Uniformly Distributed Load of Intensity q

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Uniformly distributed load of intensity qLinearly distributed load of maximum intensity q0

Concentrated load of magnitude P

Couple of magnitute M0

Example
http://en.wikipedia.org/wiki/File:Fem4.pnghttp://en.wikipedia.org/wiki/File:Fem2.pnghttp://en.wikipedia.org/wiki/File:Fem3.pnghttp://en.wikipedia.org/wiki/File:Fem1.png


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Example

The statically indeterminate beam shown in the figure is to be analysed.

Members AB, BC, CD have the same length.

Flexural rigidities are EI, 2EI, EI respectively.

Concentrated load of magnitude acts at a distance from the support A.

Uniform load of intensity acts on BC.

Member CD is loaded at its midspan with a concentrated load of magnitude.

In the following calcuations, counterclockwise moments are positive.

]edit] Fixed-end moments

]edit] Distribution factors
http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9


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The distribution factors of joints A and D are DAB = 1,DDC = 0.

]edit] Carryover factors

The carryover factors are , except for the carryover factor from D (fixed support) to C which is zero.

]edit] Moment distribution

Joint A Joint B Joint C Joint D

Distrib.factors

0 1 0.2727 0.7273 0.6667 0.3333 0 0

Fixed-end

moments

14.700 -6.300 8.333 -8.333 12.500 -12.500

Step 1 -14.700 -7.350

Step 2 1.450 3.867 1.934

Step 3 -2.034 -4.067 -2.034 -1.017

Step 4 0.555 1.479 0.739

Step 5 -0.246 -0.493 -0.246 -0.123

Step 6 0.067 0.179 0.090
http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12http://en.wikipedia.org/wiki/File:MomentDistributionMethod2.jpghttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12


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Step 7 -0.030 -0.060 -0.030 -0.015

Step 8 0.008 0.022 0.011

Step 9 -0.004 -0.007 -0.004 -0.002

Step 10 0.001 0.003

Sum ofmoment

s0 -11.569 11.569 -10.186 10.186 -13.657

Numbers in grey are balaced moments; arrows ( / ) represent the carry-over of moment from one end

to the other end of a member.

]edit] Result

Moments at joints determined by the moment distribution method

The conventional engineer's sign convention is used here, i.e. positive moments cause elongation atthe bottom part of a beam member.

For comparison purposes, the following are the results generated using a matrix method. Note that in theanalysis above, the iterative process was carried to >0.01 precision. The fact that the matrix analysis results

and the moment distribution analysis results match to 0.001 precision is mere coincidence.

Moments at joints determined by the matrix method

The complete shear and bending moment diagrams are as shown. Note that the moment distribution methodonly determines the moments at the joints. Developing complete bending moment diagrams require

additional calculations using the determined joint moments and internal section equilibrium.

SFD and BMD
http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/wiki/Matrix_methodhttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/wiki/Matrix_method


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Shear force diagram Bending moment diagram

]edit] Notes

Example

Example








http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9


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The distribution factors of joints A and D are DAB = 1,DDC = 0.]edit] Carryover factors




Distrib.factors

0 1 0.2727 0.7273 0.6667 0.3333 0 0

Fixed-end

moment

14.700 -6.300 8.333 -8.333 12.500 -12.500
http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12http://en.wikipedia.org/wiki/File:MomentDistributionMethod2.jpghttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12


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s

Step 1 -14.700 -7.350

Step 2 1.450 3.867 1.934

Step 3 -2.034 -4.067 -2.034 -1.017

Step 4 0.555 1.479 0.739

Step 5 -0.246 -0.493 -0.246 -0.123

Step 6 0.067 0.179 0.090

Step 7 -0.030 -0.060 -0.030 -0.015

Step 8 0.008 0.022 0.011

Step 9 -0.004 -0.007 -0.004 -0.002

Step 10 0.001 0.003

Sum ofmoment

s0 -11.569 11.569 -10.186 10.186 -13.657

Numbers in grey are balaced moments; arrows ( / ) represent the carry-over of moment from one endto the other end of a member.

]edit] Result


http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13


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SFD and BMD


Example

Example




http://en.wikipedia.org/wiki/Matrix_methodhttp://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/Matrix_method


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http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12


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Distrib.factors

0 1 0.2727 0.7273 0.6667 0.3333 0 0

Fixed-end

moments

14.700 -6.300 8.333 -8.333 12.500 -12.500

Step 1 -14.700 -7.350

Step 2 1.450 3.867 1.934

Step 3 -2.034 -4.067 -2.034 -1.017

Step 4 0.555 1.479 0.739

Step 5 -0.246 -0.493 -0.246 -0.123

Step 6 0.067 0.179 0.090

Step 7 -0.030 -0.060 -0.030 -0.015

Step 8 0.008 0.022 0.011

Step 9 -0.004 -0.007 -0.004 -0.002

Step 10 0.001 0.003

Sum ofmoment

0 -11.569 11.569 -10.186 10.186 -13.657
http://en.wikipedia.org/wiki/File:MomentDistributionMethod2.jpg


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s


]edit] Result



For comparison purposes, the following are the results generated using a matrix method. Note that in the

analysis above, the iterative process was carried to >0.01 precision. The fact that the matrix analysis resultsand the moment distribution analysis results match to 0.001 precision is mere coincidence.




SFD and BMD

Shear force diagram Bending moment

Example
http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/wiki/Matrix_methodhttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/wiki/Matrix_method


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Example











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Distrib.factors

0 1 0.2727 0.7273 0.6667 0.3333 0 0

Fixed-end

moments

14.700 -6.300 8.333 -8.333 12.500 -12.500

Step 1 -14.700 -7.350

Step 2 1.450 3.867 1.934

Step 3 -2.034 -4.067 -2.034 -1.017

Step 4 0.555 1.479 0.739

Step 5 -0.246 -0.493 -0.246 -0.123

Step 6 0.067 0.179 0.090


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Step 7 -0.030 -0.060 -0.030 -0.015

Step 8 0.008 0.022 0.011

Step 9 -0.004 -0.007 -0.004 -0.002

Step 10 0.001 0.003

Sum ofmoment

s0 -11.569 11.569 -10.186 10.186 -13.657



]edit] Result








SFD and BMD


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Example

Example








http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9


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Distrib.factors

0 1 0.2727 0.7273 0.6667 0.3333 0 0


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Fixed-end

moments

14.700 -6.300 8.333 -8.333 12.500 -12.500

Step 1 -14.700 -7.350

Step 2 1.450 3.867 1.934

Step 3 -2.034 -4.067 -2.034 -1.017

Step 4 0.555 1.479 0.739

Step 5 -0.246 -0.493 -0.246 -0.123

Step 6 0.067 0.179 0.090

Step 7 -0.030 -0.060 -0.030 -0.015

Step 8 0.008 0.022 0.011

Step 9 -0.004 -0.007 -0.004 -0.002

Step 10 0.001 0.003

Sum ofmoment

s

0 -11.569 11.569 -10.186 10.186 -13.657


]edit] Result

http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13


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SFD and BMD


Example
http://en.wikipedia.org/wiki/Matrix_methodhttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/Matrix_method


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Example











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Distrib.factors

0 1 0.2727 0.7273 0.6667 0.3333 0 0

Fixed-end

moments

14.700 -6.300 8.333 -8.333 12.500 -12.500

Step 1 -14.700 -7.350

Step 2 1.450 3.867 1.934

Step 3 -2.034 -4.067 -2.034 -1.017

Step 4 0.555 1.479 0.739

Step 5 -0.246 -0.493 -0.246 -0.123

Step 6 0.067 0.179 0.090


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Step 7 -0.030 -0.060 -0.030 -0.015

Step 8 0.008 0.022 0.011

Step 9 -0.004 -0.007 -0.004 -0.002

Step 10 0.001 0.003

Sum ofmoment

s0 -11.569 11.569 -10.186 10.186 -13.657



]edit] Result








SFD and BMD


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Example

Example




http://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpg


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http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=12


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Distrib.factors

0 1 0.2727 0.7273 0.6667 0.3333 0 0

Fixed-end

moments

14.700 -6.300 8.333 -8.333 12.500 -12.500

Step 1 -14.700 -7.350

Step 2 1.450 3.867 1.934

Step 3 -2.034 -4.067 -2.034 -1.017

Step 4 0.555 1.479 0.739

Step 5 -0.246 -0.493 -0.246 -0.123

Step 6 0.067 0.179 0.090

Step 7 -0.030 -0.060 -0.030 -0.015

Step 8 0.008 0.022 0.011

Step 9 -0.004 -0.007 -0.004 -0.002

Step 10 0.001 0.003

Sum ofmoment

0 -11.569 11.569 -10.186 10.186 -13.657
http://en.wikipedia.org/wiki/File:MomentDistributionMethod2.jpg


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s


]edit] Result



For comparison purposes, the following are the results generated using a matrix method. Note that in the

analysis above, the iterative process was carried to >0.01 precision. The fact that the matrix analysis resultsand the moment distribution analysis results match to 0.001 precision is mere coincidence.




SFD and BMD

http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/wiki/Matrix_methodhttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=13http://en.wikipedia.org/wiki/Matrix_method


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Example

Example


Members AB, BC, CD have the same length.Flexural rigidities are EI, 2EI, EI respectively.





http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethod.jpghttp://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Moment_distribution_method&action=edit&section=9


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Distrib.factors

0 1 0.2727 0.7273 0.6667 0.3333 0 0

Fixed-end

moments

14.700 -6.300 8.333 -8.333 12.500 -12.500

Step 1 -14.700 -7.350


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Step 2 1.450 3.867 1.934

Step 3 -2.034 -4.067 -2.034 -1.017

Step 4 0.555 1.479 0.739

Step 5 -0.246 -0.493 -0.246 -0.123

Step 6 0.067 0.179 0.090

Step 7 -0.030 -0.060 -0.030 -0.015

Step 8 0.008 0.022 0.011

Step 9 -0.004 -0.007 -0.004 -0.002

Step 10 0.001 0.003

Sum ofmoments

0 -11.569 11.569 -10.186 10.186 -13.657


]edit] Result







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SFD and BMD

http://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodBMD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpghttp://en.wikipedia.org/wiki/File:MomentDistributionMethodSFD.jpg

Uniformly Distributed Load of Intensity q

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