外文翻譯---瑞利的比例阻尼方法在對橋梁的非線性動態(tài)分析的研究

1、<p><b>　　附錄</b></p><p>　　Study on the Propriety of Rayleigh’s Damping in Nonlinear Dynamic Analysis of Bridges</p><p><b>　　Summary </b></p><p>　　In this

2、 paper, based on the results of comparison of nonlinear dynamic analysis of 5 span bridges constructed on the I 、Ⅱ、Ⅲ regions by the two method (e.g. strain-energy proportional damping method & Rayleigh’s damping

3、method), parametric study were made to the maximum bending curvature of the piers. As the results, it is found that the Rayleigh’s damping method brings a comparatively small approach to the bending curvature in many cas

4、es comparing with the strain-energy proportional damping </p><p>　　Keyword: nonlinear, dynamic analysis, the strain-energy proportional damping</p><p>　　Rayleigh’s damping, bridges, the

5、 maximum bending curvature </p><p>　　1. Introduction </p><p>　　After the new aseismic code of bridge design published in two years ago in Japan , the Rayleigh’s damping method are often used

6、 in non-linear dynamic analysis of bridges since the assumption of damping matrix C=αK+βM, the time of calculating is shorter than other method . But through the computation to the many engineering problem, it shows th

7、at the results by Rayleigh’s damping method are various with the definition of the parameter. And also, it is found that the discontinues of bending</p><p>　　2.The Study to the Common Type of Bridges </p

8、><p>　　2.1 Analysis Model </p><p>　　The 3 bridges to be analyzed were selected according to the conditions of foundation situated in the different regions. Both of them were 5 span continues bridg

9、es with the rubber bearing. The basic conditions of the bridges were showed in Fig. 1. The standard earthquake wave from bridge code of Japan were input at the base of pier. The input earthquake directions were in the d

10、irection of longitude of bridge. </p><p>　　The Takeda tri-linear model was used in the nonlinear dynamic analysis. At the bottom of the piers, less than 0.5m of mesh size were applied. The damping ratios of

11、structures were shown in Table 1. The Newmark-β method(β=1/4)were used in the analyses and the Δｔ＝0.002sec. If the stabilization of result is unavailable, the Δｔ＝0.001,ｔ＝0.0005 were applied. </p><p>　　I

12、t is very important to make selection of the 2 parameters in the Rayleigh’s damping method. The common method ① which shown in Fig. 2 use the 2 main mode of the structure. But for the bridges on foundation of Ⅲ regions (

13、soft clay foundation), it lead to a the minus parameters phenomena. The method ②, points were selected by using the main mode and a mode which have a comparatively lager effective mass. </p><p>　　2.2 Th

14、e Comparison of Analysis Results </p><p>　　The safety verification for the piers was carried out on bending moment, shearing force and residual displacement of piers. Considering the maximum bending curvatur

15、e at the bottom of the piers in Fig. 3(b), the horizontal axis is the ratio of maximum bending curvature φs which was obtained from strain-energy proportional damping method and φyo(yielding bending curvature), the verti

16、cal axis is the ratio of maximum bending curvature φr which was obtained from Rayleigh’s damping method and φs. It </p><p>　　For the analytical results of common types of bridges, it is found that the Rayle

17、igh’s damping method lead to a comparatively lower evaluation to the bending curvature than the strain-energy proportional damping method. </p><p>　　3.The Study to the Complex Type of Bridges </p>&

18、lt;p>　　3.1 Analysis Model </p><p>　　The complex type of bridge that was showed in Fig.4 was a 14 span continues steel girders and the bridge between P15 and A1 is 3 spans continue box types girders. The p

19、iers are reinforced concrete structures. The foundation is on classification of region I. Strain-energy proportional damping method & Rayleigh’s damping method were carried out on the bridge.</p><p>　　3.

20、2 The Comparison of Analysis Results </p><p>　　The total shaking mode is 796. Fig-5 shows the shaking mode from 2 to 4. The table 2 shows the characteristic values of shaking mode from 1 to 10. It is very im

21、portant to determine the two parameters on Raleigh damping method . Usually, the 2 main mode are used to make the determination of the Raleigh damping ratios. For the complex types of bridge in the paper, there are many

22、methods to make the determinations. Table 2 shows the shaking modes and its damping ratios from mode 1 to 20. </p><p>　　Fig. 6 shows the 4 kind’s method of determination. The earthquake wave showed in Fig.7

23、 was used in the calculations. By using the strain-energy proportional damping method and Rayleigh’s damping method comparative were made. The results are showed in Fig. 8 and 9. </p><p>　　3.2 The Comparis

24、on of Strain-energy Proportional Damping Method and Rayleigh’s Damping method </p><p>　　The verification of safety was carried out on bending moment shearing force and residual displacement of piers. Table 3

25、 shows the comparison of maximum bending rotating angels from the strain-energy proportional damping method and Rayleigh’s damping method. It is found that the maximum bending rotating angels from the strain-energy propo

26、rtional damping method is less than that from Rayleigh’s damping method. For the most severe cases, the P4-P6 pies lead the error of the 52%. </p><p>　　Since the damping of strain-energy proportional damping

27、 method are determined by the equation:</p><p>　　and that of Rayleigh’s damping method are determined by the equation: </p><p>　　The Rayleigh’s damping method, usually the α and β were determine

28、d by the mode of lower frequency areas. For the simple structures, the results deference is comparatively small and for the complex structures, the using of strain-energy proportional damping method is desirable. </p

29、><p>　　瑞利的比例阻尼方法在對橋梁的非線性動態(tài)分析的研究</p><p><b>　　摘要：</b></p><p>　　在本文里，基于比較非線性動力學分析修建在I,Ⅱ,Ⅲ區(qū)的五跨徑橋梁通過兩種方法的結果(如應變能量比例阻尼法和瑞利阻尼法),對受最大彎曲的橋墩做了參數研究. 作為結果, 結果發(fā)現在許多情況下瑞利阻尼法相對于應變能量比例阻尼的方

30、法帶來了比較少的方法來彎曲,并且彎曲通常發(fā)生在比較大的地區(qū). 同時,相當大的部分做成了復雜的橋梁結構,例如: 十四跨徑橋梁和三跨徑彎橋.在實際中有人建議使用瑞利的比例阻尼方法. </p><p>　　關鍵詞：非線性，動態(tài)分析，應變能源比例阻尼瑞利阻尼，橋，最大彎曲</p><p><b>　　簡介</b></p><p>　　于兩年前日本刊登了

31、新的抗震設計規(guī)范橋梁設計，在非線性動力分析的橋梁中往往是使用瑞利的阻尼方法,由于假定的阻尼系數C=αK+βM,的計算時間比其它方法短. 但通過計算許多工程的問題時，它表明瑞利阻尼方法的結果是不同的定義參數. 此外,他還發(fā)現在一些具體的分析發(fā)生中斷彎矩. 文章中，基于比較非線性動力學分析修建在I,Ⅱ,Ⅲ區(qū)的五跨徑橋梁通過兩種方法的結果(如應變能量比例阻尼法和瑞利阻尼法),對受最大彎曲的橋墩做了參數研究。同時,相當大的部分做成了復雜的橋梁結

32、構,例如: 十四跨徑橋梁和三跨徑彎橋。</p><p><b>　　對同一類型橋的研究</b></p><p><b>　　2．1分析模型</b></p><p>　　根據不同地區(qū)地基的不同狀況分析選擇了三種橋梁。它們都是有橡膠墊的五跨度連續(xù)橋梁。橋的基本狀況如圖1所示。來自日本橋梁準則的標準地震波被用于橋墩的基礎。投入的

33、地震指示的方向是延橋梁的縱向位置。 </p><p>　　非線性動態(tài)分析中使用紘三線性模型。在橋墩的底部，使用小于0.5米的網格大小。阻尼比例結構見表1。該新標示-β法(β=1/4)被用于分析和Δt=0.002秒。 如果結果的穩(wěn)定是無效的,Δt=0.001,t=0.0005將被使用。在瑞利阻尼方法中選擇使用兩種參數是非常重要的。常用方法一如圖. 2所示使用兩個主要模式的結構。 但對于Ⅲ區(qū)地基的橋梁(軟粘土地基),

34、它導致一個負的參數的現象。方法二，采用的主要方式和有相對較大有效質量的模式設點。 </p><p>　　2.2比較分析結果 </p><p>　　對橋墩的安全驗證包括進行彎矩,剪力和橋墩的殘余位移。在橋墩的底部考慮最大彎曲如圖3(b)示，橫軸是來自應變能成正比的阻尼方法和Φyo(屈服彎曲曲率)的比例最高可彎曲的Φs,豎軸是比例最高彎曲Φr是從瑞利阻尼和ΦS得到。 結果發(fā)現, 當ΦS被限制在

35、一個很小的地區(qū)時Φr/Φs幾乎等于1, 隨著S的增長,Φr/Φs迅速減少,在方法一中它達到0.3接近Φa(可彎曲)和在方法二中達到0.6。同樣的現象還發(fā)生在地區(qū)Ⅲ的橋中。</p><p>　　對于常見類型的橋梁分析結果，結果發(fā)現瑞利阻尼法比應變能成正比的阻尼方法導致一個相對較低評價的彎曲。</p><p>　　3．對復合型橋梁的研究</p><p><b>

36、;　　3.1分析模型 </b></p><p>　　復雜的橋型如圖.4示是一個十四跨度連續(xù)鋼梁與處于p15和A1的橋是三跨連續(xù)箱型梁橋。橋墩是鋼筋混凝土結構。該基礎是在區(qū)域I，橋梁中使用應變能量比例阻尼法和瑞利阻尼方法。</p><p><b>　　3.2比較分析結果</b></p><p>　　總振動模式是796。圖-5顯示了震動

37、模式由2至4情況。表2顯示了震動模式從1至10的特有的價值。這是非常重要的,用洛麗阻尼方法確定這兩個參數。在通常情況下,采用兩個主要模式測定的瑞利阻尼比率。文章中對于復雜類型橋梁,有很多方法可以作出裁決。表2顯示振動模式及其阻尼比率從模式1到20。 </p><p>　　圖6 顯示決心4 種類的方法。地震波如圖7所示被用于計算中。由使用張力能量比例阻止的方法和瑞利阻止的方法比較使用。結果如圖8 和9所示。 <

38、;/p><p>　　3.2 張力能量比例阻止的方法和瑞利阻止方法比較</p><p>　　安全驗證被用于彎曲的剪切力和橋墩的殘余位移。表3 顯示最大彎曲的轉角從張力能量比例阻止的方法和瑞利阻止方法得到。結果發(fā)現最大值彎曲的轉角來自張力能量比例阻止的方法而不是瑞利阻止方法。最嚴重的事件是P4-P6 導致52%的錯誤 。</p><p>　　3.2 張力能量比例阻止的方法和

39、瑞利阻止方法比較</p><p>　　安全驗證被用于彎曲的剪切力和橋墩的殘余位移。表3 顯示最大彎曲的轉角從張力能量比例阻止的方法和瑞利阻止方法得到。結果發(fā)現最大值彎曲的轉角來自張力能量比例阻止的方法而不是瑞利阻止方法。最嚴重的事件是P4-P6 導致52%的錯誤 。</p><p>　　由于阻尼的應變能成正比的阻尼方法是由公式: </p><p>　　而瑞利阻尼方法