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    xmlns:md="http://cnx.rice.edu/mdml/0.4"
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    id="m10761">
    <name>Flange Local Buckling</name>
    <metadata xmlns:md="http://cnx.rice.edu/mdml/0.4">
  <md:version>2.3</md:version>
  <md:created>2002/07/29</md:created>
  <md:revised>2003/07/25 10:04:34 GMT-5</md:revised>
  <md:authorlist>
      <md:author id="terk">
      <md:firstname>Michael</md:firstname>
      
      <md:surname>Terk</md:surname>
      <md:email>terk@rice.edu</md:email>
    </md:author>
  </md:authorlist>

  <md:maintainerlist>
    <md:maintainer id="Joanna">
      <md:firstname>Joanna</md:firstname>
      
      <md:surname>Gonzalez</md:surname>
      <md:email>joannag@owlnet.rice.edu</md:email>
    </md:maintainer>
    <md:maintainer id="terk">
      <md:firstname>Michael</md:firstname>
      
      <md:surname>Terk</md:surname>
      <md:email>terk@rice.edu</md:email>
    </md:maintainer>
    <md:maintainer id="jsilv">
      <md:firstname>Jeffrey</md:firstname>
      
      <md:surname>Silverman</md:surname>
      <md:email>jsilv@rice.edu</md:email>
    </md:maintainer>
  </md:maintainerlist>
  
  <md:keywordlist>
    <md:keyword>beam</md:keyword>
    <md:keyword>buckling</md:keyword>
    <md:keyword>flange</md:keyword>
    <md:keyword>limit state</md:keyword>
    <md:keyword>local</md:keyword>
  </md:keywordlist>

  <md:abstract>(Blank Abstract)</md:abstract>
</metadata>

    <content>

      <section id='intro'>
	<name>Introduction to Flange Local Buckling</name>
	<para 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 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>&lambda;</m:mi>
		<m:mi>p</m:mi>
	      </m:msub>
	    </m:ci>
	  </m:math> 
	  and 
	  <m:math>
	    <m:ci>
	      <m:msub>
		<m:mi>&lambda;</m:mi>
		<m:mi>r</m:mi>
	      </m:msub>
	    </m:ci>
	  </m:math>
	  .  Then the maximum moment can be calculated.
	</para>
      </section>

      <section id='eqns'>
	<name>Equations to determine FLB</name>
	<figure id='eqns1'>
	  <media type="image/bmp" src="compactgraph.bmp"/>
	  <caption>The graph illustrates the options for FLB</caption>
	</figure>

	<para id='eqns2'>
	  When,
	  <equation 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>&lambda;</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 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 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 id='eqns6'>
	  When,
	  <equation id='eqns7'>
	    <m:math>
	      <m:apply>
		<m:lt/>
		<m:ci>
		  <m:msub>
		    <m:mi>&lambda;</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>&lambda;</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 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>&lambda;</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>Manual</emphasis> (page 16.1-96).
	</para>
      </section>


    </content>
    
  </document>

