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    <name xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Flange Local Buckling</name>
    <metadata xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">
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  <md:created xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">2002/07/29</md:created>
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      <md:author xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="terk">
      <md:firstname xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Michael</md:firstname>
      
      <md:surname xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Terk</md:surname>
      <md:email xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">terk@rice.edu</md:email>
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      <md:firstname xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Joanna</md:firstname>
      
      <md:surname xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Gonzalez</md:surname>
      <md:email xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">joannag@owlnet.rice.edu</md:email>
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      <md:firstname xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Michael</md:firstname>
      
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    <md:maintainer xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="jsilv">
      <md:firstname xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Jeffrey</md:firstname>
      
      <md:surname xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Silverman</md:surname>
      <md:email xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">jsilv@rice.edu</md:email>
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    <md:keyword xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">beam</md:keyword>
    <md:keyword xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">buckling</md:keyword>
    <md:keyword xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">flange</md:keyword>
    <md:keyword xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">limit state</md:keyword>
    <md:keyword xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">local</md:keyword>
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  <md:abstract xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">(Blank Abstract)</md:abstract>
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    <content xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">

      <section xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="intro">
	<name xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Introduction to Flange Local Buckling</name>
	<para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="intro2">
	  The fourth limit state for beams is Flange Local Buckling,
	  or FLB for short.  It is exactly the same as Web Local
	  Buckling, except the width-thickness ratio is in terms of
	  the flange and not the web.  This type of buckling occurs
	  when the width-thickness ratio is not large enough to
	  withstand the moment on the beam.  The way to prevent this
	  type of buckling is to limit the with-thickness ratio.
	</para>

	<para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="intro3">
	  The limits can be computed for flange local buckling.  The
	  width-thickness ratio is compared to
	  <m:math>
	    <m:ci>
	      <m:msub>
		<m:mi>λ</m:mi>
		<m:mi>p</m:mi>
	      </m:msub>
	    </m:ci>
	  </m:math> 
	  and 
	  <m:math>
	    <m:ci>
	      <m:msub>
		<m:mi>λ</m:mi>
		<m:mi>r</m:mi>
	      </m:msub>
	    </m:ci>
	  </m:math>
	  .  Then the maximum moment can be calculated.
	</para>
      </section>

      <section xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns">
	<name xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Equations to determine FLB</name>
	<figure xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns1">
	  <media xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" type="image/bmp" src="compactgraph.bmp"/>
	  <caption xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">The graph illustrates the options for FLB</caption>
	</figure>

	<para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns2">
	  When,
	  <equation xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns3"> 
	    <m:math>
	      <m:apply>
		<m:lt/>
		<m:apply>
		  <m:divide/>
		  <m:ci>b</m:ci>
		  <m:ci>t</m:ci>
		</m:apply>
		<m:ci>
		  <m:msub>
		    <m:mi>λ</m:mi>
		    <m:mi>p</m:mi>
		  </m:msub>
		</m:ci>
	      </m:apply>
	    </m:math>
	  </equation>
	  there is no FLB and the cross section is compact because,
	  <equation xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns4">
	    <m:math>
	      <m:apply>
		<m:eq/>
		<m:ci>
		  <m:msub>
		    <m:mi>M</m:mi>
		    <m:mi>n</m:mi>
		  </m:msub>
		</m:ci>
		<m:ci>
		  <m:msub>
		    <m:mi>M</m:mi>
		    <m:mi>p</m:mi>
		  </m:msub>
		</m:ci>    
	      </m:apply>
	    </m:math>
	  </equation>

	  <equation xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns5">
	    <m:math>
	      <m:apply>
		<m:leq/>
		<m:apply>
		  <m:eq/>
		  <m:ci>
		    <m:msub>
		      <m:mi>M</m:mi>
		      <m:mi>p</m:mi>
		    </m:msub>
		  </m:ci>
		  <m:apply>
		    <m:times/>
		    <m:ci>
		      <m:msub>
			<m:mi>F</m:mi>
			<m:mi>y</m:mi>
		      </m:msub>
		    </m:ci>
		    <m:ci>Z</m:ci>
		  </m:apply>
		</m:apply>
		<m:apply>
		  <m:times/>
		  <m:cn>1.5</m:cn>
		  <m:ci>
		    <m:msub>
		      <m:mi>M</m:mi>
		      <m:mi>y</m:mi>
		    </m:msub>
		  </m:ci>
		</m:apply>
	      </m:apply>
	    </m:math>
	  </equation>

	  from the yielding state.
	</para>
	
	<para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns6">
	  When,
	  <equation xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns7">
	    <m:math>
	      <m:apply>
		<m:lt/>
		<m:ci>
		  <m:msub>
		    <m:mi>λ</m:mi>
		    <m:mi>p</m:mi>
		  </m:msub>
		</m:ci>
		<m:apply>
		  <m:divide/>
		  <m:ci>b</m:ci>
		  <m:ci>t</m:ci>
		</m:apply>
		<m:ci>
		  <m:msub>
		    <m:mi>λ</m:mi>
		    <m:mi>r</m:mi>
		  </m:msub>
		</m:ci>
	      </m:apply>
	    </m:math>
	  </equation>
	  the graph is linear, and therefore a linear interpolation between
	  <m:math>
	    <m:ci>
	      <m:msub>
		<m:mi>M</m:mi>
		<m:mi>p</m:mi>
	      </m:msub>
	    </m:ci>
	  </m:math>
	  and
	  <m:math>
	    <m:ci>
	      <m:msub>
		<m:mi>M</m:mi>
		<m:mi>y</m:mi>
	      </m:msub>
	    </m:ci>
	  </m:math>
	  is used for the maximum moment.
	</para>
	
	<para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="eqns8">
	  And finally, when
	  <m:equation id="eqns9">
	    <m:math>
	      <m:apply>
		<m:gt/>
		<m:apply>
		  <m:divide/>
		  <m:ci>b</m:ci>
		  <m:ci>t</m:ci>
		</m:apply>
		<m:ci>
		  <m:msub>
		    <m:mi>λ</m:mi>
		    <m:mi>r</m:mi>
		  </m:msub>
		</m:ci>
	      </m:apply>
	    </m:math>
	  </m:equation>
	  the graph is non-linear, the flange is non-slender, and
	  there is an equation to find the maximum moment in Appendix
	  F of the Specification section of the
	  <emphasis xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Manual</emphasis> (page 16.1-96).
	</para>
      </section>


    </content>
    
  </document>
