LU Decomposition-More Examples: Civil Engineering



Chapter 04.07

LU Decomposition – More Examples

Civil Engineering

Example 1

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

[pic]

In the compound cylinder, the inner cylinder has an internal radius of [pic], and an outer radius [pic], while the outer cylinder has an internal radius of [pic] and an outer radius of [pic]. Given [pic]psi, [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 LU decomposition.

Solution

[pic]

The [pic] matrix is the same as the one found at the end of the forward elimination steps of the naïve Gauss elimination method.

Forward Elimination of Unknowns

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

[pic]

First step

Divide Row 1 by [pic] and multiply it by [pic], that is, multiply Row 1 by [pic]. Then subtract the result from Row 2.

[pic]

Divide Row 1 by [pic] and multiply it by [pic], that is, multiply Row 1 by [pic]. Then subtract the result from Row 3.

[pic]

Divide Row 1 by [pic] and multiply it by 0, that is, multiply Row 1 by [pic]. Then subtract the result from Row 4.

[pic]

Second step

Divide Row 2 by [pic] and multiply it by [pic], that is, multiply Row 2 by [pic]. Then subtract the result from Row 3.

[pic]

Divide Row 2 by [pic] and multiply it by [pic] that is, multiply Row 2 by [pic]. Then subtract the result from Row 4.

[pic]

Third step

Divide Row 3 by [pic] and multiply it by [pic] that is, multiply Row 3 by [pic]. Then subtract the result from Row 4.

[pic]The coefficient matrix after the completion of the forward elimination steps is

[pic]

Now find [pic].

[pic]

From the first step of forward elimination,

[pic]

[pic]

[pic]

From the second step of forward elimination,

[pic]

[pic]

From the third step of forward elimination,

[pic]

Hence

[pic]

Now that [pic] and [pic] are known, solve [pic]

[pic]

to give

[pic]

[pic]

[pic]

[pic]

Forward substitution starting from the first equation gives

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Hence

[pic]

Now solve

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

From the fourth equation,

[pic]

[pic]

[pic]

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

[pic]

[pic]

[pic]

[pic]

Substituting the values of [pic] into the second equation,

[pic]

[pic] [pic]

[pic]

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

[pic]

[pic]

[pic]

[pic]

The solution vector is

[pic]

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

[pic]

|SIMULTANEOUS LINEAR EQUATIONS | |

|Topic |LU Decomposition – More Examples |

|Summary |Examples of LU decomposition |

|Major |Civil Engineering |

|Authors |Autar Kaw |

|Date |August 8, 2009 |

|Web Site | |

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