Derivation of First Storey Shear Rigidity for a Rigid Frame

[Scotty Boy]

I am wondering if anyone is familiar with any of Bryan Stafford-Smith and Alexander Coull's work?

I have their book Tall Building Structures, Analysis and Design.

For rigid frames they express an parameter for the racking stiffness called the shear rigidity (GA).

For intermediate storeys they give the shear rigidity as:

GA = Qh/delta

Where:

Q = the storey shear
h = storey height
delta = the lateral deflection

To calculate delta the points of contraflexure are assumed to be mid-storey height. From this we get the deflection due to column flexure as Qh^2/(12*E*C), and the deflection due to girder flexure as Qh^2/(12*E*G).

Where:

C = Ic/h
G = Ig/L

Substituting this into GA

GA = Qh/(Qh^2/12E) (1/C + 1/G)

Which gives:

GA = 12E/(h(1/C + 1/G)

A quite simple derivation I think.

Now, because it is inaccurate to assume points of contraflexure at mid height for the bottom storeys, Stafford-Smith and Coull provide a tweaked equation for the shear rigidity. It comes in the form:

GA = 12E(1 + C/2G)/(h[1/C + 2/3G])

Unfortunately, they don't give a derivation for this, and I need to know this. Does anyone know how to derive it? I assume that they have taken the points of contraflexure as 2/3 from the base.

Here is the page from the book that shows how they derive the shear rigidity for the intermediate storeys.




The diagrams demonstrate the joint rotation due to girder flexure, the storey drift due girder flexure, and storey drift due to column flexure respectively.

If you need any more information, please ask.

Thanks.

 

[Scotty Boy]

Are we all clueless on this?

 

[HTURKAK]

Are we all clueless on this?

I have more than clue .. trying to remember.
My points ;
- Your expression GA = 12E/(h(1/C + 1/G) apparently covers girder flexure and column flexure . For total drift, you need to add story drift due to overall bending.
- The first floor ( also top floor ) differs . The point of contraflexure no more at h/2, for first and top floors,
- For the first floor , assume the column is fixed to the foundation ( rigid base conn.) then calculate the rotational stiffness .

It is strange for me that , your question is for approximate methods of 50 yrs ago. Probably you are asking just for curiosity.
Nowadays , with high-speed computers, and enhanced softwares no need to use approximate methods.

 

[Scotty Boy]

Great! I am very glad you have a clue!

I'm glad you mentioned that about the top storey, because I want to derive the expression for GA in the top storey. The point of contraflexure, from my observations, can be a as small as 1/4 the length of the column (going upwards) for the top storey. If I can work out how Stafford-Smith and Coull derived the GA for the bottom storey, I can hopefully derive it for the top storey.

You says about the rotational stiffness - is this not just 6EI/h^2?

If you could help me further (maybe with some steps) I would really appreciate it.

Thanks, HTURKAK.