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<a name = "hj-top"> </a><table class = "table1" id = "table11"><tr><td><table class = "DocHeader"><tr><td class = "DocHeader1" colspan = "2"><h1>Effects of Geometric Nonlinearities</h1></td></tr><tr><td class = "DocHeader4" colspan = "2"/></tr><tr><td class = "DocHeader3" colspan = "2"><table class = "DocThemeIntro" id = "table12"><tr><td class = "Intro1Only"><p class = "header"><p class = "abstract">
<span class = "shortdesc"> This section describes the effects of geometric nonlinearities that are taken into
      account during a structural analysis when using NLGEOM=ON. The effects are shown using
      examples that are in the <span class = "ph">Tosca Structure</span> Examples Guide. </span>

</p>
<p>This page discusses: </p><ul><li><a href = "#tso-c-usr-sizing-NonLin-Effects-Geom__" id = "toc_rg" title = "">Nonlinear Geometry with a Bending Plate</a></li><li><a href = "#tso-c-usr-sizing-NonLin-Effects-Geom__" id = "toc_rg" title = "">See Also</a></li></ul>
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<div class = "section"><h2 class = "title sectiontitle">Nonlinear Geometry with a Bending Plate</h2>  <br/><img class = "image" src = "../TsoExampleImages/sizingBendPlateModel.png" width = "378"/><br/> A horizontal plate has one edge encastered and the opposite
      edge subjected to a force <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">F</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">z</mi>
          </mrow>
        </msub>
      </math>
      </span> in negative Z-direction as well as a force <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">F</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">x</mi>
          </mrow>
        </msub>
      </math>
      </span> in positive X-direction where <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">F</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">Z</mi>
          </mrow>
        </msub>
      </math>
      </span> &gt;&gt; <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">F</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">x</mi>
          </mrow>
        </msub>
      </math>
      </span>. When taking only linear kinematics into account, the displacement of the
      loaded edge is only in the negative Z-direction because of the load in Z and superposed with
      the displacement in X causes by the X-load. Thus, the membrane load in X only contributes to
      the total displacement <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">u</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">x</mi>
          </mrow>
        </msub>
      </math>
      </span> in X: <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">u</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">x</mi>
          </mrow>
        </msub>
      </math>
      </span> = <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">u</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">xM</mi>
          </mrow>
        </msub>
      </math>
      </span>. The plate elongates, which the graphic below shows (right side): <br/><img class = "image" src = "../TsoUserImages/NonLinSizing_EffectNLGEOM.jpg" width = "756"/><br/>. By taking geometric nonlinearities into account
      (NLGEOM=YES), there is an additional displacement <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">u</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">xB</mi>
          </mrow>
        </msub>
      </math>
      </span> in the negative X-direction due to the bending movement of the
      plate. Thus, the total displacement of the right edge is the sum of displacement because of
      the loads in X and Z, respectively. In this example, the bending load in Z is larger than the
      tensile force in X. This leads to a high bending displacement and the total displacement <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">u</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">x</mi>
          </mrow>
        </msub>
      </math>
      </span> = <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">u</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">xB</mi>
          </mrow>
        </msub>
      </math>
      </span> + <span class = "ph inlineequation">
      <math width = "20" height = "17" class = "- topic/foreign ">
        <msub class = "- topic/foreign ">
          <mrow class = "- topic/foreign ">
            <mi mathvariant = "normal" class = "- topic/foreign ">u</mi>
          </mrow>
          <mrow class = "- topic/foreign ">
            <mi class = "- topic/foreign ">xM</mi>
          </mrow>
        </msub>
      </math>
      </span> is negative, which can be seen in the graphic above (left side). </div>

<div class = "section"><p><map name = "FPMap1"><area href = "#hj-top" title = "Back to Top" shape = "rect" coords = "416, 0, 435, 10"/></map><span class = "itemsprite"/></p><h2 class = "title sectiontitle">See Also</h2>
 
  The optimization for this example and further
explanations can be found in the <span class = "ph">Tosca Structure</span> Examples Guide (see
<a class = "xref" href = "../TsoExampleMap/tso-m-example-sizing-NonLinExamples-Nlgeom-sb.htm#tso-m-example-sizing-NonLinExamples-Nlgeom-sb" title = "Some applications with nonlinear geometry are described within this section.">Examples with Geometric Nonlinearities</a>).
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