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225 lines
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HTML
225 lines
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HTML
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<!-- #BeginEditable "doctitle" -->
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<title>Moments of Area</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Moments of Area</h1>
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<p class="center">First and Second Moment of Area</p>
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<h2>Moment in Physics</h2>
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<p>In Physics <a href="../physics/moment-torque.html">Moment (or Torque)</a> is <b>force times distance</b>:</p>
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<p class="center"><img src="../physics/images/moment-wrench.svg" alt="moment force at right angles on wrench"></p>
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<p>But there are other Moments, read on!</p>
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<h2>First Moment of Area</h2>
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<p>First Moment of Area is <b>area times distance</b> (to some reference line):</p>
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<p class="center"><img src="images/moment-area-1.svg" alt="First Moment of Area"><br>
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First Moment of Area = A x d</p>
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<p>For this simple case we can multiply the whole area by the distance (from its middle to the reference line).</p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center"><img src="images/moment-area-1-ex1.svg" alt="First Moment of Area"></p>
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<p>Area is 20 mm x 10 mm = 200 mm<sup>2</sup></p>
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<p>First Moment of Area (relative to the bottom line) = 200 mm<sup>2</sup> x 25 mm = 5000 mm<sup>3</sup></p>
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<p>(Note the unit is mm<sup>3</sup>, but is not a volume!)</p>
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</div>
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<p>One of it's great uses is to find the <a href="centroid.html">centroid</a> , which is the average position of all the points of an object:</p>
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<p class="center"><img src="images/centroid.svg" alt="Centroid"><br>
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A plane shape cut from a piece of card will balance perfectly on its centroid.</p>
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<p>To find the distance to the centroid from any axis, we divide the First Moment of Area by the Total Area:</p>
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<div class="center larger">Distance to Centroid = <span class="intbl"><em>First Moment of Area</em><strong>Total Area</strong></span></div>
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<p>When we do that for both the x-axis and y-axis we get the centroid.</p>
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<p>We can <b>estimate</b> where the centroid is using squares:</p>
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<script src="images/area-estim.js"></script>
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<script>areaestimMain('mom1');</script>
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<h2>Second Moment of Area</h2>
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<p>For the <b>Second Moment of Area</b> we multiply the area by the distance <b>squared</b>:</p>
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<p class="center"><img src="images/moment-area-2.svg" alt="Second Moment of Area"><br>
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(need infinitely many tiny squares)</p>
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<p>But be careful! We need to multiply every tiny bit of area by its distance squared, because area further away has a bigger effect (due to the distance being squared).</p>
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<p>It is called "Second" moment because we square the distance "x<sup>2</sup>"</p>
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<p>It is also called the <b>area moment of inertia</b>.</p>
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<p>We can <b>estimate</b> the second moment using squares, but it is very inaccurate:</p>
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<iframe src="iarea-estim-2.html" scrolling="no" style="width:544px; height:464px; overflow:hidden; margin:auto; display:block; border: none;"></iframe>
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<p>We can use the x-axis or y-axis as the reference line, or we can use the <b>centroid for the reference line</b>. You can try that option above.</p>
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<div class="def">
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<p><b>Notation</b>: The symbol is an "I" followed by a little "x" or "y" for the reference axis.</p>
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<p class="center"><span class="larger">I<sub>x</sub></span> is in relation to the <b>x</b> axis (and we use <b>y</b> distances times area)<br><span class="larger">I<sub>y</sub></span> is in relation to the <b>y</b> axis (and we use <b>x</b> distances times area)</p>
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<p>The letter <b>I</b> refers to <b>Inertia</b> in "area moment of inertia".</p></div>
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<p>Where possible use an accurate formula such as:</p>
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<div class="tbl">
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<div class="row">
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<div class="lt"><img src="images/moment-area-2-rect.svg" alt="Second Moment of Area Rectangle"></div>
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<div class="rt" style="vertical-align:middle;">
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I<sub>x</sub> = <span class="intbl"><em>bh<sup>3</sup></em><strong>3</strong></span>
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<!-- I_x = bh^3/3 -->
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<br>
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I<sub>y</sub> = <span class="intbl"><em>b<sup>3</sup>h</em><strong>3</strong></span>
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<!-- I_y = b^3h/3 -->
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</div>
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</div>
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<div class="row">
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<div class="lt"><img src="images/moment-area-2-rect2.svg" alt="Second Moment of Area Rectangle at Centroid"></div>
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<div class="rt" style="vertical-align:middle;">
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I<sub>x</sub> = <span class="intbl"><em>bh<sup>3</sup></em><strong>12</strong></span>
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<!-- I_x = bh^3/12 -->
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<br>
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I<sub>y</sub> = <span class="intbl"><em>b<sup>3</sup>h</em><strong>12</strong></span>
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<!-- I_y = b^3h/12 -->
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</div>
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</div>
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<div class="row">
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<div class="lt"><img src="images/moment-area-2-tri.svg" alt="Second Moment of Area Triangle"></div>
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<div class="rt" style="vertical-align:middle;">
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I<sub>x</sub> = <span class="intbl"><em>bh<sup>3</sup></em><strong>12</strong></span>
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<br>
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I<sub>y</sub> = <span class="intbl"><em>b<sup>3</sup>h+b<sup>2</sup>ha+bha<sup>2</sup></em><strong>12</strong></span>
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</div>
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</div>
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<div class="row">
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<div class="lt"><img src="images/moment-area-2-circ.svg" alt="Second Moment of Area Circle"></div>
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<div class="rt" style="vertical-align:middle;">
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I<sub>x</sub> = <span class="intbl"><em><span class="times">π</span>r<sup>4</sup></em><strong>4</strong></span>
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<br>
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I<sub>y</sub> = <span class="intbl"><em><span class="times">π</span>r<sup>4</sup></em><strong>4</strong></span>
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</div>
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</div>
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<div class="row">
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<div class="lt">Many more!</div>
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<div class="rt" style="vertical-align:middle;">
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</div>
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</div>
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</div>
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<p>
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Engineers use the second moment of area to work out how rigid (hard to bend) a beam is. </p>
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<div class="example">
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<h3>
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Example: A beam that is 100 mm by 24 mm</h3>
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<p>
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Lying flat it looks like this:</p>
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<p class="center"><img src="images/moment-area-2-ex1.svg" alt="Second Moment of Area Beam"></p>
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<p class="center">I<sub>x</sub> = <span class="intbl"><em>bh<sup>3</sup></em><strong>12</strong></span>
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= <span class="intbl"><em>100 × 24<sup>3</sup></em><strong>12</strong></span>
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= <b>115,200</b> mm<sup>4</sup></p>
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<p>But sitting upright it is:</p>
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<p class="center"><img src="images/moment-area-2-ex2.svg" alt="Second Moment of Area Beam"></p>
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<p class="center">I<sub>x</sub> = <span class="intbl"><em>bh<sup>3</sup></em><strong>12</strong></span>
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= <span class="intbl"><em>24 × 100<sup>3</sup></em><strong>12</strong></span>
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= <b>2,000,000</b> mm<sup>4</sup></p>
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<p>It is nearly 20 times as rigid sitting upright!</p>
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<p>And that is why beams sit up like this:</p>
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<p class="center"><img src="images/structure.jpg" alt="structure"></p>
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<p>Try bending a ruler about each axis to experience it for yourself:</p>
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<p class="center"><img src="images/ruler.jpg" alt="ruler"></p>
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</div>
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<h2>Engineers Love I-Beams</h2>
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<p>Here we have two equal-sized beams, but one is solid, the other shaped like an "I"</p>
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<p class="center"><img src="images/moment-area-beam.svg" alt="Second Moment of Area Beam"></p>
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<p>The solid beam is a bit stiffer against bending (I<sub>x</sub> = 333 vs 205) but <b>very much heavier</b> (40kg vs 16kg).</p>
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<p>In practice we could have a slightly bigger I-Beam and still save a lot of money in steel, transport and handling.</p>
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<p class="center"><img src="images/i-beams.jpg" alt="i-beams"></p>
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<p><br></p>
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<div class="related">
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<a href="index.html">Geometry Index</a>
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</div>
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