Question

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2 LI2 A three-node bar element shown below is uniform and is allowed only axial displacement u. Its degrees of freedoms are us, U and us The shape functions for this bar are given by 2x-1x N L-31x+2x and N 4(Lx-x b.Using the properties of shape a function, determine which of the above shape functions are N, N2 and Na

Answer 1

The three nodes of the bar are at 0,L/2 and L. Each shape functions has it own unique node. The corresponding shape function has values one at it's own node and in other nodes it is zero. So we will put the above node value to each of the shape function and find them.

First Node ; x=0

L2 0, N Ni0, N = 1 L2

Hence shape function of first node x=0 ,

k

Second node; X=L/2

212/4 L2/2 L/2- L2/2 Ni 0 L2 4(L2/2 L2/4) L2 4(L2/4) Ni 1 L2 L2 3L2/2 2L2/4 L2 - 0 12

Thus for second node,

N2N

Third mode; X=L

2L2 L2 1 Ni = L2 4(L2 L2) = 0 Ni L2 L2 3L22L NK 0 L2

So for the third mode,

3