Lect15
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1) The document discusses natural coordinates for a 1D bar element in finite element analysis. 2) It defines the natural coordinates (L1, L2) of any point P on the bar element in terms of the Cartesian coordinates (x1, x2) of the two nodes and the length L of the element. 3) The natural coordinates at node 1 are derived to be (1, 0) and at node 2 to be (0, 1).
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The document discusses the natural coordinates of a 1D bar element in finite element analysis. It defines the natural coordinate ξ as the ratio of the distance from any point P to the origin, over the maximum distance from the origin to node 2. This yields expressions for ξ at each node. At node 1, ξ = 0, and at node 2, ξ = 1. Therefore, the natural coordinates of a 1D bar element are ξ, which ranges from 0 at node 1 to 1 at node 2.
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1) The document derives the shape functions for a two-noded bar element in finite element analysis.
2) It expresses the displacement function in terms of the nodal displacements using the coordinates of the two nodes.
3) The shape functions are obtained by equating the displacement function to the product of the shape function matrix and the nodal displacement vector.
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- 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 of 1D bar element (x-y coordinate system) 2 1 1 2 , x x x x L L L L
- 2. Let us consider a two nodded bar/line element (1D). Node 1 and 2 have Cartesian coordinates 1 x and 2 x respectively. Cartesian coordinate of any point ‘P’ is ‘ x ’. Natural coordinates of any point ‘P’ are (L1, L2). Natural coordinates of 1D bar element (x-y coordinate system)
- 3. From the definition of line element In matrix form, 1 1 2 1 2 1 1 1 L L x x x 1 2 2 2 1 1 2 1 1 ( ) 1 1 1 1 ( ) L x x x L x x x x x x L 1 2 1 2 1 1 1 L L x x x 1 2 1 1 2 2 1 L L L x L x x
- 4. Therefore natural coordinates at node 1 is (1, 0) and natural coordinates at node 2 is (0, 1). 2 1 x x L L 1 2 x x L L