AERSP 301 Plate Theory

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					          AERSP 301
            BUCKLING
EULER & COLUMN/LOAD IMPERFECTIONS


           Jose Palacios
             August 2008
                             Today

•   BUCKLING
    – EULER BUCKLING
    – COLUMN IMPERFECTIONS AND LOAD MISALIGNMENT
    – ENERGY METHODS AND APPROXIMATE SOLUTIONS

    FINAL:


    Thursday, August 14 from 10:00 am – 12 noon @ RCOE

    Tentative Schedule:

    M – Beam Buckling
    T – Plate Theory
    W – Hw # 7 Review
    R – Intro to Vibration
    F – Final Exam Review
                STRUCTURAL INSTABILITY

•   STRUCTURAL MEMBERS IN COMPRESSION ARE SUSCEPTIBLE TO
    FAILURE BY BUCKLING WHEN THE COMPRESSIVE LOAD EXCEEDS A
    CRITICAL LOAD (BUCKLING LOAD)
    – THERE ARE MULTIPLE TYPES OF BUCKING


EULER BUCKLING OF COLUMNS
• FOR SMALL, ELASTIC DEFLECTIONS OF PERFECT, SLENDER
  COLUMNS

•   VARIETY OF BOUNDARY CONDITIONS

•   PHYSICALLY – IF YOU APPLY A COMPRESSIVE LOAD TO A COLUMN,
    AT SOME VALUE OF LOAD IT WILL SUDDENLY BOW (OR BUCKLE)
             STRUCTURAL INSTABILITY (EULER)

•   IN THEORY – FOR A PERFECT COLUMN LOADED PERFECTLY ALONG
    THE CENTROIDAL AXIS:
    – THERE WILL ONLY BE A SHORTENING, NO BOWING (BUCKLING).
    – BUT WHAT HAPPENS IF A SMALL LATERAL LOAD IS APPLIED?

    – DEPENDS ON THE LEVEL OF THE COMPRESSIVE LOAD…

    – FOR:   P  Pcr P  Pcr             P  Pcr
        • ADDITION OF LATERAL LOAD RESULTS IN DIFFERENT BEHAVOIR

        • EULER BUCKLING – BEFORE AND AT CRITICAL LOAD, COLUMN IS RELATIVELY
          UNDEFORMED

        • WHEN BUCKLING LOAD IS SURPASSED, SUDDEN, LARGE, DEFORMATION
          OCCURS
         STRUCTURAL INSTABILTY (EULER)


• DETERMINATION OF BUCKLING LOAD FOR A PINNED-PINNED
  COLUMN:

              z
                          w
                                               Pcr
                  x




• AT THE CRITICAL LOAD, Pcr, ANY ADDITIONAL LOAD WILL
  BUCKLE THE COLUMN AS SHOWN
           STRUCTURAL INSTABILITY (EULER)

•   FROM BUCKLED SHAPE  BENDING MOMENT AT ANY X LOCATION
    (show this)
                      w    2
             M   EI 2  Pcr w
                     x
                   w2
               EI 2  Pcr w  0
                  x
                     w Pcr
                     2
               or            w0
                    x 2
                           EI
           STRUCTURAL INSTABILITY (EULER)

              Pcr
•   SET    
            2

              EI                 Eigenvalue Problem

                             w  2 w  0
•   SOLUTION TO THIS HOMOGENEOUS ODE IS OF THE FORM:


          w  A sin x   B cosx 
    – w – LATERAL DISPLACEMENT
    – A, B – CONSTANTS
           STRUCTURAL INSTABILITY (EULER)

              w  A sin x   B cosx 
•   USE BOUNDARY CONDITIONS TO DETERMINE CONSTANTS A & B:

            @ x  0, w  0  B  0

                    w  A sin x 

       @ x  L, w  0  A sin L   0
          STRUCTURAL INSTABILITY (EULER)

                       A sin L   0

• POSSIBLE SOLUTIONS:
  – A = 0 TRIVIAL SOLUTION
  – OR SIN(λL) = 0: Non-Trivial Solution

           sin L   0  L  n
                    n
                       n  1, 2, 3...
                     L
  – λ: EIGENVALUES (ALL POSSIBLE SOLUTIONS TO ODE)
             STRUCTURAL INSTABILITY (EULER)

    sin L   0  L  n             Pcr
             n                      
                                     2
                n  1, 2, 3...       EI
              L
•    THEN:


                      n 
                               2
                              Pcr
                          
                      L     EI
                           n
                            2 2
                     Pcr  2 EI
                            L
            STRUCTURAL INSTABILITY (EULER)
                                 Is called the buckling
•   NOW:                          mode shape

               n                   n         
    w  A sin              x   sin            x
               L                    L          

                    2 EI
n 1       Pcr 
                  L2
                 4 2 EI
n2        Pcr 
                   L2
                 9 2 EI
n3        Pcr 
                   L2
           STRUCTURAL INSTABILITY (EULER)

•   IN REALITY, BUCKLING OCCURS AT THE LOWEST VALUE


                                 2 EI
               n 1     Pcr 
                                  L2
•   HIGHER MODES WILL BE OBSERVED ONLY IF THERE ARE RESTRAINTS
    AT NODES OF THOSE MODES


                                                      4 2 EI
                                         n2    Pcr 
                                                        L2
         STRUCTURAL INSTABILITY (EULER)


• LATERAL RESTRAINT AT MID-POINT SUPPRESSES THE 1ST
  MODE AND CRITICAL BUCKLING LOAD

• LATERAL RESTRAINTS AT L/3 AND 2L/3 SUPPRESSES THE 1ST
  AND 2ND MODES AND CRITICAL BUCKLING LOAD IS
  INCREASED TO


                              9 EI
                                  2
              n3       Pcr     2
                                L
        STRUCTURAL INSTABILITY (EULER)


• DETERMINATION OF BUCKLING LOAD FOR A CLAMPED-FREE
  COLUMN




               What will the moment be?
         STRUCTURAL INSTABILITY (EULER)

• BENDING MOMENT AT X (show this):

  M   Pcr   w
  where is the tip lateral displacement
• EQUILIBRIUM EQUATION:

             w 2
          EI 2   M  Pcr   w
            x
                 w
                 2
          or EI 2  Pcr w  Pcr
                x
           STRUCTURAL INSTABILITY (EULER)
      2
    d w Pcr          Pcr
               w                               Pcr
    dx  2
             EI      EI                        
                                                2

       2 w  2                              EI
    w
•   NON-HOMOGENOUS ODE SOLUTION (2 PARTS):
    – COMPLIMENTARY SOLUTION (SOLUTION TO HOMOGENOUS PART):



                      w   w  0
                               2



           wh  A sin x   B cosx 
            STRUCTURAL INSTABILITY (EULER)

     – PARTICULAR SOLUTION:   wp  
     – FULL SOLUTION:



    w  wh  w p  A sin x   B cosx   

•   APPLY BOUNDARY CONDITIONS:


    w  A cosx   B sin x 

     @x 0              w0      w  0
            STRUCTURAL INSTABILITY (EULER)

                @x 0         w0      w  0
                 w  B    0  B  
                 w  A  0  A  0
•   ALSO, w(L) = δ


                          1 cos L 
     – THIS IMPLIES:


                          cosL  0
               STRUCTURAL INSTABILITY (EULER)

•    SO,                     cosL  0
     0
    cosL   0
                             
    L  2n  1
                              2           2n  1 
                                  n L          
                                          2 
                                           Pcr 2  2n  1  2
                                                         2

           For n = 1, 2, 3, 4,…   n L   L  
                                        2
                                                           
                                           EI     2 
           STRUCTURAL INSTABILITY (EULER)



           2n  1  2 EI
                     2

    Pcr           2             For n = 1, 2, 3, 4,…

           2         L

•   BUCKLING LOAD – LOWEST VALUE FOR CLAMPED-FREE BEAM:



                        EI
                Pcr     2
                              2

                        4L
           STRUCTURAL INSTABILITY (EULER)

•   SIMILARLY, IT CAN BE SHOWN THAT FOR A

                                     EI
    Clamped-Clamped Beam:   Pcr  4 2
                                     2

                                     L
                                            EI
    Clamped-Pinned Beam:    Pcr  2.046   22

                                            L
•   FROM THE ABOVE RESULTS, WE CAN WRITE:


                             • FOR ANY COLUMN, WHERE THE
               EI
       Pcr   22            EQUIVALENT LENGTH, Le, DEPENDS ON
                             THE BOUNDARY CONDITIONS
               Le
           STRUCTURAL INSTABILITY (EULER)

•   Le DEPENDS ON BOUNDARY CONDITIONS:



    For a pinned-pinned: Le = L
    For a clamped-clamped: Le = L/2
    For a clamped-free: Le = 2L
    For a clamped-pinned: Le = 0.7L

              EI
      Pcr   2 2

              Le
           STRUCTURAL INSTABILITY (EULER)

•   WE COULD ALSO WRITE:


             EI
    Pcr  C 2 2                  C: COEFFICIENT OF
                                  CONSTRAINT OR END
             L                    FIXITY FACTOR




    For a pinned-pinned: C = 1
    For a clamped-clamped: C = 4
    For a clamped-free: C = 0.25
    For a clamped-pinned: C = 2.046
          STRUCTURAL INSTABILITY (IMPERFECTIONS)


COLUMN IMPERFECTIONS & LOAD MISALIGNMENT




•   FORCE IS P, NOT Pcr

•   UNLIKE PERFECTLY STRAIGHT COLUMN (WHERE BENDING OCCURS
    ONLY AFTER Pcr), WITH IMPERFECTIONS BENDING OCCURS
    IMMEDIATLEY ON APPLICATION OF COMPRESSIVE FORCE (DUE TO
    ITS OFFSET FROM THE SLIGHTLY CURVED CENTER LINE).
         STRUCTURAL INSTABILITY (IMPERFECTIONS)

•   BENDING MOMENT ALONG COLUMN:

                              2
                       d w
               M   EI 2  Pwtot
                       dx
                          2
    wtot  w0  w  2 
                       d w P
                                  w0  w  0
                       dx     EI
                                 P
               setting    2

                                EI
                 2
               d w 2
                   2
                       w   w0
                                2

               dx
           STRUCTURAL INSTABILITY (IMPERFECTIONS)

•    INITIAL SHAPE OF THE COLUMN IS A SINE FUNCTION:

                                  (aoL IS THE AMPLITUDE. a0 IS THE

                       x       DIMENSIONLESS IMPERFECTION
        w0  a0 L sin           AMPLITUDE –VERY SMALL

                      L
                                  NUMBER)


                                x 
       w   w   a0 L sin  
               2         2

                               L
•    SOLUTION TO THIS NON-HOMOGENEOUS ODE:

                                    L
                                     2 2
                                                   x 
    w  A sin x   B cosx   2      a L sin  
                                    L
                                       2 2 0
                                                   L
            Homogenous Solution
                                           Particular Solution
          STRUCTURAL INSTABILITY (IMPERFECTIONS)


•   APPLY BOUNDARY CONDITIONS TO DETERMINE A & B:
    @x 0 w 0  B  0

                       2 L2           x 
    w  A sin x   2       a L sin  
                       L2 2 0
                                       L

    @ x  L w  0  A sin L   0  A  0

         2 L2           x 
     w 2       a L sin  
         L2 2 0
                         L
             STRUCTURAL INSTABILITY (IMPERFECTIONS)


•   SINCE:


             wtot  w  w0
                            x 
             w0  ao L sin  
                           L
                     2 L2                  x 
                       2 L2  1a0 L sin  L 
             wtot   2            
                                            
          STRUCTURAL INSTABILITY (IMPERFECTIONS)


•   @ X = L/2, LATERAL DEFLECTION TAKES ITS MAX. VALUE
    (CALL IT )

                   2
                            a0 L
            L
            2           2 2

                                                  2
•   USING
            
            2  Pcr                                   2
                                                           a0 L
                                              PL
               EI                          
                                           2

                                              EI
     EI            2
          aL
   EI  PL
      2    2 0
           STRUCTURAL INSTABILITY (IMPERFECTIONS)


•    RECALL, FOR PERFECT COLUMN, EULER’S CRITICAL BUCKLING LOAD
     WAS:

                      EI
                       2
             Pcr          2
                                EI  Pcr L
                                     2          2

                       L
                                 2
                       Pcr L
                            aL
                   Pcr L  PL
                        2    2 0


                        δ                   1
    Defining :        a ;           a             a0
                        L               1  P / Pcr
           STRUCTURAL INSTABILITY (IMPERFECTIONS)

•    a  NON-DIMENSIONAL MID-PT. DISP

             1                          a      1
      a             a0                             a0
         1  P / Pcr                    a0 1  P/Pcr
       P        a
or          1                  MAGNITUDE OF INITIAL IMPERFECTION,
                                 ao, AFFECTS THE AMPLITUDE OF
       Pcr      a0               DEFLECTION, BUT NOT THE LIMITING
                                 (BUCKLING) LOAD


                                 IF ao = 0, BUCKLES LIKE EULER
                                 COLUMN (NO BENDING UNTIL LOAD
                                 PASSES Pcr)
       STRUCTURAL INSTABILITY (IMPERFECTIONS)


• PREVIOUS COLUMN, LOADED PERFECTLY BUT
  GEOMETRICALLY IMPERFECT

• NOW COLUMN IS GEOMETRICALLY PERFECT, BUT
  COMPRESSIVE LOAD P IS NOT ALIGNED WITH CENTROIDAL
  AXIS (LOAD IMPERFECTION, OFFSET BY ECCENTRICITY, e)
       STRUCTURAL INSTABILITY (IMPERFECTIONS)



• BENDING MOMENT ON COLUMN:




• DETERMINE SOLUTION TO NON-HOMOGENEOUS ODE
             STRUCTURAL INSTABILITY
                (IMPERFECTIONS)


• ODE SOLUTION:



• @ X = 0,
             STRUCTURAL INSTABILITY
                (IMPERFECTIONS)


• @ X = L,
          STRUCTURAL INSTABILITY
             (IMPERFECTIONS)




• MAX LATERAL DEFLECTION, , AT THE MID-PT. (X=L/2)
         STRUCTURAL INSTABILITY
            (IMPERFECTIONS)



• a, ae,  DIMENSIONLESS MID-SPAN DEFLECTION AND
  ECCENTRICITY
        STRUCTURAL INSTABILITY
           (IMPERFECTIONS)




• SOLVING FOR
         STRUCTURAL INSTABILITY
            (IMPERFECTIONS)




• FIGURE SHOWS THAT EVEN IF THERE IS A SMALL LOAD
  ECCENTRICITY, THE LOAD CAPACITY OF THE COLUMN IS
  DECREASED.

				
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