# Definition of stiffness for structural dynamics: How would you find the stiffness of an unusual beam directly?

I have a structure which is essentially a curved beam (not an arch or anything simple) which I have modeled in microstran/spacegass. It looks something like this:

Or for something more simple, imagine I have a beam like this:

The definition of stiffness as I know it is: k = F/d

ie the stiffness of a structure in a direction is proportional to the Force applied to the location divided by the deflection of the structure at that location due to that force.

However in the case of these structures, the stiffness will vary depending on the direction I apply the load.

I have applied a downward load of 1 N to the center of each of the 'arches' and they deflect accordingly. However, they obviously deflect differently (in magnitude) when I apply an upward load at the same location. Note: I am doing a non-linear analysis.

So which is the correct deflection that I need to use in my calculation of k = F/d?

I am using this stiffness for structural dynamics purposes, I am trying to find the natural frequency at that location. The structure will therefore vibrate in both the upward and downward direction.

My initial thought was to apply a load of 0.5N in both directions and add the deflections. Would this be correct?

• I just made a model on Ftool with a two-beam "arch" with total span 20m, height of 2m and vertical force of +-100kN. The beams have E=100GPa, A=0.5m2, I=0.07m4. The deflections are equal (in modulus) regardless of the force's direction (+-0.24mm). – Wasabi Jan 13 '17 at 2:27
• You probably did a linear analysis. Sorry I forgot to mention I was referring to a non-linear analysis. I will add that in now. – Noobie Jan 13 '17 at 8:25

## 1 Answer

k=F/d is a linear relationship. If you're doing a non-linear analysis, you shouldn't expect a linear response.

For this beam, a geometrically non-linear analysis is appropriate if your deflections are any way significant. This is because a downwards load (if it is big enough) will cause snap-through buckling behaviour, whereas an upwards load will not.

Therefore you cannot use k=F/d as a) the downwards deflection is not directly proportional to the applied load, and b) deflection due to upwards loads behaves differently to deflection due to downwards loads.

My knowledge of dynamics is extremely limited. Possibly a linear analysis might be an appropriate approximation as long as deflections are small.