https://en.formulasearchengine.com/api.php?action=feedcontributions&user=115.111.221.150&feedformat=atom formulasearchengine - User contributions [en] 2020-11-29T22:49:57Z User contributions MediaWiki 1.36.0-alpha https://en.formulasearchengine.com/index.php?title=Moment-area_theorem&diff=29321 Moment-area theorem 2013-11-14T07:02:12Z <p>115.111.221.150: </p> <hr /> <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 /> <br /> &lt;math&gt;\theta=\int\left(\frac{M}{EI}\right)dx&lt;/math&gt;<br /> <br /> ==Theorem 1==<br /> <br /> 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 /> <br /> &lt;math&gt;\theta_{AB} = {\int_A}^B \frac{M}{EI}\;dx&lt;/math&gt;<br /> <br /> where,<br /> <br /> * M = moment<br /> * EI = flexural rigidity<br /> * &lt;math&gt;\theta_{AB}&lt;/math&gt; = change in slope between points A and B<br /> * A, B = points on the elastic curve&lt;ref&gt;{{cite web|title=Hibbeler, R.C. (2012). Structural Analysis. Upper Saddle River, NJ: Pearson}}&lt;/ref&gt;<br /> <br /> ==Theorem 2==<br /> <br /> 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 /> <br /> &lt;math&gt;t_{A/B} = {\int_A}^B \frac{M}{EI} \bar{x} \;dx&lt;/math&gt;<br /> <br /> where,<br /> <br /> * M = moment<br /> * EI = flexural rigidity<br /> * &lt;math&gt;t_{A/B}&lt;/math&gt; = deviation of tangent at point B with respect to the tangent at point A<br /> * &lt;math&gt;\bar{x}&lt;/math&gt; = centroid of M/EI diagram measured horizontally from point A<br /> * A, B = points on the elastic curve&lt;ref&gt;Hibbeler, R.C. (2012). Structural Analysis. Upper Saddle River, NJ: Pearson. pp.&amp;nbsp;316–325.&lt;/ref&gt;<br /> <br /> ==Rule of Sign Convention==<br /> <br /> 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.&lt;ref&gt;[http://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/area-moment-method-beam-deflections Moment-Area Method Beam Deflection]&lt;/ref&gt;<br /> <br /> ==Procedure for Analysis==<br /> <br /> 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 /> <br /> * 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 /> <br /> ==References==<br /> {{reflist}}<br /> <br /> ==External links==<br /> *[http://www.engr.mun.ca/~katna/5931/Deflections_Area-moment2p.pdf Area-Moment Method. (n.d.)]<br /> <br /> [[Category:Structural analysis]]</div> 115.111.221.150