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<?xml version ="1.0" encoding ="UTF-8"?> <!DOCTYPE html PUBLIC "-//WC3//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtm11/DTD/xhtm11-transitional.dtd"> <html xmlns = "http://www.w3.org/1999/xhtml" xml:lang ="en" lang = "en">    <head> <title>Continuous & Built In Beams</title> <meta name = "keywords" CONTENT = "Continous beams, mult span beams"> <meta name = "Description" CONTENT = "Notes on continuous beams" > <link rel=StyleSheet href="../../table.css" type="text/css"> <style type = "text/css"> td.a {background :#FEB8F7} td.b {background :#80FF80} td.c {background :#00FFFF} p.a {margin-left :5%} </style> </head> <body > <table align = "center" border = "1" cellspacing ="0" cellpadding = "2" width = "85%" bordercolor = "#000000" > <tr><td> Disclaimer: The information on this page has not been checked by an independent person.   Use this information at your own risk. </td> </tr> </table> <div align = "center" >ROYMECH</div>  <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  <a href="../../index3.htm">Home</a> <a href="Beam_Index.html#Links">Beams Index</a> <h2>Continuous Beams</h2> <table border = "1" bordercolor ="#000000" width="95%" cellspacing="0" align = "center" > <tr> <td class = "wide" > Introduction Built -in (Encastre) beams are fixed at both ends.  Continuous beam which are beams with more than two supports and covering more than one span.  These beams are not statically determinate using the static equilibrium laws.   Nomenclature e = strain σ = stress (N/m2) E = Young's Modulus = σ /e (N/m2) y = distance of surface from neutral surface (m). R = Radius of neutral axis (m). I = Moment of Inertia (m4 - more normally cm4) Z = section modulus = I/ymax(m3 - more normally cm3) M = Moment (Nm) w = Distrubuted load on beam (kg/m) or (N/m as force units) W = total load on beam (kg ) or (N as force units) F= Concentrated force on beam (N) L = length of beam (m) x = distance along beam (m) Built in beams A built in beam is normally considered to be horizontal with both ends built-in at the same level and with zero slope at both ends.   A loaded built in beam has a moment at both ends and normally the maximum moments at at one or both of the two end joints.   A built in beam is generally much stronger than a simply supported beam of the same geometry.  The bending moment reduces along the beam and changes sign at points of contraflexure between the supports and the load.  A typical built-in beam is shown below. <img src="../../images11/beam_59.gif" width="300" height="218" alt="" border="0"> It is not normally possible to determine the bending moments and the resulting stress using static equilibrium and deflection calculations are often used to enable the moments to be determined. Using the above beam as an example... <img src="../../images11/beam_60.gif" width="552" height="271" alt="" border="0"> Using the above equations the bending moment, shear force, deflection, slope can be determined at any point along the beam. <big> M = EI d 2y/dx 2 = w(- 6x2+6lx -l2)/12 at x = 0 & l then M = -wl2 /12 and at x = l/2 then M = wl2 /24 S = EI d 3y/dx 3 = w(l/2 - x) at x = 0 then S = w.l/2 at x = l then S = -w.l/2</big> <hr align="center" width="70%" noshade color="#FF6600"> Continuous Beams This type of beam is normally considered using the Clapeyron's Theorem ( Three Moments theorem) <img src="../../images11/beam_26.gif" width="445" height="196" alt="" border="0"> The three moments theorem identifies the relationship between the bending moments found at three consecutive supports in a continuous beam.  This is achieved by evaluating the slope of of the beam at the end where the two spans join.  The slopes are expressed in terms of the three moments and the supported loads which are then equated and the resulting equations solved. This relationship for spans with supports at the same height and with spans of constant section results in the following expression. <big>MA.L1 + 2.MB(L1 + L2)+ MC = 6(A1 . x1 /L1   +  A1 . x2 /L2 )</big> If the beams has a different section for each span then the more general expression applies as shown below <big>MA.L1/I1 + 2.MB(L1/I1 + L2/I2 ) + MC = 6 [A1 . x1 /(L1 .I1 )  +  A1 . x2 / (L2 .I2 ) ]</big> Example Areas and x1 value calculations <table align = "center" width = "70%" border = "1" cellspacing = "0" > <tr><td> <img src="../../images11/beam_27.gif" width="327" height="126" alt="" border="0"> </td></tr> <tr><td> <img src="../../images11/beam_28.gif" width="280" height="125" alt="" border="0"> </td></tr> </table> Example using theorem. This simple example is a two span continuous beam with the ends simple supported,therefore with no moments at the support points.. <img src="../../images11/beam_29.gif" width="337" height="70" alt="" border="0"> <img src="../../images11/beam_30.gif" width="266" height="68" alt="" border="0"> </td> </tr> </table> Sites Providing Relevant Information <ol> <li><a href="http://www.duke.edu/~hpgavin/ce130/three-mom.pdf">The Moment equation for continous beam analysis</a>...Very advanced paper download</li> <li> <a href="http://www.Mitcalc.com">Mitcalc</a>...Excel based software including coded beam calculations</li> <li><a href="http://www.grantadesign.com/resources/shapes/solutions/index.htm">Granta -Solutions to Standard Problems</a>...Very accessible notes for beams and sections</li> <li><a href="http://www.engineerstoolbox.com/index.html">ETBX Engineers Toolbox</a>...A number of ver useful mechanical analysis tools with useful background notes ..Needs Java </li> </ol> <h2 align = "center">This page is being developed</h2> <a href="../../index3.htm">Home</a> <a href="Beam_Index.html#Links">Beams Index</a> Please Send Comments to <a href="mailto:Roy@roymech.co.uk"> Roy@roymech.co.uk</a> Last Updated 29/11/2005 </body> </html>