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

1. Dec 30, 2015

### kirakun

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
$$M = M_{s} - M_{i}$$
where Ms = Moment in basic determinate structure and Mi = End moment

From analogous column stress derivations;

$$M_{i} = \frac{\text{P}}{\text{A}} + \frac{{M_{y}}}{{I_{yy}}}x + \frac{{M_{x}}}{{I_{xx}}}y$$

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, $$\theta _{a}$$ 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. Jan 4, 2016