Moment capacity calculation - concrete column 2

[Pretty Girl]

This was taken from the book "Reinforced concrete design to eurocodes - by prab bhatt" at page 353. I'm trying to understand this equation. And trying to calculate moment capacity for a compression area of a uni-axial concrete column.

In that equation, the moment capacity should be "Cc x lever arm". Lever arm is mentioned as "0.5 x (h-lambda x". Isn't that lever arm should be the centroid from the compression block of concrete to the centroid of the tensile reinforcement? but that "0.5 x (h-lambda x" never reaches the tensile r/f. So I'm confused.

 

[hardbutmild]

According to the book that you mentioned, just above the equation this is written "Taking moments about the mid-depth of the section, the moment of the compression force Cc is"
So clearly, this is not meant to be the bending moment of the section, but the component by compression force. There are many ways of coming to the same solution. In that same book a few pages below this equation is provided so Mc is just a part of the solution.

 

[Pretty Girl]

@hardbutmild
Thank you for the reply. But still I had the problem where does the "0.5 (h-lambda x)" come from? I realised, no matter what it just ends up in the neutral axis of the section. It's a weird equation though. Now I understand.

 

[Celt83]

Moments are being taken about the center of the section not the neutral axis:

 

[Pretty Girl]

@Celt83
Thank you for the reply. It extended my understanding on this. Now I can clearly see what's happening.

 

[dik]

What happens if you take the moment about the NA?

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[Celt83]

@dik:
If there is an applied axial force on the section then you need to include that force and it's location to achieve the capacity value.

Also for a column if you are taking moments about the NA then the P-M Curve will be built with a variable reference frame making the resulting graph nearly impossible to interpret correctly. A constant frame of reference should be established for the generation of P-M curves, and in my opinion the most ideal option is the gross cross section centroid as that is typically how the member would be modeled and how the member forces would be given by an analysis program.

 

[Pretty Girl]

@Celt83
Isn't it required to assume that the moment taking axis as the neutral axis as well to build the chart? or else how can you calculate "lamda x" if the actual neutral axis is unknown.

I feel like we have to assume the "x" as the distance to the neutral axis? (also the 0.5h)

If we assume the "x" as the neutral axis, we can calculate the 'lambda x' as 'lambda x' is usually "0.8x"

 

[hardbutmild]

I made a quick example so I'm going to attach a photo. Maybe it will help. I am sorry for not being very tidy, but I hope you will understand what I did. It is really not that complicated - you make a force equilibrium and a moment equilibrium about any point really... if you want to find the required reinforcement you have a quadratic equation (first part of my example). If you want to find the moment capacity of a section with a known reinforcement it's even easier (second part of my example)... the only thing where I kind of cheated you is that I knew I am at the load level where compression reinforcement does not really matter (since the axial load is low), so some differences are possible but the point is the same. Always force and moment equilibrium.

 

[Pretty Girl]

@hardbutmild
Thank you for the calculation. It's very helpful to understand the concept for practical applications.