<description/> <author>jon</author> <company/> <keywords/> <revisedBy>Jon</revisedBy> </userData> <identityInfo> <revision>11</revision> <documentID>A638E3FB-6EFB-4021-83BC-EC17AC55E098</documentID> <versionID>722F2126-D7C5-4834-8920-EA868736D4EE</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="MS Sans Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="narrow-dot" derivative="derivative" literal-subscript="small" definition="colon-equal" global-definition="triple-equal" local-definition="left-arrow" equality="bold-equal" symbolic-evaluation="right-arrow"/> <mathStyles> <mathStyle name="Variables" font-family="MS Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Constants" font-family="MS Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 1" font-family="MS Sans Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 2" font-family="Courier" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 3" font-family="System" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 4" font-family="Script" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 5" font-family="Roman" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 6" font-family="Modern" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 7" font-family="MS Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="MS Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="charge" thermodynamic-temperature="temperature" luminous-intensity="luminosity" amount-of-substance="substance" display="false"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="86.4" right="86.25002" top="72" bottom="72"/> <header use-full-page-width="true"/> <footer use-full-page-width="true"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="us" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="223" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="3" left="6" top="9.75" width="321.75" height="18" align-x="6" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <sp count="29"/> <f size="14"> <sp count="3"/>COLUMN BUCKLING</f> </text> </region> <region region-id="6" left="6" top="44.25" width="25.5" height="12" align-x="6" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Req:<sp count="2"/> </text> </region> <region region-id="137" left="66" top="44.25" width="355.5" height="24" align-x="66" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Find the Critical Buckling Load for an Euler or Johnson column (dependent on the slenderness ratio.<sp count="2"/>With either a circular or square cross-section. </text> </region> <region region-id="165" left="66" top="80.25" width="185.25" height="12" align-x="66" align-y="90" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Shigley - "Mech Eng Design"<sp count="2"/>(pp142-149) </text> </region> <region region-id="7" left="6" top="110.25" width="27" height="24" align-x="6" align-y="120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Given: </text> </region> <region region-id="8" left="6" top="158.25" width="52.5" height="24" align-x="6" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Assumption: </text> </region> <region region-id="88" left="18" top="176.25" width="72" height="12" align-x="18" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> End Conditions: </text> </region> <region region-id="96" left="102" top="176.25" width="227.25" height="12" align-x="102" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Fixed - Free:<sp count="55"/>C=.25 </text> </region> <region region-id="97" left="102" top="194.25" width="251.25" height="12" align-x="102" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Rounded(pinned) - Rounded(pinned):<sp count="5"/>C=1.0 </text> </region> <region region-id="98" left="102" top="212.25" width="278.25" height="12" align-x="102" align-y="222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Fixed - Rounded(pinned):<sp count="29"/>C=1.2 </text> </region> <region region-id="99" left="102" top="230.25" width="208.5" height="12" align-x="102" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Fixed - Fixed:<sp count="53"/>C=1.2 </text> </region> <region region-id="9" left="6" top="272.25" width="38.25" height="12" align-x="6" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="MS Sans Serif" charset="0"> <inlineAttr line-through="false">Soln:</inlineAttr> </f> </text> </region> <region region-id="20" left="54" top="272.25" width="79.5" height="24" align-x="54" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="MS Sans Serif" charset="0"> <inlineAttr line-through="false">Circular Section </inlineAttr> </f> </text> </region> <region region-id="173" left="174" top="272.25" width="33" height="13.5" align-x="187.5" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">C</ml:id> <ml:real>2.5</ml:real> </ml:define> </math> <rendering item-idref="1"/> </region> <region region-id="30" left="6" top="308.25" width="343.5" height="15" align-x="6" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="MS Sans Serif" charset="0"> <inlineAttr line-through="false">Euler Column:</inlineAttr> </f> <sp count="2"/>(where the slenderness ratio L/k > (L/k)cr<f size="5"> <sp count="2"/> </f> ) </text> </region> <region region-id="10" left="6" top="338.25" width="32.25" height="12" align-x="6" align-y="348" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Length: </text> </region> <region region-id="17" left="60" top="338.25" width="43.5" height="13.5" align-x="72.75" align-y="348" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult/> <ml:real>4.0</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="25" left="144" top="334.5" width="64.5" height="19.5" align-x="159.75" align-y="348" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="y">S</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>40</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="18" left="60" top="362.25" width="49.5" height="13.5" align-x="74.25" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:mult/> <ml:real>.375</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="28" left="150" top="358.5" width="68.25" height="17.25" align-x="162.75" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">E</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>28.0</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="19" left="60" top="380.25" width="42.75" height="13.5" align-x="72" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">d</ml:id> <ml:apply> <ml:mult/> <ml:real>.00</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="21" left="12" top="408.75" width="75" height="29.25" align-x="25.5" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">d</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="22" left="114" top="408.75" width="74.25" height="29.25" align-x="124.5" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">I</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>64</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">d</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="23" left="210" top="408.75" width="64.5" height="35.25" align-x="222" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">k</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">d</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>4</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="47" left="348" top="422.25" width="58.5" height="13.5" align-x="358.5" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">k</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">in</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.093749999999999986</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="inch"/> </unitMonomial> </unitedValue> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="5" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="10"/> </region> <region region-id="139" left="12" top="452.25" width="218.25" height="15" align-x="12" align-y="462" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Slenderness Ratio:<sp count="19"/>L/kcritical:<sp count="23"/> </text> </region> <region region-id="24" left="12" top="474.75" width="53.25" height="29.25" align-x="26.25" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:div/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>42.666666666666671</ml:real> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="138" left="96" top="477.75" width="9" height="18" align-x="96" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="MS Sans Serif" charset="0" size="14"> <inlineAttr line-through="false">></inlineAttr> </f> </text> </region> <region region-id="26" left="138" top="482.25" width="51.75" height="15" align-x="138" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> (L/k)cr = </text> </region> <region region-id="185" left="198" top="468.75" width="84" height="37.5" align-x="222.75" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">Lk</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="y">S</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="186" left="330" top="482.25" width="66.75" height="15.75" align-x="353.25" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">Lk</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>185.85912784636849</ml:real> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="141" left="6" top="524.25" width="102.75" height="12" align-x="24" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Critical Buckling Load:<sp count="2"/> </text> </region> <region region-id="31" left="6" top="543" width="72" height="52.5" align-x="25.5" align-y="564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">P</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:div/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="32" left="306" top="556.5" width="84" height="19.5" align-x="324" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">P</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">lbf</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>41915.318674397444</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pound_force"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="15"/> </region> <region region-id="187" left="12" top="620.25" width="186.75" height="12" align-x="27" align-y="630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Column Type Test__1=Euler, 0=Johnson:<sp count="2"/> </text> </region> <region region-id="188" left="12" top="642.75" width="147.75" height="29.25" align-x="58.5" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Condition</ml:id> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="cr">Lk</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:real font="0">1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="189" left="306" top="650.25" width="63.75" height="13.5" align-x="351" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:provenance expr-id="1"> <originRef doc-id="35FC8E52-9D76-4971-902A-58B94E2E7CD7" version-id="9BB96CF7-91B8-479D-9EAE-843A1AA27452" branch-id="00000000-0000-0000-0000-000000000000" revision-num="327221076" is-modified="false" region-id="0" href="F:\Jon_At_MBARI\MCadfile_14\Column_Euler-Johnson.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="35FC8E52-9D76-4971-902A-58B94E2E7CD7" version-id="9BB96CF7-91B8-479D-9EAE-843A1AA27452" branch-id="00000000-0000-0000-0000-000000000000" revision-num="327220956" is-modified="false" region-id="0" href="F:\Jon_At_MBARI\MCadfile_14\Column_Euler-Johnson.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">c83afc3453e71f958aa1bb24facaa817</contentHash> <ml:id xml:space="preserve">Condition</ml:id> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="17"> <element-image-map> <box left="1.5" top="0.75" width="39" height="12" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="33" left="6" top="698.25" width="399" height="27" align-x="6" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="MS Sans Serif" charset="0"> <inlineAttr line-through="false">Johnson Column:</inlineAttr> </f> <sp count="2"/>(where the slenderness ratio L/k < (L/k)cr ) : where<sp count="2"/>P<f size="5"> cr </f>/A = S<f size="5"> y </f>/2 is assumed! </text> </region> <region region-id="34" left="12" top="726.75" width="149.25" height="35.25" align-x="31.5" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">P</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="y">S</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="y">S</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:div/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="35" left="312" top="736.5" width="84" height="19.5" align-x="330" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">P</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">lbf</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>4301.4541964491464</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pound_force"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="191" left="12" top="818.25" width="255.75" height="12" align-x="34.5" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Eccentrically Loaded Euler Column: <sp count="32"/> </text> </region> <region region-id="194" left="12" top="842.25" width="45.75" height="12" align-x="12" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Where:<sp count="6"/> </text> </region> <region region-id="196" left="66" top="842.25" width="148.5" height="12" align-x="66" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> e = eccentricity of the load<sp count="15"/> </text> </region> <region region-id="198" left="66" top="860.25" width="183.75" height="12" align-x="66" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> c = cetroidal distance of the load<sp count="18"/> </text> </region> <region region-id="200" left="66" top="878.25" width="364.5" height="24" align-x="66" align-y="888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> K = effective length factor =<sp count="5"/>0.5<sp count="21"/>1,<sp count="20"/>.669,<sp count="21"/>2<sp count="14"/> <sp count="46"/>fixed-fixed<sp count="6"/>Pined-Pined<sp count="8"/>fixed-pined<sp count="9"/>fixed-free<sp count="8"/> </text> </region> <region region-id="207" left="66" top="914.25" width="228" height="12" align-x="66" align-y="924" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> σ.crit-max = maximum compressive stress<sp count="19"/> </text> </region> <region region-id="202" left="12" top="950.25" width="66.75" height="12" align-x="12" align-y="960" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Given Values:<sp count="2"/> </text> </region> <region region-id="203" left="12" top="968.25" width="26.25" height="13.5" align-x="25.5" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">K</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="204" left="72" top="968.25" width="47.25" height="13.5" align-x="84" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIConstant" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult/> <ml:real>.180</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="205" left="144" top="968.25" width="46.5" height="13.5" align-x="155.25" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">c</ml:id> <ml:apply> <ml:mult/> <ml:real>.180</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="209" left="210" top="968.25" width="86.25" height="15.75" align-x="250.5" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="crit_max">σ</ml:id> <ml:apply> <ml:mult/> <ml:real>50000</ml:real> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="212" left="6" top="992.25" width="300.75" height="12" align-x="6" align-y="1002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Where P must be solved iteratively to appraoch σ.crit_max<sp count="19"/> </text> </region> <region region-id="223" left="12" top="1024.5" width="118.5" height="17.25" align-x="88.5" align-y="1038" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>800</ml:real> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> <ml:real>.601</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">in</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">lbf</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>961.6</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pound_force"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="220" left="12" top="1058.25" width="48.75" height="13.5" align-x="24.75" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:mult/> <ml:real>962</ml:real> <ml:id xml:space="preserve">lbf</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="221" left="12" top="1084.5" width="204.75" height="35.25" align-x="71.25" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">CriticalStress</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">A</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sec</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">K</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="222" left="264" top="1090.5" width="124.5" height="17.25" align-x="321.75" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">CriticalStress</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">psi</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>52696.944452096643</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pounds_per_sq_inch"/> </unitMonomial> </unitedValue> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="27"/> </region> </regions> <binaryContent> <item 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Mathcad Professional 14.0 <description/> <author>jon</author> <company/> <keywords/> <revisedBy>Jon</revisedBy> </userData> <identityInfo> <revision>11</revision> <documentID>A638E3FB-6EFB-4021-83BC-EC17AC55E098</documentID> <versionID>722F2126-D7C5-4834-8920-EA868736D4EE</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="MS Sans Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="narrow-dot" derivative="derivative" literal-subscript="small" definition="colon-equal" global-definition="triple-equal" local-definition="left-arrow" equality="bold-equal" symbolic-evaluation="right-arrow"/> <mathStyles> <mathStyle name="Variables" font-family="MS Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Constants" font-family="MS Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 1" font-family="MS Sans Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 2" font-family="Courier" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 3" font-family="System" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 4" font-family="Script" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 5" font-family="Roman" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 6" font-family="Modern" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 7" font-family="MS Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="MS Serif" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="charge" thermodynamic-temperature="temperature" luminous-intensity="luminosity" amount-of-substance="substance" display="false"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="86.4" right="86.25002" top="72" bottom="72"/> <header use-full-page-width="true"/> <footer use-full-page-width="true"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="us" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="223" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="3" left="6" top="9.75" width="321.75" height="18" align-x="6" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <sp count="29"/> <f size="14"> <sp count="3"/>COLUMN BUCKLING</f> </text> </region> <region region-id="6" left="6" top="44.25" width="25.5" height="12" align-x="6" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Req:<sp count="2"/> </text> </region> <region region-id="137" left="66" top="44.25" width="355.5" height="24" align-x="66" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Find the Critical Buckling Load for an Euler or Johnson column (dependent on the slenderness ratio.<sp count="2"/>With either a circular or square cross-section. </text> </region> <region region-id="165" left="66" top="80.25" width="185.25" height="12" align-x="66" align-y="90" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Shigley - "Mech Eng Design"<sp count="2"/>(pp142-149) </text> </region> <region region-id="7" left="6" top="110.25" width="27" height="24" align-x="6" align-y="120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Given: </text> </region> <region region-id="8" left="6" top="158.25" width="52.5" height="24" align-x="6" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Assumption: </text> </region> <region region-id="88" left="18" top="176.25" width="72" height="12" align-x="18" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> End Conditions: </text> </region> <region region-id="96" left="102" top="176.25" width="227.25" height="12" align-x="102" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Fixed - Free:<sp count="55"/>C=.25 </text> </region> <region region-id="97" left="102" top="194.25" width="251.25" height="12" align-x="102" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Rounded(pinned) - Rounded(pinned):<sp count="5"/>C=1.0 </text> </region> <region region-id="98" left="102" top="212.25" width="278.25" height="12" align-x="102" align-y="222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Fixed - Rounded(pinned):<sp count="29"/>C=1.2 </text> </region> <region region-id="99" left="102" top="230.25" width="208.5" height="12" align-x="102" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Fixed - Fixed:<sp count="53"/>C=1.2 </text> </region> <region region-id="9" left="6" top="272.25" width="38.25" height="12" align-x="6" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="MS Sans Serif" charset="0"> <inlineAttr line-through="false">Soln:</inlineAttr> </f> </text> </region> <region region-id="20" left="54" top="272.25" width="79.5" height="24" align-x="54" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="MS Sans Serif" charset="0"> <inlineAttr line-through="false">Circular Section </inlineAttr> </f> </text> </region> <region region-id="173" left="174" top="272.25" width="33" height="13.5" align-x="187.5" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">C</ml:id> <ml:real>2.5</ml:real> </ml:define> </math> <rendering item-idref="1"/> </region> <region region-id="30" left="6" top="308.25" width="343.5" height="15" align-x="6" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="MS Sans Serif" charset="0"> <inlineAttr line-through="false">Euler Column:</inlineAttr> </f> <sp count="2"/>(where the slenderness ratio L/k > (L/k)cr<f size="5"> <sp count="2"/> </f> ) </text> </region> <region region-id="10" left="6" top="338.25" width="32.25" height="12" align-x="6" align-y="348" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Length: </text> </region> <region region-id="17" left="60" top="338.25" width="43.5" height="13.5" align-x="72.75" align-y="348" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult/> <ml:real>4.0</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="25" left="144" top="334.5" width="64.5" height="19.5" align-x="159.75" align-y="348" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="y">S</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>40</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="18" left="60" top="362.25" width="49.5" height="13.5" align-x="74.25" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:mult/> <ml:real>.375</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="28" left="150" top="358.5" width="68.25" height="17.25" align-x="162.75" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">E</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>28.0</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="19" left="60" top="380.25" width="42.75" height="13.5" align-x="72" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">d</ml:id> <ml:apply> <ml:mult/> <ml:real>.00</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="21" left="12" top="408.75" width="75" height="29.25" align-x="25.5" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">d</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="22" left="114" top="408.75" width="74.25" height="29.25" align-x="124.5" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">I</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>64</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">d</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="23" left="210" top="408.75" width="64.5" height="35.25" align-x="222" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">k</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">d</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>4</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="47" left="348" top="422.25" width="58.5" height="13.5" align-x="358.5" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">k</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">in</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.093749999999999986</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="inch"/> </unitMonomial> </unitedValue> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="5" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="10"/> </region> <region region-id="139" left="12" top="452.25" width="218.25" height="15" align-x="12" align-y="462" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Slenderness Ratio:<sp count="19"/>L/kcritical:<sp count="23"/> </text> </region> <region region-id="24" left="12" top="474.75" width="53.25" height="29.25" align-x="26.25" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:div/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>42.666666666666671</ml:real> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="138" left="96" top="477.75" width="9" height="18" align-x="96" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="MS Sans Serif" charset="0" size="14"> <inlineAttr line-through="false">></inlineAttr> </f> </text> </region> <region region-id="26" left="138" top="482.25" width="51.75" height="15" align-x="138" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> (L/k)cr = </text> </region> <region region-id="185" left="198" top="468.75" width="84" height="37.5" align-x="222.75" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">Lk</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="y">S</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="186" left="330" top="482.25" width="66.75" height="15.75" align-x="353.25" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">Lk</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>185.85912784636849</ml:real> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="141" left="6" top="524.25" width="102.75" height="12" align-x="24" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Critical Buckling Load:<sp count="2"/> </text> </region> <region region-id="31" left="6" top="543" width="72" height="52.5" align-x="25.5" align-y="564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">P</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:div/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="32" left="306" top="556.5" width="84" height="19.5" align-x="324" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">P</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">lbf</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>41915.318674397444</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pound_force"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="15"/> </region> <region region-id="187" left="12" top="620.25" width="186.75" height="12" align-x="27" align-y="630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Column Type Test__1=Euler, 0=Johnson:<sp count="2"/> </text> </region> <region region-id="188" left="12" top="642.75" width="147.75" height="29.25" align-x="58.5" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Condition</ml:id> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="cr">Lk</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:real font="0">1</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="189" left="306" top="650.25" width="63.75" height="13.5" align-x="351" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:provenance expr-id="1"> <originRef doc-id="35FC8E52-9D76-4971-902A-58B94E2E7CD7" version-id="9BB96CF7-91B8-479D-9EAE-843A1AA27452" branch-id="00000000-0000-0000-0000-000000000000" revision-num="327221076" is-modified="false" region-id="0" href="F:\Jon_At_MBARI\MCadfile_14\Column_Euler-Johnson.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="35FC8E52-9D76-4971-902A-58B94E2E7CD7" version-id="9BB96CF7-91B8-479D-9EAE-843A1AA27452" branch-id="00000000-0000-0000-0000-000000000000" revision-num="327220956" is-modified="false" region-id="0" href="F:\Jon_At_MBARI\MCadfile_14\Column_Euler-Johnson.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">c83afc3453e71f958aa1bb24facaa817</contentHash> <ml:id xml:space="preserve">Condition</ml:id> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="17"> <element-image-map> <box left="1.5" top="0.75" width="39" height="12" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="33" left="6" top="698.25" width="399" height="27" align-x="6" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="MS Sans Serif" charset="0"> <inlineAttr line-through="false">Johnson Column:</inlineAttr> </f> <sp count="2"/>(where the slenderness ratio L/k < (L/k)cr ) : where<sp count="2"/>P<f size="5"> cr </f>/A = S<f size="5"> y </f>/2 is assumed! </text> </region> <region region-id="34" left="12" top="726.75" width="149.25" height="35.25" align-x="31.5" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">P</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="y">S</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="y">S</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:div/> <ml:id xml:space="preserve">L</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="35" left="312" top="736.5" width="84" height="19.5" align-x="330" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">P</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">lbf</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>4301.4541964491464</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pound_force"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="191" left="12" top="818.25" width="255.75" height="12" align-x="34.5" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Eccentrically Loaded Euler Column: <sp count="32"/> </text> </region> <region region-id="194" left="12" top="842.25" width="45.75" height="12" align-x="12" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Where:<sp count="6"/> </text> </region> <region region-id="196" left="66" top="842.25" width="148.5" height="12" align-x="66" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> e = eccentricity of the load<sp count="15"/> </text> </region> <region region-id="198" left="66" top="860.25" width="183.75" height="12" align-x="66" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> c = cetroidal distance of the load<sp count="18"/> </text> </region> <region region-id="200" left="66" top="878.25" width="364.5" height="24" align-x="66" align-y="888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> K = effective length factor =<sp count="5"/>0.5<sp count="21"/>1,<sp count="20"/>.669,<sp count="21"/>2<sp count="14"/> <sp count="46"/>fixed-fixed<sp count="6"/>Pined-Pined<sp count="8"/>fixed-pined<sp count="9"/>fixed-free<sp count="8"/> </text> </region> <region region-id="207" left="66" top="914.25" width="228" height="12" align-x="66" align-y="924" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> σ.crit-max = maximum compressive stress<sp count="19"/> </text> </region> <region region-id="202" left="12" top="950.25" width="66.75" height="12" align-x="12" align-y="960" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Given Values:<sp count="2"/> </text> </region> <region region-id="203" left="12" top="968.25" width="26.25" height="13.5" align-x="25.5" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">K</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="204" left="72" top="968.25" width="47.25" height="13.5" align-x="84" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIConstant" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult/> <ml:real>.180</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="205" left="144" top="968.25" width="46.5" height="13.5" align-x="155.25" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">c</ml:id> <ml:apply> <ml:mult/> <ml:real>.180</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="209" left="210" top="968.25" width="86.25" height="15.75" align-x="250.5" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="crit_max">σ</ml:id> <ml:apply> <ml:mult/> <ml:real>50000</ml:real> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="212" left="6" top="992.25" width="300.75" height="12" align-x="6" align-y="1002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Where P must be solved iteratively to appraoch σ.crit_max<sp count="19"/> </text> </region> <region region-id="223" left="12" top="1024.5" width="118.5" height="17.25" align-x="88.5" align-y="1038" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>800</ml:real> <ml:id xml:space="preserve">psi</ml:id> </ml:apply> <ml:real>.601</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">in</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">lbf</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>961.6</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pound_force"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="220" left="12" top="1058.25" width="48.75" height="13.5" align-x="24.75" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:mult/> <ml:real>962</ml:real> <ml:id xml:space="preserve">lbf</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="221" left="12" top="1084.5" width="204.75" height="35.25" align-x="71.25" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">CriticalStress</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">P</ml:id> <ml:id xml:space="preserve">A</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">e</ml:id> <ml:id 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