高性能の矩形平面板曲げ要素の開発とその検証

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  • DEVELOPMENT AND VERIFICATION OF HIGH PERFORMANCE RECTANGULAR PLATE BENDING ELEMENT WITH FOUR-NODE
  • コウセイノウ ノ クケイ ヘイメン イタマゲ ヨウソ ノ カイハツ ト ソノ ケンショウ

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<p> 1. Introduction The plate bending problem of flat plate is the basis of static analysis and vibration analysis of three dimensional wall structure and folded plate structure. Development of elements having high accuracy with few elements is one of the essence of the finite element method. A rectangular flat plate bending element is easy to use and its shape function is determined in a simple way. The generalized coordinates of each node of the bending plane plate element are three degrees of freedom, displacement, and rotation in two directions. A rectangular four-node element has a total of 12 degrees of freedom. The ACM element defines the shape function using the low order polynomial of Pascal's triangle. Several adaptive elements with the same degree of freedom have been developed, but it is hard to find out which gives solutions with good precision . Since weak points of rectangular bending flat plate elements with 4 nodes and 12 degrees of freedom are conforming conditions between neighbouring elements, this paper imposes the following two conditions to improve these conditions, and aims to advance the element performance.</p><p> Condition 1 :The rotation in the normal direction of each side of the quadrilateral element coincides with the adjacent element at both ends. Here, we introduce the condition that the rotation in the normal direction at the midpoint of the side is the average value of the rotation at both ends.</p><p> Condition 2 :Completely eliminates the influence of the node displacement of this element on the rotation in the normal direction of each side of this element.</p><p> 2., 3. Characteristics of a rectangular flat plate bending element with 12 degrees of freedom</p><p> 4. Numerical analyses and verification A shape function of a rectangular flat plate bending element having 4 nodes and 12 degrees of freedom satisfying these two conditions can be constructed by using a basis function expressed by the formula (10.2). The resulting shape function is shown in equation (11.1). In order to demonstrate the performance of the rectangular element developed here, four benchmark tests are carried out using six element meshes. Considering the symmetric condition, 1/4 of the analysis model is set as the analysis target. For element division, the same elements 1 × 1, 2 × 2, 3 × 3, 5 × 5, 7 × 7, and 9 × 9 were used in two directions.</p><p> 4.1. Simply supported square plate under a uniform vertical load q</p><p> 4.2. Simply supported square plate under loading a force P at the central point.</p><p> 4.3. Natural frequencies of a simply supported square plate</p><p> 4.4. Deflection of square plate with built-in edges under loading force P at the central point.</p><p> This analysis was done to compare accuracy with the solution BCIZ-SQ element developed recently and ANDES 4 element. 1 × 1, 2 × 2, 4 × 4, 8 × 8, and 16 × 16 element meshes were used. This discussion is shown in section 4.4.</p><p> 5. Conclusion We analyzed the incompatibility of the existing rectangular flat plate bending element with 4 nodes and 12 degrees of freedom and found two conditions to improve incompatibility and aimed for advanced element performance. These conditions lead to improvement of element basis functions, and developed high precision elements with identical degrees of freedom. Static analysis and natural frequency were analyzed, and the performance obtained from the existing solution was compared with the performance. We verified that the developed element will be a rectangular flat plate bending element with high performance.</p>

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