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Column analogy (Hardy Cross) - Carry Over Factor and stiffness

  1. Dec 30, 2015 #1
    Hi all,
    1. The problem statement, all variables and given/known data

    Book example:
    Determine using the column analogy method, the carry over factor from A to B and the stiffness at A for a propped cantilever.
    (Propped end is defined as A, and fixed end is defined as B)

    2. Relevant principles
    1. Moment at any point
    [tex] M = M_{s} - M_{i} [/tex]
    where Ms = Moment in basic determinate structure and Mi = End moment

    From analogous column stress derivations;

    [tex] M_{i} = \frac{\text{P}}{\text{A}} + \frac{{M_{y}}}{{I_{yy}}}x + \frac{{M_{x}}}{{I_{xx}}}y [/tex]

    where : 1st term represents the axial stress in column,
    2nd term bending stress about y-axis
    3rd term bending stress about x-axis

    Note: x-axis passes through the centroid and is parallel to the span L of the beam. y-axis is perpendicular to x-axis.

    3. The attempt at a solution

    1. Determine the M/EI diagram for the basic determinate structure. The basic structure is arbitrarily chosen as a cantilever with a moment MA at end A.

    2. The M/EI diagram represents the load on the analogous column.

    3. Construct an analogous column of width (1/EI ) and length = span of beam.

    4. Ms is obtained from the moment diagram of basic structure.

    5. Determine the eccentricity of the center of area of M/EI diagram with respect to the centroid of the analogous column. For a beam eccentricity is about the y-axis only. From this eccentricity calculate My and hence Mi.

    6. Use 1st equation to compute end moment...

    4. My problem

    I cannot actually figure the load distribution on the analogous column. To me the moment diagram for the cantilever is a constant moment throughout. So the load on the column should be a U.D.L .

    However the book puts a point load of rotation at A, [tex] \theta _{a} [/tex] at end A and performs the computations. Can anyone explain this part to me? I'm clueless as to why this was done.

    Thank you! And early Happy new year!
     
  2. jcsd
  3. Jan 4, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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