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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