The Finite Element Method for the Analysis of Non-Linear and ...
The Finite Element Method for the Analysis of Non-Linear and ...
The Finite Element Method for the Analysis of Non-Linear and ...
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Truss <strong>and</strong> Cable (Bar) <strong>Element</strong>s<br />
<strong>The</strong> nodal point coordinates determine <strong>the</strong> spatial configuration <strong>of</strong> <strong>the</strong> bar<br />
at time 0 <strong>and</strong> t using:<br />
n∑<br />
n∑<br />
n∑<br />
0 x 1 (r) = N 0 k x k 1<br />
0 x 2 (r) = N 0 k x k 2<br />
0 x 3 (r) = N 0 k x k 3<br />
<strong>and</strong> t x 1 (r) =<br />
Shape Functions<br />
k=1<br />
n∑<br />
N t k x k 1<br />
k=1<br />
t x 2 (r) =<br />
n = 2 nodes at r 1 = −1, r 2 = 1<br />
k=1<br />
n∑<br />
N t k x k 2<br />
k=1<br />
N1 a = 1 2 (1 − r), N 2 a = 1 (1 + r)<br />
2<br />
n = 3 nodes at r 1 = −1, r 2 = 1, r 3 = 0<br />
t x 3 (r) =<br />
N b 1 = N a 1 − 1 2 (1 − r2 ), N b 2 = N a 2 − 1 2 (1 − r2 ), N b 3 = (1 − r 2 )<br />
k=1<br />
n∑<br />
N t k x k 3<br />
k=1<br />
n = 4 nodes at r 1 = −1, r 2 = 1, r 3 = − 1 3 , r4 = 1 3<br />
N c 1 = N b 1 + 1 16 (−9r3 + r 2 + 9r − 1), N c 2 = N b 2 + 1 16 (9r3 + r 2 − 9r − 1),<br />
N c 3 = N b 3 + 1 16 (27r3 + 7r 2 − 27r − 7), N c 4 = 1 16 (−27r3 − 9r 2 + 27r + 9)<br />
Institute <strong>of</strong> Structural Engineering <strong>Method</strong> <strong>of</strong> <strong>Finite</strong> <strong>Element</strong>s II 15