二分之一空間固體內部受集中力作用時的三種解析解與有限元素法的比較

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Title:	二分之一空間固體內部受集中力作用時的三種解析解與有限元素法的比較
Other Titles:	The Comparison of Three Analytical Solutions and F.E.M. Solutions of Concentrate Forces Acting Inside a Half-Space Solid
Authors:	邵德文;倪永寬
Contributors:	淡江大學機械與機電工程學系
Keywords:	半空間固體;三維彈性力學;Mindlin問題;有限元素法;Half-Space Solid;Three-Dimensional Elasticity;Mindlin Problem;Finite Element Method
Date:	2000-12
Issue Date:	2014-03-07 11:27:08 (UTC+8)
Abstract:	從1936年R. D. Mindlin發表集中力作用在二分之一空間固體內部解析解後，就很少人對此問題去探討它的正確性，但這問題是三維彈性力學基本型式的題目，正確與否有關後續三維彈性力學的研究。本文中除了用Mindlin原始的方法分析Mindlin問題外又增加了兩種不同的解析方法，在Mindlin的原始方法中，從拍伯克維支-紐伯-布斯尼斯克函數B/sub x/、B/sub y/、B/sub z/、β四項函數中選擇B/sub z/、β二項，而本文中選擇B/sub x/、B/sub z/、β三項函數。除此之外我們用有限元素法解Mindlin問題，在有限元素法中我們使用COSMOS/M軟體。而有限元素法隨著元素的增加可以驗證解析解。然而在實際應用所得到的解，我們需要提供一種以上的解析法來驗證。我們比較三種解析方法與有限元素法得到的解，發現四種方法的每個方向的位移和每個面每個方向的應力圖形與物理意義都非常的相似，同時四種方法的最大值比較也很相近，因此這些解是相近且準確可以接受的。所以，我們可以下結論，Mindlin原始的方法及本文所提兩種不同解析方法是可以接受的。 In 1936, R. D. Mindlin published the analytical solution ofconcentrated force acting inside a half-space solid. Since then, theaccuracy of that solution seems seldom been explored. This problem isone of the basic type of problems in the three-dimensional elasticity.Its accuracy has great influence on the research of three-dimensionalelasticity. In this work, we analyzed the Mindlin's Problem by twodifferent analytical methods in addition to Mindlin's originalmethods. In Mindlin's original method, only two functions B/sub z/ andβ are chosen out of four Papkovitch-Neuber-Boussinesq functions B/subx/, B/sub y/, B/sub z/ and β. In this work, we choose three functionsB/sub x/, B/sub z/ and β out of four functions. Also, we model theMindlin's Problem by finite element method. The package of COSMOS/Mfor the finite element method was used for the solution. The finiteelement method with increasing number of elements can offer a checkfor the analytical solutions. However, for practical application ofthe results, more than one method of analysis is suggested. Then, wecompared the solutions by the three analytical methods and the finiteelement solution. We find that the results of displacement componentsand stress components have similar graphs and similar physicalsimilarity for all four methods used. The maximum values for the fourmethods are compared also. We get similar answers with an accuracythat may be acceptable. So, we may conclude that Mindlin's originalwork is acceptable. The two different analytical methods proposed inthis thesis are also acceptable.
Relation:	中國機械工程學會第十七屆全國學術研討會論文集〈第三冊下集固力與設計〉=Proceedings of the 17th National Conference on Mechanical Engineering the Chinese Society of Mechanical Engineers，頁603-610
Appears in Collections:	[Graduate Institute & Department of Mechanical and Electro-Mechanical Engineering] Proceeding

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