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

[kirakun]

Hi all,
1. Homework Statement
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 M_s = Moment in basic determinate structure and M_i = 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.

The Attempt at a Solution

[/B]
1. Determine the M/EI diagram for the basic determinate structure. The basic structure is arbitrarily chosen as a cantilever with a moment M_A 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. M_s 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 M_y and hence M_i.

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!

 

[KataKoniK]

Hello! I can help explain the reasoning behind using a point load of rotation at end A instead of a U.D.L. on the analogous column.

Firstly, let's consider the basic determinate structure (the cantilever beam) and its moment diagram. As you correctly stated, the moment diagram for a cantilever beam is a constant moment throughout. However, this is only true for the internal forces within the beam itself. The external forces acting on the beam, such as the support at end B, will create a different moment diagram on the beam as a whole.

Now, let's think about the analogous column. We are trying to use this column to represent the behavior of the cantilever beam, which means it should have a similar moment diagram. If we were to apply a U.D.L. on the column, we would essentially be applying a distributed load throughout the entire length of the column. This would not accurately represent the behavior of the cantilever beam, as the external forces acting on the beam are not distributed evenly throughout its length.

Instead, by applying a point load at end A, we are simulating the rotation at that point caused by the external forces acting on the beam. This allows us to more accurately represent the behavior of the cantilever beam and obtain more accurate results for the carry over factor and stiffness at end A.

I hope this helps clarify the reasoning behind using a point load of rotation at end A. Happy new year to you as well!