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<div>The '''moment-area theorem''' is an engineering tool to derive the slope, rotation and deflection of beams and frames. This theorem was developed by [[Christian Otto Mohr|Mohr]] and later stated namely by Charles E. Greene in 1873. This method is advantageous when we solve problems involving beams, especially for those subjected to a series of concentrated loadings or having segments with different [[moments of inertia]]. If we draw the [[moment diagram]] for the beam and then divided it by the flexural rigidity(EI), the 'M/EI diagram' results by the following equation<br />
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<math>\theta=\int\left(\frac{M}{EI}\right)dx</math><br />
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==Theorem 1==<br />
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The change in slope between any two points on the elastic curve equals the area of the M/EI diagram between these two points.<br />
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<math>\theta_{AB} = {\int_A}^B \frac{M}{EI}\;dx</math><br />
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where,<br />
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* M = moment<br />
* EI = flexural rigidity<br />
* <math>\theta_{AB}</math> = change in slope between points A and B<br />
* A, B = points on the elastic curve<ref>{{cite web|title=Hibbeler, R.C. (2012). Structural Analysis. Upper Saddle River, NJ: Pearson}}</ref><br />
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==Theorem 2==<br />
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The vertical deviation of the tangent at a point A on an elastic curve with respect to the tangent which is extended from another point B equals the moment of the area under the M/EI diagram between those two points (A and B). This moment is computed about point A where the deviation from B to A is to be determined.<br />
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<math>t_{A/B} = {\int_A}^B \frac{M}{EI} \bar{x} \;dx</math><br />
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where,<br />
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* M = moment<br />
* EI = flexural rigidity<br />
* <math>t_{A/B}</math> = deviation of tangent at point B with respect to the tangent at point A<br />
* <math>\bar{x}</math> = centroid of M/EI diagram measured horizontally from point A<br />
* A, B = points on the elastic curve<ref>Hibbeler, R.C. (2012). Structural Analysis. Upper Saddle River, NJ: Pearson. pp.&nbsp;316–325.</ref><br />
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==Rule of Sign Convention==<br />
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The deviation at any point on the elastic curve is positive if the point lies above the tangent, negative if the point is below the tangent; we measured it from left tangent, if θ is counterclockwise direction, the change in slope is positive, negative if θ is clockwise direction.<ref>[http://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/area-moment-method-beam-deflections Moment-Area Method Beam Deflection]</ref><br />
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==Procedure for Analysis==<br />
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The following procedure provides a method that may be used to determine the displacement and slope at a point on the elastic curve of a beam using the moment-area theorem.<br />
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* Determine the reaction forces of a structure and draw the M/EI diagram of the structure.<br />
* If there are only concentrated loads on the structure, the problem will be easy to draw M/EI diagram which will results a series of triangular shapes.<br />
* If there are mixed with distributed loads and concentrated, the moment diagram (M/EI) will results parabolic curves, cubic and etc.<br />
* Then, assume and draw the deflection shape of the structure by looking at M/EI diagram.<br />
* Find the rotations, change of slopes and deflections of the structure by using the geometric mathematics.<br />
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==References==<br />
{{reflist}}<br />
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==External links==<br />
*[http://www.engr.mun.ca/~katna/5931/Deflections_Area-moment2p.pdf Area-Moment Method. (n.d.)]<br />
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[[Category:Structural analysis]]</div>115.111.221.150