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<document xmlns="http://cnx.rice.edu/cnxml" 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="m10758">
    <name>Web Local Buckling</name>
    <metadata>
  <md:version>2.1</md:version>
  <md:created>2002/07/26</md:created>
  <md:revised>2003/01/13</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:maintainerlist>
  
  <md:keywordlist>
    <md:keyword>beam</md:keyword>
    <md:keyword>buckling</md:keyword>
    <md:keyword>local</md:keyword>
    <md:keyword>web</md:keyword>
  </md:keywordlist>

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

    <content>
      <section id="intro">
	<name>Introduction to Web Local Buckling</name>
	<para id="intro2">
	  The third limit state for beams is Web Local Buckling, or
	  WLB for short.  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 web 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 id="eqns">
	<name>Equations to determine WLB</name>
	<figure id="eqns1">
	  <media type="image/bmp" src="compactgraph.bmp"/>
	  <caption>The graph illustrates the options for WLB</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>λ</m:mi>
		    <m:mi>p</m:mi>
		  </m:msub>
		</m:ci>
	      </m:apply>
	    </m:math>
	  </equation>
	  there is no WLB 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>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 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 web 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>
