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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Special Considerations-Geometric Stiffness<br />
Cable <strong>Element</strong><br />
<strong>The</strong> fundamental equations <strong>for</strong> <strong>the</strong> geometric stiffness <strong>for</strong> a rod or a cable are<br />
very simple to derive. Consider <strong>the</strong> horizontal cable, <strong>of</strong> length L with an initial<br />
tension T. If <strong>the</strong> cable is subjected to lateral displacements, v i <strong>and</strong> v j , at both<br />
DYNAMIC ANALYSIS OF STRUCTURES<br />
ends, as shown, <strong>the</strong>n additional <strong>for</strong>ces, F i <strong>and</strong> F j , must be developed <strong>for</strong> <strong>the</strong><br />
cable element to be in equilibrium in its displaced position.<br />
T<br />
Fi<br />
´<br />
De<strong>for</strong>med Position<br />
F j<br />
T<br />
v i<br />
´<br />
i<br />
L<br />
´<br />
´<br />
j<br />
v j<br />
T<br />
T<br />
Note that we haveFigure assumed 11.1. all <strong>for</strong>ces Forces <strong>and</strong>Acting displacements on a Cable are positive <strong>Element</strong> in <strong>the</strong> up<br />
direction. We have also made <strong>the</strong> assumption that <strong>the</strong> displacements are small<br />
Taking <strong>and</strong> do not moments changeabout <strong>the</strong> tension point j in in <strong>the</strong> <strong>the</strong> cable. de<strong>for</strong>med position, <strong>the</strong> following equilibrium<br />
equation can be written:<br />
Institute <strong>of</strong> Structural Engineering <strong>Method</strong> <strong>of</strong> <strong>Finite</strong> <strong>Element</strong>s II 50
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