Moment distribution method: Difference between revisions

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{{Short description|Structural analysis technique for statically indeterminate structures}}

{{Distinguish|Moment redistribution}}

The '''moment distribution method''' is a [[structural analysis]] methomethod for [[statically indeterminate]] [[Beam (structure)|beam]]s and [[Framing (construction)|frames]] developed by [[Hardy Cross]]. It was published in 1930 in an [[American Society of Civil Engineers|ASCE]] journal.<ref name="asce1">{{Cite news|first=Hardy|last=Cross|year=1930|title=Analysis of Continuous Frames by Distributing Fixed-End Moments|periodical=Proceedings of the American Society of Civil Engineers|publisher=ASCE|pages=919–928}}</ref> The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until [[computers]] began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.

== Introduction ==

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=== Fixed end moments ===

[[Fixed end moments]] are the moments produced at member ends by external loads. Spanwise calculation is carried out assuming each support to be fixed and implementing formulas as per the nature of load ,i.e. point load (mid span or unequal), udl, uvl or couple.

=== Bending stiffness ===

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Numbers <span style="background-color:#F8F8F8; border-style:solid; border-width:1px; border-color:#AAAAAA;">in grey</span> are balanced moments; arrows (<span style="border-style:solid; border-width:1px; border-color:#AAAAAA;">&nbsp;→&nbsp;/&nbsp;←&nbsp;</span>) represent the carry-over of moment from one end to the other end of a member.

* Step 1: As joint A is released, balancing moment of magnitude equal to the fixed end moment <math>M_{AB}^{f} = 14.700 \mathrm{\,kN \,m}</math> develops and is carried-over from joint A to joint B.
* Step 2: The unbalanced moment at joint B now is the summation of the fixed end moments <math>M_{BA}^{f}</math>, <math>M_{BC}^{f}</math> and the carry-over moment from joint A. This unbalanced moment is distributed to members BA and BC in accordance with the distribution factors <math>D_{BA} = 0.2727</math> and <math>D_{BC} = 0.7273</math>. Step 2 ends with carry-over of balanced moment <math>M_{BC}=3.867 \mathrm{\,kN \,m}</math> to joint C. Joint A is a roller support which has no rotational restraint, so moment carryover from joint B to joint A is zero.
* Step 3: The unbalanced moment at joint C now is the summation of the fixed end moments <math>M_{CB}^{f}</math>, <math>M_{CD}^{f}</math> and the carryover moment from joint B. As in the previous step, this unbalanced moment is distributed to each member and then carried over to joint D and back to joint B. Joint D is a fixed support and carried-over moments to this joint will not be distributed nor be carried over to joint C.
* Step 4: Joint B still has balanced moment which was carried over from joint C in step 3. Joint B is released once again to induce moment distribution and to achieve equilibrium.
* Steps 5 - 10: Joints are released and fixed again until every joint has unbalanced moments of size zero or neglectably small in required precision. Arithmetically summing all moments in each respective columns gives the final moment values.

=== Result ===

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=== Result via displacements method ===

As the Hardy Cross method provides only approximate results, with a margin of error inversely proportionate to the number of iterations, it is important{{citation needed|date=September 2012}} to have an idea of how accurate this method might be. With this in mind, here is the result obtained by using an exact method: the [[displacementdirect stiffness method]] (displacement method).

For this, the displacements method equation assumes the following form: