1 側向剛度的框架梁-柱的D值法</p><p>　　以標準層框架結構的中間關節(jié)作為一個例子，由D值法計算出梁-柱的側向位移剛度：</p><p>　　:關節(jié)旋轉的影響系數；</p><p>　　:框架柱的線性剛度;</p><p><b>　　:層高；&l

10、t;/b></p><p>　　:梁-柱的平均線性剛度;</p><p><b>　　:梁柱的線性剛度;</b></p><p>　　假設所有的梁-柱線性剛度都是，所以，</p><p>　　考慮到框架梁鑄在原址層的鋼筋混凝土框架結構受限的影響，中間框架梁的轉動慣量[6]；</p><p>

11、　　是梁截面的轉動慣量，所以，</p><p>　　因此，標準的框架柱側移剛度得出以下公式：</p><p>　　因為,側移剛度列進一步推導出下述公式：</p><p>　　2.2 基于拉格朗日乘數法得到合理線剛度比</p><p>　　為了獲得框架柱的側移剛度最大D值，我們需要找到目標函數：</p><p><

12、b>　?。?）</b></p><p>　　假設截面框架梁是和截面框架柱是，在材料總量是A的前提下，在材料總量是A的前提下，公式滿足這個約束條件，得：</p><p><b>　　（2）</b></p><p>　　通過拉格朗日數乘法來獲得目標函數：</p><p><b>　　因為</

13、b></p><p><b>　　所以</b></p><p><b>　　（3）</b></p><p>　　E : 混凝土的彈性模量；</p><p>　　同理，柱線性剛度可以用下面的公式推導：</p><p><b>　　因此我們得到：</b>

14、;</p><p><b>　　（4）</b></p><p>　　把公式（3）和公式（4）代入公式（2），得，</p><p><b>　　可以進一步推導：</b></p><p><b>　　（5）</b></p><p><b>　　(T

15、是定值)</b></p><p>　　在一定約束條件下，根據拉格朗日數乘法，目標函數可以由公式（5）得到：</p><p><b>　?。?）</b></p><p>　　分別對求各自的偏導，并且令其偏導數為0，得：</p><p><b>　　整理上述方程，得，</b></p&g

16、t;<p><b>　?。?）</b></p><p>　　從等式（7）對 開根號，得：</p><p><b>　?。?）</b></p><p>　　上面的公式是在框架結構標準層中間接頭的柱的側移剛度是最大時的梁-柱線性剛度比，即合理梁柱線剛度比。</p><p>　　同理，我們可

17、以在框架結構柱的側移剛度是最大時，得出標準層側接頭的合理線剛度比。</p><p><b>　?。?）</b></p><p>　　3 工程梁柱合理線剛度的應用</p><p>　　3.1多層和高層框架結構的合理線剛度應用</p><p>　　在消耗材料量是一個定值前提下，框架結構的標準框架層的中間關節(jié)作為梁-柱線性剛度

18、比滿足式（8），且框架結構標準層框架側接頭作為線性剛度比滿足公式（9），那么框架結構側向位移剛度將一直保持最大值。</p><p>　　顯然，那么總的側向位移是最小的結構就在這時[7]; 其工程應用價值不言而喻。</p><p>　　在一般的框架結構中，柱高和梁跨度滿足下面的公式：</p><p>　　梁-柱的截面高度比滿足下式[8]：</p><

19、;p>　　對于框架結構的標準層的中間節(jié)點，我們可以由等式（8）推出：</p><p><b>　　（10）</b></p><p>　　上文中的計算公式在合理線剛度比的梁-柱框架結構的標準層中間接頭的應用范圍。</p><p>　　結果表明，梁-柱的截面尺寸可被相應的計算出來，如果梁-柱線性剛度比滿足等式（10）時，框架柱的側向位移則是最

20、大值。</p><p><b>　　3.2 例子</b></p><p>　　在一般負載下，一個10層的、4跨度的鋼筋混凝土澆筑在原址框架結構，每層的高度是3.6米，梁跨度為7.2m，梁-柱的混凝土強度等級是一樣的。在材料量是定值的前提下，中間框的標準層梁 - 柱中間關節(jié)的截面尺寸通用估計方法估計，然后將估計值與由式8和式9計算的截面尺寸的結果相比較。根據一般方法估算

21、梁-柱的截面尺寸：</p><p><b>　　梁:</b></p><p><b>　　柱:</b></p><p>　　則梁-柱材料的量一共是：</p><p>　　那么，柱側向位移的剛度比是：</p><p>　　然而，線性剛度比是在等式10的應用范圍之內。</p

22、><p>　　基于等式10，在A是定值的條件下，計算梁柱的截面尺寸，且調整梁柱的截面尺寸，如下：</p><p><b>　　然后，</b></p><p>　　現在側向位移剛度是：</p><p><b>　　然后，顯然，</b></p><p><b>　　此時，&

23、lt;/b></p><p>　　梁柱的線性剛度比在等式10的范圍之內，所以它就是合理線性剛度比并且在工程應用范圍內。</p><p><b>　　4 結論</b></p><p>　　（1）在上述的例子可以得出：在梁柱的材料消耗總量A是定值前提下，在標準框架結構的最初設計階段，如果梁柱的截面尺寸可以調整，梁寬保持不變。在這個例子中，梁寬

24、保持不變，把柱高從650毫米調整到600毫米，然后把柱高從500毫米調整到560毫米，如果柱截面寬度保持不變，則會得到柱側向位移剛度的最大值。</p><p>　　證明梁柱的線性剛度比此時滿足等式（10），所以在標準框架結構的最初設計階段，得到的梁柱截面尺寸在應用范圍內的合理線性剛度比內。</p><p>　　(2) 該研究方法通過拉格朗日數乘法來獲得柱側向位移剛度最大值被廣泛應用于研究類

25、似的框架結構。例如，可以用于研究中間框架底部、側向框架、類似工程結構的合理線性剛度比</p><p>　　(3) 這個研究的結論可以為框架結構其他方面的研究提供一定的參考。例如，通過調整截面尺寸獲得柱側向位移剛度的最大值在框架結構的抗震設計中變得越來越重要。增加柱的截面尺寸可以有效地控制軸壓比，剪壓縮比，從而提高結構延性和減少地震災害造成的損失。</p><p><b>　　參考

26、文獻</b></p><p>　　[1] Tao Ji, Zhixiong Huang, Multi-story and High-rise Reinforced Concrete Structure Design, Michanical Industry Publishing House, Beijing, 2007.</p><p>　　[2] Shihua Bao, Hi

27、gh-rise Building Structure of New Edition.Water Resource and Hydropower Publishing House of China, Beijing, 2005.</p><p>　　[3] Ahmed Ghobarah and A. Said, “Shear strengthening of Beam-column Joints”, Journal

28、 of Engineering Structures,2002, 24（7）, pp.881-888.</p><p>　　[4] Ahmed Ghobarah, Seism Resistance Design & Seism Resistance Methods [M], Maruzen Company, Limited. 1963.</p><p>　　[5] Aichuan.

29、 Jiang, Structural Optimization Design, Qinghua Publishing House, Beijing, 1986.</p><p>　　[6] Xi’an Zhao, High-rise Reinforced Concrete Structure Design, Architecture＆Building Press Beijing, China, 2003.<

30、/p><p>　　[7] P.G. Bakir and H.M. Boduro.lu, “A New Design Equation for Predicting the Joint Shear Strength of Monotonically Loaded Exterior Beam-column Joints”, Journal of Engineering Structures, 2002, 24（8）, p

31、p.1105-1117.</p><p>　　[8] Huanling Meng and Pusheng Shen, “Research on Behaviors of Frame-shear Wall Structures Based on Stiffness Degradation”, Journal of Railway Science and Engineering,2006, 3(1), pp.12–1

32、7.</p><p>　　Study on Reasonable Linear Stiffness Ratio in Frame Structure Based on Lagrange Multiplier Method</p><p>　　Zhiqin LIU,Guoliang BAI. Study on Reasonable Linear Stiffness Ratio in Fram

33、e Structure Based on Lagrange Multiplier Method[J].Information Management,Innovation Management and Industrial Engineering,2008.ICIII’08.International Conference on,2008,9(6):368-371.</p><p><b>　　Abstr

34、act</b></p><p>　　Frame 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 optimizati

35、on 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 theor

36、y and on the premise that total material quantity of framework beam-column is definite. Differen</p><p>　　1 Introduction</p><p>　　During the preliminary design of concrete frame structures, gene

37、rally, 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 compres

38、sion 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 c<

39、/p><p>　　Meanwhile, column lateral displacement stiffness value is directly related with linear stiffness of frame beam-column. </p><p>　　The purpose of the thesis is to get a reasonable linear sti

40、ffness 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-in

41、flexion 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 an

42、d</p><p>　　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

43、 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 meth

44、od of calculating the column lateral displacement stiffness because it focuses on mu</p><p>　　Research on reasonable linear stiffness ratio of frame beam-column is few at home and abroad, only Liang Qizhi de

45、rives 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

46、-rise frame structure, supplements and modifies the column lateral stiffness calculation formula on bottom and top layer;applying Smith & Coull method, Yanxin Tian der</p><p>　　The thesis calculates the

47、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 th

48、e 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

49、section dimension of frame beam-co</p><p>　　2 Reasonable Linear Stiffness Ratio of Frame Beam-column Calculated by Constraint Lagrange Multiplier Method</p><p>　　2.1 Lateral Displacement Stiffne

50、ss of Frame Beam-Column by D Value Method</p><p>　　Taking the middle joint of standard floor frame structure as an example, lateral displacement stiffness of beam-column calculated by D value method:</p&g

51、t;<p>　　: Influence coefficient of joint rotation；</p><p>　　: Linear stiffness of frame column; </p><p>　　: Story height; </p><p>　　: Average linear stiffness of floor beam-c

52、olumn;: Linear stiffness of beam-column；Suppose all linear stiffness of beam-column is ,Then,</p><p>　　Considering the restriction effect on frame beam from cast-in-situ floor of reinforced concrete frame st

53、ructures, the inertia moment of the middle frame beam [6];</p><p>　　is the inertia moment of the beam section, then,</p><p>　　Thus, lateral displacement stiffness of standard frame column is der

54、ived by the following formula: </p><p>　　For , lateral displacement stiffness of column</p><p>　　is further derived from the following formula:</p><p>　　2.2 Deriving the Reasonable

55、Linear Stiffness Ratio Based On Lagrange Multiplier Method</p><p>　　In order to get the maximal lateral displacement stiffness of frame column D value, we need to find the objective function:</p><

56、p><b>　　(1)</b></p><p>　　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 con

57、dition, then, </p><p><b>　　(2)</b></p><p>　　To find objective function by Lagrange Multiplier Method:</p><p><b>　　For, </b></p><p><b>　　T

58、hen,</b></p><p><b>　　(3)</b></p><p>　　E : Elastic modulus of concrete; </p><p>　　In the same way, the linear stiffness of column is derived by the following formul

59、a:</p><p>　　Therefore we get：</p><p><b>　　(4)</b></p><p>　　It can be further derived:</p><p><b>　　(5)</b></p><p>　　(T is a definite

60、 value) According to the Lagrange Multiplier Method</p><p>　　under certain constraints, the objective function can be derived by formula (5):</p><p><b>　　(6)</b></p><p>

61、　　Calculation the partial derivatives of ,</p><p>　　respectively and make the results equal to 0, then，</p><p>　　Further derive the above formula, then，</p><p><b>　　(7)</b&

62、gt;</p><p>　　Derive from Eq.7, </p><p><b>　　(8)</b></p><p>　　The above formula is linear stiffness ratio of beam-column standard floor middle joint in frame structure w

63、hen 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 t

64、he frame structure when the lateral displacement stiffness of column is maximal.</p><p><b>　　(9)</b></p><p>　　3 Application of Reasonable Linear Stiffness of Beam-column in Engineeri

65、ng</p><p>　　3.1 Application of Reasonable Linear Stiffness in Multi-layer and High-rise Frame Structures</p><p>　　On the premise that consumed material amount is a definite value, to the middle

66、joints of standard frame floor of frame structures as linear stiffness ratio of beam-column satisfies the formula (8) and to side joint of frame standard floor of frame structures as linear stiffness ratio satisfies the

67、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 momen</p><p>

68、;　　The section height ratio of beam-column satisfies the formula [8]:</p><p>　　To standard floor middle joint of frame structure, it can be derived from Eq.8,</p><p><b>　　(10)</b><

69、;/p><p>　　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-co

70、lumn 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.</p><p>　　3.2 Example</p>

71、;<p>　　A 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 pre

72、mise 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 di

73、mension by the results of Eq.8 and Eq.9. Estimate the section dimension of</p><p>　　Beam: Column:</p><p>　　Then the material amount of beam-column is </p><p>　　Therefore, the lat

74、eral displacement stiffness of column is</p><p>　　However, 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

75、condition that A is a definite value, and adjust the section dimension of beam-column, as:,Then,</p><p>　　Now the lateral displacement stiffness is </p><p>　　Then, obviously, </p><p&

76、gt;<b>　　Now, </b></p><p>　　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.</p>&l

77、t;p>　　4 Conclusion</p><p>　　(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 desi

78、gn 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 65

79、0mm to 600mm and then adjust column height from 500mm to 560mm if the column dimension width remains unc</p><p>　　(2) The research method obtaining the maximum value of column lateral displacement stiffness

80、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 e

81、ngineering structures.</p><p>　　(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 colum

82、n 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 r

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