<?xml version="1.0" encoding="utf-8" standalone="yes"?>
<?application progid="SMath Studio Desktop" version="1.0.8206.25180"?>
<worksheet xmlns="http://smath.info/schemas/worksheet/1.0">
  <settings ppi="96">
    <identity>
      <id>143e54cf-285b-41b4-9e54-be5ca735da6d</id>
      <revision>2</revision>
    </identity>
    <calculation>
      <precision>2</precision>
      <exponentialThreshold>5</exponentialThreshold>
      <trailingZeros>true</trailingZeros>
      <significantDigitsMode>false</significantDigitsMode>
      <roundingMode>0</roundingMode>
      <approximateEqualAccuracy>1</approximateEqualAccuracy>
      <fractions>decimal</fractions>
    </calculation>
    <pageModel active="false" viewMode="2" printGrid="false" printAreas="true" simpleEqualsOnly="false" printBackgroundImages="true">
      <paper id="9" orientation="Portrait" width="827" height="1169" />
      <margins left="39" right="39" top="49" bottom="49" />
      <header alignment="Center" color="#a9a9a9">&amp;[DATE] &amp;[TIME] - &amp;[FILENAME]</header>
      <footer alignment="Center" color="#a9a9a9">&amp;[PAGENUM] / &amp;[COUNT]</footer>
      <backgrounds />
    </pageModel>
    <dependencies>
      <assembly name="SMath Studio Desktop" version="1.0.8206.25180" guid="a37cba83-b69c-4c71-9992-55ff666763bd" />
      <assembly name="MathRegion" version="1.11.8206.25180" guid="02f1ab51-215b-466e-a74d-5d8b1cf85e8d" />
      <assembly name="PictureRegion" version="1.10.8206.25180" guid="06b5df04-393e-4be7-9107-305196fcb861" />
      <assembly name="TextRegion" version="1.11.8206.25180" guid="485d28c5-349a-48b6-93be-12a35a1c1e39" />
    </dependencies>
  </settings>
  <regions type="content">
    <region left="99" top="0" width="446" height="38" color="#000000" fontSize="20">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p style="font-size: 20px;">Chapter 12 Example Problems</p>
        </content>
      </text>
    </region>
    <region left="99" top="45" width="638" height="77" color="#000000" fontSize="15">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p style="font-size: 15px;">12.3.5 Thin Shells of Revolution Under Concentrated or Discontinuous Loadings Producing Bending and Membrane Stresses: Design of a Pressure Vessel </p>
        </content>
      </text>
    </region>
    <region left="162" top="126" width="598" height="83" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Two partial spheres of aluminum are to be welded together as shown in Fig. 12.6 to form a pressure vessel to withstand an internal pressure of 200 lbf/in2. The mean radius of each sphere is 2 ft, and the wall thickness is 0.5 in. Calculate the stresses at the seam. Given: E = 10(106) lbf/in2 and n = 0.33.</p>
        </content>
      </text>
    </region>
    <region left="639" top="198" width="86" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Fig. 12.6</p>
        </content>
      </text>
    </region>
    <region left="0" top="216" 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="243" width="214" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Modulus of elasticity, E:</p>
        </content>
      </text>
    </region>
    <region left="396" top="243" width="123" height="51" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">E</e>
          <e type="operand">10</e>
          <e type="operand">10</e>
          <e type="operand">6</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</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="585" top="243" width="183" height="137" color="#000000">
      <picture>
        <raw format="png" encoding="base64"> </raw>
      </picture>
    </region>
    <region left="162" top="297" width="166" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Poisson's ratio, n:</p>
        </content>
      </text>
    </region>
    <region left="396" top="297" width="70" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ν</e>
          <e type="operand">0.33</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="396" top="342" width="96" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">q</e>
          <e type="operand">200</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</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="162" top="351" width="182" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Internal pressure, q:</p>
        </content>
      </text>
    </region>
    <region left="594" top="378" width="122" height="135" color="#000000">
      <picture>
        <raw format="png" encoding="base64"> </raw>
      </picture>
    </region>
    <region left="162" top="396" width="262" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Mean radius of each sphere, R2:</p>
        </content>
      </text>
    </region>
    <region left="396" top="414" width="73" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">R.2</e>
          <e type="operand">2</e>
          <e type="operand" style="unit">ft</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="468" width="158" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Wall thickness, t:</p>
        </content>
      </text>
    </region>
    <region left="396" top="468" width="82" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">t</e>
          <e type="operand">0.5</e>
          <e type="operand" style="unit">in</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="9" top="513" width="150" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 28, Case 3a</p>
        </content>
      </text>
    </region>
    <region left="162" top="513" width="254" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Theta (Table 28, Case 3a), q0:</p>
        </content>
      </text>
    </region>
    <region left="396" top="522" width="97" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.0</e>
          <e type="operand">120</e>
          <e type="operand" style="unit">deg</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="9" top="576" width="150" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 30, Case 1a</p>
        </content>
      </text>
    </region>
    <region left="162" top="576" width="254" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Theta (Table 30, Case 1a), q1:</p>
        </content>
      </text>
    </region>
    <region left="396" top="576" width="89" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">θ.1</e>
          <e type="operand">30</e>
          <e type="operand" style="unit">deg</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="630" width="310" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Phi, omega (Table 30, Case 1a); f, w:</p>
        </content>
      </text>
    </region>
    <region left="396" top="648" width="150" height="27" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ϕ</e>
          <e type="operand">ω</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand">120</e>
          <e type="operand">0</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="operand" style="unit">deg</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="0" top="675" width="238" height="23" color="#000000" fontSize="10">
      <text lang="eng" width="238" fontFamily="Courier New" fontSize="10">
        <content>
          <p style="font-weight: bold;">Calculation procedure</p>
        </content>
      </text>
    </region>
    <region left="162" top="702" width="598" height="128" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>The edge loading will be considered in three parts, as shown in Fig. 12.6b. The tangential edge force T will be applied to balance the internal pressure and, together with the pressure, will cause only membrane stresses and the accompanying change in circumferential radius DR; this loading will produce no rotation of the meridian. Owing to the symmetry of the two shells, there is no resultant radial load on the edge, so Qo is added to eliminate that component of T. Mo is needed to ensure no edge rotation.</p>
        </content>
      </text>
    </region>
    <region left="9" top="837" width="150" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 28, Case 3a</p>
        </content>
      </text>
    </region>
    <region left="162" top="837" width="398" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>First apply the formulas from Table 28, Case 3a:</p>
        </content>
      </text>
    </region>
    <region left="207" top="882" width="364" height="53" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.10</e>
          <e type="operand">σ.20</e>
          <e type="operand">σ'.10</e>
          <e type="operand">σ'.20</e>
          <e type="operand">1</e>
          <e type="operand">4</e>
          <e type="function" args="6">mat</e>
          <e type="operand">q</e>
          <e type="operand">R.2</e>
          <e type="operator" args="2">*</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">q</e>
          <e type="operand">R.2</e>
          <e type="operator" args="2">*</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">0</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</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="operand">0</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</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="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="207" top="945" width="409" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.10</e>
          <e type="operand">σ.20</e>
          <e type="operand">σ'.10</e>
          <e type="operand">σ'.20</e>
          <e type="operand">1</e>
          <e type="operand">4</e>
          <e type="function" args="6">mat</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">4800.00</e>
          <e type="operand">4800.00</e>
          <e type="operand">0.00</e>
          <e type="operand">0.00</e>
          <e type="operand">1</e>
          <e type="operand">4</e>
          <e type="function" args="6">mat</e>
        </result>
      </math>
    </region>
    <region left="207" top="999" width="219" height="57" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR.0</e>
          <e type="operand">q</e>
          <e type="operand">R.2</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">1</e>
          <e type="operand">ν</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">θ.0</e>
          <e type="function" args="1">sin</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">t</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="1026" width="102" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR.0</e>
        </input>
        <contract>
          <e type="operand" style="unit">in</e>
        </contract>
        <result action="numeric">
          <e type="operand">0.01</e>
        </result>
      </math>
    </region>
    <region left="207" top="1080" width="79" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">T</e>
          <e type="operand">σ.10</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="1080" width="124" height="41" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">T</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">2400.00</e>
        </result>
      </math>
    </region>
    <region left="207" top="1116" width="81" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ψ.0</e>
          <e type="operand">0</e>
          <e type="operand" style="unit">rad</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="9" top="1152" width="150" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 30, Case 1a</p>
        </content>
      </text>
    </region>
    <region left="162" top="1152" width="278" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Next apply Case 1a from Table 30:</p>
        </content>
      </text>
    </region>
    <region left="207" top="1188" width="119" height="32" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">Q.o</e>
          <e type="operand">T</e>
          <e type="operand">θ.1</e>
          <e type="function" args="1">sin</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="1188" width="130" height="41" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">Q.o</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">1200.00</e>
        </result>
      </math>
    </region>
    <region left="207" top="1215" width="206" height="69" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">β</e>
          <e type="operand">3</e>
          <e type="operand">1</e>
          <e type="operand">ν</e>
          <e type="operand">2</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="operand">R.2</e>
          <e type="operand">t</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="bracket">(</e>
          <e type="operand">0.25</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="1242" width="68" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">β</e>
        </input>
        <result action="numeric">
          <e type="operand">8.86</e>
        </result>
      </math>
    </region>
    <region left="162" top="1287" width="110" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>At the edge:</p>
        </content>
      </text>
    </region>
    <region left="207" top="1305" width="68" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ω</e>
        </input>
        <result action="numeric">
          <e type="operand">0.00</e>
        </result>
      </math>
    </region>
    <region left="207" top="1323" width="217" height="43" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">K.1</e>
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="operand">ν</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">-</e>
          <e type="operand">2</e>
          <e type="operand">β</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="bracket">(</e>
          <e type="operand">ϕ</e>
          <e type="operand">ω</e>
          <e type="operator" args="2">-</e>
          <e type="function" args="1">cot</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">-</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="207" top="1359" width="219" height="43" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">K.2</e>
          <e type="operand">1</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="operand">ν</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operand">2</e>
          <e type="operand">β</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="bracket">(</e>
          <e type="operand">ϕ</e>
          <e type="operand">ω</e>
          <e type="operator" args="2">-</e>
          <e type="function" args="1">cot</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">-</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="207" top="1404" width="165" height="33" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">K.1</e>
          <e type="operand">K.2</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
        </input>
        <result action="numeric">
          <e type="operand">1.01</e>
          <e type="operand">1.05</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
        </result>
      </math>
    </region>
    <region left="207" top="1431" width="271" height="63" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR.1</e>
          <e type="operand">Q.o</e>
          <e type="operand">R.2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">β</e>
          <e type="operator" args="2">*</e>
          <e type="operand">ϕ</e>
          <e type="function" args="1">sin</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">E</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operand">K.1</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operand">1</e>
          <e type="operand">K.1</e>
          <e type="operand">K.2</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="513" top="1458" width="102" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR.1</e>
        </input>
        <contract>
          <e type="operand" style="unit">in</e>
        </contract>
        <result action="numeric">
          <e type="operand">0.08</e>
        </result>
      </math>
    </region>
    <region left="207" top="1494" width="168" height="62" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ψ.1</e>
          <e type="operand">Q.o</e>
          <e type="operand">2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">β</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">ϕ</e>
          <e type="function" args="1">sin</e>
          <e type="operator" args="2">*</e>
          <e type="operand">E</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operand">K.1</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="1521" width="74" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ψ.1</e>
        </input>
        <result action="numeric">
          <e type="operand">0.03</e>
        </result>
      </math>
    </region>
    <region left="207" top="1557" width="129" height="48" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.11</e>
          <e type="operand">Q.o</e>
          <e type="operand">ϕ</e>
          <e type="function" args="1">cos</e>
          <e type="operator" args="2">*</e>
          <e type="operand">t</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="1557" width="151" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.11</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">1200.00</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="207" top="1611" width="101" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.11</e>
          <e type="operand">0</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</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="207" top="1665" width="254" height="55" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.21</e>
          <e type="operand">Q.o</e>
          <e type="operand">β</e>
          <e type="operator" args="2">*</e>
          <e type="operand">ϕ</e>
          <e type="function" args="1">sin</e>
          <e type="operator" args="2">*</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="operand">K.1</e>
          <e type="operator" args="2">/</e>
          <e type="operand">K.1</e>
          <e type="operator" args="2">+</e>
          <e type="operand">K.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="513" top="1665" width="147" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.21</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">37224.22</e>
        </result>
      </math>
    </region>
    <region left="207" top="1710" width="178" height="62" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.21</e>
          <e type="operand">Q.o</e>
          <e type="operator" args="1">-</e>
          <e type="operand">β</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">ϕ</e>
          <e type="function" args="1">cos</e>
          <e type="operator" args="2">*</e>
          <e type="operand">K.1</e>
          <e type="operand">R.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="513" top="1728" width="147" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.21</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">1940.53</e>
        </result>
      </math>
    </region>
    <region left="9" top="1800" width="150" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Table 30, Case 1b</p>
        </content>
      </text>
    </region>
    <region left="162" top="1800" width="478" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Now apply Case 1b from Table 30. Define Mo as some moment:</p>
        </content>
      </text>
    </region>
    <region left="207" top="1827" width="93" height="41" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.o</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand" style="unit">in</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="207" top="1863" width="176" height="62" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR.2</e>
          <e type="operand">M.o</e>
          <e type="operand">2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">β</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">ϕ</e>
          <e type="function" args="1">sin</e>
          <e type="operator" args="2">*</e>
          <e type="operand">E</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operand">K.1</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="1863" width="175" height="47" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR.2</e>
        </input>
        <contract>
          <e type="operand">M.o</e>
          <e type="operand" style="unit">in</e>
          <e type="operator" args="2">*</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">2.69</e>
          <e type="operand">10</e>
          <e type="operand">5</e>
          <e type="operator" args="1">-</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
        </result>
      </math>
    </region>
    <region left="207" top="1917" width="127" height="62" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ψ.2</e>
          <e type="operand">M.o</e>
          <e type="operand">4</e>
          <e type="operator" args="2">*</e>
          <e type="operand">β</e>
          <e type="operand">3</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">E</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operand">R.2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">K.1</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="1926" width="158" height="53" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ψ.2</e>
          <e type="operand">M.o</e>
          <e type="operator" args="2">/</e>
        </input>
        <contract>
          <e type="operand">1</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">2.29</e>
          <e type="operand">10</e>
          <e type="operand">5</e>
          <e type="operator" args="1">-</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
        </result>
      </math>
    </region>
    <region left="207" top="1971" width="58" height="53" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">r</e>
          <e type="operand">ψ.2</e>
          <e type="operand">M.o</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="162" top="2025" width="398" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>Since the combined edge rotation y must be zero:</p>
        </content>
      </text>
    </region>
    <region left="207" top="2052" width="134" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">0</e>
          <e type="operand">ψ.0</e>
          <e type="operand">ψ.1</e>
          <e type="operator" args="2">+</e>
          <e type="operand">r</e>
          <e type="operand">M.o</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">≡</e>
        </input>
      </math>
    </region>
    <region left="207" top="2079" width="120" height="49" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.o</e>
          <e type="operand">ψ.0</e>
          <e type="operator" args="1">-</e>
          <e type="operand">ψ.1</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">r</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="2079" width="163" height="41" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.o</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand" style="unit">in</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">*</e>
        </contract>
        <result action="numeric">
          <e type="operand">1407.70</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="162" top="2142" width="38" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>and</p>
        </content>
      </text>
    </region>
    <region left="207" top="2169" width="176" height="62" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR.2</e>
          <e type="operand">M.o</e>
          <e type="operand">2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">β</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">ϕ</e>
          <e type="function" args="1">sin</e>
          <e type="operator" args="2">*</e>
          <e type="operand">E</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operand">K.1</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="207" top="2250" width="152" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR</e>
          <e type="operand">ΔR.0</e>
          <e type="operand">ΔR.1</e>
          <e type="operator" args="2">+</e>
          <e type="operand">ΔR.2</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="2250" width="96" height="24" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">ΔR</e>
        </input>
        <contract>
          <e type="operand" style="unit">in</e>
        </contract>
        <result action="numeric">
          <e type="operand">0.05</e>
        </result>
      </math>
    </region>
    <region left="207" top="2295" width="93" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.12</e>
          <e type="operand">0</e>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</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="207" top="2358" width="108" height="56" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.12</e>
          <e type="operand">6</e>
          <e type="operator" args="1">-</e>
          <e type="operand">M.o</e>
          <e type="operator" args="2">*</e>
          <e type="operand">t</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="513" top="2367" width="155" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.12</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">33784.84</e>
        </result>
      </math>
    </region>
    <region left="207" top="2421" width="116" height="62" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.22</e>
          <e type="operand">M.o</e>
          <e type="operand">2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">β</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">*</e>
          <e type="operand">R.2</e>
          <e type="operand">K.1</e>
          <e type="operator" args="2">*</e>
          <e type="operand">t</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="2430" width="159" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.22</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">18211.23</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="207" top="2493" width="321" height="53" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.2</e>
          <e type="operand">M.o</e>
          <e type="operand">2</e>
          <e type="operand">ν</e>
          <e type="operator" args="2">*</e>
          <e type="operand">K.1</e>
          <e type="operator" args="2">*</e>
          <e type="operator" args="2">/</e>
          <e type="operand">1</e>
          <e type="operand">ν</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operand">K.1</e>
          <e type="operand">K.2</e>
          <e type="operator" args="2">+</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">*</e>
          <e type="operand">2</e>
          <e type="operand">K.2</e>
          <e type="operator" args="2">*</e>
          <e type="bracket">(</e>
          <e type="operator" args="1">-</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="513" top="2529" width="155" height="41" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">M.2</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand" style="unit">in</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">*</e>
        </contract>
        <result action="numeric">
          <e type="operand">383.69</e>
          <e type="operator" args="1">-</e>
        </result>
      </math>
    </region>
    <region left="207" top="2583" width="108" height="56" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.22</e>
          <e type="operand">6</e>
          <e type="operator" args="1">-</e>
          <e type="operand">M.2</e>
          <e type="operator" args="2">*</e>
          <e type="operand">t</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="513" top="2592" width="147" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.22</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">9208.47</e>
        </result>
      </math>
    </region>
    <region left="162" top="2637" width="438" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>The superimposed stresses at the joint are, therefore</p>
        </content>
      </text>
    </region>
    <region left="207" top="2664" width="145" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.1</e>
          <e type="operand">σ.10</e>
          <e type="operand">σ.11</e>
          <e type="operator" args="2">+</e>
          <e type="operand">σ.12</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="2664" width="132" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.1</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">3600.00</e>
        </result>
      </math>
    </region>
    <region left="207" top="2709" width="145" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.2</e>
          <e type="operand">σ.20</e>
          <e type="operand">σ.21</e>
          <e type="operator" args="2">+</e>
          <e type="operand">σ.22</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="2709" width="140" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.2</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">23812.99</e>
        </result>
      </math>
    </region>
    <region left="207" top="2754" width="177" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.1</e>
          <e type="operand">σ'.10</e>
          <e type="operand">σ'.11</e>
          <e type="operator" args="2">+</e>
          <e type="operand">σ'.12</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="2754" width="148" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.1</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">33784.84</e>
        </result>
      </math>
    </region>
    <region left="207" top="2799" width="177" height="30" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.2</e>
          <e type="operand">σ'.20</e>
          <e type="operand">σ'.21</e>
          <e type="operator" args="2">+</e>
          <e type="operand">σ'.22</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region left="513" top="2799" width="148" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ'.2</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">11149.00</e>
        </result>
      </math>
    </region>
    <region left="162" top="2844" width="182" height="23" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>The maximum stress is</p>
        </content>
      </text>
    </region>
    <region left="198" top="2871" width="298" height="50" color="#000000" fontSize="10">
      <math>
        <input>
          <e type="operand">σ.1</e>
          <e type="operand">σ'.1</e>
          <e type="operator" args="2">+</e>
          <e type="operand">σ.2</e>
          <e type="operand">σ'.2</e>
          <e type="operator" args="2">+</e>
          <e type="operand">1</e>
          <e type="operand">2</e>
          <e type="function" args="4">mat</e>
          <e type="function" args="1">max</e>
        </input>
        <contract>
          <e type="operand" style="unit">lbf</e>
          <e type="operand" style="unit">in</e>
          <e type="operand">2</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">/</e>
        </contract>
        <result action="numeric">
          <e type="operand">37384.84</e>
        </result>
      </math>
    </region>
    <region left="162" top="2934" width="550" height="38" color="#000000" fontSize="10">
      <text lang="eng" fontFamily="Courier New" fontSize="10">
        <content>
          <p>at the joint. A further consideration would be given to any stress concentrations due to the shape of the weld cross section.</p>
        </content>
      </text>
    </region>
  </regions>
</worksheet>