Question

Figures 1 (a) and (b) show a linear square plane stress element in st coordinates and its distorted form mapped into quadrilateral in x-y coordinates, respectively. Edge - 1 u (*3. y) (-1,1) 1 1 (1,1) Edge PNJA 5 -Edge s = 1 (XY) 2 (xz. Y2) 2 Edge 1 = -1 (-1,-1) (1, -1) (b) Figure 1 (a) Specify the interpolation functions for the four-node isoparametric quadrilateral element. (4 marks) (b) Discuss how the isoparametric formulation is applied into quadrilateral in x-y coordinates. (4 marks) (C) What is the advantages of using isoparametric elements, which have been adopted in most of the finite element analysis software? (2 marks)

Answer 1

All the 3 parts have been answered below

a) Futerpolation function for four-bode isoparametro quadrilateral element 4,V Edget- 5=/ + 3.(3.45) v1 T (-1,04 (74,44) 4. Edge S--- 14 P(X4) + 4 -- s For Edge s=1 +2 (22,42) 1 Edge t - (1,0) x, u Pra) To get this mapping, . The parent Rectangular element heus 4 nodes 1,2,3 $4 with co-ordinates (-1,-1), (1,-1), (1, 1) and (-1,1). The shape function of this element are; OLI N = (1-&)(1-2) N₂ = (1+8)(1-2) 4. N3 = (1+8)(1+) (-43c14m) ; N4 4 point with 4 so the co-ordinate with (&, n). b) Now consider global coordinate system (X,Y): In global system, the co-padinates of nodal points are: (*19), (X2,42), (43,93), (X4,44) we define the co-ordinate of point & as P(x,y): (x= x,N, + X₂N₂ + x3 3 + x4 N4) and (y=y,N, + y 2 N2 + Y3N3 + 44N4) 11 At nodal co-ordinates (nodes) shape function are such that Ni=1 and rest othess are zero, c) Advantages q Iso-parametric elements : i) Iso parametric elements are more versatille elements for modelling irregular / complicated geometry shapes. un These elements can be used for modelling curved boundaries, ii) Isoparametaic elements can also be used for transition from coarse to fine mesh so as to find out concentration of stress in critical regions.