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simplify-units="true" 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="9" orientation="portrait" print-single-page-width="true" page-width="595.5" page-height="841.5"> <margins left="59.4" right="45.28799" top="56.592" bottom="42.48"/> <header use-full-page-width="false"> <left>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fbidi\fprq6\fcharset0 Arial;}{\f1\fnil\fprq15\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\lang11274\f0\fs18 Ejercicio resuelto - Axil (Williot)\lang1033\f1\par } </left> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang11274{\fonttbl{\f0\fbidi\fprq6\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18 Ing. Lopez Rivarola\par } </right> </header> <footer use-full-page-width="false"> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang11274{\fonttbl{\f0\fbidi\fprq6\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18\{n\}/\{nn\}\par }</right> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="1" 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="true" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC12" local-assignment="MC12"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="true"/> <customUnitSystem base-unit-system="si"> <baseUnits> <unit unit-name="mole" quantity-name="amount_of_substance"/> <unit 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quantity-name="torque"/> <unit unit-name="kilonewton" quantity-name="force"/> </derivedUnits> </customUnitSystem> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="true" ruler-unit="cm"/> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="5352" 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="4678" left="6" top="8.25" width="477" height="12" align-x="30" 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"> <f size="11"> Enunciado: </f> <f size="11"> <inlineAttr font-weight="normal" underline="false"> <sp count="2"/>Para la siguiente estructura, calcular el desplazamiento del punto A.</inlineAttr> </f> </text> </region> <region region-id="4764" left="18" top="30" width="211.5" height="159.75" align-x="18" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{720DB9AF-D62C-4ED0-A377-429C22312852}" type="embedded" item-idref="1"/> <rendering item-idref="2"/> </region> <region region-id="4765" left="282" top="38.25" width="44.25" height="12" align-x="293.25" align-y="48" 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 size="11"> Datos: </f> </text> </region> <region region-id="4766" left="324" top="39" width="38.25" height="16.5" align-x="342.75" align-y="48" 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="H">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="5173" left="522" top="39" width="33" height="12.75" align-x="531.75" align-y="48" 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">º</ml:id> <ml:id xml:space="preserve">deg</ml:id> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="4767" left="324" top="69" width="37.5" height="16.5" align-x="342" align-y="78" 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="V">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="4768" left="324" top="93" width="52.5" height="18.75" align-x="338.25" align-y="108" 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">A</ml:id> <ml:apply> <ml:mult/> <ml:real>2.5</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="4769" left="324" top="129" width="50.25" height="12.75" align-x="336.75" align-y="138" 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 style="auto-select"/> <ml:real>210</ml:real> <ml:id xml:space="preserve">GPa</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="4770" left="324" top="159" width="40.5" height="12.75" align-x="336" align-y="168" 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 style="auto-select"/> <ml:real>50</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="4774" left="6" top="212.25" width="93" height="12" align-x="17.25" 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"> <f size="11"> Resolución: </f> </text> </region> <region region-id="4856" left="534" top="231" width="55.5" height="16.5" align-x="552.75" align-y="240" 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="1">N</ml:id> <ml:apply> <ml:mult/> <ml:real>-30</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="4857" left="606" top="231" width="55.5" height="16.5" align-x="624.75" align-y="240" 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="2">N</ml:id> <ml:apply> <ml:mult/> <ml:real>-30</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="4776" left="6" top="236.25" width="473.25" height="36" align-x="14.25" align-y="246" 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"> Para poder calcular el desplazamiento del punto P debemos primero calcular las tensiones y deformaciones especificas de las barras. Para poder calcular esto requerimos conocer el valor de los esfuerzos axiles de las barras. </text> </region> <region region-id="4924" left="6" top="290.25" width="33" height="12" align-x="19.5" align-y="300" 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"> Siendo: </text> </region> <region region-id="4994" left="54" top="284.25" width="46.5" height="27.75" align-x="85.5" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="4995" left="120" top="284.25" width="48" height="27.75" align-x="153" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>5</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="12"/> </region> <region region-id="4996" left="186" top="284.25" width="45.75" height="27.75" align-x="216.75" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>5</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="13"/> </region> <region region-id="5001" left="264" top="284.25" width="96" height="27.75" align-x="276.75" align-y="300" 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">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">acos</ml:id> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>5</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.64350110879328426</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="4830" left="6" top="326.25" width="378.75" height="12" align-x="28.5" align-y="336" 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"> Asumiendo los esfuerzos de las barras de tracción, calculamos el equilibrio del nudo A: </text> </region> <region region-id="4931" left="504" top="333" width="24" height="12.75" align-x="515.25" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="15"/> </region> <region region-id="4831" left="18" top="348.75" width="27" height="32.25" align-x="30.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:real>.</ml:real> </ml:boundVars> <ml:id xml:space="preserve" subscript="h">F</ml:id> </ml:lambda> </ml:apply> </math> <rendering item-idref="16"/> </region> <region region-id="4845" left="96" top="351" width="110.25" height="16.5" align-x="105.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="4833" left="18" top="372.75" width="26.25" height="32.25" align-x="30" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:real>.</ml:real> </ml:boundVars> <ml:id xml:space="preserve" subscript="v">F</ml:id> </ml:lambda> </ml:apply> </math> <rendering item-idref="18"/> </region> <region region-id="4846" left="96" top="375" width="132" height="16.5" align-x="105.75" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">P</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="4875" left="498" top="381" width="94.5" height="36.75" align-x="529.5" align-y="402" 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:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="2">N</ml:id> </ml:matrix> <ml:apply> <ml:Find auto-method="true" method="linear" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="2">N</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="4935" left="6" top="404.25" width="93.75" height="12" align-x="12.75" align-y="414" 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"> Del cual despejamos: </text> </region> <region region-id="4936" left="108" top="405" width="69" height="16.5" align-x="125.25" align-y="414" 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="1">N</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-31.249999999886313</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilonewton"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="4937" left="198" top="405" width="69" height="16.5" align-x="215.25" align-y="414" 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="2">N</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-31.249999999886313</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilonewton"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="4943" left="6" top="446.25" width="53.25" height="12" align-x="12.75" align-y="456" 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"> Por lo tanto: </text> </region> <region region-id="4944" left="72" top="436.5" width="91.5" height="31.5" align-x="89.25" align-y="456" 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="1">σ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve">A</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">MPa</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-124.99999999954525</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="megapascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="4976" left="222" top="436.5" width="91.5" height="31.5" align-x="239.25" align-y="456" 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="2">σ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:id xml:space="preserve">A</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">MPa</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-124.99999999954525</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="megapascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="4982" left="72" top="484.5" width="114" height="31.5" align-x="88.5" align-y="504" 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="1">ε</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">σ</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-0.00059523809523592971</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="4983" left="222" top="484.5" width="114" height="31.5" align-x="238.5" align-y="504" 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="2">ε</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="2">σ</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-0.00059523809523592971</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="5007" left="516" top="507" width="36" height="16.5" align-x="532.5" 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:id xml:space="preserve" subscript="1">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="5011" left="576" top="507" width="36" height="16.5" align-x="592.5" 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:id xml:space="preserve" subscript="2">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="5005" left="72" top="537" width="108" height="16.5" align-x="96.75" align-y="546" 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="1">ΔL</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">ε</ml:id> <ml:id xml:space="preserve" subscript="1">L</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-2.9761904761796485</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="5006" left="222" top="537" width="108" height="16.5" align-x="246.75" align-y="546" 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="2">ΔL</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">ε</ml:id> <ml:id xml:space="preserve" subscript="2">L</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-2.9761904761796485</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="5022" left="372" top="536.25" width="33" height="12" align-x="385.5" align-y="546" 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"> Siendo: </text> </region> <region region-id="5023" left="414" top="537" width="55.5" height="16.5" align-x="450.75" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve" subscript="1">L</ml:id> <ml:id xml:space="preserve" subscript="2">L</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="31"/> </region> <region region-id="5041" left="288" top="570" width="197.25" height="166.5" align-x="288" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{720DB9AF-D62C-4ED0-A377-429C22312852}" type="embedded" item-idref="32"/> <rendering item-idref="33"/> </region> <region region-id="5034" left="6" top="584.25" width="260.25" height="72" align-x="27" align-y="594" 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"> Finalmente, por Williot, cada barra puede deformarse (acortarse o alargarse) en la dirección de su eje, y luego girar perpendicular a este. Por lo tanto el punto A se mueve al punto A' como se ve en el esquema. Dado que el sistema es totalmente simétrico, el punto A sufre únicamente un desplazamiento vertical, hacia arriba, de valor: </text> </region> <region region-id="5036" left="54" top="676.5" width="93" height="31.5" align-x="65.25" align-y="696" 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">δ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve" subscript="1">ΔL</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>3.7202380952245608</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="5042" left="6" top="752.25" width="477" height="12" align-x="30" align-y="762" 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 size="11"> Enunciado: </f> <f size="11"> <inlineAttr font-weight="normal" underline="false"> <sp count="2"/>Para la siguiente estructura, calcular el desplazamiento del punto A.</inlineAttr> </f> </text> </region> <region region-id="5155" left="36" top="774" width="163.5" height="175.5" align-x="36" align-y="774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{720DB9AF-D62C-4ED0-A377-429C22312852}" type="embedded" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="5138" left="270" top="782.25" width="44.25" height="12" align-x="281.25" align-y="792" 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 size="11"> Datos: </f> </text> </region> <region region-id="5146" left="318" top="783" width="42.75" height="16.5" align-x="341.25" align-y="792" 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="H1">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="5168" left="396" top="777" width="57" height="22.5" align-x="414.75" align-y="792" 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="1">A</ml:id> <ml:apply> <ml:mult/> <ml:real>2.5</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="5180" left="318" top="819" width="42.75" height="16.5" align-x="341.25" align-y="828" 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="H2">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="5181" left="396" top="813" width="50.25" height="22.5" align-x="414.75" align-y="828" 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="2">A</ml:id> <ml:apply> <ml:mult/> <ml:real>5</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="5182" left="318" top="855" width="37.5" height="16.5" align-x="336" align-y="864" 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="V">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="5183" left="396" top="855" width="50.25" height="12.75" align-x="408.75" align-y="864" 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 style="auto-select"/> <ml:real>210</ml:real> <ml:id xml:space="preserve">GPa</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="5184" left="318" top="891" width="33" height="12.75" align-x="330.75" align-y="900" 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">γ</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>50</ml:real> <ml:id xml:space="preserve">º</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="5185" left="396" top="891" width="45" height="12.75" align-x="408" align-y="900" 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 style="auto-select"/> <ml:real>100</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="5050" left="6" top="956.25" width="93" height="12" align-x="17.25" align-y="966" 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 size="11"> Resolución: </f> </text> </region> <region region-id="5051" left="534" top="975" width="55.5" height="16.5" align-x="552.75" align-y="984" 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="1">N</ml:id> <ml:apply> <ml:mult/> <ml:real>-30</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="5052" left="606" top="975" width="55.5" height="16.5" align-x="624.75" align-y="984" 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="2">N</ml:id> <ml:apply> <ml:mult/> <ml:real>-30</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="5053" left="6" top="980.25" width="473.25" height="36" align-x="14.25" align-y="990" 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"> Para poder calcular el desplazamiento del punto P debemos primero calcular las tensiones y deformaciones especificas de las barras. Para poder calcular esto requerimos conocer el valor de los esfuerzos axiles de las barras. </text> </region> <region region-id="5054" left="6" top="1034.25" width="33" height="12" align-x="19.5" align-y="1044" 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"> Siendo: </text> </region> <region region-id="5189" left="48" top="1028.25" width="94.5" height="27.75" align-x="60.75" align-y="1044" 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">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">atan</ml:id> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.64350110879328437</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="5207" left="156" top="1028.25" width="94.5" height="27.75" align-x="168.75" align-y="1044" 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">β</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">atan</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.24497866312686414</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="48"/> </region> <region region-id="5059" left="6" top="1070.25" width="378.75" height="12" align-x="28.5" align-y="1080" 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"> Asumiendo los esfuerzos de las barras de tracción, calculamos el equilibrio del nudo A: </text> </region> <region region-id="5060" left="504" top="1077" width="24" height="12.75" align-x="515.25" align-y="1086" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="49"/> </region> <region region-id="5061" left="18" top="1092.75" width="27" height="32.25" align-x="30.75" 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:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:real>.</ml:real> </ml:boundVars> <ml:id xml:space="preserve" subscript="h">F</ml:id> </ml:lambda> </ml:apply> </math> <rendering item-idref="50"/> </region> <region region-id="5062" left="96" top="1095" width="156" height="16.5" align-x="105.75" 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:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="51"/> </region> <region region-id="5063" left="18" top="1116.75" width="26.25" height="32.25" align-x="30" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:real>.</ml:real> </ml:boundVars> <ml:id xml:space="preserve" subscript="v">F</ml:id> </ml:lambda> </ml:apply> </math> <rendering item-idref="52"/> </region> <region region-id="5064" left="96" top="1119" width="162.75" height="16.5" align-x="105.75" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="53"/> </region> <region region-id="5065" left="498" top="1125" width="94.5" height="36.75" align-x="529.5" align-y="1146" 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:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="2">N</ml:id> </ml:matrix> <ml:apply> <ml:Find auto-method="true" method="linear" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="2">N</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="54"/> </region> <region region-id="5066" left="6" top="1148.25" width="93.75" height="12" align-x="12.75" align-y="1158" 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"> Del cual despejamos: </text> </region> <region region-id="5067" left="108" top="1149" width="72.75" height="16.5" align-x="125.25" align-y="1158" 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="1">N</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>115.84266819182804</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilonewton"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="5068" left="198" top="1149" width="73.5" height="16.5" align-x="215.25" align-y="1158" 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="2">N</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-29.269281142537903</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilonewton"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="5069" left="6" top="1190.25" width="53.25" height="12" align-x="12.75" align-y="1200" 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"> Por lo tanto: </text> </region> <region region-id="5070" left="72" top="1180.5" width="102" height="35.25" align-x="89.25" align-y="1200" 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="1">σ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="1">A</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">MPa</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>463.37067276731216</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="megapascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="57"/> </region> <region region-id="5071" left="222" top="1180.5" width="102.75" height="35.25" align-x="239.25" align-y="1200" 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="2">σ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:id xml:space="preserve" subscript="2">A</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">MPa</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-58.538562285075805</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="megapascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="5072" left="72" top="1222.5" width="108.75" height="31.5" align-x="88.5" align-y="1242" 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="1">ε</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">σ</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.0022065270131776771</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="5073" left="222" top="1222.5" width="114" height="31.5" align-x="238.5" align-y="1242" 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="2">ε</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="2">σ</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-0.00027875505850036096</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="60"/> </region> <region region-id="5223" left="6" top="1268.25" width="33" height="12" align-x="19.5" align-y="1278" 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"> Siendo: </text> </region> <region region-id="5224" left="54" top="1261.5" width="120.75" height="24" align-x="70.5" align-y="1278" 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="1">L</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>5</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="61"/> </region> <region region-id="5236" left="198" top="1261.5" width="136.5" height="24" align-x="214.5" align-y="1278" 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="2">L</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>4.1231056256176606</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="62"/> </region> <region region-id="5232" left="54" top="1299" width="107.25" height="16.5" align-x="78.75" align-y="1308" 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="1">ΔL</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">ε</ml:id> <ml:id xml:space="preserve" subscript="1">L</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>11.032635065888385</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="63"/> </region> <region region-id="5237" left="198" top="1299" width="108" height="16.5" align-x="222.75" align-y="1308" 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="2">ΔL</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">ε</ml:id> <ml:id xml:space="preserve" subscript="2">L</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-1.1493365498722183</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="64"/> </region> <region region-id="5349" left="294" top="1320" width="187.5" height="161.25" align-x="294" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{720DB9AF-D62C-4ED0-A377-429C22312852}" type="embedded" item-idref="65"/> <rendering item-idref="66"/> </region> <region region-id="5297" left="546" top="1317" width="38.25" height="16.5" align-x="562.5" align-y="1326" 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">a</ml:id> <ml:id xml:space="preserve">mm</ml:id> </ml:define> </math> <rendering item-idref="67"/> </region> <region region-id="5298" left="600" top="1317" width="36.75" height="16.5" align-x="615" align-y="1326" 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="x">a</ml:id> <ml:id xml:space="preserve">mm</ml:id> </ml:define> </math> <rendering item-idref="68"/> </region> <region region-id="5272" left="6" top="1334.25" width="260.25" height="60" align-x="27" align-y="1344" 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"> Finalmente, por Williot, cada barra puede deformarse (acortarse o alargarse) en la dirección de su eje, y luego girar perpendicular a este. Por lo tanto el punto A se mueve al punto A' como se ve en el esquema. Definiendo como positivo hacia abajo y hacia la derecha: </text> </region> <region region-id="5295" left="540" top="1347" width="24" height="12.75" align-x="551.25" align-y="1356" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="69"/> </region> <region region-id="5281" left="42" top="1401" width="120" height="16.5" align-x="65.25" 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:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="1">ΔL</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="x">a</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="y">a</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="70"/> </region> <region region-id="5350" left="42" top="1425" width="125.25" height="16.5" align-x="65.25" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="2">ΔL</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve" subscript="x">a</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="y">a</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="71"/> </region> <region region-id="5296" left="540" top="1419" width="86.25" height="36.75" align-x="569.25" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="x">a</ml:id> <ml:id xml:space="preserve" subscript="y">a</ml:id> </ml:matrix> <ml:apply> <ml:Find auto-method="true" method="linear" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="x">a</ml:id> <ml:id xml:space="preserve" subscript="y">a</ml:id> </ml:sequence> </ml:apply> 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name="User 6" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 7" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="Times New Roman" font-charset="0" font-size="14" font-weight="normal" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="current" 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="true" simplify-units="true" 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="9" orientation="portrait" print-single-page-width="true" page-width="595.5" page-height="841.5"> <margins left="59.4" right="45.28799" top="56.592" bottom="42.48"/> <header use-full-page-width="false"> <left>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fbidi\fprq6\fcharset0 Arial;}{\f1\fnil\fprq15\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\lang11274\f0\fs18 Ejercicio resuelto - Axil (Williot)\lang1033\f1\par } </left> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang11274{\fonttbl{\f0\fbidi\fprq6\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18 Ing. Lopez Rivarola\par } </right> </header> <footer use-full-page-width="false"> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang11274{\fonttbl{\f0\fbidi\fprq6\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18\{n\}/\{nn\}\par }</right> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="1" 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="true" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC12" local-assignment="MC12"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="true"/> <customUnitSystem base-unit-system="si"> <baseUnits> <unit unit-name="mole" quantity-name="amount_of_substance"/> <unit unit-name="ampere" quantity-name="current"/> <unit unit-name="meter" quantity-name="length"/> <unit unit-name="candela" quantity-name="luminous_intensity"/> <unit unit-name="kilogram" quantity-name="mass"/> <unit unit-name="kelvin" quantity-name="thermodynamic_temperature"/> <unit unit-name="second" quantity-name="time"/> </baseUnits> <derivedUnits> <unit unit-name="farad" quantity-name="capacitance"/> <unit unit-name="coulomb" quantity-name="charge"/> <unit unit-name="volt" quantity-name="electric_potential"/> <unit unit-name="lux" quantity-name="illuminosity"/> <unit unit-name="henry" quantity-name="inductance"/> <unit unit-name="weber" quantity-name="magnetic_flux"/> <unit unit-name="tesla" quantity-name="magnetic_flux_density"/> <unit unit-name="watt" quantity-name="power"/> <unit unit-name="pascal" quantity-name="pressure"/> <unit unit-name="ohm" quantity-name="resistance"/> <unit unit-name="liter" quantity-name="volume"/> <unit unit-name="kiloNewton_meter" quantity-name="torque"/> <unit unit-name="kilonewton" quantity-name="force"/> </derivedUnits> </customUnitSystem> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="true" ruler-unit="cm"/> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="5352" 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="4678" left="6" top="8.25" width="477" height="12" align-x="30" 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"> <f size="11"> Enunciado: </f> <f size="11"> <inlineAttr font-weight="normal" underline="false"> <sp count="2"/>Para la siguiente estructura, calcular el desplazamiento del punto A.</inlineAttr> </f> </text> </region> <region region-id="4764" left="18" top="30" width="211.5" height="159.75" align-x="18" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{720DB9AF-D62C-4ED0-A377-429C22312852}" type="embedded" item-idref="1"/> <rendering item-idref="2"/> </region> <region region-id="4765" left="282" top="38.25" width="44.25" height="12" align-x="293.25" align-y="48" 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 size="11"> Datos: </f> </text> </region> <region region-id="4766" left="324" top="39" width="38.25" height="16.5" align-x="342.75" align-y="48" 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="H">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="5173" left="522" top="39" width="33" height="12.75" align-x="531.75" align-y="48" 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">º</ml:id> <ml:id xml:space="preserve">deg</ml:id> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="4767" left="324" top="69" width="37.5" height="16.5" align-x="342" align-y="78" 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="V">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="4768" left="324" top="93" width="52.5" height="18.75" align-x="338.25" align-y="108" 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">A</ml:id> <ml:apply> <ml:mult/> <ml:real>2.5</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="4769" left="324" top="129" width="50.25" height="12.75" align-x="336.75" align-y="138" 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 style="auto-select"/> <ml:real>210</ml:real> <ml:id xml:space="preserve">GPa</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="4770" left="324" top="159" width="40.5" height="12.75" align-x="336" align-y="168" 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 style="auto-select"/> <ml:real>50</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="4774" left="6" top="212.25" width="93" height="12" align-x="17.25" 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"> <f size="11"> Resolución: </f> </text> </region> <region region-id="4856" left="534" top="231" width="55.5" height="16.5" align-x="552.75" align-y="240" 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="1">N</ml:id> <ml:apply> <ml:mult/> <ml:real>-30</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="4857" left="606" top="231" width="55.5" height="16.5" align-x="624.75" align-y="240" 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="2">N</ml:id> <ml:apply> <ml:mult/> <ml:real>-30</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="4776" left="6" top="236.25" width="473.25" height="36" align-x="14.25" align-y="246" 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"> Para poder calcular el desplazamiento del punto P debemos primero calcular las tensiones y deformaciones especificas de las barras. Para poder calcular esto requerimos conocer el valor de los esfuerzos axiles de las barras. </text> </region> <region region-id="4924" left="6" top="290.25" width="33" height="12" align-x="19.5" align-y="300" 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"> Siendo: </text> </region> <region region-id="4994" left="54" top="284.25" width="46.5" height="27.75" align-x="85.5" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="4995" left="120" top="284.25" width="48" height="27.75" align-x="153" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>5</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="12"/> </region> <region region-id="4996" left="186" top="284.25" width="45.75" height="27.75" align-x="216.75" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>5</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="13"/> </region> <region region-id="5001" left="264" top="284.25" width="96" height="27.75" align-x="276.75" align-y="300" 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">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">acos</ml:id> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>5</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.64350110879328426</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="4830" left="6" top="326.25" width="378.75" height="12" align-x="28.5" align-y="336" 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"> Asumiendo los esfuerzos de las barras de tracción, calculamos el equilibrio del nudo A: </text> </region> <region region-id="4931" left="504" top="333" width="24" height="12.75" align-x="515.25" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="15"/> </region> <region region-id="4831" left="18" top="348.75" width="27" height="32.25" align-x="30.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:real>.</ml:real> </ml:boundVars> <ml:id xml:space="preserve" subscript="h">F</ml:id> </ml:lambda> </ml:apply> </math> <rendering item-idref="16"/> </region> <region region-id="4845" left="96" top="351" width="110.25" height="16.5" align-x="105.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="4833" left="18" top="372.75" width="26.25" height="32.25" align-x="30" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:real>.</ml:real> </ml:boundVars> <ml:id xml:space="preserve" subscript="v">F</ml:id> </ml:lambda> </ml:apply> </math> <rendering item-idref="18"/> </region> <region region-id="4846" left="96" top="375" width="132" height="16.5" align-x="105.75" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">P</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="4875" left="498" top="381" width="94.5" height="36.75" align-x="529.5" align-y="402" 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:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="2">N</ml:id> </ml:matrix> <ml:apply> <ml:Find auto-method="true" method="linear" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="2">N</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="4935" left="6" top="404.25" width="93.75" height="12" align-x="12.75" align-y="414" 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"> Del cual despejamos: </text> </region> <region region-id="4936" left="108" top="405" width="69" height="16.5" align-x="125.25" align-y="414" 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="1">N</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-31.249999999886313</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilonewton"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="4937" left="198" top="405" width="69" height="16.5" align-x="215.25" align-y="414" 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="2">N</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-31.249999999886313</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilonewton"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="4943" left="6" top="446.25" width="53.25" height="12" align-x="12.75" align-y="456" 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"> Por lo tanto: </text> </region> <region region-id="4944" left="72" top="436.5" width="91.5" height="31.5" align-x="89.25" align-y="456" 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="1">σ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve">A</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">MPa</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-124.99999999954525</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="megapascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="4976" left="222" top="436.5" width="91.5" height="31.5" align-x="239.25" align-y="456" 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="2">σ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:id xml:space="preserve">A</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">MPa</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-124.99999999954525</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="megapascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="4982" left="72" top="484.5" width="114" height="31.5" align-x="88.5" align-y="504" 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="1">ε</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">σ</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-0.00059523809523592971</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="4983" left="222" top="484.5" width="114" height="31.5" align-x="238.5" align-y="504" 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="2">ε</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="2">σ</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-0.00059523809523592971</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="5007" left="516" top="507" width="36" height="16.5" align-x="532.5" 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:id xml:space="preserve" subscript="1">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="5011" left="576" top="507" width="36" height="16.5" align-x="592.5" 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:id xml:space="preserve" subscript="2">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="5005" left="72" top="537" width="108" height="16.5" align-x="96.75" align-y="546" 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="1">ΔL</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">ε</ml:id> <ml:id xml:space="preserve" subscript="1">L</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-2.9761904761796485</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="5006" left="222" top="537" width="108" height="16.5" align-x="246.75" align-y="546" 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="2">ΔL</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">ε</ml:id> <ml:id xml:space="preserve" subscript="2">L</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-2.9761904761796485</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="5022" left="372" top="536.25" width="33" height="12" align-x="385.5" align-y="546" 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"> Siendo: </text> </region> <region region-id="5023" left="414" top="537" width="55.5" height="16.5" align-x="450.75" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve" subscript="1">L</ml:id> <ml:id xml:space="preserve" subscript="2">L</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="31"/> </region> <region region-id="5041" left="288" top="570" width="197.25" height="166.5" align-x="288" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{720DB9AF-D62C-4ED0-A377-429C22312852}" type="embedded" item-idref="32"/> <rendering item-idref="33"/> </region> <region region-id="5034" left="6" top="584.25" width="260.25" height="72" align-x="27" align-y="594" 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"> Finalmente, por Williot, cada barra puede deformarse (acortarse o alargarse) en la dirección de su eje, y luego girar perpendicular a este. Por lo tanto el punto A se mueve al punto A' como se ve en el esquema. Dado que el sistema es totalmente simétrico, el punto A sufre únicamente un desplazamiento vertical, hacia arriba, de valor: </text> </region> <region region-id="5036" left="54" top="676.5" width="93" height="31.5" align-x="65.25" align-y="696" 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">δ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve" subscript="1">ΔL</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>3.7202380952245608</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="5042" left="6" top="752.25" width="477" height="12" align-x="30" align-y="762" 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 size="11"> Enunciado: </f> <f size="11"> <inlineAttr font-weight="normal" underline="false"> <sp count="2"/>Para la siguiente estructura, calcular el desplazamiento del punto A.</inlineAttr> </f> </text> </region> <region region-id="5155" left="36" top="774" width="163.5" height="175.5" align-x="36" align-y="774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{720DB9AF-D62C-4ED0-A377-429C22312852}" type="embedded" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="5138" left="270" top="782.25" width="44.25" height="12" align-x="281.25" align-y="792" 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 size="11"> Datos: </f> </text> </region> <region region-id="5146" left="318" top="783" width="42.75" height="16.5" align-x="341.25" align-y="792" 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="H1">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="5168" left="396" top="777" width="57" height="22.5" align-x="414.75" align-y="792" 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="1">A</ml:id> <ml:apply> <ml:mult/> <ml:real>2.5</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="5180" left="318" top="819" width="42.75" height="16.5" align-x="341.25" align-y="828" 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="H2">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="5181" left="396" top="813" width="50.25" height="22.5" align-x="414.75" align-y="828" 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="2">A</ml:id> <ml:apply> <ml:mult/> <ml:real>5</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="5182" left="318" top="855" width="37.5" height="16.5" align-x="336" align-y="864" 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="V">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="5183" left="396" top="855" width="50.25" height="12.75" align-x="408.75" align-y="864" 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 style="auto-select"/> <ml:real>210</ml:real> <ml:id xml:space="preserve">GPa</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="5184" left="318" top="891" width="33" height="12.75" align-x="330.75" align-y="900" 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">γ</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>50</ml:real> <ml:id xml:space="preserve">º</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="5185" left="396" top="891" width="45" height="12.75" align-x="408" align-y="900" 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 style="auto-select"/> <ml:real>100</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="5050" left="6" top="956.25" width="93" height="12" align-x="17.25" align-y="966" 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 size="11"> Resolución: </f> </text> </region> <region region-id="5051" left="534" top="975" width="55.5" height="16.5" align-x="552.75" align-y="984" 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="1">N</ml:id> <ml:apply> <ml:mult/> <ml:real>-30</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="5052" left="606" top="975" width="55.5" height="16.5" align-x="624.75" align-y="984" 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="2">N</ml:id> <ml:apply> <ml:mult/> <ml:real>-30</ml:real> <ml:id xml:space="preserve">kN</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="5053" left="6" top="980.25" width="473.25" height="36" align-x="14.25" align-y="990" 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"> Para poder calcular el desplazamiento del punto P debemos primero calcular las tensiones y deformaciones especificas de las barras. Para poder calcular esto requerimos conocer el valor de los esfuerzos axiles de las barras. </text> </region> <region region-id="5054" left="6" top="1034.25" width="33" height="12" align-x="19.5" align-y="1044" 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"> Siendo: </text> </region> <region region-id="5189" left="48" top="1028.25" width="94.5" height="27.75" align-x="60.75" align-y="1044" 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">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">atan</ml:id> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.64350110879328437</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="5207" left="156" top="1028.25" width="94.5" height="27.75" align-x="168.75" align-y="1044" 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">β</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">atan</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.24497866312686414</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="48"/> </region> <region region-id="5059" left="6" top="1070.25" width="378.75" height="12" align-x="28.5" align-y="1080" 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"> Asumiendo los esfuerzos de las barras de tracción, calculamos el equilibrio del nudo A: </text> </region> <region region-id="5060" left="504" top="1077" width="24" height="12.75" align-x="515.25" align-y="1086" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="49"/> </region> <region region-id="5061" left="18" top="1092.75" width="27" height="32.25" align-x="30.75" 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:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:real>.</ml:real> </ml:boundVars> <ml:id xml:space="preserve" subscript="h">F</ml:id> </ml:lambda> </ml:apply> </math> <rendering item-idref="50"/> </region> <region region-id="5062" left="96" top="1095" width="156" height="16.5" align-x="105.75" 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:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="51"/> </region> <region region-id="5063" left="18" top="1116.75" width="26.25" height="32.25" align-x="30" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:real>.</ml:real> </ml:boundVars> <ml:id xml:space="preserve" subscript="v">F</ml:id> </ml:lambda> </ml:apply> </math> <rendering item-idref="52"/> </region> <region region-id="5064" left="96" top="1119" width="162.75" height="16.5" align-x="105.75" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">P</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="53"/> </region> <region region-id="5065" left="498" top="1125" width="94.5" height="36.75" align-x="529.5" align-y="1146" 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:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="2">N</ml:id> </ml:matrix> <ml:apply> <ml:Find auto-method="true" method="linear" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="2">N</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="54"/> </region> <region region-id="5066" left="6" top="1148.25" width="93.75" height="12" align-x="12.75" align-y="1158" 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"> Del cual despejamos: </text> </region> <region region-id="5067" left="108" top="1149" width="72.75" height="16.5" align-x="125.25" align-y="1158" 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="1">N</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>115.84266819182804</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilonewton"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="5068" left="198" top="1149" width="73.5" height="16.5" align-x="215.25" align-y="1158" 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="2">N</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-29.269281142537903</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilonewton"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="5069" left="6" top="1190.25" width="53.25" height="12" align-x="12.75" align-y="1200" 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"> Por lo tanto: </text> </region> <region region-id="5070" left="72" top="1180.5" width="102" height="35.25" align-x="89.25" align-y="1200" 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="1">σ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">N</ml:id> <ml:id xml:space="preserve" subscript="1">A</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">MPa</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>463.37067276731216</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="megapascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="57"/> </region> <region region-id="5071" left="222" top="1180.5" width="102.75" height="35.25" align-x="239.25" align-y="1200" 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="2">σ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="2">N</ml:id> <ml:id xml:space="preserve" subscript="2">A</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">MPa</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-58.538562285075805</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="megapascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="5072" left="72" top="1222.5" width="108.75" height="31.5" align-x="88.5" align-y="1242" 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="1">ε</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">σ</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.0022065270131776771</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="5073" left="222" top="1222.5" width="114" height="31.5" align-x="238.5" align-y="1242" 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="2">ε</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="2">σ</ml:id> <ml:id xml:space="preserve">E</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-0.00027875505850036096</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="60"/> </region> <region region-id="5223" left="6" top="1268.25" width="33" height="12" align-x="19.5" align-y="1278" 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"> Siendo: </text> </region> <region region-id="5224" left="54" top="1261.5" width="120.75" height="24" align-x="70.5" align-y="1278" 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="1">L</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>5</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="61"/> </region> <region region-id="5236" left="198" top="1261.5" width="136.5" height="24" align-x="214.5" align-y="1278" 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="2">L</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>4.1231056256176606</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="62"/> </region> <region region-id="5232" left="54" top="1299" width="107.25" height="16.5" align-x="78.75" align-y="1308" 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="1">ΔL</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">ε</ml:id> <ml:id xml:space="preserve" subscript="1">L</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>11.032635065888385</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="63"/> </region> <region region-id="5237" left="198" top="1299" width="108" height="16.5" align-x="222.75" align-y="1308" 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="2">ΔL</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">ε</ml:id> <ml:id xml:space="preserve" subscript="2">L</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-1.1493365498722183</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millimeter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="64"/> </region> <region region-id="5349" left="294" top="1320" width="187.5" height="161.25" align-x="294" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{720DB9AF-D62C-4ED0-A377-429C22312852}" type="embedded" item-idref="65"/> <rendering item-idref="66"/> </region> <region region-id="5297" left="546" top="1317" width="38.25" height="16.5" align-x="562.5" align-y="1326" 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">a</ml:id> <ml:id xml:space="preserve">mm</ml:id> </ml:define> </math> <rendering item-idref="67"/> </region> <region region-id="5298" left="600" top="1317" width="36.75" height="16.5" align-x="615" align-y="1326" 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="x">a</ml:id> <ml:id xml:space="preserve">mm</ml:id> </ml:define> </math> <rendering item-idref="68"/> </region> <region region-id="5272" left="6" top="1334.25" width="260.25" height="60" align-x="27" align-y="1344" 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"> Finalmente, por Williot, cada barra puede deformarse (acortarse o alargarse) en la dirección de su eje, y luego girar perpendicular a este. Por lo tanto el punto A se mueve al punto A' como se ve en el esquema. Definiendo como positivo hacia abajo y hacia la derecha: </text> </region> <region region-id="5295" left="540" top="1347" width="24" height="12.75" align-x="551.25" align-y="1356" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="69"/> </region> <region region-id="5281" left="42" top="1401" width="120" height="16.5" align-x="65.25" 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:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="1">ΔL</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="x">a</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="y">a</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="70"/> </region> <region region-id="5350" left="42" top="1425" width="125.25" height="16.5" align-x="65.25" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="2">ΔL</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve" subscript="x">a</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="y">a</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="71"/> </region> <region region-id="5296" left="540" top="1419" width="86.25" height="36.75" align-x="569.25" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="x">a</ml:id> <ml:id xml:space="preserve" subscript="y">a</ml:id> </ml:matrix> <ml:apply> <ml:Find auto-method="true" method="linear" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="x">a</ml:id> <ml:id xml:space="preserve" subscript="y">a</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="72"/> </region> <region region-id="5324" left="6" top="1460.25" width="38.25" height="12" align-x="21.75" align-y="1470" 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"> Despejo: </text> </region> <region region-id="5341" left="60" top="1461" width="57.75" height="16.5" align-x="75" align-y="1470" 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="y">a</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">mm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.5591667091202197</ml:real> <unitMonomial 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