# Stiffness of column when fixed versus pin fixed

1. Aug 2, 2012

Guys when i study earthquake , they told me that the column stiffness is 12EI/L3 if column is fixed -fixed and 3EI/L3 if column is pin -fixed, so how they get this, i try to study stiffness Matrix and it gives you big matrix for frames not Only one factor like 12EI/L3

2. Aug 2, 2012

### PhanthomJay

Re: Stiffness

They are not talking about the stiffness of the joints, just the stiffness of the member framing into the joint with a lateral load is applied at the joint. If both ends are fixed at the joints (translation but no relative rotation), K = 12EI/L^3, which is the inverse of its end deflection for a fixed-guided beam subject to point load at the joint, and if it's fixed pinned, then K = 3EI/L^3, the inverse of the deflection of a cantilever with a point load at joint

3. Aug 2, 2012

Re: Stiffness

Thank you very much. i didnt get it ,could uou provide me with something to read , in order to understand it.Please

4. Aug 2, 2012

Re: Stiffness

What do you mean by the inverse of the deflection of a cantilever with a point load at joint?? and how it is equivelent please expalin

5. Aug 2, 2012

### PhanthomJay

Re: Stiffness

The deflection of a cantilever with a point load F at the free end is FL^3/3EI. It's stiffness, K, or spring constant if you will, per Hooke's Law F=kx, is

F=kx

k = F/x
k = F/(FL^3/3EI)
k = 3EI/L^3
which is the inverse of the deflection under a unit load.
You are asking why, I think, you use the cantilever stiffness for a fixed pinned column in a frame with a load applied at the joint. That's because the joint translates, and the deflected sahpe is equivalent to the cantilever's deflected shape. For the fixed-fixed case with joint translation, the stiffness is equivalent to that of a beam fixed at one end and free to translate but not rotate at the other, in which case K = 12EI/L^3, which youcan find in beam tables.