ECIV 320 Structural Analysis I by vbf10787

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									ECIV 320 Structural Analysis I




        Deflections
             Last Time
Deflection Diagrams & Elastic Curve

                  ASSUMPTION
          Linear Elastic Material Response
A structure subjected to a load will return to its original
       undeformed position after load is removed

                  OBJECTIVE
       Calculate Slope and Deflection due to
         Bending at any point on a Beam

                   METHODS
          • Double Integration Method
          • Moment Area Theorems
          • Conjugate Beam
Last Time - Double Integration Method


     Relate Moments to Deflections

   d 2u M
      2
        
   dx     EI                         Integration
           du    M x 
   ( x)               dx          Constants
           dx    EI ( x )
                                   Use Boundary
                 M x  2      Conditions to Evaluate
   u( x )              dx   Integration Constants
                 EI ( x )
    Conjugate Beam Method


Developed by H. Muller Breslau - 1865


 Use to find slopes and deflection due to
            bending of beams



         Method is based on
      Principles of Statics Only
            Conjugate Beam


Internal Loadings     Beam Theory

 dV                      d M
     w                   
 dx                      dx EI


d 2 M dV               d 2u d M
    2
           w           2
                               
 dx     dx             dx     dx EI
                 Conjugate Beam


  Internal Loadings            Beam Theory


   V    wdx                       M     
                                        dx
                                      EI   


             
M     wdx dx
                                      M  
                               u     dxdx
                                       EI  

                          M
             Let      w
                          EI
                       Conjugate Beam


   A “fictitious” beam of the same length as the real beam
        loaded with the real beam’s M/EI diagram…


Real Beam = VConjugate Beam



uReal Beam = MConjugate Beam


                                However...
             Conjugate Beam Supports


Shear and Moment at Supports of Conjugate Beam should
account for the corresponding slope and deflection of real
                   beam at its supports
             Conjugate Beam Supports


Shear and Moment at Supports of Conjugate Beam should
account for the corresponding slope and deflection of real
                   beam at its supports
             Conjugate Beam Supports


Shear and Moment at Supports of Conjugate Beam should
account for the corresponding slope and deflection of real
                   beam at its supports
Examples of Conjugate Beam Supports
Examples of Conjugate Beam Supports
Examples of Conjugate Beam Supports

								
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