Gaussian Elimination-More Examples: Civil Engineering



Chapter 04.06

Gaussian Elimination – More Examples

Civil Engineering

Example 1

To find the maximum stresses in a compound cylinder, the following four simultaneous linear equations need to be solved.

[pic]

In the compound cylinder, the inner cylinder has an internal radius of [pic] and an outer radius of [pic], while the outer cylinder has an internal radius of [pic] and an outer radius of [pic]. Given [pic], [pic], and that the hoop stress in the outer cylinder is given by

[pic],

find the stress on the inside radius of the outer cylinder.

Find the values of [pic], [pic], [pic] and [pic] using naïve Gauss elimination.

Solution

Forward Elimination of Unknowns

Since there are four equations, there will be three steps of forward elimination of unknowns.

First step

Divide Row 1 by [pic]and then multiply it by [pic], that is, multiply Row 1 by [pic].

[pic]

Subtract the result from Row 2 to get

[pic]

Divide Row 1 by [pic] and then multiply it by [pic], that is, multiply Row 1 by [pic].

[pic]

Subtract the result from Row 3 to get

[pic]Divide Row 1 by [pic] and then multiply it by 0, that is, multiply Row 1 by [pic].

[pic]

Subtract the result from Row 4 to get

[pic]

Second step

We now divide Row 2 by [pic] and then multiply it by [pic], that is, multiply Row 2 by [pic].

[pic]

Subtract the result from Row 3 to get

[pic]Divide Row 2 by [pic] and then multiply by [pic], that is, multiply Row 2 by [pic].

[pic]

Subtract the result from Row 4 to get

[pic]

Third step

We now divide Row 3 by [pic] and then multiply by [pic], that is, multiply Row 2 by [pic].

[pic]Subtract the result from Row 4 to get

[pic]

Back substitution

From the fourth equation,

[pic]

[pic]

Substituting the value of [pic] in the third equation,

[pic]

[pic]

[pic]

[pic]

Substituting the value of [pic] in the second equation,

[pic]

[pic]

[pic]

[pic]

Substituting the values of [pic], [pic] in the first equation,

[pic]

[pic]

[pic]

[pic]

Hence the solution vector is

[pic]

The stress on the inside radius of the outer cylinder is then given by

[pic]

|SIMULTANEOUS LINEAR EQUATIONS | |

|Topic |Gaussian Elimination – More Examples |

|Summary |Examples of Gaussian elimination |

|Major |Civil Engineering |

|Authors |Autar Kaw |

|Date |August 8, 2009 |

|Web Site | |

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