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      <header alignment="Center" color="#a9a9a9">&amp;[DATE] &amp;[TIME] - &amp;[FILENAME]</header>
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    <region left="99" top="0" width="270" height="38" color="#000000" fontSize="20">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p style="font-size: 20px;">Example Problems</p>
        </content>
      </text>
    </region>
    <region left="99" top="36" width="602" height="54" color="#000000" fontSize="15">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p style="font-size: 15px;">7.8.2 Beams of Variable Section: Maximum Bending Stress and Deflection Under Load</p>
        </content>
      </text>
    </region>
    <region left="162" top="90" width="598" height="203" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>A machine part is an 800-mm-long straight beam with a variable wide-flangecross-section and a constant thickness of 2 mm.The flange varies from a minimum of 4 mm at the left end, A, to a maximum of 8 mm at the right end, B.The single central web has a constant thickness of 1.5 mm but a linearly varying depth from 6 mm at the left end to 10 mm at the right end.The flange has a constant thickness with a base of 4 mm at A and 8 mm at B.The web flanges are welded together continuously over the entire length and are also welded to supporting structures at each end to provide fixed ends.A concentrated lateral load of 100 N acts normal to the central axis of the beam parallel to the web at a distance of 300 mm from the left end.The maximum bending stress and the deflection under the load are desired.The modulus of elasticity is 70 GPa, or 70,000 N/mm2. </p>
        </content>
      </text>
    </region>
    <region left="9" top="288" width="334" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p style="font-weight: bold;">Enter dimensions, properties and loading</p>
        </content>
      </text>
    </region>
    <region left="162" top="324" width="206" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Machine part length, L: </p>
        </content>
      </text>
    </region>
    <region left="360" top="333" width="82" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">L</e>
          <e type="operand">800</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="360" width="166" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Flange dimensions: </p>
        </content>
      </text>
    </region>
    <region left="180" top="387" width="238" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>thickness, basemin, basemax:</p>
        </content>
      </text>
    </region>
    <region left="324" top="423" width="194" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">t</e>
          <e type="operand">b.min</e>
          <e type="operand">b.max</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operand">2</e>
          <e type="operand">4</e>
          <e type="operand">8</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="459" width="134" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Web dimensions:</p>
        </content>
      </text>
    </region>
    <region left="189" top="486" width="254" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>thickness, depthmin, depthmax:</p>
        </content>
      </text>
    </region>
    <region left="288" top="522" width="225" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">t.ω</e>
          <e type="operand">d.min</e>
          <e type="operand">d.max</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operand">1.5</e>
          <e type="operand">6</e>
          <e type="operand">10</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="558" width="142" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Lateral load, W:</p>
        </content>
      </text>
    </region>
    <region left="387" top="558" width="74" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">W</e>
          <e type="operand">100</e>
          <e type="operand" style="unit">N</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="594" width="326" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Distance of lateral load from left end:</p>
        </content>
      </text>
    </region>
    <region left="387" top="621" width="82" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">a</e>
          <e type="operand">300</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="648" width="294" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Value of"a"for end load conditions:</p>
        </content>
      </text>
    </region>
    <region left="387" top="666" width="73" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">a.1</e>
          <e type="operand">0</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="702" width="190" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Modulus of elasticity:</p>
        </content>
      </text>
    </region>
    <region left="387" top="702" width="112" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">E</e>
          <e type="operand">70000</e>
          <e type="operand" style="unit">N</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="9" top="774" width="110" height="38" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Calculation <br />procedure</p>
        </content>
      </text>
    </region>
    <region left="162" top="774" width="590" height="53" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>First determine the left-end reaction force and moment for a beam of constant cross-section with the given loading and restraints.From Table 3, Case 1d:</p>
        </content>
      </text>
    </region>
    <region left="9" top="828" width="70" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 3</p>
        </content>
      </text>
    </region>
    <region left="162" top="873" width="207" height="52" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">R.A</e>
          <e type="operand">W</e>
          <e type="operand">L</e>
          <e type="operand">3</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">L</e>
          <e type="operand">2</e>
          <e type="operand">a</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="378" top="882" width="94" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">R.A</e>
        </input>
        <contract>
          <e type="operand" style="unit">N</e>
        </contract>
        <result action="numeric">
          <e type="operand">68.36</e>
        </result>
      </math>
    </region>
    <region left="162" top="945" width="155" height="52" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A</e>
          <e type="operand">W</e>
          <e type="operator" args="1">-</e>
          <e type="operand">a</e>
          <e type="operator" args="2">*</e>
          <e type="operand">L</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="378" top="954" width="151" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A</e>
        </input>
        <contract>
          <e type="operand" style="unit">N</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
        </contract>
        <result action="numeric">
          <e type="operand">11718.75</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="162" top="1008" width="398" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>The beam depth and width as a function of x are:</p>
        </content>
      </text>
    </region>
    <region left="306" top="1044" width="204" height="47" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">x</e>
          <e type="function" args="1">d</e>
          <e type="operand">d.min</e>
          <e type="operand">d.max</e>
          <e type="operand">d.min</e>
          <e type="operator" args="2">-</e>
          <e type="operand">L</e>
          <e type="operator" args="2">/</e>
          <e type="operand">x</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="1053" width="62" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Depth:</p>
        </content>
      </text>
    </region>
    <region left="306" top="1107" width="204" height="47" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">x</e>
          <e type="function" args="1">b</e>
          <e type="operand">b.min</e>
          <e type="operand">b.max</e>
          <e type="operand">b.min</e>
          <e type="operator" args="2">-</e>
          <e type="operand">L</e>
          <e type="operator" args="2">/</e>
          <e type="operand">x</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="1116" width="62" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Width:</p>
        </content>
      </text>
    </region>
    <region left="108" top="1170" width="582" height="38" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>The moments of inertia at the left end, IA, the right end, IB, and the middle IM can be found from the equation I(x) below.</p>
        </content>
      </text>
    </region>
    <region left="162" top="1206" width="363" height="58" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">x</e>
          <e type="function" args="1">I</e>
          <e type="operand">x</e>
          <e type="function" args="1">b</e>
          <e type="operand">2</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operand">x</e>
          <e type="function" args="1">d</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operand">3</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">12</e>
          <e type="operator" args="2">/</e>
          <e type="operand">x</e>
          <e type="function" args="1">b</e>
          <e type="operand">t.ω</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">x</e>
          <e type="function" args="1">d</e>
          <e type="operand">3</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">12</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">-</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="1278" width="266" height="46" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.A</e>
          <e type="operand">I.B</e>
          <e type="operand">I.M</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operand">L</e>
          <e type="operand">L</e>
          <e type="operator" args="2">-</e>
          <e type="function" args="1">I</e>
          <e type="operand">L</e>
          <e type="function" args="1">I</e>
          <e type="operand">L</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="function" args="1">I</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="1341" width="311" height="38" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.A</e>
          <e type="operand">I.B</e>
          <e type="operand">I.M</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
        </contract>
        <result action="numeric">
          <e type="operand">288.33</e>
          <e type="operand">1287.67</e>
          <e type="operand">672.00</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
        </result>
      </math>
    </region>
    <region left="108" top="1395" width="590" height="98" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Since we already know RA and MA from Table 3 (calculated above), we may now find the multipliers that correct for tapered beams by using Table 13.Set the first five variables in Table 13, Case 1d to the following values.Do not change the order of the values. (note:If your loading or restraints differ from the ones used here, jump to Table 13 and use those equations below):</p>
        </content>
      </text>
    </region>
    <region left="9" top="1431" width="78" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 13</p>
        </content>
      </text>
    </region>
    <region left="171" top="1530" width="482" height="59" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.B</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">I.A</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">I.M</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">a</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">/</e>
          <e type="operand">L</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">/</e>
          <e type="operand">1</e>
          <e type="operand">5</e>
          <e type="function" args="7">mat</e>
        </input>
        <result action="numeric">
          <e type="operand">1287.67</e>
          <e type="operand">288.33</e>
          <e type="operand">672.00</e>
          <e type="operand">300.00</e>
          <e type="operand">800.00</e>
          <e type="operand">1</e>
          <e type="operand">5</e>
          <e type="function" args="7">mat</e>
        </result>
      </math>
    </region>
    <region left="108" top="1593" width="542" height="38" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>For the given conditions, Table 13, Case 1d, yields the RA and MA multipliers (one must interpolate for n=2.86 and a/L=0.375):</p>
        </content>
      </text>
    </region>
    <region left="162" top="1638" width="260" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">R.A1dmult</e>
          <e type="operand">M.A1dmult</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">0.878</e>
          <e type="operand">0.751</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="1674" width="124" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">R.A</e>
          <e type="operand">R.A</e>
          <e type="operand">R.A1dmult</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="360" top="1674" width="94" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">R.A</e>
        </input>
        <contract>
          <e type="operand" style="unit">N</e>
        </contract>
        <result action="numeric">
          <e type="operand">60.02</e>
        </result>
      </math>
    </region>
    <region left="360" top="1701" width="151" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A</e>
        </input>
        <contract>
          <e type="operand" style="unit">N</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
        </contract>
        <result action="numeric">
          <e type="operand">11718.75</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="162" top="1710" width="124" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A</e>
          <e type="operand">M.A</e>
          <e type="operand">M.A1dmult</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="108" top="1746" width="462" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>The use of static equilibrium now gives the value of MB:</p>
        </content>
      </text>
    </region>
    <region left="162" top="1773" width="104" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">L</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
        </contract>
        <result action="numeric">
          <e type="operand">800.00</e>
        </result>
      </math>
    </region>
    <region left="387" top="1773" width="104" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">a</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
        </contract>
        <result action="numeric">
          <e type="operand">300.00</e>
        </result>
      </math>
    </region>
    <region left="162" top="1809" width="292" height="31" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A</e>
          <e type="operand">R.A</e>
          <e type="operand">L</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operand">W</e>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">-</e>
          <e type="operand">M.B</e>
          <e type="operator" args="2">-</e>
          <e type="operand">0</e>
          <e type="operand">newton</e>
          <e type="operator" args="2">*</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">≡</e>
        </input>
      </math>
    </region>
    <region left="162" top="1854" width="200" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.B</e>
          <e type="operand">M.A</e>
          <e type="operand">R.A</e>
          <e type="operand">L</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operand">W</e>
          <e type="operand">L</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">-</e>
          <e type="operand">W</e>
          <e type="operand">a</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="1854" width="151" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.B</e>
        </input>
        <contract>
          <e type="operand" style="unit">N</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
        </contract>
        <result action="numeric">
          <e type="operand">10785.16</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="162" top="1890" width="230" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>and the moment at the load:</p>
        </content>
      </text>
    </region>
    <region left="162" top="1935" width="124" height="31" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.a</e>
          <e type="operand">M.A</e>
          <e type="operand">R.A</e>
          <e type="operand">a</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="1935" width="85" height="30" color="#000000" fontSize="10">
      <math error="2">
        <input>
          <e type="operand">M.a</e>
        </input>
        <contract>
          <e type="operand">N</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
        </contract>
        <result action="numeric">
          <e type="operand">#</e>
        </result>
      </math>
    </region>
    <region left="162" top="2088" width="146" height="43" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">x</e>
          <e type="function" args="1">c</e>
          <e type="operand">x</e>
          <e type="function" args="1">d</e>
          <e type="operand">2</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="117" top="2142" width="582" height="38" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Therefore, the distance to the neutral axis at the left end, the load, and the right end is:</p>
        </content>
      </text>
    </region>
    <region left="162" top="2205" width="262" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">c.A</e>
          <e type="operand">c.a</e>
          <e type="operand">c.B</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operand">L</e>
          <e type="operand">L</e>
          <e type="operator" args="2">-</e>
          <e type="function" args="1">c</e>
          <e type="operand">a</e>
          <e type="function" args="1">c</e>
          <e type="operand">L</e>
          <e type="function" args="1">c</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="2250" width="245" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">c.A</e>
          <e type="operand">c.a</e>
          <e type="operand">c.B</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
        </contract>
        <result action="numeric">
          <e type="operand">5.00</e>
          <e type="operand">5.75</e>
          <e type="operand">7.00</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
        </result>
      </math>
    </region>
    <region left="117" top="2304" width="334" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Therefore, themaximum bending stress is:</p>
        </content>
      </text>
    </region>
    <region left="162" top="2340" width="114" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M</e>
          <e type="operand">M.A</e>
          <e type="operand">M.a</e>
          <e type="operand">M.B</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="324" top="2340" width="114" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">c</e>
          <e type="operand">c.A</e>
          <e type="operand">c.a</e>
          <e type="operand">c.B</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="486" top="2340" width="139" height="34" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.m</e>
          <e type="operand">I.A</e>
          <e type="operand">a</e>
          <e type="function" args="1">I</e>
          <e type="operand">I.B</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="171" top="2376" width="70" height="53" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ</e>
          <e type="operand">M</e>
          <e type="operand">c</e>
          <e type="operator" args="2">*</e>
          <e type="operand">I.m</e>
          <e type="operator" args="2">/</e>
          <e type="function" args="1">vectorize</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="324" top="2394" width="244" height="27" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ</e>
        </input>
        <contract>
          <e type="operand" style="unit">MPa</e>
        </contract>
        <result action="numeric">
          <e type="operand">152.61</e>
          <e type="operator" args="1">-</e>
          <e type="operand">95.12</e>
          <e type="operand">58.63</e>
          <e type="operator" args="1">-</e>
          <e type="operand">1</e>
          <e type="operand">3</e>
          <e type="function" args="5">mat</e>
        </result>
      </math>
    </region>
    <region left="162" top="2439" width="143" height="26" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ</e>
          <e type="function" args="1">max</e>
        </input>
        <contract>
          <e type="operand" style="unit">MPa</e>
        </contract>
        <result action="numeric">
          <e type="operand">95.12</e>
        </result>
      </math>
    </region>
    <region left="117" top="2475" width="598" height="143" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>No correction coefficients for deflections are included for Case 1d in Table 13.The deflection at the load position can be found from the information in this table, however, if either end of the beam is isolated and treated as a cantilever with an end load and an end moment.The moment of inertia at the load position C (Note the definitions.This location is defined as "a" in the given parameters section to be consistent with Table 13.It is defined as C here to be consistent with the sixth edition of Roark's Formulas for Stress and Strain.) is given by: </p>
        </content>
      </text>
    </region>
    <region left="162" top="2628" width="80" height="31" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.C</e>
          <e type="operand">a</e>
          <e type="function" args="1">I</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="2628" width="120" height="39" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.C</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
        </contract>
        <result action="numeric">
          <e type="operand">556.44</e>
        </result>
      </math>
    </region>
    <region left="117" top="2664" width="238" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Store the values of L and a.</p>
        </content>
      </text>
    </region>
    <region left="162" top="2709" width="190" height="27" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">store1</e>
          <e type="operand">store2</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="117" top="2745" width="582" height="53" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Treat the left portion of the beam as an a-long cantilever. Use Table 13, Case 1a with an end load of RA and Case 3a with an end moment of Ma.The moment of inertia at the midpoint will be IM:</p>
        </content>
      </text>
    </region>
    <region left="162" top="2808" width="84" height="43" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.M</e>
          <e type="operand">a</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="function" args="1">I</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="2808" width="120" height="39" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.M</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
        </contract>
        <result action="numeric">
          <e type="operand">408.45</e>
        </result>
      </math>
    </region>
    <region left="117" top="2862" width="278" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>The correction coefficients for: </p>
        </content>
      </text>
    </region>
    <region left="162" top="2898" width="48" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="2898" width="54" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">a</e>
          <e type="operand">a.1</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="2925" width="458" height="59" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.A</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">I.C</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">I.M</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">a</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">/</e>
          <e type="operand">L</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">/</e>
          <e type="operand">1</e>
          <e type="operand">5</e>
          <e type="function" args="7">mat</e>
        </input>
        <result action="numeric">
          <e type="operand">288.33</e>
          <e type="operand">556.44</e>
          <e type="operand">408.45</e>
          <e type="operand">0.00</e>
          <e type="operand">300.00</e>
          <e type="operand">1</e>
          <e type="operand">5</e>
          <e type="function" args="7">mat</e>
        </result>
      </math>
    </region>
    <region left="9" top="3006" width="78" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 13</p>
        </content>
      </text>
    </region>
    <region left="117" top="3006" width="590" height="53" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>are found in Table 13, Case 1a and Table 13, Case 3a.Set the first five variables in Table 13, Case 1a and Table 13, Case 3a to the values above.Do not change the order of the values.</p>
        </content>
      </text>
    </region>
    <region left="117" top="3060" width="422" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>The coefficients from Table 13 are interpolated as:</p>
        </content>
      </text>
    </region>
    <region left="162" top="3096" width="260" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.A1amult</e>
          <e type="operand">y.A3amult</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">1.650</e>
          <e type="operand">1.564</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="3114" width="15" height="23" color="#000000" fontSize="10" />
    <region left="162" top="3141" width="206" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Thus for the given beam:</p>
        </content>
      </text>
    </region>
    <region left="162" top="3186" width="190" height="27" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">store1</e>
          <e type="operand">store2</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="3231" width="298" height="62" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.C</e>
          <e type="operand">R.A</e>
          <e type="operator" args="1">-</e>
          <e type="operand">a</e>
          <e type="operand">3</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">y.A1amult</e>
          <e type="operator" args="2">*</e>
          <e type="operand">3</e>
          <e type="operand">E</e>
          <e type="operator" args="2">*</e>
          <e type="operand">I.C</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operand">M.a</e>
          <e type="operand">a</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">y.A3amult</e>
          <e type="operator" args="2">*</e>
          <e type="operand">2</e>
          <e type="operand">E</e>
          <e type="operator" args="2">*</e>
          <e type="operand">I.C</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="3303" width="107" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.C</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
        </contract>
        <result action="numeric">
          <e type="operand">6.25</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="108" top="3348" width="598" height="53" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>This deflection coefficient at the load can be checked by repeating the above procedure by using the right-hand portion of the beam.The slope of the beam at the load can also be used as a check.</p>
        </content>
      </text>
    </region>
    <region left="108" top="3420" width="606" height="98" color="#000000" fontSize="10">
      <text lang="eng" width="590" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Alternative solution.The solution just presented was intended to illustrate appropriate methods of interpolation with the limited load positions shown in the tables.There is also an alternative solution involving superposition of cases.Remove the fixity at end A and treat the (L-a)-mm-long right portion as a cantilever with an end load of W. <br />The moment of inertia at the midpoint is:</p>
        </content>
      </text>
    </region>
    <region left="162" top="3537" width="133" height="43" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.M</e>
          <e type="operand">L</e>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operator" args="2">-</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">-</e>
          <e type="function" args="1">I</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="3537" width="120" height="39" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.M</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
        </contract>
        <result action="numeric">
          <e type="operand">872.79</e>
        </result>
      </math>
    </region>
    <region left="99" top="3609" width="398" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Define L and aafter storing the original values:</p>
        </content>
      </text>
    </region>
    <region left="162" top="3654" width="190" height="27" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">store1</e>
          <e type="operand">store2</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="3690" width="141" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operator" args="2">-</e>
          <e type="operand">a.1</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="99" top="3735" width="62" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Given:</p>
        </content>
      </text>
    </region>
    <region left="162" top="3771" width="466" height="59" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.B</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">I.C</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">I.M</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">a</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">/</e>
          <e type="operand">L</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">/</e>
          <e type="operand">1</e>
          <e type="operand">5</e>
          <e type="function" args="7">mat</e>
        </input>
        <result action="numeric">
          <e type="operand">1287.67</e>
          <e type="operand">556.44</e>
          <e type="operand">872.79</e>
          <e type="operand">0.00</e>
          <e type="operand">500.00</e>
          <e type="operand">1</e>
          <e type="operand">5</e>
          <e type="function" args="7">mat</e>
        </result>
      </math>
    </region>
    <region left="9" top="3843" width="78" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 13</p>
        </content>
      </text>
    </region>
    <region left="99" top="3843" width="574" height="53" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 13, Case 1a yields the following factors when the first five variables in Table 13, Case 1a are defined as the values in the above matrix.Do not change the order of the values. </p>
        </content>
      </text>
    </region>
    <region left="162" top="3915" width="234" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.Amult</e>
          <e type="operand">θ.Amult</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">0.554</e>
          <e type="operand">0.592</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="3951" width="174" height="65" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.C</e>
          <e type="operand">W</e>
          <e type="operand">L</e>
          <e type="bracket">(</e>
          <e type="operand">3</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">y.Amult</e>
          <e type="operator" args="2">*</e>
          <e type="bracket">(</e>
          <e type="operator" args="1">-</e>
          <e type="operand">3</e>
          <e type="operand">E</e>
          <e type="operator" args="2">*</e>
          <e type="operand">I.C</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="3969" width="115" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.C</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
        </contract>
        <result action="numeric">
          <e type="operand">59.26</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="162" top="4014" width="161" height="64" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.C</e>
          <e type="operand">W</e>
          <e type="operand">L</e>
          <e type="bracket">(</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">θ.Amult</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">2</e>
          <e type="operand">E</e>
          <e type="operator" args="2">*</e>
          <e type="operand">I.C</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="4041" width="74" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.C</e>
        </input>
        <result action="numeric">
          <e type="operand">0.19</e>
        </result>
      </math>
    </region>
    <region left="99" top="4086" width="534" height="38" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Since the left portion is unloaded and remains straight, the end deflection and slope are: </p>
        </content>
      </text>
    </region>
    <region left="162" top="4149" width="190" height="27" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">L</e>
          <e type="operand">a</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">store1</e>
          <e type="operand">store2</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="4185" width="122" height="32" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.A</e>
          <e type="operand">y.C</e>
          <e type="operand">a</e>
          <e type="operand">θ.C</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">-</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="4185" width="123" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.A</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
        </contract>
        <result action="numeric">
          <e type="operand">116.26</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="162" top="4239" width="60" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.A</e>
          <e type="operand">θ.C</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="4239" width="74" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.A</e>
        </input>
        <result action="numeric">
          <e type="operand">0.19</e>
        </result>
      </math>
    </region>
    <region left="99" top="4311" width="598" height="38" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Next treat the complete beam as a cantilever under an end load RA and an end moment MA:</p>
        </content>
      </text>
    </region>
    <region left="162" top="4374" width="84" height="43" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.M</e>
          <e type="operand">L</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="function" args="1">I</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="4374" width="120" height="39" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.M</e>
        </input>
        <contract>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
        </contract>
        <result action="numeric">
          <e type="operand">672.00</e>
        </result>
      </math>
    </region>
    <region left="162" top="4428" width="88" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">store1</e>
          <e type="operand">a</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="387" top="4428" width="54" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">a</e>
          <e type="operand">a.1</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="4473" width="466" height="59" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">I.B</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">I.A</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">I.M</e>
          <e type="operand" style="unit">mm</e>
          <e type="operand">4</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
          <e type="operand">a</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">/</e>
          <e type="operand">L</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">/</e>
          <e type="operand">1</e>
          <e type="operand">5</e>
          <e type="function" args="7">mat</e>
        </input>
        <result action="numeric">
          <e type="operand">1287.67</e>
          <e type="operand">288.33</e>
          <e type="operand">672.00</e>
          <e type="operand">0.00</e>
          <e type="operand">800.00</e>
          <e type="operand">1</e>
          <e type="operand">5</e>
          <e type="function" args="7">mat</e>
        </result>
      </math>
    </region>
    <region left="162" top="4545" width="88" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">a</e>
          <e type="operand">store1</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="9" top="4599" width="78" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 13</p>
        </content>
      </text>
    </region>
    <region left="99" top="4599" width="590" height="53" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Replacing the first five variables in Table 13, Case 1a and Table 13, Case 3a with the values in the above matrix yields the factors shown in the matrix below.Do not change the order of the values.</p>
        </content>
      </text>
    </region>
    <region left="153" top="4671" width="448" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.A1mult</e>
          <e type="operand">θ.A1mult</e>
          <e type="operand">y.A3mult</e>
          <e type="operand">θ.A3mult</e>
          <e type="operand">1</e>
          <e type="operand">4</e>
          <e type="function" args="6">mat</e>
          <e type="operand">0.331</e>
          <e type="operand">0.378</e>
          <e type="operand">0.378</e>
          <e type="operand">0.494</e>
          <e type="operand">1</e>
          <e type="operand">4</e>
          <e type="function" args="6">mat</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="4707" width="86" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Defining:</p>
        </content>
      </text>
    </region>
    <region left="162" top="4725" width="137" height="63" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.1</e>
          <e type="operand">L</e>
          <e type="operand">3</e>
          <e type="operator" args="2">^</e>
          <e type="operand">y.A1mult</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">3</e>
          <e type="operand">E</e>
          <e type="operator" args="2">*</e>
          <e type="operand">I.A</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="351" top="4725" width="137" height="63" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.2</e>
          <e type="operand">L</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="bracket">(</e>
          <e type="operand">θ.A1mult</e>
          <e type="operator" args="2">*</e>
          <e type="operand">2</e>
          <e type="operand">E</e>
          <e type="operator" args="2">*</e>
          <e type="operand">I.A</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="4770" width="137" height="63" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.1</e>
          <e type="operand">L</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operand">y.A3mult</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">2</e>
          <e type="operand">E</e>
          <e type="operator" args="2">*</e>
          <e type="operand">I.A</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="351" top="4770" width="139" height="55" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.2</e>
          <e type="operand">L</e>
          <e type="bracket">(</e>
          <e type="operand">θ.A3mult</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">E</e>
          <e type="operand">I.A</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="90" top="4833" width="54" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Thus:</p>
        </content>
      </text>
    </region>
    <region left="162" top="4860" width="199" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">y.1</e>
          <e type="operand">R.A2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">y.2</e>
          <e type="operand">M.A2</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operand">y.A</e>
          <e type="operator" args="2">+</e>
          <e type="operand">0</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">≡</e>
        </input>
      </math>
    </region>
    <region left="162" top="4896" width="158" height="55" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A2</e>
          <e type="operand">y.1</e>
          <e type="operand">R.A2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">y.A</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="1">-</e>
          <e type="operand">y.2</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">≡</e>
        </input>
      </math>
    </region>
    <region left="162" top="4950" width="46" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>and:</p>
        </content>
      </text>
    </region>
    <region left="162" top="4986" width="190" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">R.A2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">θ.2</e>
          <e type="operand">M.A2</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">-</e>
          <e type="operand">θ.A</e>
          <e type="operator" args="2">+</e>
          <e type="operand">0</e>
          <e type="operator" args="2">≡</e>
        </input>
      </math>
    </region>
    <region left="162" top="5031" width="274" height="55" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">R.A2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">θ.2</e>
          <e type="operand">y.1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">R.A2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">y.A</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">y.2</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">-</e>
          <e type="operand">θ.A</e>
          <e type="operator" args="2">+</e>
          <e type="operand">0</e>
          <e type="operator" args="2">≡</e>
        </input>
      </math>
    </region>
    <region left="162" top="5094" width="157" height="103" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">R.A2</e>
          <e type="operand">θ.2</e>
          <e type="operand">y.2</e>
          <e type="operator" args="2">/</e>
          <e type="operand">y.A</e>
          <e type="operator" args="2">*</e>
          <e type="operand">θ.A</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="1">-</e>
          <e type="operand">θ.1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">θ.2</e>
          <e type="operand">y.2</e>
          <e type="operator" args="2">/</e>
          <e type="operand">y.1</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="153" top="5220" width="198" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Solving for RA2 yields:</p>
        </content>
      </text>
    </region>
    <region left="378" top="5220" width="111" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">R.A2</e>
        </input>
        <contract>
          <e type="operand">newton</e>
        </contract>
        <result action="numeric">
          <e type="operand">#</e>
        </result>
      </math>
    </region>
    <region left="153" top="5265" width="94" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Similarly:</p>
        </content>
      </text>
    </region>
    <region left="153" top="5310" width="158" height="55" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A2</e>
          <e type="operand">y.1</e>
          <e type="operand">R.A2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">y.A</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="1">-</e>
          <e type="operand">y.2</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">≡</e>
        </input>
      </math>
    </region>
    <region left="153" top="5373" width="242" height="128" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A2</e>
          <e type="operand">y.1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">θ.2</e>
          <e type="operator" args="1">-</e>
          <e type="operand">y.2</e>
          <e type="operator" args="2">/</e>
          <e type="operand">y.A</e>
          <e type="operator" args="2">*</e>
          <e type="operand">θ.A</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">θ.1</e>
          <e type="operator" args="1">-</e>
          <e type="operand">θ.2</e>
          <e type="operand">y.2</e>
          <e type="operator" args="2">/</e>
          <e type="operand">y.1</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">*</e>
          <e type="operand">y.A</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">y.2</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="153" top="5526" width="150" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.A2</e>
        </input>
        <contract>
          <e type="operand" style="unit">N</e>
          <e type="operand" style="unit">mm</e>
          <e type="operator" args="2">*</e>
        </contract>
        <result action="numeric">
          <e type="operand">8736.32</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
  </regions>
</worksheet>