A triangle element with 3 nodes is shown in the figure 1. The coordinates of Node...

 

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A triangle element with 3 nodes is shown in the figure 1. The coordinates of Node 1, 2 and 3 are (0, 1), (0, 0) and (1, 0), respectively. The deflections along x and y axis of Nodes 1, 2, 3 are ((, ), (, V) and (3, V3), respectively. (1) Try to use the nodal deflections of these 3 nodes to build deflection functions of this triangle element, that is, u(x, y) andv(x, y). Hints: Assume u(x, y) and v(x, y) are a linear function of x and y, that is, u(x, y) = a + bx+cy and v(x, y) = a + bx+cy. Then use the deflections of 3 nodes to determine a, b and c. (2) Express the deflection functions u(x, y) and v(x, y) by shape functions. Hints: u(x, y) = N + NU +NU v(x, y) = Nv + NV + NV3 (3) If ( = 0, v = 0.01), (u = 0, v =0) and (u3 =0.02, v3 = 0), calculate the strain & and at point (0.5,0.5) Hints: use equations & = du(x, y) dx Node 1 (u, V) (0,1) dv (x, y) dy Node 3 (u,,V) (1,0) (0,0) Node 2 (1) Figure 1 A 3 node triangle element A triangle element with 3 nodes is shown in the figure 1. The coordinates of Node 1, 2 and 3 are (0, 1), (0, 0) and (1, 0), respectively. The deflections along x and y axis of Nodes 1, 2, 3 are ((, ), (, V) and (3, V3), respectively. (1) Try to use the nodal deflections of these 3 nodes to build deflection functions of this triangle element, that is, u(x, y) andv(x, y). Hints: Assume u(x, y) and v(x, y) are a linear function of x and y, that is, u(x, y) = a + bx+cy and v(x, y) = a + bx+cy. Then use the deflections of 3 nodes to determine a, b and c. (2) Express the deflection functions u(x, y) and v(x, y) by shape functions. Hints: u(x, y) = N + NU +NU v(x, y) = Nv + NV + NV3 (3) If ( = 0, v = 0.01), (u = 0, v =0) and (u3 =0.02, v3 = 0), calculate the strain & and at point (0.5,0.5) Hints: use equations & = du(x, y) dx Node 1 (u, V) (0,1) dv (x, y) dy Node 3 (u,,V) (1,0) (0,0) Node 2 (1) Figure 1 A 3 node triangle element

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