Lect17
- 1. Dr. A. S. Sayyad Professor & Head Department of Structural Engineering Sanjivani College of Engineering, Kopargaon 423603. (An Autonomous Institute, Affiliated to Savitribai Phule Pune University, Pune) Finite Element Method In Civil Engineering Natural Coordinates for CST Element or Area coordinates of CST Element 3 1 2 1 2 3 A A A L L L A A A
- 2. Natural Coordinates for CST Element Or Area coordinates of CST Element Let us consider three nodded triangular element as shown in figure. (x1, y1), (x2, y2), (x3, y3) are the Cartesian coordinates of nodes 1, 2, 3 respectively. (u1, v1), (u2, v2), (u3, v3) are the displacements of nodes 1, 2, 3 respectively. Let ‘P’ be the any point on the element having Cartesian coordinates (x, y) and natural coordinates (L1, L2, L3)
- 3. According to definition of natural coordinates 1 2 3 1 1 2 2 3 3 1 1 2 2 3 3 1 L L L L x L x L x x L y L y L y y In matrix form 1 2 1 2 3 3 1 2 3 1 1 2 1 2 3 3 1 2 3 1 1 1 1 1 1 1 1 L L x x x x L y y y y L L x x x x L y y y y
- 4. 1 1 1 1 2 2 2 2 3 3 3 3 1 1 L a b c L a b c x D L a b c y where 2 3 3 2 3 1 1 3 1 2 2 1 1 2 3 3 2 1 3 1 1 3 1 1 2 2 1 2 3 2 2 3 1 2 1 2 3 3 1 3 1 3 3 2 1 D x y x y x y x y x y x y a x y x y b x y x y c x y x y a y y b y y c y y a x x b x x c x x 1 1 1 1 2 2 2 2 3 3 3 3 1 1 2 L a b c L a b c x A L a b c y 3 3 3 1 1 1 2 2 2 1 2 3 ; and 2 2 2 a b x c y a b x c y a b x c y L L L A A A
- 5. Let divide the total area of triangle into three parts (A1, A2, A3). Now, let us consider any point P shifted to node 1: 1 2 3 0 A A A A 2 3 2 3 2 3 3 2 3 3 2 2 1 1 1 1 1 1 1 1 2 2 D x x x y y y D x y x y xy x y xy x y A a b x c y A
- 6. Similarly, if any point P is shifted to node 2 and 3, we get 2 2 2 2 3 3 3 3 2 and 2 a b x c y A a b x c y A Therefore, natural coordinates are written as 3 3 1 1 2 2 1 2 3 2 2 2 and 2 2 2 A A A A A A L L L A A A A A A The variation of natural coordinates at each node is represented as