<description/> <author>Amit Varma</author> <company>Parametric Technology Corporation</company> <keywords/> <revisedBy>Lisa Choe</revisedBy> </userData> <identityInfo> <revision>6</revision> <documentID>E5808E9B-8974-4D2F-8AD7-F97C9D02F6A3</documentID> <versionID>400628A5-24DB-44B6-BD22-5BEB4F83C34E</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="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" 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<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="true" strings-use-origin="false" zero-over-zero="error"> <compatibility multiple-assignment="MC12" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <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="1675" 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="1161" left="6" top="8.25" width="255.75" height="12" align-x="9" align-y="18" 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"> 1. Shape function matrix w.r.t. the generalized coordinates: </text> </region> <region region-id="920" left="300" top="62.25" width="35.25" height="12" align-x="315" align-y="72" 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"> Answer </text> </region> <region region-id="515" left="126" top="55.5" width="131.25" height="29.25" align-x="166.5" align-y="72.75" 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:function> <ml:id xml:space="preserve" subscript="s">N</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="2" cols="6"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y</ml:id> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:matrix> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="1158" left="6" top="110.25" width="220.5" height="12" align-x="9" 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"> 2. Shape function matrix w.r.t nodal displacement: </text> </region> <region region-id="1159" left="12" top="134.25" width="95.25" height="12" align-x="37.5" align-y="144" 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"> Interpolation approach </text> </region> <region region-id="908" left="204" top="201" width="32.25" height="12.75" align-x="219.75" align-y="210" 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">x1</ml:id> <ml:real>12</ml:real> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="909" left="270" top="201" width="32.25" height="12.75" align-x="285.75" align-y="210" 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">x2</ml:id> <ml:real>18</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="910" left="336" top="201" width="32.25" height="12.75" align-x="351.75" align-y="210" 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">x3</ml:id> <ml:real>12</ml:real> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="911" left="48" top="171.75" width="44.25" height="95.25" align-x="59.25" align-y="222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This variable is undefined." optimize="false" 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<ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:id xml:space="preserve">x1</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">x2</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">x3</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y1</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y2</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y3</ml:id> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:id xml:space="preserve">x1</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">x2</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">x3</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y1</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y2</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y3</ml:id> </ml:matrix> </ml:define> </math> <rendering item-idref="11"/> </region> <region 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<ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:real>3</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>6</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>6</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="925" left="390" top="506.25" width="35.25" height="12" align-x="405" align-y="516" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="1141" left="6" top="626.25" width="261" height="12" align-x="9" align-y="636" 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"> 3. Strain displacement matrix w.r.t. generalized coordinates: </text> </region> <region region-id="935" left="120" top="664.5" width="103.5" height="45.75" align-x="132.75" align-y="690" 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">B</ml:id> <ml:matrix rows="3" cols="6"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="936" left="252" top="680.25" width="35.25" height="12" align-x="267" align-y="690" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="1134" left="6" top="740.25" width="242.25" height="12" align-x="9" align-y="750" 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"> 4. Strain displacement matrix w.r.t. nodal displacement: </text> </region> <region region-id="946" left="114" top="774" width="205.5" height="90.75" align-x="155.25" align-y="822" 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:function> <ml:id xml:space="preserve">B1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:symEval style="default" hide-keywords="false" hide-lhs="false"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> <ml:symResult> <ml:matrix rows="3" cols="6"> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:matrix> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="951" left="348" top="812.25" width="35.25" height="12" align-x="363" align-y="822" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="953" left="18" top="878.25" width="141" height="12" align-x="21" 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"> 5. Plane stress elasticity matrix: </text> </region> <region region-id="956" left="54" top="909" width="58.5" height="12.75" align-x="65.25" align-y="918" 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:real>29000</ml:real> <ml:id xml:space="preserve">ksi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="957" left="150" top="909" width="30.75" height="12.75" align-x="162" align-y="918" 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">ν</ml:id> <ml:real>0.3</ml:real> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="958" left="222" top="909" width="46.5" height="12.75" align-x="233.25" align-y="918" 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">h</ml:id> <ml:apply> <ml:mult/> <ml:real>0.25</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="959" left="54" top="956.25" width="293.25" height="93" align-x="66.75" align-y="1020" 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:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">E</ml:id> <ml:matrix rows="3" cols="3"> <ml:real>1</ml:real> <ml:id xml:space="preserve">ν</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">ν</ml:id> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">ν</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:apply> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">ν</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">ksi</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="3" cols="3"> <ml:real>31868.13186813187</ml:real> <ml:real>9560.43956043956</ml:real> <ml:real>0</ml:real> <ml:real>9560.43956043956</ml:real> <ml:real>31868.13186813187</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>11153.846153846151</ml:real> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kips_per_sq_inch"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="1130" left="6" top="1082.25" width="153" height="12" align-x="9" align-y="1092" 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"> 6. Stress-displacement matrix [DB] </text> </region> <region region-id="1265" left="6" top="1122" width="324" height="90.75" align-x="25.5" align-y="1170" 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">DB</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult style="thin-space"/> <ml:matrix rows="3" cols="3"> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3.187</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>9.56</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>9.56</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3.187</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.115</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:matrix> <ml:id xml:space="preserve">ksi</ml:id> </ml:apply> <ml:matrix rows="3" cols="6"> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:matrix> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="1266" left="6" top="1249.5" width="377.25" height="63.75" align-x="24.75" align-y="1284" 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">DB</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ksi</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="3" cols="6"> <ml:real>-5311.6666666666661</ml:real> <ml:real>-1593.3333333333333</ml:real> <ml:real>-5575</ml:real> <ml:real>-4780</ml:real> <ml:real>-15935</ml:real> <ml:real>-1858.3333333333333</ml:real> <ml:real>5311.6666666666661</ml:real> <ml:real>1593.3333333333333</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1858.3333333333333</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>5575</ml:real> <ml:real>4780</ml:real> <ml:real>15935</ml:real> <ml:real>0</ml:real> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kips_per_sq_inch"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="1267" left="384" top="1322.25" width="35.25" height="12" align-x="399" align-y="1332" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="1163" left="12" top="1364.25" width="169.5" height="12" align-x="15" align-y="1374" 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"> 7. Generalized element stiffness matrix </text> </region> <region region-id="1168" left="30" top="1394.25" width="125.25" height="27.75" align-x="56.25" align-y="1410" 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">Area</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>1</ml:real> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x2</ml:id> <ml:id xml:space="preserve">x1</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">y3</ml:id> <ml:id xml:space="preserve">y1</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="1412" left="30" top="1431" width="43.5" height="12.75" align-x="55.5" align-y="1440" 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">Area</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>6</ml:real> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="1482" left="30" top="1459.5" width="209.25" height="63.75" align-x="50.25" align-y="1494" 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="m">D</ml:id> <ml:apply> <ml:mult style="thin-space"/> <ml:matrix rows="3" cols="3"> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3.187</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>9.56</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>9.56</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3.187</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.115</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:matrix> <ml:id xml:space="preserve">ksi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="1523" left="36" top="1551.75" width="108" height="21.75" align-x="62.25" align-y="1566" 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="bar">K</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">Area</ml:id> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">B</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="m">D</ml:id> </ml:apply> <ml:id xml:space="preserve">B</ml:id> </ml:apply> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="1524" left="198" top="1556.25" width="77.25" height="12" align-x="204.75" align-y="1566" 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"> per unit thickness </text> </region> <region region-id="1525" left="18" top="1599.75" width="286.5" height="119.25" align-x="43.5" align-y="1662" 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="bar">K</ml:id> <ml:unitOverride> <ml:apply> <ml:div/> <ml:id xml:space="preserve">kip</ml:id> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="6" cols="6"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>191220</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>57360</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>66900</ml:real> <ml:real>0</ml:real> <ml:real>66900</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>66900</ml:real> <ml:real>0</ml:real> <ml:real>66900</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>57360</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>191220</ml:real> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="inch" power-numerator="-1"/> <unitReference unit="kip"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="26"/> </region> <region region-id="1526" left="372" top="1652.25" width="35.25" height="12" align-x="387" align-y="1662" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="1553" left="12" top="1784.25" width="57" height="12" align-x="15" align-y="1794" 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"> If h = 0.25 in, </text> </region> <region region-id="1554" left="36" top="1815.75" width="124.5" height="21.75" align-x="67.5" align-y="1830" 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="bar1">K</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">Area</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">B</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="m">D</ml:id> </ml:apply> <ml:id xml:space="preserve">B</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="1667" left="6" top="1869.75" width="300.75" height="119.25" align-x="36.75" align-y="1932" 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="bar1">K</ml:id> <ml:unitOverride> <ml:apply> <ml:div/> <ml:id xml:space="preserve">kip</ml:id> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="6" cols="6"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> 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Generalized element stiffness matrix: </text> </region> <region region-id="1670" left="12" top="2043.75" width="113.25" height="32.25" align-x="24.75" align-y="2064" 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:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:transpose/> <ml:parens> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="bar">K</ml:id> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering 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<miscellaneous> <handbook handbook-region-tag-ub="1675" 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="1161" left="6" top="8.25" width="255.75" height="12" align-x="9" align-y="18" 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"> 1. Shape function matrix w.r.t. the generalized coordinates: </text> </region> <region region-id="920" left="300" top="62.25" width="35.25" height="12" align-x="315" align-y="72" 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"> Answer </text> </region> <region region-id="515" left="126" top="55.5" width="131.25" height="29.25" align-x="166.5" align-y="72.75" 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:function> <ml:id xml:space="preserve" subscript="s">N</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="2" cols="6"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y</ml:id> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:matrix> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="1158" left="6" top="110.25" width="220.5" height="12" align-x="9" 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"> 2. 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<ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>-2</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>-2</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="12"/> </region> <region region-id="918" left="36" top="483.75" width="339.75" height="59.25" align-x="93.75" align-y="516" 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:function> <ml:id xml:space="preserve" subscript="nodal">N</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:symEval style="default" hide-keywords="false" hide-lhs="false"> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" subscript="s">N</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> <ml:symResult> <ml:matrix rows="2" cols="6"> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:real>3</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:real>3</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>6</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>6</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="925" left="390" top="506.25" width="35.25" height="12" align-x="405" align-y="516" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="1141" left="6" top="626.25" width="261" height="12" align-x="9" align-y="636" 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"> 3. Strain displacement matrix w.r.t. generalized coordinates: </text> </region> <region region-id="935" left="120" top="664.5" width="103.5" height="45.75" align-x="132.75" align-y="690" 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">B</ml:id> <ml:matrix rows="3" cols="6"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="936" left="252" top="680.25" width="35.25" height="12" align-x="267" align-y="690" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="1134" left="6" top="740.25" width="242.25" height="12" align-x="9" align-y="750" 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"> 4. Strain displacement matrix w.r.t. nodal displacement: </text> </region> <region region-id="946" left="114" top="774" width="205.5" height="90.75" align-x="155.25" align-y="822" 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:function> <ml:id xml:space="preserve">B1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:symEval style="default" hide-keywords="false" hide-lhs="false"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> <ml:symResult> <ml:matrix rows="3" cols="6"> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:matrix> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="951" left="348" top="812.25" width="35.25" height="12" align-x="363" align-y="822" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="953" left="18" top="878.25" width="141" height="12" align-x="21" 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"> 5. Plane stress elasticity matrix: </text> </region> <region region-id="956" left="54" top="909" width="58.5" height="12.75" align-x="65.25" align-y="918" 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:real>29000</ml:real> <ml:id xml:space="preserve">ksi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="957" left="150" top="909" width="30.75" height="12.75" align-x="162" align-y="918" 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">ν</ml:id> <ml:real>0.3</ml:real> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="958" left="222" top="909" width="46.5" height="12.75" align-x="233.25" align-y="918" 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">h</ml:id> <ml:apply> <ml:mult/> <ml:real>0.25</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="959" left="54" top="956.25" width="293.25" height="93" align-x="66.75" align-y="1020" 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:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">E</ml:id> <ml:matrix rows="3" cols="3"> <ml:real>1</ml:real> <ml:id xml:space="preserve">ν</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">ν</ml:id> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">ν</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:apply> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">ν</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">ksi</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="3" cols="3"> <ml:real>31868.13186813187</ml:real> <ml:real>9560.43956043956</ml:real> <ml:real>0</ml:real> <ml:real>9560.43956043956</ml:real> <ml:real>31868.13186813187</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>11153.846153846151</ml:real> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kips_per_sq_inch"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="1130" left="6" top="1082.25" width="153" height="12" align-x="9" align-y="1092" 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"> 6. Stress-displacement matrix [DB] </text> </region> <region region-id="1265" left="6" top="1122" width="324" height="90.75" align-x="25.5" align-y="1170" 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">DB</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult style="thin-space"/> <ml:matrix rows="3" cols="3"> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3.187</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>9.56</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>9.56</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3.187</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.115</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:matrix> <ml:id xml:space="preserve">ksi</ml:id> </ml:apply> <ml:matrix rows="3" cols="6"> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>6</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:matrix> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="1266" left="6" top="1249.5" width="377.25" height="63.75" align-x="24.75" align-y="1284" 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">DB</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ksi</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="3" cols="6"> <ml:real>-5311.6666666666661</ml:real> <ml:real>-1593.3333333333333</ml:real> <ml:real>-5575</ml:real> <ml:real>-4780</ml:real> <ml:real>-15935</ml:real> <ml:real>-1858.3333333333333</ml:real> <ml:real>5311.6666666666661</ml:real> <ml:real>1593.3333333333333</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1858.3333333333333</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>5575</ml:real> <ml:real>4780</ml:real> <ml:real>15935</ml:real> <ml:real>0</ml:real> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kips_per_sq_inch"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="1267" left="384" top="1322.25" width="35.25" height="12" align-x="399" align-y="1332" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="1163" left="12" top="1364.25" width="169.5" height="12" align-x="15" align-y="1374" 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"> 7. Generalized element stiffness matrix </text> </region> <region region-id="1168" left="30" top="1394.25" width="125.25" height="27.75" align-x="56.25" align-y="1410" 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">Area</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>1</ml:real> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x2</ml:id> <ml:id xml:space="preserve">x1</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">y3</ml:id> <ml:id xml:space="preserve">y1</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="1412" left="30" top="1431" width="43.5" height="12.75" align-x="55.5" align-y="1440" 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">Area</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>6</ml:real> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="1482" left="30" top="1459.5" width="209.25" height="63.75" align-x="50.25" align-y="1494" 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="m">D</ml:id> <ml:apply> <ml:mult style="thin-space"/> <ml:matrix rows="3" cols="3"> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3.187</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>9.56</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>9.56</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3.187</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.115</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> </ml:matrix> <ml:id xml:space="preserve">ksi</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="1523" left="36" top="1551.75" width="108" height="21.75" align-x="62.25" align-y="1566" 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="bar">K</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">Area</ml:id> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">B</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="m">D</ml:id> </ml:apply> <ml:id xml:space="preserve">B</ml:id> </ml:apply> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="1524" left="198" top="1556.25" width="77.25" height="12" align-x="204.75" align-y="1566" 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"> per unit thickness </text> </region> <region region-id="1525" left="18" top="1599.75" width="286.5" height="119.25" align-x="43.5" align-y="1662" 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="bar">K</ml:id> <ml:unitOverride> <ml:apply> <ml:div/> <ml:id xml:space="preserve">kip</ml:id> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="6" cols="6"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>191220</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>57360</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>66900</ml:real> <ml:real>0</ml:real> <ml:real>66900</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>66900</ml:real> <ml:real>0</ml:real> <ml:real>66900</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>57360</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>191220</ml:real> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="inch" power-numerator="-1"/> <unitReference unit="kip"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="26"/> </region> <region region-id="1526" left="372" top="1652.25" width="35.25" height="12" align-x="387" align-y="1662" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" 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<ml:transpose/> <ml:id xml:space="preserve">B</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="m">D</ml:id> </ml:apply> <ml:id xml:space="preserve">B</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="1667" left="6" top="1869.75" width="300.75" height="119.25" align-x="36.75" align-y="1932" 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="bar1">K</ml:id> <ml:unitOverride> <ml:apply> <ml:div/> <ml:id xml:space="preserve">kip</ml:id> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="6" cols="6"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>47804.999999999993</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>14340</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>16724.999999999996</ml:real> <ml:real>0</ml:real> <ml:real>16724.999999999996</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>16724.999999999996</ml:real> <ml:real>0</ml:real> <ml:real>16724.999999999996</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>14340</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>47804.999999999993</ml:real> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="inch" power-numerator="-1"/> <unitReference unit="kip"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="28"/> </region> <region region-id="1668" left="348" top="1922.25" width="35.25" height="12" align-x="363" align-y="1932" 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"> <inlineAttr font-style="normal">Answer</inlineAttr> </text> </region> <region region-id="1669" left="12" top="2018.25" width="172.5" height="12" align-x="15" align-y="2028" 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"> 7. Generalized element stiffness matrix: </text> </region> <region region-id="1670" left="12" top="2043.75" width="113.25" height="32.25" align-x="24.75" align-y="2064" 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:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:transpose/> <ml:parens> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="bar">K</ml:id> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering 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xml:space="preserve" subscript="1">K</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:transpose/> <ml:parens> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve" subscript="bar1">K</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="1656" left="18" top="2133.75" width="390" height="131.25" align-x="30" align-y="2202" 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:apply> <ml:div/> <ml:id xml:space="preserve">kip</ml:id> <ml:id xml:space="preserve">in</ml:id> 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tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">K</ml:id> <ml:unitOverride> <ml:apply> <ml:div/> <ml:id xml:space="preserve">kip</ml:id> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:matrix rows="6" cols="6"> <ml:real>5509.1666666666661</ml:real> <ml:real>2588.7499999999995</ml:real> <ml:real>-1327.9166666666663</ml:real> <ml:real>-1393.7499999999998</ml:real> <ml:real>-4181.2499999999991</ml:real> <ml:real>-1195</ml:real> <ml:real>2588.7499999999995</ml:real> <ml:real>12415.833333333332</ml:real> <ml:real>-1195</ml:real> <ml:real>-464.58333333333326</ml:real> <ml:real>-1393.7499999999998</ml:real> <ml:real>-11951.249999999998</ml:real> <ml:real>-1327.9166666666663</ml:real> <ml:real>-1195</ml:real> <ml:real>1327.9166666666663</ml:real> 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