Andrew Ferguson 1/30/2008 Two Rod (Truss) 2D Structure 413 A

Kmod for the original truss structure: 10

5.625 10

4

2.16 10

4

Kmod

2.88 10

4

4

2.16 10

10

5.625 10

4

2.16 10

4

4

2.88 10

4

2.16 10

4

8.505 10

4

4

2.16 10

4

1.62 10

4 4

1.62 10

2.16 10

0

0

5.625 10

1.62 10 0

0

0

0

0

4

0

0

0 4

2.16 10

2.16 10

0

5.625 10

0

0

0 10

5.625 10

0 10

0

5.625 10

Kmod for the modified truss structure (with only two fixities): 10

5.625 10

4

2.16 10

4

Kmod

2.88 10

4

4

2.16 10

4

1.62 10

4

2.16 10

4

4

2.88 10

4

2.16 10

4

8.505 10

4

4

2.16 10

4

1.62 10

4

2.16 10

4

2.16 10

1.62 10

2.16 10

1.62 10

0

0

5.625 10

0

0

0

0

0

4

0

0

0

0 4

5.625 10

0

0

0 10

5.625 10 0

0 0

When the fixities are changed so there are only two, the truss structure is now underdefined. The determinant of Kmod and its Condition Number are both zero when the truss is underdefined. When the truss is defined, the Kmod and its Condition Number are both non-zero (but not necessarily equal to each other).

Kmod matrix for the new truss as shown in the above figure: 10

5.625 10

4

2.16 10

4

Kmod

2.88 10

4

4

2.16 10

10

5.625 10

4

2.16 10

4

4

2.88 10

4

2.16 10

4

8.505 10

4

4

2.16 10

4

1.62 10

4

2.16 10

0

0

0 4

5.625 10

0

1.62 10

0

0

4

2.16 10

0

2.16 10

1.62 10

0

0

5.625 10

0

5.625 10

0

0

0

0

0

4

4

0 10

5.625 10

The determinant of Kmod is non-zero while the Condition Number is zero.

Kmod matrix for the truss figured above: 10

7.5 10

4

2.16 10

4

2.88 10

4

Kmod

4

2.16 10

10

7.5 10

4

2.16 10

4

2.16 10

1.62 10

0

0

0

4

2.88 10

4

2.16 10

4

8.505 10

4

4

2.16 10

4

1.62 10

4

2.16 10

4

2.16 10

1.62 10 4

0

0

0

0

0

7.5 10

0

0

5.625 10

0

0

0

0

0

0

0

4

4

10

5.625 10

0

7.5 10

0

0

4.32 10

0

0

0

0

5.76 10

0

0

0

0

4

The determinant of Kmod and CN are both zero.

7.5 10

4 4

4

4.32 10

4

4.32 10

10

7.5 10

4

4.32 10

4

3.24 10

4

5.76 10

4

4.32 10

4

5.76 10

4

4.32 10

4

4.32 10

4

3.24 10

4

4.32 10

4

3.24 10

The Kmod matrix for truss system above: 10

7.5 10

4

4.32 10

Kmod

4

4.32 10

10

7.5 10

0

0

0

0

0

0

7.5 10

4.32 10

5.625 10

0

5.76 10

0

0

4.32 10

5

4

0

0

1.139 10

0

0

4.32 10

0

0

5.625 10

0

0

7.5 10

0

0

4

4

5.76 10

4

4.32 10

4

4.32 10

4

3.24 10

4

4

5.76 10

4

4.32 10

4.32 10 3.24 10

4

4

4

0

4

4

10

7.5 10

4

4.32 10 4

4.32 10

4

3.24 10

5.76 10

4

5.76 10

4

4.32 10

4 4 4

4

4.32 10

10

7.5 10

4

4.32 10

4

3.24 10

4

5.76 10

4

4.32 10

5

1.728 10

4

4.32 10

4

4.32 10

4

3.24 10

4

4.32 10

4

3.24 10

4

4.32 10

4

3.24 10

4

4.32 10

Because the truss between nodes 1 and 2 is now broken into two elements, the truss system meets the required number of elements and is now defined. Kmod CN

0.22

62

4.305 10

4

9.72 10

Test

Test using In class Tutorial 7