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Advanced mechanics of materials (1993-John Willey) Episode 14

Tham khảo tài liệu 'advanced mechanics of materials (1993-john willey) episode 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 762 19 THE FINITE ELEMENT METHOD W 1 4 2 n4 4 19.51 After the element is mapped from natural to physical coordinates the Ỉ and r axes need not remain orthogonal. The principal reason for using isoparametric elements is to avoid integrating in physical coordinates. However the general expression for the stiffness matrix Eq. 19.31 is expressed in terms of physical coordinates. Therefore the differential lengths dx and dy must be expressed in terms of the natural coordinate differentials dị and dr . In addition strain is defined in terms of the derivatives of the shape functions with respect to physical coordinates. These derivatives are the elements in the B matrix and they must be converted to derivatives with respect to natural coordinates. The differentials dx dy are related to the differentials dị dr by means of Eq. 19.49a . Thus . dx dx - dx - dị -r-dr oq dr og or 19.52 where dx dNi ôx dNi dr dr dy dNị dy _ y dNị dr dr The coordinate derivatives are combined in matrix form as 19.53 where J is the Jacobian of the transformation Courant 1950 . Equations 19.52 and 19.53 relate the differentials of the two coordinate systems as dx r r T dyj L J dr 19.54 19.4 THE LINEAR ISOPARAMETRIC QUADRILATERAL 763 In a like manner derivatives of the shape function for node i are related by ôNị dNị ÔX õNị J 1 ôNị . dy. ÔrỊ 19.55 If J -1 exists then the area mapping from Ỉ rj coordinates to x y coordinates is unique and reversible. A physical interpretation of J can be obtained by comparing the area of the element in x y coordinates to that in ỉ fị coordinates. If the determinate IJI 0 then the area of the element is preserved and the mapping is physically meaningful. In precise terms J is the differential area ratio Axy A at any point in the element. This physical interpretation of J leads to a change in the differential volume for a constant thickness plane elasticity element from tdxdy to t J dỊdrj. The limits of integration are 1 to 1 in Ỉ and 1 to Í in TỊ. So the integral of .